All smooth morphisms f:X\to S are equivalent to morphisms locally of finite presentation which are formally smooth. Hence formal smoothness is a slight generalization of smooth morphisms.
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This also proves this morphism is not smooth from the equivalence between formally smooth morphisms locally of finite presentation and smooth morphisms.
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More special kinds of morphisms that induce adjoint mappings in the other direction are the morphisms usually considered for frames (or locales).
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A category C consists of objects and morphisms between these objects. The morphisms reflect relations between the objects. In many situations, it is meaningful to replace C by another category C' in which certain morphisms are forced to be isomorphisms. This process is called localization.
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We choose . This operation on morphisms is called cartesian product of morphisms. Second, consider the general product functor. For families we should find a morphism .
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All the morphisms that can serve as composition of two given morphisms are related to each other by higher order invertible morphisms (2-simplices thought of as "homotopies"). These higher order morphisms can also be composed, but again the composition is well-defined only up to still higher order invertible morphisms, etc. The idea of higher category theory (at least, higher category theory when higher morphisms are invertible) is that, as opposed to the standard notion of a category, there should be a mapping space (rather than a mapping set) between two objects. This suggests that a higher category should simply be a topologically enriched category.
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The idea of a local homeomorphism can be formulated in geometric settings different from that of topological spaces. For differentiable manifolds, we obtain the local diffeomorphisms; for schemes, we have the formally étale morphisms and the étale morphisms; and for toposes, we get the étale geometric morphisms.
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Let be a morphism of a category containing two objects A and B. Associated with these objects are the identity morphisms and . By composing these with f, we construct two morphisms: :, and :. Both are morphisms between the same objects as f. We have, accordingly, the following coherence statement: :.
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In mathematics (especially category theory), a multicategory is a generalization of the concept of category that allows morphisms of multiple arity. If morphisms in a category are viewed as analogous to functions, then morphisms in a multicategory are analogous to functions of several variables. Multicategories are also sometimes called operads, or colored operads.
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The localization of a category introduces new morphisms to turn several of the original category's morphisms into isomorphisms. This tends to increase the number of morphisms between objects, rather than decrease it as in the case of quotient categories. But in both constructions it often happens that two objects become isomorphic that weren't isomorphic in the original category.
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Explicitly, a pullback of the morphisms and consists of an object and two morphisms and for which the diagram :125px commutes. Moreover, the pullback must be universal with respect to this diagram.Mitchell, p. 9 That is, for any other such triple where and are morphisms with , there must exist a unique such that :p_2 \circ u=q_2, \qquad p_1\circ u=q_1.
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For more general rings such as local rings, it is no longer true that morphisms of rings induce morphisms of the maximal spectra, and the use of prime ideals rather than maximal ideals gives a cleaner theory.
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This allows a relation between such morphisms and covering maps in topology.
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In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms and with a common codomain. The pullback is often written : and comes equipped with two natural morphisms and . The pullback of two morphisms and need not exist, but if it does, it is essentially uniquely defined by the two morphisms. In many situations, may intuitively be thought of as consisting of pairs of elements with in , in , and .
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How this functor maps objects is obvious. Mapping of morphisms is subtle, because the product of morphisms defined above does not fit. First, consider the binary product functor, which is a bifunctor. For we should find a morphism .
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A category C enriched in a monoidal category M replaces the notion of a set of morphisms between pairs of objects in C with the notion of an M-object of morphisms between every two objects in C.
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Quasi-separated morphisms are important for algebraic spaces and algebraic stacks, where many natural morphisms are quasi-separated but not separated. The condition that a morphism is quasi-separated often occurs together with the condition that it is quasi- compact.
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In algebraic geometry, a morphism of schemes f from X to Y is called quasi- separated if the diagonal map from X to X×YX is quasi-compact (meaning that the inverse image of any quasi-compact open set is quasi compact). A scheme X is called quasi-separated if the morphism to Spec Z is quasi-separated. Quasi- separated algebraic spaces and algebraic stacks and morphisms between them are defined in a similar way, though some authors include the condition that X is quasi-separated as part of the definition of an algebraic space or algebraic stack X. Quasi-separated morphisms were introduced by as a generalization of separated morphisms. All separated morphisms (and all morphisms of Noetherian schemes) are automatically quasi-separated.
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The algebraic function fields over k form a category; the morphisms from function field K to L are the ring homomorphisms f : K → L with f(a) = a for all a in k. All these morphisms are injective. If K is a function field over k of n variables, and L is a function field in m variables, and n > m, then there are no morphisms from K to L.
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Let BordM be the category whose morphisms are n-dimensional submanifolds of M and whose objects are connected components of the boundaries of such submanifolds. Regard two morphisms as equivalent if they are homotopic via submanifolds of M, and so form the quotient category hBordM: The objects in hBordM are the objects of BordM, and the morphisms of hBordM are homotopy equivalence classes of morphisms in BordM. A TQFT on M is a symmetric monoidal functor from hBordM to the category of vector spaces. Note that cobordisms can, if their boundaries match, be sewn together to form a new bordism.
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This functor takes an object c′ of C and gives back all of the morphisms c′ → c. A subfunctor of gives back only some of the morphisms. Such a subfunctor is called a sieve, and it is usually used when defining Grothendieck topologies.
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In mathematics, specifically in category theory, a quasi-abelian category is a pre-abelian category in which the pushout of a kernel along arbitrary morphisms is again a kernel and, dually, the pullback of a cokernel along arbitrary morphisms is again a cokernel.
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Several kinds of homomorphisms have a specific name, which is also defined for general morphisms.
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One construction of the localization is done by declaring that its objects are the same as those in C, but the morphisms are enhanced by adding a formal inverse for each morphism in W. Under suitable hypotheses on W, the morphisms between two objects X, Y are given by roofs :X \stackrel f \leftarrow X' \rightarrow Y (where X' is an arbitrary object of C and f is in the given class W of morphisms), modulo certain equivalence relations. These relations turn the map going in the "wrong" direction into an inverse of f. This procedure, however, in general yields a proper class of morphisms between X and Y. Typically, the morphisms in a category are only allowed to form a set. Some authors simply ignore such set-theoretic issues.
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In the mathematical field of category theory, FinSet is the category whose objects are all finite sets and whose morphisms are all functions between them. FinOrd is the category whose objects are all finite ordinal numbers and whose morphisms are all functions between them.
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In mathematics, a doctrine is simply a 2-category which is heuristically regarded as a system of theories. For example, algebraic theories, as invented by William Lawvere, is an example of a doctrine, as are multi-sorted theories, operads, categories, and toposes. The objects of the 2-category are called theories, the 1-morphisms f\colon A\rightarrow B are called models of the in , and the 2-morphisms are called morphisms between models. The distinction between a 2-category and a doctrine is really only heuristic: one does not typically consider a 2-category to be populated by theories as objects and models as morphisms.
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A functor between preadditive categories is additive if it is an abelian group homomorphism on each hom-set in C. If the categories are additive, then a functor is additive if and only if it preserves all biproduct diagrams. That is, if is a biproduct of in C with projection morphisms and injection morphisms , then should be a biproduct of in D with projection morphisms and injection morphisms . Almost all functors studied between additive categories are additive. In fact, it is a theorem that all adjoint functors between additive categories must be additive functors (see here), and most interesting functors studied in all of category theory are adjoints.
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They are variously defined, for example, as sheaves of sets or sheaves of rings, depending on the type of data assigned to open sets. There are also maps (or morphisms) from one sheaf to another; sheaves (of a specific type, such as sheaves of abelian groups) with their morphisms on a fixed topological space form a category. On the other hand, to each continuous map there is associated both a direct image functor, taking sheaves and their morphisms on the domain to sheaves and morphisms on the codomain, and an inverse image functor operating in the opposite direction. These functors, and certain variants of them, are essential parts of sheaf theory.
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This follows from the fact that the only ideals in a field F are the zero ideal and F itself. One can then view morphisms in Field as field extensions. The category of fields is not connected. There are no morphisms between fields of different characteristic.
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Category theory deals with morphisms instead of functions. Morphisms are arrows from one object to another. The domain of any morphism is the object from which an arrow starts. In this context, many set theoretic ideas about domains must be abandoned—or at least formulated more abstractly.
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A faithful functor need not be injective on objects or morphisms. That is, two objects X and X′ may map to the same object in D (which is why the range of a full and faithful functor is not necessarily isomorphic to C), and two morphisms f : X → Y and f′ : X′ → Y′ (with different domains/codomains) may map to the same morphism in D. Likewise, a full functor need not be surjective on objects or morphisms. There may be objects in D not of the form FX for some X in C. Morphisms between such objects clearly cannot come from morphisms in C. A full and faithful functor is necessarily injective on objects up to isomorphism. That is, if F : C → D is a full and faithful functor and F(X)\cong F(Y) then X \cong Y.
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Some important properties P of morphisms of schemes are preserved under arbitrary base change. That is, if X → Y has property P and Z → Y is any morphism of schemes, then the base change X xY Z → Z has property P. For example, flat morphisms, smooth morphisms, proper morphisms, and many other classes of morphisms are preserved under arbitrary base change.. The word descent refers to the reverse question: if the pulled-back morphism X xY Z → Z has some property P, must the original morphism X → Y have property P? Clearly this is impossible in general: for example, Z might be the empty scheme, in which case the pulled- back morphism loses all information about the original morphism. But if the morphism Z → Y is flat and surjective (also called faithfully flat) and quasi- compact, then many properties do descend from Z to Y. Properties that descend include flatness, smoothness, properness, and many other classes of morphisms.. These results form part of Grothendieck's theory of faithfully flat descent.
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Category theory is a branch of mathematics that formalizes the notion of a special function via arrows or morphisms. A category is an algebraic object that (abstractly) consists of a class of objects, and for every pair of objects, a set of morphisms. A partial (equiv. dependently typed) binary operation called composition is provided on morphisms, every object has one special morphism from it to itself called the identity on that object, and composition and identities are required to obey certain relations.
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A factorization system (E, M) for a category C consists of two classes of morphisms E and M of C such that: #E and M both contain all isomorphisms of C and are closed under composition. #Every morphism f of C can be factored as f=m\circ e for some morphisms e\in E and m\in M. #The factorization is functorial: if u and v are two morphisms such that vme=m'e'u for some morphisms e, e'\in E and m, m'\in M, then there exists a unique morphism w making the following diagram commute: center Remark: (u,v) is a morphism from me to m'e' in the arrow category.
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Hence Ω(X) is not an arbitrary complete lattice but a complete Heyting algebra (also called frame or locale – the various names are primarily used to distinguish several categories that have the same class of objects but different morphisms: frame morphisms, locale morphisms and homomorphisms of complete Heyting algebras). Now an obvious question is: To what extent is a topological space characterized by its locale of open sets? As already hinted at above, one can go even further. The category Top of topological spaces has as morphisms the continuous functions, where a function f is continuous if the inverse image f −1(O) of any open set in the codomain of f is open in the domain of f.
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In this manner, "the 'arrows' or the 'structural language' can then be interpreted as morphisms which conserve this unique distinction". If more than one distinction is considered, however, the model becomes much more complex, and the interpretation of distinction states as events, or morphisms as processes, is much less straightforward.
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In categories with zero morphisms, one can define a cokernel of a morphism f as the coequalizer of f and the parallel zero morphism. In preadditive categories it makes sense to add and subtract morphisms (the hom- sets actually form abelian groups). In such categories, one can define the coequalizer of two morphisms f and g as the cokernel of their difference: :coeq(f, g) = coker(g – f). A stronger notion is that of an absolute coequalizer, this is a coequalizer that is preserved under all functors.
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In category theory, n-ary functions generalise to n-ary morphisms in a multicategory. The interpretation of an n-ary morphism as an ordinary morphisms whose domain is some sort of product of the domains of the original n-ary morphism will work in a monoidal category. The construction of the derived morphisms of one variable will work in a closed monoidal category. The category of sets is closed monoidal, but so is the category of vector spaces, giving the notion of bilinear transformation above.
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2009 This is a mathematical structure consisting of objects, and for any pair of objects, a set of morphisms between them. In most examples, the objects are mathematical structures (such as sets, vector spaces, or topological spaces) and the morphisms are functions between these structures.A basic reference on category theory is Mac Lane 1998. One can also consider categories where the objects are D-branes and the morphisms between two branes \alpha and \beta are states of open strings stretched between \alpha and \beta.
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An elaborate treatise of the theory of quasi-categories has been expounded by . Quasi-categories are certain simplicial sets. Like ordinary categories, they contain objects (the 0-simplices of the simplicial set) and morphisms between these objects (1-simplices). But unlike categories, the composition of two morphisms need not be uniquely defined.
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Limits and colimits can also be defined for collections of objects and morphisms without the use of diagrams. The definitions are the same (note that in definitions above we never needed to use composition of morphisms in J). This variation, however, adds no new information. Any collection of objects and morphisms defines a (possibly large) directed graph G. If we let J be the free category generated by G, there is a universal diagram F : J → C whose image contains G. The limit (or colimit) of this diagram is the same as the limit (or colimit) of the original collection of objects and morphisms. Weak limit and weak colimits are defined like limits and colimits, except that the uniqueness property of the mediating morphism is dropped.
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If the automorphisms of an object form a set (instead of a proper class), then they form a group under composition of morphisms. This group is called the automorphism group of . ;Closure: Composition of two automorphisms is another automorphism. ;Associativity: It is part of the definition of a category that composition of morphisms is associative.
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Monoids can be viewed as a special class of categories. Indeed, the axioms required of a monoid operation are exactly those required of morphism composition when restricted to the set of all morphisms whose source and target is a given object. That is, : A monoid is, essentially, the same thing as a category with a single object. More precisely, given a monoid , one can construct a small category with only one object and whose morphisms are the elements of M. The composition of morphisms is given by the monoid operation •.
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In mathematics, in particular in category theory, the lifting property is a property of a pair of morphisms in a category. It is used in homotopy theory within algebraic topology to define properties of morphisms starting from an explicitly given class of morphisms. It appears in a prominent way in the theory of model categories, an axiomatic framework for homotopy theory introduced by Daniel Quillen. It is also used in the definition of a factorization system, and of a weak factorization system, notions related to but less restrictive than the notion of a model category.
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In mathematics, the category FdHilb has all finite-dimensional Hilbert spaces for objects and the linear transformations between them as morphisms.
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When the letter A is falling (i.e. a subscript), h_A assigns to an object X the morphisms from X into A .
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Maps preserving the structure are then the morphisms, and the symmetry group is the automorphism group of the object in question.
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To be specific, the equaliser of the morphisms f and g is the kernel of the difference g − f. In symbols: :eq (f, g) = ker (g − f). It is because of this fact that binary equalisers are called "difference kernels", even in non- preadditive categories where morphisms cannot be subtracted. Every kernel, like any other equaliser, is a monomorphism.
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The zero ring serves as both an initial and terminal object in Rng (that is, it is a zero object). It follows that Rng, like Grp but unlike Ring, has zero morphisms. These are just the rng homomorphisms that map everything to 0. Despite the existence of zero morphisms, Rng is still not a preadditive category.
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In mathematics, and particularly category theory, a coherence condition is a collection of conditions requiring that various compositions of elementary morphisms are equal. Typically the elementary morphisms are part of the data of the category. A coherence theorem states that, in order to be assured that all these equalities hold, it suffices to check a small number of identities.
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If the given category is finite (has finitely many objects and morphisms), then the following two definitions of the category algebra agree.
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In category theory, string diagrams are a way of representing morphisms in monoidal categories, or more generally 2-cells in 2-categories.
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In this setting, the category of linear representations of over is the functor category → Vect, which has natural transformations as its morphisms.
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We can then "compose" these "bimorphisms" both horizontally and vertically, and we require a 2-dimensional "exchange law" to hold, relating the two composition laws. In this context, the standard example is Cat, the 2-category of all (small) categories, and in this example, bimorphisms of morphisms are simply natural transformations of morphisms in the usual sense. Another basic example is to consider a 2-category with a single object; these are essentially monoidal categories. Bicategories are a weaker notion of 2-dimensional categories in which the composition of morphisms is not strictly associative, but only associative "up to" an isomorphism.
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A subgroupoid is a subcategory that is itself a groupoid. A groupoid morphism is simply a functor between two (category- theoretic) groupoids. The category whose objects are groupoids and whose morphisms are groupoid morphisms is called the groupoid category, or the category of groupoids, denoted Grpd. It is useful that this category is, like the category of small categories, Cartesian closed.
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The cdh topology is the smallest Grothendieck topology whose covering morphisms include those of the proper cdh topology and those of the Nisnevich topology.
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In category theory the usage of "left" is "right" has some algebraic resemblance, but refers to left and right sides of morphisms. See adjoint functors.
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If and are group homomorphisms, then so is . This shows that the class of all groups, together with group homomorphisms as morphisms, forms a category.
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Mathematically, branes can be described using the notion of a category.Aspinwall et al. 2009 This is a mathematical structure consisting of objects, and for any pair of objects, a set of morphisms between them. In most examples, the objects are mathematical structures (such as sets, vector spaces, or topological spaces) and the morphisms are functions between these structures.A basic reference on category theory is Mac Lane 1998.
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In category theory, the category of sets, denoted Set, is the category consisting of the collection of all sets as objects and the collection of all functions between sets as morphisms, with the composition of functions as the composition of the morphisms. In Set, an isomorphism between two sets is precisely a bijection, and two sets are equinumerous precisely if they are isomorphic as objects in Set.
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In mathematics, a categorical ring is, roughly, a category equipped with addition and multiplication. In other words, a categorical ring is obtained by replacing the underlying set of a ring by a category. For example, given a ring R, let C be a category whose objects are the elements of the set R and whose morphisms are only the identity morphisms. Then C is a categorical ring.
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This construction differs slightly from the one in (Macdonald, 1979). That construction only uses the surjective morphisms ρn without mentioning the injective morphisms φn: it constructs the homogeneous components of ΛR separately, and equips their direct sum with a ring structure using the ρn. It is also observed that the result can be described as an inverse limit in the category of graded rings. That description however somewhat obscures an important property typical for a direct limit of injective morphisms, namely that every individual element (symmetric function) is already faithfully represented in some object used in the limit construction, here a ring R[X1,...,Xd]Sd.
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One can use the nerve construction to recover mapping spaces, and even get "higher-homotopical" information about maps. Let D be a category, and let X and Y be objects of D. One is often interested in computing the set of morphisms X -> Y. We can use a nerve construction to recover this set. Let C = C(X,Y) be the category whose objects are diagrams :X \longleftarrow U \longrightarrow V \longleftarrow Y such that the morphisms U -> X and Y -> V are isomorphisms in D. Morphisms in C(X, Y) are diagrams of the following shape: : Image:Mappings-as-moduli.png Here, the indicated maps are to be isomorphisms or identities.
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In the following we assume all manifolds are differentiable manifolds of class Cr for a fixed r ≥ 1, and all morphisms are differentiable of class Cr.
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In category theory, an automorphism is an endomorphism (i.e., a morphism from an object to itself) which is also an isomorphism (in the categorical sense of the word). This is a very abstract definition since, in category theory, morphisms aren't necessarily functions and objects aren't necessarily sets. In most concrete settings, however, the objects will be sets with some additional structure and the morphisms will be functions preserving that structure.
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In mathematics, a comma category (a special case being a slice category) is a construction in category theory. It provides another way of looking at morphisms: instead of simply relating objects of a category to one another, morphisms become objects in their own right. This notion was introduced in 1963 by F. W. Lawvere (Lawvere, 1963 p. 36), although the technique did not become generally known until many years later.
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The quiver itself can be considered a category, where the vertices are objects and paths are morphisms. Then a representation of Q is just a covariant functor from this category to the category of finite dimensional vector spaces. Morphisms of representations of Q are precisely natural transformations between the corresponding functors. For a finite quiver Γ (a quiver with finitely many vertices and edges), let KΓ be its path algebra.
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Smooth morphisms are supposed to geometrically correspond to smooth submersions in differential geometry; that is, they are smooth locally trivial fibrations over some base space (by Ehresmann's theorem).
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A Riemann surface is a connected complex analytic manifold of one complex dimension, which makes it a connected real manifold of two dimensions. It is compact if it is compact as a topological space. There is a triple equivalence of categories between the category of smooth irreducible projective algebraic curves over C (with non- constant regular maps as morphisms), the category of compact Riemann surfaces (with non-constant holomorphic maps as morphisms), and the opposite of the category of algebraic function fields in one variable over C (with field homomorphisms that fix C as morphisms). This means that in studying these three subjects we are in a sense studying one and the same thing.
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In the case of groups, the morphisms are the group homomorphisms. A group homomorphism between two groups "preserves the group structure" in a precise sense; informally it is a "process" taking one group to another, in a way that carries along information about the structure of the first group into the second group. The study of group homomorphisms then provides a tool for studying general properties of groups and consequences of the group axioms. A similar type of investigation occurs in many mathematical theories, such as the study of continuous maps (morphisms) between topological spaces in topology (the associated category is called Top), and the study of smooth functions (morphisms) in manifold theory.
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In this language, the definition of the étale topology is succinct but abstract: It is the topology generated by the pretopology whose covering families are jointly surjective families of étale morphisms. The small étale site of X is the category O(Xét) whose objects are schemes U with a fixed étale morphism U → X. The morphisms are morphisms of schemes compatible with the fixed maps to X. The big étale site of X is the category Ét/X, that is, the category of schemes with a fixed map to X, considered with the étale topology. The étale topology can be defined using slightly less data. First, notice that the étale topology is finer than the Zariski topology.
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Different ways of constructing homology could be shown to coincide: for example in the case of a simplicial complex the groups defined directly would be isomorphic to those of the singular theory. What cannot easily be expressed without the language of natural transformations is how homology groups are compatible with morphisms between objects, and how two equivalent homology theories not only have the same homology groups, but also the same morphisms between those groups.
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Sigmundur Gudmundsson (born 1960) is an Icelandic-Swedish mathematician working at Lund UniversityFaculty profile, Lund University, retrieved 2015-02-02. in the fields of differential geometry and global analysis. He is mainly interested in the geometric aspects of harmonic maps and harmonic morphisms. His work is partially devoted to the existence theory of complex- valued harmonic morphisms from Riemannian homogeneous spaces of various types, such as symmetric spaces and semisimple, solvable and nilpotent Lie groups.
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In category theory, a branch of mathematics, a closed category is a special kind of category. In a locally small category, the external hom (x, y) maps a pair of objects to a set of morphisms. So in the category of sets, this is an object of the category itself. In the same vein, in a closed category, the (object of) morphisms from one object to another can be seen as lying inside the category.
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A spectral map f: X → Y between spectral spaces X and Y is a continuous map such that the preimage of every open and compact subset of Y under f is again compact. The category of spectral spaces, which has spectral maps as morphisms, is dually equivalent to the category of bounded distributive lattices (together with morphisms of such lattices). In this anti-equivalence, a spectral space X corresponds to the lattice K\circ(X).
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The product is a special case of a limit. This may be seen by using a discrete category (a family of objects without any morphisms, other than their identity morphisms) as the diagram required for the definition of the limit. The discrete objects will serve as the index of the components and projections. If we regard this diagram as a functor, it is a functor from the index set considered as a discrete category.
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In mathematics, especially in order theory, a complete Heyting algebra is a Heyting algebra that is complete as a lattice. Complete Heyting algebras are the objects of three different categories; the category CHey, the category Loc of locales, and its opposite, the category Frm of frames. Although these three categories contain the same objects, they differ in their morphisms, and thus get distinct names. Only the morphisms of CHey are homomorphisms of complete Heyting algebras.
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It is also fruitful to consider examples of morphisms as examples of schemes since they demonstrate their technical effectiveness for encapsulating many objects of study in algebraic and arithmetic geometry.
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252, Theorem 10.1. Usually the morphisms induced by inclusion in this theorem are not themselves injective, and the more precise version of the statement is in terms of pushouts of groups.
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Schematic representation of a category with objects X, Y, Z and morphisms f, g, g ∘ f. (The category's three identity morphisms 1X, 1Y and 1Z, if explicitly represented, would appear as three arrows, from the letters X, Y, and Z to themselves, respectively.) Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms). A category has two basic properties: the ability to compose the arrows associatively, and the existence of an identity arrow for each object. The language of category theory has been used to formalize concepts of other high-level abstractions such as sets, rings, and groups.
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In category theory, a branch of mathematics, a pushout (also called a fibered coproduct or fibered sum or cocartesian square or amalgamated sum) is the colimit of a diagram consisting of two morphisms f : Z -> X and g : Z -> Y with a common domain. The pushout consists of an object P along with two morphisms X -> P and Y -> P that complete a commutative square with the two given morphisms f and g. In fact, the defining universal property of the pushout (given below) essentially says that the pushout is the "most general" way to complete this commutative square. Common notations for the pushout are P = X \sqcup_Z Y and P = X +_Z Y. The pushout is the categorical dual of the pullback.
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In mathematics, the category of topological vector spaces is the category whose objects are topological vector spaces and whose morphisms are continuous linear maps between them. This is a category because the composition of two continuous linear maps is again a continuous linear map. The category is often denoted TVect or TVS. Fixing a topological field K, one can also consider the subcategory TVectK of topological vector spaces over K with continuous K-linear maps as the morphisms.
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One can proceed to prove theorems about groups by making logical deductions from the set of axioms defining groups. For example, it is immediately proven from the axioms that the identity element of a group is unique. Instead of focusing merely on the individual objects (e.g., groups) possessing a given structure, category theory emphasizes the morphisms – the structure-preserving mappings – between these objects; by studying these morphisms, one is able to learn more about the structure of the objects.
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Mitchell, 1965). A functor associates to every object of one category an object of another category, and to every morphism in the first category a morphism in the second. As a result, this defines a category of categories and functors – the objects are categories, and the morphisms (between categories) are functors. Studying categories and functors is not just studying a class of mathematical structures and the morphisms between them but rather the relationships between various classes of mathematical structures.
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In algebra, epimorphisms are often defined as surjective homomorphisms. On the other hand, in category theory, epimorphisms are defined as right cancelable morphisms. This means that a (homo)morphism f: A \to B is an epimorphism if, for any pair g, h of morphisms from B to any other object C, the equality g \circ f = h \circ f implies g = h. A surjective homomorphism is always right cancelable, but the converse is not always true for algebraic structures.
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In category theory, a faithful functor (respectively a full functor) is a functor that is injective (respectively surjective) when restricted to each set of morphisms that have a given source and target.
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The category whose objects are double groupoids and whose morphisms are double groupoid homomorphisms that are double groupoid diagram (D) functors is called the double groupoid category, or the category of double groupoids.
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In category theory, a strict 2-category is a category with "morphisms between morphisms", that is, where each hom-set itself carries the structure of a category. It can be formally defined as a category enriched over Cat (the category of categories and functors, with the monoidal structure given by product of categories). The concept of 2-category was first introduced by Charles Ehresmann in his work on enriched categories, in 1965.Charles Ehresmann, Catégories et structures, Dunod, Paris 1965.
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The ontology of NBG provides scaffolding for speaking about "large objects" without risking paradox. For instance, in some developments of category theory, a "large category" is defined as one whose objects and morphisms make up a proper class. On the other hand, a "small category" is one whose objects and morphisms are members of a set. Thus, we can speak of the "category of all sets" or "category of all small categories" without risking paradox since NBG supports large categories.
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The Alexandroff extension can be viewed as a functor from the category of topological spaces with proper continuous maps as morphisms to the category whose objects are continuous maps c\colon X \rightarrow Y and for which the morphisms from c_1\colon X_1 \rightarrow Y_1 to c_2\colon X_2 \rightarrow Y_2 are pairs of continuous maps f_X\colon X_1 \rightarrow X_2, \ f_Y\colon Y_1 \rightarrow Y_2 such that f_Y \circ c_1 = c_2 \circ f_X. In particular, homeomorphic spaces have isomorphic Alexandroff extensions.
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If C and D are two categories and F and G are two functors from C to D, the inserter category Ins(F, G) is the category whose objects are pairs (X, f) where X is an object of C and f is a morphism in D from F(X) to G(X) and whose morphisms from (X, f) to (Y, g) are morphisms h in C from X to Y such that G(h) \circ f = g \circ F(h).
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In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory.
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On the other hand, some authors have no use for this distinction of morphisms (especially since the emerging concepts of "complete semilattice morphisms" can as well be specified in general terms). Consequently, complete meet-semilattices have also been defined as those meet-semilattices that are also complete partial orders. This concept is arguably the "most complete" notion of a meet-semilattice that is not yet a lattice (in fact, only the top element may be missing). This discussion is also found in the article on semilattices.
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Many of the above concepts, especially equivalence of categories, adjoint functor pairs, and functor categories, can be situated into the context of higher-dimensional categories. Briefly, if we consider a morphism between two objects as a "process taking us from one object to another", then higher-dimensional categories allow us to profitably generalize this by considering "higher-dimensional processes". For example, a (strict) 2-category is a category together with "morphisms between morphisms", i.e., processes which allow us to transform one morphism into another.
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In mathematics, localization of a category consists of adding to a category inverse morphisms for some collection of morphisms, constraining them to become isomorphisms. This is formally similar to the process of localization of a ring; it in general makes objects isomorphic that were not so before. In homotopy theory, for example, there are many examples of mappings that are invertible up to homotopy; and so large classes of homotopy equivalent spaces. Calculus of fractions is another name for working in a localized category.
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In homological algebra, a δ-functor between two abelian categories A and B is a collection of functors from A to B together with a collection of morphisms that satisfy properties generalising those of derived functors. A universal δ-functor is a δ-functor satisfying a specific universal property related to extending morphisms beyond "degree 0". These notions were introduced by Alexander Grothendieck in his "Tohoku paper" to provide an appropriate setting for derived functors.Grothendieck 1957 In particular, derived functors are universal δ-functors.
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In EGA III, Grothendieck calls the following statement which does not involve connectedness a "Main theorem" of Zariski : :If f:X->Y is a quasi-projective morphism of Noetherian schemes then the set of points that are isolated in their fiber is open in X. Moreover the induced scheme of this set is isomorphic to an open subset of a scheme that is finite over Y. In EGA IV, Grothendieck observed that the last statement could be deduced from a more general theorem about the structure of quasi-finite morphisms, and the latter is often referred to as the "Zariski's main theorem in the form of Grothendieck". It is well known that open immersions and finite morphisms are quasi-finite. Grothendieck proved that under the hypothesis of separatedness all quasi-finite morphisms are compositions of such : :if Y is a quasi-compact separated scheme and f: X \to Y is a separated, quasi-finite, finitely presented morphism then there is a factorization into X \to Z \to Y, where the first map is an open immersion and the second one is finite. The relation between this theorem about quasi-finite morphisms and Théorème 4.4.
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In the 1960s, Grothendieck defined the notion of a site, meaning a category equipped with a Grothendieck topology. A site C axiomatizes the notion of a set of morphisms Vα → U in C being a covering of U. A topological space X determines a site in a natural way: the category C has objects the open subsets of X, with morphisms being inclusions, and with a set of morphisms Vα → U being called a covering of U if and only if U is the union of the open subsets Vα. The motivating example of a Grothendieck topology beyond that case was the étale topology on schemes. Since then, many other Grothendieck topologies have been used in algebraic geometry: the fpqc topology, the Nisnevich topology, and so on. The definition of a sheaf works on any site.
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In category theory, two categories C and D are isomorphic if there exist functors F : C → D and G : D → C which are mutually inverse to each other, i.e. FG = 1D (the identity functor on D) and GF = 1C. This means that both the objects and the morphisms of C and D stand in a one-to-one correspondence to each other. Two isomorphic categories share all properties that are defined solely in terms of category theory; for all practical purposes, they are identical and differ only in the notation of their objects and morphisms.
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In mathematics, the category of topological spaces, often denoted Top, is the category whose objects are topological spaces and whose morphisms are continuous maps. This is a category because the composition of two continuous maps is again continuous, and the identity function is continuous. The study of Top and of properties of topological spaces using the techniques of category theory is known as categorical topology. N.B. Some authors use the name Top for the categories with topological manifolds or with compactly generated spaces as objects and continuous maps as morphisms.
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Particular kinds of morphisms of groupoids are of interest. A morphism p: E \to B of groupoids is called a fibration if for each object x of E and each morphism b of B starting at p(x) there is a morphism e of E starting at x such that p(e)=b. A fibration is called a covering morphism or covering of groupoids if further such an e is unique. The covering morphisms of groupoids are especially useful because they can be used to model covering maps of spaces.
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Given a topological space X, let G_0 be the set X. The morphisms from the point p to the point q are equivalence classes of continuous paths from p to q, with two paths being equivalent if they are homotopic. Two such morphisms are composed by first following the first path, then the second; the homotopy equivalence guarantees that this composition is associative. This groupoid is called the fundamental groupoid of X, denoted \pi_1(X) (or sometimes, \Pi_1(X)). The usual fundamental group \pi_1(X,x) is then the vertex group for the point x.
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In category theory, a branch of mathematics, compact closed categories are a general context for treating dual objects. The idea of a dual object generalizes the more familiar concept of the dual of a finite-dimensional vector space. So, the motivating example of a compact closed category is FdVect, the category having finite-dimensional vector spaces as objects and linear maps as morphisms, with tensor product as the monoidal structure. Another example is Rel, the category having sets as objects and relations as morphisms, with Cartesian monoidal structure.
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The category hTop, where the objects are topological spaces and the morphisms are homotopy classes of continuous functions, is an example of a category that is not concretizable. While the objects are sets (with additional structure), the morphisms are not actual functions between them, but rather classes of functions. The fact that there does not exist any faithful functor from hTop to Set was first proven by Peter Freyd. In the same article, Freyd cites an earlier result that the category of "small categories and natural equivalence-classes of functors" also fails to be concretizable.
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The set of events can then be structured in the same way as invariance of causal structure, or local-to- global causal connections or even formal properties of global causal connections. The morphisms between (non-trivial) objects could be viewed as representing causal connections leading from one event to another one. For example, the morphism f above leads form event s1 to event s2. The sequences or "paths" of morphisms for which there is no inverse morphism, could then be interpreted as defining horismotic or chronological precedence relations.
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Explicitly, the pushout of the morphisms f and g consists of an object P and two morphisms i1 : X -> P and i2 : Y -> P such that the diagram :125px commutes and such that (P, i1, i2) is universal with respect to this diagram. That is, for any other such set (Q, j1, j2) for which the following diagram commutes, there must exist a unique u : P -> Q also making the diagram commute: :225px As with all universal constructions, the pushout, if it exists, is unique up to a unique isomorphism.
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In category theory, a branch of mathematics, a diagram is the categorical analogue of an indexed family in set theory. The primary difference is that in the categorical setting one has morphisms that also need indexing. An indexed family of sets is a collection of sets, indexed by a fixed set; equivalently, a function from a fixed index set to the class of sets. A diagram is a collection of objects and morphisms, indexed by a fixed category; equivalently, a functor from a fixed index category to some category.
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This is not a group because two operations A and B can only be composed if the empty point after carrying out A is the empty point at the beginning of B. It is in fact a groupoid (a category such that every morphism is invertible) whose 13 objects are the 13 points, and whose morphisms from x to y are the operations taking the empty point from x to y. The morphisms fixing the empty point form a group isomorphic to the Mathieu group M12 with 12×11×10×9×8 elements.
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Nowadays the phrase GAGA-style result is used for any theorem of comparison, allowing passage between a category of objects from algebraic geometry, and their morphisms, to a well-defined subcategory of analytic geometry objects and holomorphic mappings.
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There are natural generalizations of arithmetic dynamics in which and are replaced by number fields and their -adic completions. Another natural generalization is to replace self-maps of or with self-maps (morphisms) of other affine or projective varieties.
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The nerve of C(X, Y) is the moduli space of maps X -> Y. In the appropriate model category setting, this moduli space is weak homotopy equivalent to the simplicial set of morphisms of D from X to Y.
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If C is an additive category and we require the congruence relation ~ on C to be additive (i.e. if f1, f2, g1 and g2 are morphisms from X to Y with f1 ~ f2 and g1 ~g2, then f1 \+ f2 ~ g1 \+ g2), then the quotient category C/~ will also be additive, and the quotient functor C -> C/~ will be an additive functor. The concept of an additive congruence relation is equivalent to the concept of a two-sided ideal of morphisms: for any two objects X and Y we are given an additive subgroup I(X,Y) of HomC(X, Y) such that for all f ∈ I(X,Y), g ∈ HomC(Y, Z) and h∈ HomC(W, X), we have gf ∈ I(X,Z) and fh ∈ I(W,Y). Two morphisms in HomC(X, Y) are congruent iff their difference is in I(X,Y).
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The analogous concept in category theory is called a diagram. A diagram is a functor giving rise to an indexed family of objects in a category C, indexed by another category J, and related by morphisms depending on two indices.
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The construction of the dual vector space mentioned in the introduction is an example of such a duality. Indeed, the set of morphisms, i.e., linear maps, forms a vector space in its own right. The map mentioned above is always injective.
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In algebraic geometry, the syntomic topology is a Grothendieck topology introduced by . Mazur defined a morphism to be syntomic if it is flat and locally a complete intersection. The syntomic topology is generated by surjective syntomic morphisms of affine schemes.
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For the general definition, a universal property is used, which essentially expresses the fact that the pullback is the "most general" way to complete the two given morphisms to a commutative square. The dual concept of the pullback is the pushout.
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In mathematics, a classifying topos for some sort of structure is a topos T such that there is a natural equivalence between geometric morphisms from a cocomplete topos E to T and the category of models for the structure in E.
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Nevertheless, composition of relations and manipulation of the operators according to Schröder rules, provides a calculus to work in the power set of A × B. In contrast to homogeneous relations, the composition of relations operation is only a partial function. The necessity of matching range to domain of composed relations has led to the suggestion that the study of heterogeneous relations is a chapter of category theory as in the category of sets, except that the morphisms of this category are relations. The objects of the category Rel are sets, and the relation-morphisms compose as required in a category.
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However, NBG does not support a "category of all categories" since large categories would be members of it and NBG does not allow proper classes to be members of anything. An ontological extension that enables us to talk formally about such a "category" is the conglomerate, which is a collection of classes. Then the "category of all categories" is defined by its objects: the conglomerate of all categories; and its morphisms: the conglomerate of all morphisms from A to B where A and B are objects.. On whether an ontology including classes as well as sets is adequate for category theory, see .
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Illusie's construction of the cotangent complex generalizes that of Michel André and Daniel Quillen to morphisms of ringed topoi. The generality of the framework makes it possible to apply the formalism to various first- order deformation problems: schemes, morphisms of schemes, group schemes and torsors under group schemes. Results concerning commutative group schemes in particular were the key tool in Grothendieck's proof of his existence and structure theorem for infinitesimal deformations of Barsotti–Tate groups, an ingredient in Gerd Faltings' proof of the Mordell conjecture. In Chapter VIII of the second volume of the thesis, Illusie introduces and studies derived de Rham complexes.
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Every morphism in a concrete category whose underlying function is injective is a monomorphism; in other words, if morphisms are actually functions between sets, then any morphism which is a one-to-one function will necessarily be a monomorphism in the categorical sense. In the category of sets the converse also holds, so the monomorphisms are exactly the injective morphisms. The converse also holds in most naturally occurring categories of algebras because of the existence of a free object on one generator. In particular, it is true in the categories of all groups, of all rings, and in any abelian category.
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Eval and apply are the two interdependent components of the eval-apply cycle, which is the essence of evaluating Lisp, described in SICP.The Metacircular Evaluator (SICP Section 4.1) In category theory, the eval morphism is used to define the closed monoidal category. Thus, for example, the category of sets, with functions taken as morphisms, and the cartesian product taken as the product, forms a Cartesian closed category. Here, eval (or, properly speaking, apply) together with its right adjoint, currying, form the simply typed lambda calculus, which can be interpreted to be the morphisms of Cartesian closed categories.
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In a concrete category (that is, a category whose objects are sets (perhaps with extra structure) and whose morphisms are structure-preserving functions), such as the category of topological spaces or categories of algebraic objects like groups, rings, and modules, an isomorphism must be bijective on the underlying sets. In algebraic categories (specifically, categories of varieties in the sense of universal algebra), an isomorphism is the same as a homomorphism which is bijective on underlying sets. However, there are concrete categories in which bijective morphisms are not necessarily isomorphisms (such as the category of topological spaces).
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The face map di drops the i-th element from such a list, and the degeneracy maps si duplicates the i-th element. A similar construction can be performed for every category C, to obtain the nerve NC of C. Here, NC([n]) is the set of all functors from [n] to C, where we consider [n] as a category with objects 0,1,...,n and a single morphism from i to j whenever i ≤ j. Concretely, the n-simplices of the nerve NC can be thought of as sequences of n composable morphisms in C: a0 -> a1 -> ... -> an. (In particular, the 0-simplices are the objects of C and the 1-simplices are the morphisms of C.) The face map d0 drops the first morphism from such a list, the face map dn drops the last, and the face map di for 0 < i < n drops ai and composes the ith and (i + 1)th morphisms.
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The same definition also applies if C is an ∞-category, provided that the above set of morphisms gets replaced by the mapping space in C (and the filtered colimits are understood in the ∞-categorical sense, sometimes also referred to as filtered homotopy colimits).
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In mathematics, specifically in category theory, hom-sets, i.e. sets of morphisms between objects, give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory and other branches of mathematics.
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Wood is a coauthor of the monograph: Harmonic Morphisms Between Riemannian Manifolds. This was published in 2003 and is still the standard text on the subject. Wood earned his Ph.D. from the University of Warwick in 1974, under the supervision of James Eells.
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In mathematics, a quotient category is a category obtained from another one by identifying sets of morphisms. Formally, it is a quotient object in the category of (locally small) categories, analogous to a quotient group or quotient space, but in the categorical setting.
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J.P. May, A Concise Course in Algebraic Topology, 1999, The University of Chicago Press (see chapter 2) It is also true that the category of covering morphisms of a given groupoid B is equivalent to the category of actions of the groupoid B on sets.
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In category theory and homotopy theory the Burnside category of a finite group G is a category whose objects are finite G-sets and whose morphisms are (equivalence classes of) spans of G-equivariant maps. It is a categorification of the Burnside ring of G.
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3 of EGA III quoted above is that if f:X→Y is a projective morphism of varieties, then the set of points that are isolated in their fiber is quasifinite over Y. Then structure theorem for quasi-finite morphisms applies and yields the desired result.
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The terms homological δ-functor and cohomological δ-functor are sometimes used to distinguish between the case where the morphisms "go down" (homological) and the case where they "go up" (cohomological). In particular, one of these modifiers is always implicit, although often left unstated.
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In functional analysis, a discipline within mathematics, an operator space is a Banach space "given together with an isometric embedding into the space B(H) of all bounded operators on a Hilbert space H.". The appropriate morphisms between operator spaces are completely bounded maps.
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Given a topological space X and a point x in that space, there is a fundamental 2-group of X at x, written Π2(X,x). As a monoidal category, the objects are loops at x, with multiplication given by concatenation, and the morphisms are basepoint-preserving homotopies between loops, with these morphisms identified if they are themselves homotopic. Conversely, given any 2-group G, one can find a unique (up to weak homotopy equivalence) pointed connected space (X,x) whose fundamental 2-group is G and whose homotopy groups πn are trivial for n > 2\. In this way, 2-groups classify pointed connected weak homotopy 2-types.
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Let F : J → C be a diagram in C. Formally, a diagram is nothing more than a functor from J to C. The change in terminology reflects the fact that we think of F as indexing a family of objects and morphisms in C. The category J is thought of as an "index category". One should consider this in analogy with the concept of an indexed family of objects in set theory. The primary difference is that here we have morphisms as well. Thus, for example, when J is a discrete category, it corresponds most closely to the idea of an indexed family in set theory.
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If we denote the -fold product of with itself by , then morphisms from to are m-by-n matrices with entries from the ring . Conversely, given any ring , we can form a category by taking objects An indexed by the set of natural numbers (including zero) and letting the hom-set of morphisms from to be the set of -by- matrices over , and where composition is given by matrix multiplication.H.D. Macedo, J.N. Oliveira, Typing linear algebra: A biproduct- oriented approach, Science of Computer Programming, Volume 78, Issue 11, 1 November 2013, Pages 2160-2191, , . Then is an additive category, and equals the -fold power .
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One may generalize the notion of the heap of a group to the case of a groupoid which has two objects A and B when viewed as a category. The elements of the heap may be identified with the morphisms from A to B, such that three morphisms x, y, z define a heap operation according to: :[x,y,z] = x y^{-1} z . This reduces to the heap of a group if a particular morphism between the two objects is chosen as the identity. This intuitively relates the description of isomorphisms between two objects as a heap and the description of isomorphisms between multiple objects as a groupoid.
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If V′ is a variety contained in Am, we say that f is a regular map from V to V′ if the range of f is contained in V′. The definition of the regular maps apply also to algebraic sets. The regular maps are also called morphisms, as they make the collection of all affine algebraic sets into a category, where the objects are the affine algebraic sets and the morphisms are the regular maps. The affine varieties is a subcategory of the category of the algebraic sets. Given a regular map g from V to V′ and a regular function f of k[V′], then .
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Recall that an ∞-groupoid \Pi(X) is an \infty-category generalization of groupoids which is conjectured to encode the data of the homotopy type of X in an algebraic formalism. The objects are the points in the space X, morphisms are homotopy classes of paths between points, and higher morphisms are higher homotopies of those points. The existence of the Whitehead product is the main reason why defining a notion of ∞-groupoids is such a demanding task. It was shown that any strict ∞-groupoid has only trivial Whitehead products, hence strict groupoids can never model the homotopy types of spheres, such as S^3.
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The more general concept of bicategory (or weak 2-category), where composition of morphisms is associative only up to a 2-isomorphism, was discovered in 1968 by Jean Bénabou.Jean Bénabou, Introduction to bicategories, in Reports of the Midwest Category Seminar, Springer, Berlin, 1967, pp. 1--77.
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In category theory, a span, roof or correspondence is a generalization of the notion of relation between two objects of a category. When the category has all pullbacks (and satisfies a small number of other conditions), spans can be considered as morphisms in a category of fractions.
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Quasi-categories can be thought of as categories in which the composition of morphisms is defined only up to homotopy, and information about the composition of higher homotopies is also retained. Quasi-categories are defined as simplicial sets satisfying one additional condition, the weak Kan condition.
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Every limit and colimit provides an example for a simple natural transformation, as a cone amounts to a natural transformation with the diagonal functor as domain. Indeed, if limits and colimits are defined directly in terms of their universal property, they are universal morphisms in a functor category.
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Any abelian category, in particular the category Ab of abelian groups, is pseudo-abelian. Indeed, in an abelian category, every morphism has a kernel. The category of associative rngs (not rings!) together with multiplicative morphisms is pseudo-abelian. A more complicated example is the category of Chow motives.
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In mathematics, the category of rings, denoted by Ring, is the category whose objects are rings (with identity) and whose morphisms are ring homomorphisms (that preserve the identity). Like many categories in mathematics, the category of rings is large, meaning that the class of all rings is proper.
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Further, diagrams may be impossible to draw (because they are infinite) or simply messy (because there are too many objects or morphisms); however, schematic commutative diagrams (for subcategories of the index category, or with ellipses, such as for a directed system) are used to clarify such complex diagrams.
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The basic definitions in this article are contained within the first few chapters of any of these books. Any monoid can be understood as a special sort of category (with a single object whose self-morphisms are represented by the elements of the monoid), and so can any preorder.
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Let C be any category. The Yoneda embedding gives a functor Hom(−, X) for each object X of C. The canonical topology is the biggest (finest) topology such that every representable presheaf, i.e. presheaf of the form Hom(−, X), is a sheaf. A covering sieve or covering family for this site is said to be strictly universally epimorphic because it consists of the legs of a colimit cone (under the full diagram on the domains of its constituent morphisms) and these colimits are stable under pullbacks along morphisms in C. A topology that is less fine than the canonical topology, that is, for which every covering sieve is strictly universally epimorphic, is called subcanonical.
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In the language of representable functors one can state the above result as follows. The contravariant functor, which associates to each k-variety T the set of families of degree 0 line bundles parametrised by T and to each k-morphism f: T → T' the mapping induced by the pullback with f, is representable. The universal element representing this functor is the pair (Av, P). This association is a duality in the sense that there is a natural isomorphism between the double dual Avv and A (defined via the Poincaré bundle) and that it is contravariant functorial, i.e. it associates to all morphisms f: A → B dual morphisms fv: Bv → Av in a compatible way.
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An Algebraic Theory T is a category whose objects are natural numbers 0, 1, 2,..., and which, for each n, has an n-tuple of morphisms: proji: n → 1, i = 1,..., n This allows interpreting n as a cartesian product of n copies of 1. Example. Let's define an algebraic theory T taking hom(n, m) to be m-tuples of polynomials of n free variables X1,..., Xn with integer coefficients and with substitution as composition. In this case proji is the same as Xi. This theory T is called the theory of commutative rings. In an algebraic theory, any morphism n → m can be described as m morphisms of signature n → 1.
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A categorical product can be characterized by a universal construction. For concreteness, one may consider the Cartesian product in Set, the direct product in Grp, or the product topology in Top, where products exist. Let X and Y be objects of a category D with finite products. The product of X and Y is an object X × Y together with two morphisms :\pi_1 : X \times Y \to X :\pi_2 : X \times Y \to Y such that for any other object Z of D and morphisms f: Z \to X and g: Z \to Y there exists a unique morphism h: Z \to X \times Y such that f = \pi_1 \circ h and g = \pi_2 \circ h.
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Dialectica spaces are a categorical way of constructing models of linear logic. They were introduced by Valeria de Paiva, Martin Hyland's student, in her doctoral thesis, as a way of modeling both linear logic and Gödel's dialectica interpretation--hence the name. Given a category C and a specific object K of C with certain (logical) properties, one can construct the category of Dialectica spaces over C, whose objects are pairs of objects of C, related by a C-morphism into the given object. Morphisms of Dialectica spaces are similar to Chu space morphisms, but instead of an equality condition, they have an inequality condition, which is read as a logical implication, the first object implies the second.
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The pair (E,M) of classes of morphisms of C is a factorization system if and only if it satisfies the following conditions: #Every morphism f of C can be factored as f=m\circ e with e\in E and m\in M. #E=M^\uparrow and M=E^\downarrow.
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If C and D are categories, one can form the product category C × D: the objects are pairs consisting of one object from C and one from D, and the morphisms are also pairs, consisting of one morphism in C and one in D. Such pairs can be composed componentwise.
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The category of small categories Cat has a forgetful functor into the quiver category Quiv: : : Cat → Quiv which takes objects to vertices and morphisms to arrows. Intuitively, "[forgets] which arrows are composites and which are identities". This forgetful functor is right adjoint to the functor sending a quiver to the corresponding free category.
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In category theory, Met is a category that has metric spaces as its objects and metric maps (continuous functions between metric spaces that do not increase any pairwise distance) as its morphisms. This is a category because the composition of two metric maps is again a metric map. It was first considered by .
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An exact sequence is a concept in mathematics, especially in group theory, ring and module theory, homological algebra, as well as in differential geometry. An exact sequence is a sequence, either finite or infinite, of objects and morphisms between them such that the image of one morphism equals the kernel of the next.
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If a complete lattice is freely generated from a given poset used in place of the set of generators considered above, then one speaks of a completion of the poset. The definition of the result of this operation is similar to the above definition of free objects, where "sets" and "functions" are replaced by "posets" and "monotone mappings". Likewise, one can describe the completion process as a functor from the category of posets with monotone functions to some category of complete lattices with appropriate morphisms that is left adjoint to the forgetful functor in the converse direction. As long as one considers meet- or join-preserving functions as morphisms, this can easily be achieved through the so-called Dedekind–MacNeille completion.
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By the above, we can define the category of cones to F as the comma category (Δ ↓ F). Morphisms of cones are then just morphisms in this category. This equivalence is rooted in the observation that a natural map between constant functors Δ(N), Δ(M) corresponds to a morphism between N and M. In this sense, the diagonal functor acts trivially on arrows. In similar vein, writing down the definition of a natural map from a constant functor Δ(N) to F yields the same diagram as the above. As one might expect, a morphism from a cone (N, ψ) to a cone (L, φ) is just a morphism N → L such that all the "obvious" diagrams commute (see the first diagram in the next section).
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For any scheme X, let Ét(X) be the category of all étale morphisms from a scheme to X. This is the analog of the category of open subsets of X (that is, the category whose objects are varieties and whose morphisms are open immersions). Its objects can be informally thought of as étale open subsets of X. The intersection of two objects corresponds to their fiber product over X. Ét(X) is a large category, meaning that its objects do not form a set. An étale presheaf on X is a contravariant functor from Ét(X) to the category of sets. A presheaf F is called an étale sheaf if it satisfies the analog of the usual gluing condition for sheaves on topological spaces.
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The above examples of localization of R-modules is abstracted in the following definition. In this shape, it applies in many more examples, some of which are sketched below. Given a category C and some class W of morphisms in C, the localization C[W−1] is another category which is obtained by inverting all the morphisms in W. More formally, it is characterized by a universal property: there is a natural localization functor C -> C[W−1] and given another category D, a functor F: C -> D factors uniquely over C[W−1] if and only if F sends all arrows in W to isomorphisms. Thus, the localization of the category is unique up to unique isomorphism of categories, provided that it exists.
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Any finite product in a preadditive category must also be a coproduct, and conversely. In fact, finite products and coproducts in preadditive categories can be characterised by the following biproduct condition: :The object B is a biproduct of the objects A1, ..., An if and only if there are projection morphisms pj: B → Aj and injection morphisms ij: Aj → B, such that (i1∘p1) \+ ··· \+ (in∘pn) is the identity morphism of B, pj∘ij is the identity morphism of Aj, and pj∘ik is the zero morphism from Ak to Aj whenever j and k are distinct. This biproduct is often written A1 ⊕ ··· ⊕ An, borrowing the notation for the direct sum. This is because the biproduct in well known preadditive categories like Ab is the direct sum.
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Like many categories, the category TVect is a concrete category, meaning its objects are sets with additional structure (i.e. a vector space structure and a topology) and its morphisms are functions preserving this structure. There are obvious forgetful functors into the category of topological spaces, the category of vector spaces and the category of sets.
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Professor John C. Wood (born 1949) is a British mathematician working at the University of Leeds.Faculty profile, University of Leeds, retrieved 2015-02-02. He is one of the leading experts on harmonic maps and harmonic morphisms in the field of differential geometry.Book Review, Bulletin of the London Mathematical Society 38 (2006), 869-872.
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Given a commutative ring R one can define the category R-Alg whose objects are all R-algebras and whose morphisms are R-algebra homomorphisms. The category of rings can be considered a special case. Every ring can be considered a Z-algebra is a unique way. Ring homomorphisms are precisely the Z-algebra homomorphisms.
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Totally ordered sets form a full subcategory of the category of partially ordered sets, with the morphisms being maps which respect the orders, i.e. maps f such that if a ≤ b then f(a) ≤ f(b). A bijective map between two totally ordered sets that respects the two orders is an isomorphism in this category.
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Any closed subgroup and image of diagonalizable groups are diagonalizable. The torsion subgroup of a diagonalizable group is dense. The category of diagonalizable groups defined over k is equivalent to the category of finitely generated abelian group with Gal(k/ks)-equivariant morphisms without p-torsion. This is an analog of Poincaré duality and motivated the terminology.
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In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed, this intuition can be formalized to define so-called functor categories.
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Namely, Cl can be considered as a functor from the category of vector spaces with quadratic forms (whose morphisms are linear maps preserving the quadratic form) to the category of associative algebras. The universal property guarantees that linear maps between vector spaces (preserving the quadratic form) extend uniquely to algebra homomorphisms between the associated Clifford algebras.
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There exist measurable spaces that are not Borel spaces, for any choice of topology on the underlying space.Jochen Wengenroth, Is every sigma-algebra the Borel algebra of a topology? Measurable spaces form a category in which the morphisms are measurable functions between measurable spaces. A function f:X \rightarrow Y is measurable if it pulls back measurable sets, i.e.
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Many authors do not require rings to have a multiplicative identity element and, accordingly, do not require ring homomorphism to preserve the identity (should it exist). This leads to a rather different category. For distinction we call such algebraic structures rngs and their morphisms rng homomorphisms. The category of all rngs will be denoted by Rng.
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All of these classes of orders can be cast into various categories of dcpos, using functions that are monotone, Scott- continuous, or even more specialized as morphisms. Finally, note that the term domain itself is not exact and thus is only used as an abbreviation when a formal definition has been given before or when the details are irrelevant.
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Spivak provides some rules of good practice for writing an olog whose morphisms have a functional nature (see the first example in the section Mathematical formalism). The text in a box should adhere to the following rules: # begin with the word "a" or "an". (Example: "an amino acid"). # refer to a distinction made and recognizable by the olog's author.
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Bijections are precisely the isomorphisms in the category Set of sets and set functions. However, the bijections are not always the isomorphisms for more complex categories. For example, in the category Grp of groups, the morphisms must be homomorphisms since they must preserve the group structure, so the isomorphisms are group isomorphisms which are bijective homomorphisms.
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We choose the product of morphisms . A category where every finite set of objects has a product is sometimes called a cartesian category (although some authors use this phrase to mean "a category with all finite limits"). The product is associative. Suppose is a cartesian category, product functors have been chosen as above, and denotes a terminal object of .
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Given two algebras of a theory T, say A and B, a homomorphism is a function f \colon A \to B such that : f(o_A(a_1, \dots, a_n)) = o_B(f(a_1), \dots, f(a_n)) for every operation o of arity n. Any theory gives a category where the objects are algebras of that theory and the morphisms are homomorphisms.
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In algebraic geometry, the tautological ring is the subring of the Chow ring of the moduli space of curves generated by tautological classes. These are classes obtained from 1 by pushforward along various morphisms described below. The tautological cohomology ring is the image of the tautological ring under the cycle map (from the Chow ring to the cohomology ring).
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In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise manner of the gluing process being specified by morphisms between the objects. Inverse limits can be defined in any category, and they are a special case of the concept of a limit in category theory.
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The precise statement is as follows: if A is a small abelian category, then there exists a ring R (with 1, not necessarily commutative) and a full, faithful and exact functor F: A → R-Mod (where the latter denotes the category of all left R-modules). The functor F yields an equivalence between A and a full subcategory of R-Mod in such a way that kernels and cokernels computed in A correspond to the ordinary kernels and cokernels computed in R-Mod. Such an equivalence is necessarily additive. The theorem thus essentially says that the objects of A can be thought of as R-modules, and the morphisms as R-linear maps, with kernels, cokernels, exact sequences and sums of morphisms being determined as in the case of modules.
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The set of morphisms (order-preserving functions) between two preorders actually has more structure than that of a set. It can be made into a preordered set itself by the pointwise relation: : (f ≤ g) ⇔ (∀x f(x) ≤ g(x)) This preordered set can in turn be considered as a category, which makes Ord a 2-category (the additional axioms of a 2-category trivially hold because any equation of parallel morphisms is true in a posetal category). With this 2-category structure, a pseudofunctor F from a category C to Ord is given by the same data as a 2-functor, but has the relaxed properties: : ∀x ∈ F(A), F(idA)(x) ≃ x, : ∀x ∈ F(A), F(g∘f)(x) ≃ F(g)(F(f)(x)), where x ≃ y means x ≤ y and y ≤ x.
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In mathematics, specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category that is enriched over the category of abelian groups, Ab. That is, an Ab-category C is a category such that every hom-set Hom(A,B) in C has the structure of an abelian group, and composition of morphisms is bilinear, in the sense that composition of morphisms distributes over the group operation. In formulas: f\circ (g + h) = (f\circ g) + (f\circ h) and (f + g)\circ h = (f\circ h) + (g\circ h), where + is the group operation. Some authors have used the term additive category for preadditive categories, but here we follow the current trend of reserving this word for certain special preadditive categories (see below).
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Diagrams and functor categories are often visualized by commutative diagrams, particularly if the index category is a finite poset category with few elements: one draws a commutative diagram with a node for every object in the index category, and an arrow for a generating set of morphisms, omitting identity maps and morphisms that can be expressed as compositions. The commutativity corresponds to the uniqueness of a map between two objects in a poset category. Conversely, every commutative diagram represents a diagram (a functor from a poset index category) in this way. Not every diagram commutes, as not every index category is a poset category: most simply, the diagram of a single object with an endomorphism or with two parallel arrows (\bullet \rightrightarrows \bullet; f,g\colon X \to Y) need not commute.
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Using the language of category theory, R. A. G. Seely introduced the notion of a locally cartesian closed category (LCCC) as the basic model of type theory. This has been refined by Hofmann and Dybjer to Categories with Families or Categories with Attributes based on earlier work by Cartmell. A category with families is a category C of contexts (in which the objects are contexts, and the context morphisms are substitutions), together with a functor T : Cop → Fam(Set). Fam(Set) is the category of families of Sets, in which objects are pairs of an "index set" A and a function B: X → A, and morphisms are pairs of functions f : A → A' and g : X → X' , such that B' ° g = f ° B in other words, f maps Ba to Bg(a).
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For this purpose, one abstractly defines a field extension as an injective ring homomorphism between two fields. Every non-zero ring homomorphism between fields is injective because fields do not possess nontrivial proper ideals, so field extensions are precisely the morphisms in the category of fields. Henceforth, we will suppress the injective homomorphism and assume that we are dealing with actual subfields.
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Another group used a hill climbing procedure that applies network morphisms, followed by short cosine-annealing optimization runs. The approach yielded competitive results, requiring resources on the same order of magnitude as training a single network. E.g., on CIFAR-10, the method designed and trained a network with an error rate below 5% in 12 hours on a single GPU.
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A morphism f:X\to S is called proper if # it is separated # of finite-type # universally closed The last condition means that given a morphism S' \to S the base change morphism S'\times_SX is a closed immersion. Most known examples of proper morphisms are in fact projective; but, examples of proper varieties which are not projective can be found using toric geometry.
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A direct system or an ind-object in a category C is defined to be a functor :F : I \to C from a small filtered category I to C. For example, if I is the category N mentioned above, this datum is equivalent to a sequence :X_0 \to X_1 \to \cdots of objects in C together with morphisms as displayed.
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The category of rings, Ring, is a nonfull subcategory of Rng. It is nonfull because there are rng homomorphisms between rings which do not preserve the identity, and are therefore not morphisms in Ring. The inclusion functor Ring → Rng has a left adjoint which formally adjoins an identity to any rng. The inclusion functor Ring → Rng respects limits but not colimits.
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Respectively, other (non-identity) automorphisms are called nontrivial automorphisms. The exact definition of an automorphism depends on the type of "mathematical object" in question and what, precisely, constitutes an "isomorphism" of that object. The most general setting in which these words have meaning is an abstract branch of mathematics called category theory. Category theory deals with abstract objects and morphisms between those objects.
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A double vector bundle consists of (E, E^H, E^V, B), where # the side bundles E^H and E^V are vector bundles over the base B, # E is a vector bundle on both side bundles E^H and E^V, # the projection, the addition, the scalar multiplication and the zero map on E for both vector bundle structures are morphisms.
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In mathematics, and especially in category theory, a commutative diagram is a diagram of objects, also known as vertices, and morphisms, also known as arrows or edges, such that when selecting two objects any directed path through the diagram leads to the same result by composition. Commutative diagrams play the role in category theory that equations play in algebra. Hasse diagram.
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A homeomorphism is a bijection that is continuous and whose inverse is also continuous. Two spaces are called homeomorphic if there exists a homeomorphism between them. From the standpoint of topology, homeomorphic spaces are essentially identical. In category theory, Top, the category of topological spaces with topological spaces as objects and continuous functions as morphisms, is one of the fundamental categories.
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The composition of quasi- compact morphisms is quasi-compact. The base change of a quasi-compact morphism is quasi-compact. An affine scheme is quasi-compact. In fact, a scheme is quasi-compact if and only if it is a finite union of open affine subschemes. Serre’s criterion gives a necessary and sufficient condition for a quasi-compact scheme to be affine.
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Since the data of a (pre-)sheaf depends on the open subsets of the base space, sheaves on different topological spaces are unrelated to each other in the sense that there are no morphisms between them. However, given a continuous map f : X → Y between two topological spaces, pushforward and pullback relate sheaves on X to those on Y and vice versa.
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Because the hom-sets in a preadditive category have zero morphisms, the notion of kernel and cokernel make sense. That is, if f: A → B is a morphism in a preadditive category, then the kernel of f is the equaliser of f and the zero morphism from A to B, while the cokernel of f is the coequaliser of f and this zero morphism. Unlike with products and coproducts, the kernel and cokernel of f are generally not equal in a preadditive category. When specializing to the preadditive categories of abelian groups or modules over a ring, this notion of kernel coincides with the ordinary notion of a kernel of a homomorphism, if one identifies the ordinary kernel K of f: A → B with its embedding K → A. However, in a general preadditive category there may exist morphisms without kernels and/or cokernels.
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A graph consists of two sets, an edge set and a vertex set, and two functions s,t between those sets, assigning to every edge e its source s(e) and target t(e). Grph is thus equivalent to the functor category SetC, where C is the category with two objects E and V and two morphisms s,t: E -> V giving respectively the source and target of each edge. The Yoneda Lemma asserts that Cop embeds in SetC as a full subcategory. In the graph example the embedding represents Cop as the subcategory of SetC whose two objects are V' as the one-vertex no-edge graph and E' as the two-vertex one-edge graph (both as functors), and whose two nonidentity morphisms are the two graph homomorphisms from V' to E' (both as natural transformations).
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It is this vocabulary that makes the theory of doctrines worth while. For example, the 2-category Cat of categories, functors, and natural transformations is a doctrine. One sees immediately that all presheaf categories are categories of models. As another example, one may take the subcategory of Cat consisting only of categories with finite products as objects and product-preserving functors as 1-morphisms.
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Let C and D be categories. The collection of all functors from C to D forms the objects of a category: the functor category. Morphisms in this category are natural transformations between functors. Functors are often defined by universal properties; examples are the tensor product, the direct sum and direct product of groups or vector spaces, construction of free groups and modules, direct and inverse limits.
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For L a subset of B∗, a finite subset T of L is a test set for L if morphisms f and g on B∗ agree on L if and only if they agree on T. The Ehrenfeucht conjecture is that any subset L has a test set: it has been proved independently by Albert and Lawrence; McNaughton; and Guba. The proofs rely on Hilbert's basis theorem.
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FinSet is a full subcategory of Set, the category whose objects are all sets and whose morphisms are all functions. Like Set, FinSet is a large category. FinOrd is a full subcategory of FinSet as by the standard definition, suggested by John von Neumann, each ordinal is the well-ordered set of all smaller ordinals. Unlike Set and FinSet, FinOrd is a small category.
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They are often called A and B branes respectively. Morphisms in the categories are given by the massless spectrum of open strings stretching between two branes. The closed string A and B models only capture the so- called topological sector—a small portion of the full string theory. Similarly, the branes in these models are only topological approximations to the full dynamical objects that are D-branes.
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This is the composition law for morphisms in the cobordism category. Since functors are required to preserve composition, this says that the linear map corresponding to a sewn together morphism is just the composition of the linear map for each piece. There is an equivalence of categories between the category of 2-dimensional topological quantum field theories and the category of commutative Frobenius algebras.
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Nonetheless, definitions of connectedness often reflect the topological meaning in some way. For example, in category theory, a category is said to be connected if each pair of objects in it is joined by a sequence of morphisms. Thus, a category is connected if it is, intuitively, all one piece. There may be different notions of connectedness that are intuitively similar, but different as formally defined concepts.
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Taking history preserving maps as morphisms in the category of prefix orders leads to a notion of product that is not the Cartesian product of the two orders since the Cartesian product is not always a prefix order. Instead, it leads to an arbitrary interleaving of the original prefix orders. The union of two prefix orders is the disjoint union, as it is with partial orders.
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The pullback is similar to the product, but not the same. One may obtain the product by "forgetting" that the morphisms and exist, and forgetting that the object exists. One is then left with a discrete category containing only the two objects and , and no arrows between them. This discrete category may be used as the index set to construct the ordinary binary product.
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In mathematics, a 2-valued morphism. is a homomorphism that sends a Boolean algebra B onto the two-element Boolean algebra 2 = {0,1}. It is essentially the same thing as an ultrafilter on B, and, in a different way, also the same things as a maximal ideal of B. 2-valued morphisms have also been proposed as a tool for unifying the language of physics.
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It follows from the three defining coherence conditions that a large class of diagrams (i.e. diagrams whose morphisms are built using \alpha, \lambda, \rho, identities and tensor product) commute: this is Mac Lane's "coherence theorem". It is sometimes inaccurately stated that all such diagrams commute. There is a general notion of monoid object in a monoidal category, which generalizes the ordinary notion of monoid from abstract algebra.
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In arithmetic geometry, a Frobenioid is a category with some extra structure that generalizes the theory of line bundles on models of finite extensions of global fields. Frobenioids were introduced by . The word "Frobenioid" is a portmanteau of Frobenius and monoid, as certain Frobenius morphisms between Frobenioids are analogues of the usual Frobenius morphism, and some of the simplest examples of Frobenioids are essentially monoids.
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Michael Artin defined an algebraic space as the quotient of a scheme by the equivalence relations that define étale morphisms. Algebraic spaces retain many of the useful properties of schemes while simultaneously being more flexible. For instance, the Keel–Mori theorem can be used to show that many moduli spaces are algebraic spaces. More general than an algebraic space is a Deligne–Mumford stack.
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The tensor product of V-modules descends to a monoidal structure on Va-Mod. An almost module R ∈ Va-Mod with a map R ⊗ R → R satisfying natural conditions, similar to a definition of a ring, is called an almost V-algebra or an almost ring if the context is unambiguous. Many standard properties of algebras and morphisms between them carry to the "almost" world.
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In category theory, a discipline within mathematics, the nerve N(C) of a small category C is a simplicial set constructed from the objects and morphisms of C. The geometric realization of this simplicial set is a topological space, called the classifying space of the category C. These closely related objects can provide information about some familiar and useful categories using algebraic topology, most often homotopy theory.
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Formally, a diagram of type J in a category C is a (covariant) functor The category J is called the index category or the scheme of the diagram D; the functor is sometimes called a J-shaped diagram. The actual objects and morphisms in J are largely irrelevant; only the way in which they are interrelated matters. The diagram D is thought of as indexing a collection of objects and morphisms in C patterned on J. Although, technically, there is no difference between an individual diagram and a functor or between a scheme and a category, the change in terminology reflects a change in perspective, just as in the set theoretic case: one fixes the index category, and allows the functor (and, secondarily, the target category) to vary. One is most often interested in the case where the scheme J is a small or even finite category.
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To every Boolean algebra B we can associate a Stone space S(B) as follows: the elements of S(B) are the ultrafilters on B, and the topology on S(B), called the Stone topology, is generated by the sets of the form {F∈S(B) : b∈F}, where b is an element of B. Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to the Boolean algebra of clopen sets of the Stone space S(B); and furthermore, every Stone space X is homeomorphic to the Stone space belonging to the Boolean algebra of clopen sets of X. These assignments are functorial, and we obtain a category-theoretic duality between the category of Boolean algebras (with homomorphisms as morphisms) and the category of Stone spaces (with continuous maps as morphisms). Stone's theorem gave rise to a number of similar dualities, now collectively known as Stone dualities.
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Many groups are simultaneously groups and examples of other mathematical structures. In the language of category theory, they are group objects in a category, meaning that they are objects (that is, examples of another mathematical structure) which come with transformations (called morphisms) that mimic the group axioms. For example, every group (as defined above) is also a set, so a group is a group object in the category of sets.
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The resulting theory is now sometimes called Serre–Grothendieck–Verdier duality, and is a basic tool in algebraic geometry. A treatment of this theory, Residues and Duality (1966) by Robin Hartshorne, became a reference. One concrete spin-off was the Grothendieck residue. To go beyond proper morphisms, as for the versions of Poincaré duality that are not for closed manifolds, requires some version of the compact support concept.
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More generally, one can consider a category C enriched over the monoidal category of modules over a commutative ring , called an -linear category. In other words, each hom-set Hom(A,B) in C has the structure of an -module, and composition of morphisms is -bilinear. When considering functors between two -linear categories, one often restricts to those that are -linear, so those that induce -linear maps on each hom-set.
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220px In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism. A monomorphism from to is often denoted with the notation . In the more general setting of category theory, a monomorphism (also called a monic morphism or a mono) is a left-cancellative morphism. That is, an arrow such that for all objects and all morphisms , : f \circ g_1 = f \circ g_2 \implies g_1 = g_2.
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Any homomorphism of lattices is necessarily monotone with respect to the associated ordering relation; see Limit preserving function. The converse is not true: monotonicity by no means implies the required preservation of meets and joins (see Pic. 9), although an order-preserving bijection is a homomorphism if its inverse is also order-preserving. Given the standard definition of isomorphisms as invertible morphisms, a lattice isomorphism is just a bijective lattice homomorphism.
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Using natural transformations as morphisms, one can form the category of all representations of G in C. This is just the functor category CG. For another example, take C = Top, the category of topological spaces. A representation of G in Top is a topological space on which G acts continuously. An equivariant map is then a continuous map f : X → Y between representations which commutes with the action of G.
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150px In category theory, a branch of mathematics, a section is a right inverse of some morphism. Dually, a retraction is a left inverse of some morphism. In other words, if f : X → Y and g : Y → X are morphisms whose composition f o g : Y → Y is the identity morphism on Y, then g is a section of f, and f is a retraction of g.Mac Lane (1978, p.19).
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In other words, is squarefree if and only if h(w) is squarefree for all squarefree of length 3. It is possible to find a squarefree morphism by brute-force search. algorithm squarefree_morphism is output: a squarefree morphism with the lowest possible rank . set k = 3 while True do set k_sf_words to the list of all squarefree words of length over a ternary alphabet for each h(0) in k_sf_words do for each h(1) in k_sf_words do for each h(2) in k_sf_words do if h(1) = h(2) then break from the current loop (advance to next h(1)) if h(0) e h(1) and h(2) e h(0) then if h(w) is squarefree for all squarefree of length then return h(0), h(1), h(2) increment by Over a ternary alphabet, there are exactly 144 uniform squarefree morphisms of rank 11 and no uniform squarefree morphisms with a lower rank than 11.
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The Karoubi envelope construction associates to an arbitrary category C a category kar(C) together with a functor :s:C\rightarrow kar(C) such that the image s(p) of every idempotent p in C splits in kar(C). When applied to a preadditive category C, the Karoubi envelope construction yields a pseudo-abelian category kar(C) called the pseudo-abelian completion of C. Moreover, the functor :C\rightarrow kar(C) is in fact an additive morphism. To be precise, given a preadditive category C we construct a pseudo-abelian category kar(C) in the following way. The objects of kar(C) are pairs (X,p) where X is an object of C and p is an idempotent of X. The morphisms :f:(X,p)\rightarrow (Y,q) in kar(C) are those morphisms :f:X\rightarrow Y such that f=q\circ f = f \circ p in C. The functor :C\rightarrow kar(C) is given by taking X to (X,id_X).
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Suppose C is a small category (i.e. the objects and morphisms form a set rather than a proper class) and D is an arbitrary category. The category of functors from C to D, written as Fun(C, D), Funct(C,D), [C,D], or D ^C, has as objects the covariant functors from C to D, and as morphisms the natural transformations between such functors. Note that natural transformations can be composed: if \mu (X) : F(X) \to G(X) is a natural transformation from the functor F : C \to D to the functor G : C \to D, and \eta(X) : G(X) \to H(X) is a natural transformation from the functor G to the functor H, then the collection \eta(X)\mu(X) : F(X) \to H(X) defines a natural transformation from F to H. With this composition of natural transformations (known as vertical composition, see natural transformation), D^C satisfies the axioms of a category.
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An associative algebra over K is given by a K-vector space A endowed with a bilinear map A × A → A having two inputs (multiplicator and multiplicand) and one output (product), as well as a morphism K → A identifying the scalar multiples of the multiplicative identity. If the bilinear map A × A → A is reinterpreted as a linear map (i. e., morphism in the category of K-vector spaces) A ⊗ A → A (by the universal property of the tensor product), then we can view an associative algebra over K as a K-vector space A endowed with two morphisms (one of the form A ⊗ A → A and one of the form K → A) satisfying certain conditions that boil down to the algebra axioms. These two morphisms can be dualized using categorial duality by reversing all arrows in the commutative diagrams that describe the algebra axioms; this defines the structure of a coalgebra.
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For any scheme X the category Et(X) is the category of all étale morphisms from a scheme to X. It is an analogue of the category of open subsets of a topological space, and its objects can be thought of informally as "étale open subsets" of X. The intersection of two open sets of a topological space corresponds to the pullback of two étale maps to X. There is a rather minor set-theoretical problem here, since Et(X) is a "large" category: its objects do not form a set. However, it is equivalent to a small category because étale morphisms are locally of finite presentation, so it is harmless to pretend that it is a small category. A presheaf on a topological space X is a contravariant functor from the category of open subsets to sets. By analogy we define an étale presheaf on a scheme X to be a contravariant functor from Et(X) to sets.
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They occur in several branches of mathematics. For example, the functions from a set into itself form a monoid with respect to function composition. More generally, in category theory, the morphisms of an object to itself form a monoid, and, conversely, a monoid may be viewed as a category with a single object. In computer science and computer programming, the set of strings built from a given set of characters is a free monoid.
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Many definitions and theorems about monoids can be generalised to small categories with more than one object. For example, a quotient of a category with one object is just a quotient monoid. Monoids, just like other algebraic structures, also form their own category, Mon, whose objects are monoids and whose morphisms are monoid homomorphisms. There is also a notion of monoid object which is an abstract definition of what is a monoid in a category.
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A commonly applied technique in mathematics is to study objects carrying a particular structure by introducing a category whose morphisms preserve this structure. Then one may ask, when are two given objects isomorphic and ask for a "particularly nice" representative in each isomorphism class. The classification of algebraic varieties, i.e. application of this idea in the case of algebraic varieties, is very difficult due to the highly non-linear structure of the objects.
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One can view the fundamental group as a category; there is one object and the morphisms from it to itself are the elements of . The selection, for each in , of a continuous path from to , allows one to use concatenation to view any path in as a loop based at . This defines an equivalence of categories between and the fundamental groupoid of . More precisely, this exhibits as a skeleton of the fundamental groupoid of .
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In mathematics, the category Ord has preordered sets as objects and order- preserving functions as morphisms. This is a category because the composition of two order-preserving functions is order preserving and the identity map is order preserving. The monomorphisms in Ord are the injective order-preserving functions. The empty set (considered as a preordered set) is the initial object of Ord, and the terminal objects are precisely the singleton preordered sets.
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Saunders Mac Lane attempted to create a distinction between epimorphisms, which were maps in a concrete category whose underlying set maps were surjective, and epic morphisms, which are epimorphisms in the modern sense. However, this distinction never caught on. It is a common mistake to believe that epimorphisms are either identical to surjections or that they are a better concept. Unfortunately this is rarely the case; epimorphisms can be very mysterious and have unexpected behavior.
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The collection of all algebraic structures of a given type will usually be a proper class. Examples include the class of all groups, the class of all vector spaces, and many others. In category theory, a category whose collection of objects forms a proper class (or whose collection of morphisms forms a proper class) is called a large category. The surreal numbers are a proper class of objects that have the properties of a field.
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The product does not necessarily exist. For example, an empty product (i.e. is the empty set) is the same as a terminal object, and some categories, such as the category of infinite groups, do not have a terminal object: given any infinite group there are infinitely many morphisms , so cannot be terminal. If is a set such that all products for families indexed with exist, then one can treat each product as a functor .
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For a fixed semigroup S, the left S-acts are the objects of a category, denoted S-Act, whose morphisms are the S-homomorphisms. The corresponding category of right S-acts is sometimes denoted by Act-S. (This is analogous to the categories R-Mod and Mod-R of left and right modules over a ring.) For a monoid M, the categories M-Act and Act-M are defined in the same way.
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The category of small dg- categories can be endowed with a model category structure such that weak equivalences are those functors that induce an equivalence of derived categories. Given a dg-category C over some ring R, there is a notion of smoothness and properness of C that reduces to the usual notions of smooth and proper morphisms in case C is the category of quasi-coherent sheaves on some scheme X over R.
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In mathematics, the Yoneda lemma is arguably the most important result in category theory. It is an abstract result on functors of the type morphisms into a fixed object. It is a vast generalisation of Cayley's theorem from group theory (viewing a group as a miniature category with just one object and only isomorphisms). It allows the embedding of any category into a category of functors (contravariant set-valued functors) defined on that category.
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Given a left G-module M, it can be turned into a right G-module by defining a·g = g−1·a. A function f : M → N is called a morphism of G-modules (or a G-linear map, or a G-homomorphism) if f is both a group homomorphism and G-equivariant. The collection of left (respectively right) G-modules and their morphisms form an abelian category G-Mod (resp. Mod-G). The category G-Mod (resp.
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This gives rise to a number of useful categorical dualities between the categories of all complete semilattices with morphisms preserving all meets or joins, respectively. Another usage of "complete meet-semilattice" refers to a bounded complete cpo. A complete meet-semilattice in this sense is arguably the "most complete" meet-semilattice that is not necessarily a complete lattice. Indeed, a complete meet-semilattice has all non-empty meets (which is equivalent to being bounded complete) and all directed joins.
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This terminology is not completely fixed, as these terms are generally not formally defined, and can be considered to be jargon. These terms may have originated as a generalization of the process of making a geographical map, which consists of mapping the Earth surface to a sheet of paper. Maps may either be functions or morphisms, though the terms share some overlap. The term map may be used to distinguish some special types of functions, such as homomorphisms.
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Informally, category theory is a general theory of functions. Several terms used in category theory, including the term "morphism", are used differently from their uses in the rest of mathematics. In category theory, morphisms obey conditions specific to category theory itself. Samuel Eilenberg and Saunders Mac Lane introduced the concepts of categories, functors, and natural transformations from 1942–45 in their study of algebraic topology, with the goal of understanding the processes that preserve mathematical structure.
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The original example of an additive category is the category of abelian groups Ab. The zero object is the trivial group, the addition of morphisms is given pointwise, and biproducts are given by direct sums. More generally, every module category over a ring is additive, and so in particular, the category of vector spaces over a field is additive. The algebra of matrices over a ring, thought of as a category as described below, is also additive.
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The idea of homotopy can be turned into a formal category of category theory. The homotopy category is the category whose objects are topological spaces, and whose morphisms are homotopy equivalence classes of continuous maps. Two topological spaces X and Y are isomorphic in this category if and only if they are homotopy-equivalent. Then a functor on the category of topological spaces is homotopy invariant if it can be expressed as a functor on the homotopy category.
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Not all mathematical structures are F-algebras. For example, a poset P may be defined in categorical terms with a morphism s:P × P -> Ω, on a subobject classifier (Ω = {0,1} in the category of sets and s(x,y)=1 precisely when x≤y). The axioms restricting the morphism s to define a poset can be rewritten in terms of morphisms. However, as the codomain of s is Ω and not P, it is not an F-algebra.
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After introducing, via the Eilenberg–Steenrod axioms, the abstract approach to homology theory, he and Eilenberg originated category theory in 1945. He is especially known for his work on coherence theorems. A recurring feature of category theory, abstract algebra, and of some other mathematics as well, is the use of diagrams, consisting of arrows (morphisms) linking objects, such as products and coproducts. According to McLarty (2005), this diagrammatic approach to contemporary mathematics largely stems from Mac Lane (1948).
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There is a convenient relationship between the kernel and cokernel and the abelian group structure on the hom-sets. Given parallel morphisms f and g, the equaliser of f and g is just the kernel of g − f, if either exists, and the analogous fact is true for coequalisers. The alternative term "difference kernel" for binary equalisers derives from this fact. A preadditive category in which all biproducts, kernels, and cokernels exist is called pre-abelian.
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The presentation here is Verdier's own, and appears, complete with octahedral diagram, in . In the following diagram, u and v are the given morphisms, and the primed letters are the cones of various maps (chosen so that every exact triangle has an X, a Y, and a Z letter). Various arrows have been marked with [1] to indicate that they are of "degree 1"; e.g. the map from Z′ to X is in fact from Z′ to X[1].
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In 1952 he had the appellation "Federigo Enriques" attached to the Institute, to commemorate his memory. This name has been maintained by the Institute, and by the Department of Mathematics (which it became) since 1982. He was a major contributor to the Enciclopedia Italiana, and from 1946–1967 editor of the journal Il periodico di matematiche. The Chisini conjecture in algebraic geometry is a uniqueness question for morphisms of generic smooth projective surfaces, branched on a cuspidal curve.
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The first isomorphism theorem can be expressed in category theoretical language by saying that the category of groups is (normal epi, mono)-factorizable; in other words, the normal epimorphisms and the monomorphisms form a factorization system for the category. This is captured in the commutative diagram in the margin, which shows the objects and morphisms whose existence can be deduced from the morphism f : G \rightarrow H. The diagram shows that every morphism in the category of groups has a kernel in the category theoretical sense; the arbitrary morphism f factors into \iota \circ \pi, where ι is a monomorphism and π is an epimorphism (in a conormal category, all epimorphisms are normal). This is represented in the diagram by an object \ker f and a monomorphism \kappa: \ker f \rightarrow G (kernels are always monomorphisms), which complete the short exact sequence running from the lower left to the upper right of the diagram. The use of the exact sequence convention saves us from having to draw the zero morphisms from \ker f to H and G / \ker f.
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The dual notions of limits and cones are colimits and co-cones. Although it is straightforward to obtain the definitions of these by inverting all morphisms in the above definitions, we will explicitly state them here: A co-cone of a diagram F : J → C is an object N of C together with a family of morphisms :\psi_X:F(X) \to N for every object X of J, such that for every morphism f : X → Y in J, we have ψY ∘ F(f) = ψX. A colimit of a diagram F : J → C is a co-cone (L, \varphi) of F such that for any other co- cone (N, ψ) of F there exists a unique morphism u : L → N such that u o \varphiX = ψX for all X in J. A universal co-cone Colimits are also referred to as universal co-cones. They can be characterized as initial objects in the category of co-cones from F. As with limits, if a diagram F has a colimit then this colimit is unique up to a unique isomorphism.
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For morphisms to Spec(R), the fiber above the special point is the special fiber, an important concept for example in reduction modulo p, monodromy theory and other theories about degeneration. The generic fiber, equally, is the fiber above the generic point. Geometry of degeneration is largely then about the passage from generic to special fibers, or in other words how specialization of parameters affects matters. (For a discrete valuation ring the topological space in question is the Sierpinski space of topologists.
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In mathematics, particularly in homotopy theory, a model category is a category with distinguished classes of morphisms ('arrows') called 'weak equivalences', 'fibrations' and 'cofibrations'. These abstract from a conventional homotopy category of topological spaces or of chain complexes (derived category theory), via the acyclic model theorem. The concept was introduced by . In recent decades, the language of model categories has been used in some parts of algebraic K-theory and algebraic geometry, where homotopy-theoretic approaches led to deep results.
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When used to represent composition of functions (g \circ f)(x) \ = \ g(f(x)) however, the text sequence is reversed to illustrate the different operation sequences accordingly. The composition is defined in the same way for partial functions and Cayley's theorem has its analogue called the Wagner–Preston theorem. The category of sets with functions as morphisms is the prototypical category. The axioms of a category are in fact inspired from the properties (and also the definition) of function composition.
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If (E, M) is a factorization system, then the morphisms in M may be regarded as the embeddings, especially when the category is well powered with respect to M. Concrete theories often have a factorization system in which M consists of the embeddings in the previous sense. This is the case of the majority of the examples given in this article. As usual in category theory, there is a dual concept, known as quotient. All the preceding properties can be dualized.
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A morphism in an allegory is called a map if it is entire (1\subseteq R^\circ R) and deterministic (RR^\circ \subseteq 1). Another way of saying this is that a map is a morphism that has a right adjoint in when is considered, using the local order structure, as a 2-category. Maps in an allegory are closed under identity and composition. Thus, there is a subcategory of with the same objects but only the maps as morphisms.
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In mathematics, a bicategory (or a weak 2-category) is a concept in category theory used to extend the notion of category to handle the cases where the composition of morphisms is not (strictly) associative, but only associative up to an isomorphism. The notion was introduced in 1967 by Jean Bénabou. Bicategories may be considered as a weakening of the definition of 2-categories. A similar process for 3-categories leads to tricategories, and more generally to weak n-categories for n-categories.
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In mathematics, a quiver is a directed graph where loops and multiple arrows between two vertices are allowed, i.e. a multidigraph. They are commonly used in representation theory: a representation V of a quiver assigns a vector space V(x) to each vertex x of the quiver and a linear map V(a) to each arrow a. In category theory, a quiver can be understood to be the underlying structure of a category, but without composition or a designation of identity morphisms.
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Objects: all topological spaces, i.e., all pairs (X,T) of set X together with a collection T of subsets of X satisfying: # The empty set and X are in T. # The union of any collection of sets in T is also in T. # The intersection of any pair of sets in T is also in T. :The sets in T are the open sets. Morphisms: all ordinary continuous functions, i.e. all functions such that the inverse image of every open set is open.
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If both F and G are contravariant, the vertical arrows in this diagram are reversed. If \eta is a natural transformation from F to G , we also write \eta : F \to G or \eta : F \implies G . This is also expressed by saying the family of morphisms \eta_X: F(X) \to G(X) is natural in X . If, for every object X in C , the morphism \eta_X is an isomorphism in D , then \eta is said to be a ' (or sometimes natural equivalence' or isomorphism of functors).
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A span is a diagram of type \Lambda = (-1 \leftarrow 0 \rightarrow +1), i.e., a diagram of the form Y \leftarrow X \rightarrow Z. That is, let Λ be the category (-1 ← 0 → +1). Then a span in a category C is a functor S : Λ → C. This means that a span consists of three objects X, Y and Z of C and morphisms f : X → Y and g : X → Z: it is two maps with common domain. The colimit of a span is a pushout.
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There is a categorical picture of paths which is sometimes useful. Any topological space X gives rise to a category where the objects are the points of X and the morphisms are the homotopy classes of paths. Since any morphism in this category is an isomorphism this category is a groupoid, called the fundamental groupoid of X. Loops in this category are the endomorphisms (all of which are actually automorphisms). The automorphism group of a point x0 in X is just the fundamental group based at x0.
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In both cases the third class of morphisms is given by a lifting condition (see below). In some cases, when the category C is a Reedy category, there is a third model structure lying in between the projective and injective. The process of forcing certain maps to become weak equivalences in a new model category structure on the same underlying category is known as Bousfield localization. For example, the category of simplicial sheaves can be obtained as a Bousfield localization of the model category of simplicial presheaves.
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The vertices of the graph are the right cosets Hg = { hg : h in H } for g in G. The edges of the graph are of the form (Hg,Hgxi). The Cayley graph of the group G with {xi : i in I} is the Schreier coset graph for H = {1G} . A spanning tree of a Schreier coset graph corresponds to a Schreier transversal, as in Schreier's subgroup lemma . The book "Categories and Groupoids" listed below relates this to the theory of covering morphisms of groupoids.
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Given a topological space , the path-connected components of are naturally encoded in its fundamental groupoid; the observation is that and are in the same path-connected component of if and only if the collection of equivalence classes of continuous paths from to is nonempty. In categorical terms, the assertion is that the objects and are in the same groupoid component if and only if the set of morphisms from to is nonempty.Spanier, section 1.7; Theorem 9. Suppose that is path-connected, and fix an element of .
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One can also consider categories where the objects are D-branes and the morphisms between two branes \alpha and \beta are states of open strings stretched between \alpha and \beta.Zaslow 2008, p. 536 A cross section of a Calabi–Yau manifold In one version of string theory known as the topological B-model, the D-branes are complex submanifolds of certain six-dimensional shapes called Calabi–Yau manifolds, together with additional data that arise physically from having charges at the endpoints of strings.Zaslow 2008, p.
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Obstructions to extending local sections may be generalized in the following manner: take a topological space and form a category whose objects are open subsets, and morphisms are inclusions. Thus we use a category to generalize a topological space. We generalize the notion of a "local section" using sheaves of abelian groups, which assigns to each object an abelian group (analogous to local sections). There is an important distinction here: intuitively, local sections are like "vector fields" on an open subset of a topological space.
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This is a category with a collection of objects A, B, C and collection of morphisms denoted f, g, , and the loops are the identity arrows. This category is typically denoted by a boldface 3. In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object.
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Multisets appeared explicitly in the work of Richard Dedekind. Other mathematicians formalized multisets and began to study them as precise mathematical structures in the 20th century. For example, Whitney (1933) described generalized sets ("sets" whose characteristic functions may take any integer value - positive, negative or zero). Monro (1987) investigated the category Mul of multisets and their morphisms, defining a multiset as a set with an equivalence relation between elements "of the same sort", and a morphism between multisets as a function which respects sorts.
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The typical diagram of the definition of a universal morphism. In category theory, a branch of mathematics, a universal property is an important property which is satisfied by a universal morphism (see Formal Definition). Universal morphisms can also be thought of more abstractly as initial or terminal objects of a comma category (see Connection with Comma Categories). Universal properties occur almost everywhere in mathematics, and hence the precise category theoretic concept helps point out similarities between different branches of mathematics, some of which may even seem unrelated.
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More generally, bundles or bundle objects can be defined in any category: in a category C, a bundle is simply an epimorphism π: E → B. If the category is not concrete, then the notion of a preimage of the map is not necessarily available. Therefore these bundles may have no fibers at all, although for sufficiently well behaved categories they do; for instance, for a category with pullbacks and a terminal object 1 the points of B can be identified with morphisms p:1→B and the fiber of p is obtained as the pullback of p and π. The category of bundles over B is a subcategory of the slice category (C↓B) of objects over B, while the category of bundles without fixed base object is a subcategory of the comma category (C↓C) which is also the functor category C², the category of morphisms in C. The category of smooth vector bundles is a bundle object over the category of smooth manifolds in Cat, the category of small categories. The functor taking each manifold to its tangent bundle is an example of a section of this bundle object.
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In mathematics, the category of magmas, denoted Mag, is the category whose objects are magmas (that is, sets equipped with a binary operation), and whose morphisms are magma homomorphisms. The category Mag has direct products, so the concept of a magma object (internal binary operation) makes sense. (As in any category with direct products.) There is an inclusion functor: as trivial magmas, with operations given by projection: . An important property is that an injective endomorphism can be extended to an automorphism of a magma extension, just the colimit of the (constant sequence of the) endomorphism.
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In order theory, arbitrary meets can be expressed in terms of arbitrary joins and vice versa (for details, see completeness (order theory)). In effect, this means that it is sufficient to require the existence of either all meets or all joins to obtain the class of all complete lattices. As a consequence, some authors use the terms complete meet-semilattice or complete join-semilattice as another way to refer to complete lattices. Though similar on objects, the terms entail different notions of homomorphism, as will be explained in the below section on morphisms.
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A morphism of varieties is finite if the inverse image of every point is finite and the morphism is proper. A morphism of varieties is birational if it restricts to an isomorphism between dense open subsets. So, for example, the cuspidal cubic curve X in the affine plane A2 defined by x2 = y3 is not normal, because there is a finite birational morphism A1 → X (namely, t maps to (t3, t2)) which is not an isomorphism. By contrast, the affine line A1 is normal: it cannot be simplified any further by finite birational morphisms.
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Every pre-abelian category is of course an additive category, and many basic properties of these categories are described under that subject. This article concerns itself with the properties that hold specifically because of the existence of kernels and cokernels. Although kernels and cokernels are special kinds of equalisers and coequalisers, a pre- abelian category actually has all equalisers and coequalisers. We simply construct the equaliser of two morphisms f and g as the kernel of their difference g − f; similarly, their coequaliser is the cokernel of their difference.
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In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any category. The way they are put together is specified by a system of homomorphisms (group homomorphism, ring homomorphism, or in general morphisms in the category) between those smaller objects. The direct limit of the objects A_i, where i ranges over some directed set I, is denoted by \varinjlim A_i .
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However, the objects of a category need not be sets, and the arrows need not be functions. Any way of formalising a mathematical concept such that it meets the basic conditions on the behaviour of objects and arrows is a valid category—and all the results of category theory apply to it. The "arrows" of category theory are often said to represent a process connecting two objects, or in many cases a "structure- preserving" transformation connecting two objects. There are, however, many applications where much more abstract concepts are represented by objects and morphisms.
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In the category of unital rings, there are no kernels in the category-theoretic sense; indeed, this category does not even have zero morphisms. Nevertheless, there is still a notion of kernel studied in ring theory that corresponds to kernels in the category of non-unital rings. In the category of pointed topological spaces, if f : X → Y is a continuous pointed map, then the preimage of the distinguished point, K, is a subspace of X. The inclusion map of K into X is the categorical kernel of f.
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Homomorphisms are also used in the study of formal languagesSeymour Ginsburg, Algebraic and automata theoretic properties of formal languages, North-Holland, 1975, , and are often briefly referred to as morphisms.T. Harju, J. Karhumӓki, Morphisms in Handbook of Formal Languages, Volume I, edited by G. Rozenberg, A. Salomaa, Springer, 1997, . Given alphabets Σ1 and Σ2, a function such that for all u and v in Σ1∗ is called a homomorphism on Σ1∗.The ∗ denotes the Kleene star operation, while Σ∗ denotes the set of words formed from the alphabet Σ, including the empty word.
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Given a quasi-category C, one can associate to it an ordinary category hC, called the homotopy category of C. The homotopy category has as objects the vertices of C. The morphisms are given by homotopy classes of edges between vertices. Composition is given using the horn filler condition for n = 2\. For a general simplicial set there is a functor \tau_1 from sSet to Cat, known as the fundamental category functor, and for a quasi-category C the fundamental category is the same as the homotopy category, i.e. \tau_1(C)=hC.
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The use of institutions makes it possible to develop concepts of specification languages (like structuring of specifications, parameterization, implementation, refinement, development), proof calculi and even tools in a way completely independent of the underlying logical system. There are also morphisms that allow to relate and translate logical systems. Important applications of this are re-use of logical structure (also called borrowing), heterogeneous specification and combination of logics. The spread of institutional model theory has generalized various notions and results of model theory, and institutions themselves have impacted the progress of universal logic.
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The algebraic approach to graph rewriting is based upon category theory. The algebraic approach is further divided into sub-approaches, the most common of which are the double-pushout (DPO) approach and the single-pushout (SPO) approach. Other sub-approaches include the sesqui-pushout and the pullback approach. From the perspective of the DPO approach a graph rewriting rule is a pair of morphisms in the category of graphs and graph homomorphisms between them: r = (L \leftarrow K \rightarrow R) (or L \supseteq K \subseteq R) where K \rightarrow L is injective.
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In algebraic geometry, dévissage is a technique introduced by Alexander Grothendieck for proving statements about coherent sheaves on noetherian schemes. Dévissage is an adaptation of a certain kind of noetherian induction. It has many applications, including the proof of generic flatness and the proof that higher direct images of coherent sheaves under proper morphisms are coherent. Laurent Gruson and Michel Raynaud extended this concept to the relative situation, that is, to the situation where the scheme under consideration is not necessarily noetherian, but instead admits a finitely presented morphism to another scheme.
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The codensity monad of a functor G: D \to C is defined to be the right Kan extension of G along itself, provided that this Kan extension exists. Thus, by definition it is in particular a functor :T^G : C \to C. The monad structure on T^G stems from the universal property of the right Kan extension. The codensity monad exists whenever D is a small category (has only a set, as opposed to a proper class, of morphisms) and C possesses all (small, i.e., set-indexed) limits.
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Infinity- categories are a variant of classical categories where composition of morphisms is not uniquely defined, but only up to contractible choice. In general, it does not make sense to say that a diagram commutes strictly in an infinity-category, but only that it commutes up to coherent homotopy. One can define an infinity-category of spectra (as done by Lurie). One can also define infinity-versions of (commutative) monoids and then define A_\infty-ring spectra as monoids in spectra and E_\infty-ring spectra as commutative monoids in spectra.
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One alternative proposal is the theory of derivators proposed in Pursuing stacks by Grothendieck in the 80spg 191, and later developed in the 90s in his manuscript on the topic. Essentially, these are a system of homotopy categories given by the diagram categories I \to M for a category with a class of weak equivalences (M, W). These categories are then related by the morphisms of diagrams I \to J. This formalism has the advantage of being able to recover the homotopy limits and colimits, which replaces the cone construction.
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When lattices with more structure are considered, the morphisms should "respect" the extra structure, too. In particular, a bounded-lattice homomorphism (usually called just "lattice homomorphism") f between two bounded lattices L and M should also have the following property: : f(0L) = 0M , and : f(1L) = 1M . In the order-theoretic formulation, these conditions just state that a homomorphism of lattices is a function preserving binary meets and joins. For bounded lattices, preservation of least and greatest elements is just preservation of join and meet of the empty set.
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The construction of free groups is a common and illuminating example. Let F : Set → Grp be the functor assigning to each set Y the free group generated by the elements of Y, and let G : Grp → Set be the forgetful functor, which assigns to each group X its underlying set. Then F is left adjoint to G: Initial morphisms. For each set Y, the set GFY is just the underlying set of the free group FY generated by Y. Let \eta_Y:Y\to GFY be the set map given by "inclusion of generators".
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Commutative diagram for the set product X1×X2. A category-theoretic product A × B in a category of sets represents the set of ordered pairs, with the first element coming from A and the second coming from B. In this context the characteristic property above is a consequence of the universal property of the product and the fact that elements of a set X can be identified with morphisms from 1 (a one element set) to X. While different objects may have the universal property, they are all naturally isomorphic.
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In algebraic topology, the fundamental group π1(X,x) of a pointed topological space (X,x) is defined as the group of homotopy classes of loops based at x. This definition works well for spaces such as real and complex manifolds, but gives undesirable results for an algebraic variety with the Zariski topology. In the classification of covering spaces, it is shown that the fundamental group is exactly the group of deck transformations of the universal covering space. This is more promising: finite étale morphisms are the appropriate analogue of covering spaces.
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The narrator describes Reena and her interactions with individuals in the city, including a man looking for the subway station. All individuals share a common connection - an ability to share spaces and take up space within the entity of the city scape. _Chapter 16_ The narrator goes off on another tangent, reflecting on body morphisms "the body is just one part of a body" and tangent stories about "Indians". _Chapter 17_ Maris heavily edits Reena's photos while the narrator describes aspects of Maris’ life, including childhood memories and internalized thoughts, making the narrator omnipresent.
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In category theory, a branch of mathematics, a subterminal object is an object X of a category C with the property that every object of C has at most one morphism into X. If X is subterminal, then the pair of identity morphisms (1X, 1X) makes X into the product of X and X. If C has a terminal object 1, then an object X is subterminal if and only if it is a subobject of 1, hence the name. The category of categories with subterminal objects and functors preserving them is not accessible.
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In representation theory, the category of representations of some algebraic structure has the representations of as objects and equivariant maps as morphisms between them. One of the basic thrusts of representation theory is to understand the conditions under which this category is semisimple; i.e., whether an object decomposes into simple objects (see Maschke's theorem for the case of finite groups). The Tannakian formalism gives conditions under which a group G may be recovered from the category of representations of it together with the forgetful functor to the category of vector spaces.
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Formally, each of the following definitions defines a concrete category, and every pair of these categories can be shown to be concretely isomorphic. This means that for every pair of categories defined below, there is an isomorphism of categories, for which corresponding objects have the same underlying set and corresponding morphisms are identical as set functions. To actually establish the concrete isomorphisms is more tedious than illuminating. The simplest approach is probably to construct pairs of inverse concrete isomorphisms between each category and the category of topological spaces Top.
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If X and Y are topoi, a geometric morphism is a pair of adjoint functors (u∗,u∗) (where u∗ : Y → X is left adjoint to u∗ : X → Y) such that u∗ preserves finite limits. Note that u∗ automatically preserves colimits by virtue of having a right adjoint. By Freyd's adjoint functor theorem, to give a geometric morphism X → Y is to give a functor u∗: Y → X that preserves finite limits and all small colimits. Thus geometric morphisms between topoi may be seen as analogues of maps of locales.
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Limits and colimits in a category C are defined by means of diagrams in C. Formally, a diagram of shape J in C is a functor from J to C: :F:J\to C. The category J is thought of as an index category, and the diagram F is thought of as indexing a collection of objects and morphisms in C patterned on J. One is most often interested in the case where the category J is a small or even finite category. A diagram is said to be small or finite whenever J is.
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Let A be a Grothendieck category (an AB5 category with a generator), G a generator of A and R be the ring of endomorphisms of G; also, let S be the functor from A to Mod-R (the category of right R-modules) defined by S(X) = Hom(G,X). Then the Gabriel–Popescu theorem states that S is full and faithful and has an exact left adjoint. This implies that A is equivalent to the Serre quotient category of Mod-R by a certain localizing subcategory C. (A localizing subcategory of Mod-R is a full subcategory C of Mod-R, closed under arbitrary direct sums, such that for any short exact sequence of modules 0\rarr M_1\rarr M_2\rarr M_3\rarr 0, we have M2 in C if and only if M1 and M3 are in C. The Serre quotient of Mod-R by any localizing subcategory is a Grothendieck category.) We may take C to be the kernel of the left adjoint of the functor S. Note that the embedding S of A into Mod-R is left-exact but not necessarily right-exact: cokernels of morphisms in A do not in general correspond to the cokernels of the corresponding morphisms in Mod-R.
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Two functors F and G are called naturally isomorphic or simply isomorphic if there exists a natural isomorphism from F to G. An infranatural transformation \eta from F to G is simply a family of morphisms \eta_X : F(X) \to G(X) , for all X in C. Thus a natural transformation is an infranatural transformation for which \eta_Y \circ F(f) = G(f) \circ \eta_X for every morphism f : X \to Y . The naturalizer of \eta , nat (\eta) , is the largest subcategory of C containing all the objects of C on which \eta restricts to a natural transformation.
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Like many categories, the category Top is a concrete category, meaning its objects are sets with additional structure (i.e. topologies) and its morphisms are functions preserving this structure. There is a natural forgetful functor :U : Top -> Set to the category of sets which assigns to each topological space the underlying set and to each continuous map the underlying function. The forgetful functor U has both a left adjoint :D : Set -> Top which equips a given set with the discrete topology, and a right adjoint :I : Set -> Top which equips a given set with the indiscrete topology.
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In a so- called concrete category, the objects are associated with mathematical structures like sets, magmas, groups, rings, topological spaces, vector spaces, metric spaces, partial orders, differentiable manifolds, uniform spaces, etc., and morphisms between two objects are associated with structure- preserving functions between them. In the examples above, these would be functions, magma homomorphisms, group homomorphisms, ring homomorphisms, continuous functions, linear transformations (or matrices), metric maps, monotonic functions, differentiable functions, and uniformly continuous functions, respectively. As an algebraic theory, one of the advantages of category theory is to enable one to prove many general results with a minimum of assumptions.
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Hence, the challenge is to define special objects without referring to the internal structure of those objects. To define the empty set without referring to elements, or the product topology without referring to open sets, one can characterize these objects in terms of their relations to other objects, as given by the morphisms of the respective categories. Thus, the task is to find universal properties that uniquely determine the objects of interest. Numerous important constructions can be described in a purely categorical way if the category limit can be developed and dualized to yield the notion of a colimit.
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Like all type classes, arrows can be thought of as a set of qualities that can be applied to any data type. In the Haskell programming language, arrows allow functions (represented in Haskell by `->` symbol) to combine in a reified form. However, the actual term "arrow" may also come from the fact that some (but not all) arrows correspond to the morphisms (also known as "arrows" in category theory) of different Kleisli categories. As a relatively new concept, there is not a single, standard definition, but all formulations are logically equivalent, feature some required methods, and strictly obey certain mathematical laws.
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Viewing a graph as a category whose objects are the vertices and whose morphisms are the paths in the graph, the cartesian product of graphs corresponds to the funny tensor product of categories. The cartesian product of graphs is one of two graph products that turn the category of graphs and graph homomorphisms into a symmetric closed monoidal category (as opposed to merely symmetric monoidal), the other being the tensor product of graphs. The internal hom [G, H] for the cartesian product of graphs has graph homomorphisms from G to H as vertices and "unnatural transformations" between them as edges.
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Because every hom-set Hom(A,B) is an abelian group, it has a zero element 0. This is the zero morphism from A to B. Because composition of morphisms is bilinear, the composition of a zero morphism and any other morphism (on either side) must be another zero morphism. If you think of composition as analogous to multiplication, then this says that multiplication by zero always results in a product of zero, which is a familiar intuition. Extending this analogy, the fact that composition is bilinear in general becomes the distributivity of multiplication over addition.
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In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets (or sometimes to another category, see Relative concreteness below). This functor makes it possible to think of the objects of the category as sets with additional structure, and of its morphisms as structure-preserving functions. Many important categories have obvious interpretations as concrete categories, for example the category of topological spaces and the category of groups, and trivially also the category of sets itself. On the other hand, the homotopy category of topological spaces is not concretizable, i.e.
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In mathematics, a multiplicative character (or linear character, or simply character) on a group G is a group homomorphism from G to the multiplicative group of a field , usually the field of complex numbers. If G is any group, then the set Ch(G) of these morphisms forms an abelian group under pointwise multiplication. This group is referred to as the character group of G. Sometimes only unitary characters are considered (characters whose image is in the unit circle); other such homomorphisms are then called quasi-characters. Dirichlet characters can be seen as a special case of this definition.
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A filter of the Boolean algebra A is a subset p such that for all x, y in p we have x ∧ y in p and for all a in A we have a ∨ x in p. The dual of a maximal (or prime) ideal in a Boolean algebra is ultrafilter. Ultrafilters can alternatively be described as 2-valued morphisms from A to the two-element Boolean algebra. The statement every filter in a Boolean algebra can be extended to an ultrafilter is called the Ultrafilter Theorem and cannot be proven in ZF, if ZF is consistent.
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In category theory, a subobject classifier is a special object Ω of a category such that, intuitively, the subobjects of any object X in the category correspond to the morphisms from X to Ω. In typical examples, that morphism assigns "true" to the elements of the subobject and "false" to the other elements of X. Therefore, a subobject classifier is also known as a "truth value object" and the concept is widely used in the categorical description of logic. Note however that subobject classifiers are often much more complicated than the simple binary logic truth values {true, false}.
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Suppose that (X, 𝒜) and (Y, ℬ) are bounded structures. A map f : X -> Y is called locally bounded or just bounded if the image under f of every 𝒜-bounded set is a ℬ-bounded set; that is, if for every A ∈ 𝒜, f(A) ∈ ℬ. Since the composition of two locally bounded map is again locally bounded, it is clear that the class of all bounded structures forms a category whose morphisms are bounded maps. An isomorphism in this category is called a bornomorphism and it is a bijective locally bounded map whose inverse is also locally bounded.
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A more sophisticated approach is to enrich the classification by remembering the isomorphisms. More precisely, on any base B one can consider the category of families on B with only isomorphisms between families taken as morphisms. One then considers the fibred category which assigns to any space B the groupoid of families over B. The use of these categories fibred in groupoids to describe a moduli problem goes back to Grothendieck (1960/61). In general, they cannot be represented by schemes or even algebraic spaces, but in many cases, they have a natural structure of an algebraic stack.
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Algebraic varieties are locally defined as the common zero sets of polynomials and since polynomials over the complex numbers are holomorphic functions, algebraic varieties over C can be interpreted as analytic spaces. Similarly, regular morphisms between varieties are interpreted as holomorphic mappings between analytic spaces. Somewhat surprisingly, it is often possible to go the other way, to interpret analytic objects in an algebraic way. For example, it is easy to prove that the analytic functions from the Riemann sphere to itself are either the rational functions or the identically infinity function (an extension of Liouville's theorem).
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Foundations for the many relations between the two theories were put in place during the early part of the 1950s, as part of the business of laying the foundations of algebraic geometry to include, for example, techniques from Hodge theory. The major paper consolidating the theory was Géometrie Algébrique et Géométrie Analytique by Jean-Pierre Serre, now usually referred to as GAGA. It proves general results that relate classes of algebraic varieties, regular morphisms and sheaves with classes of analytic spaces, holomorphic mappings and sheaves. It reduces all of these to the comparison of categories of sheaves.
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In the mathematical field of category theory, an allegory is a category that has some of the structure of the category Rel of sets and binary relations between them. Allegories can be used as an abstraction of categories of relations, and in this sense the theory of allegories is a generalization of relation algebra to relations between different sorts. Allegories are also useful in defining and investigating certain constructions in category theory, such as exact completions. In this article we adopt the convention that morphisms compose from right to left, so means "first do , then do ".
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As stated earlier, an adjunction between categories C and D gives rise to a family of universal morphisms, one for each object in C and one for each object in D. Conversely, if there exists a universal morphism to a functor G : C → D from every object of D, then G has a left adjoint. However, universal constructions are more general than adjoint functors: a universal construction is like an optimization problem; it gives rise to an adjoint pair if and only if this problem has a solution for every object of D (equivalently, every object of C).
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In higher-dimensional algebra (HDA), a double groupoid is a generalisation of a one-dimensional groupoid to two dimensions, and the latter groupoid can be considered as a special case of a category with all invertible arrows, or morphisms. Double groupoids are often used to capture information about geometrical objects such as higher-dimensional manifolds (or n-dimensional manifolds). In general, an n-dimensional manifold is a space that locally looks like an n-dimensional Euclidean space, but whose global structure may be non-Euclidean. Double groupoids were first introduced by Ronald Brown in 1976, in ref.
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The five lemma is often applied to long exact sequences: when computing homology or cohomology of a given object, one typically employs a simpler subobject whose homology/cohomology is known, and arrives at a long exact sequence which involves the unknown homology groups of the original object. This alone is often not sufficient to determine the unknown homology groups, but if one can compare the original object and sub object to well- understood ones via morphisms, then a morphism between the respective long exact sequences is induced, and the five lemma can then be used to determine the unknown homology groups.
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Two linear maps S and T in induce the same map between P(V) and P(W) if and only if they differ by a scalar multiple, that is if for some . Thus if one identifies the scalar multiples of the identity map with the underlying field K, the set of K-linear morphisms from P(V) to P(W) is simply . The automorphisms can be described more concretely. (We deal only with automorphisms preserving the base field K). Using the notion of sheaves generated by global sections, it can be shown that any algebraic (not necessarily linear) automorphism must be linear, i.e.
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The collection of all G-modules is a category (the morphisms are group homomorphisms f with the property f(gx) = g(f(x)) for all g in G and x in M). Sending each module M to the group of invariants M^G yields a functor from the category of G-modules to the category Ab of abelian groups. This functor is left exact but not necessarily right exact. We may therefore form its right derived functors.This uses that the category of G-modules has enough injectives, since it is isomorphic to the category of all modules over the group ring \Z[G].
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A far-reaching generalization of the theorem as formulated above can be stated as follows: the inclusion of pro-reductive groups into all linear algebraic groups, where morphisms G \to H in both categories are taken up to conjugation by elements in H(k), admits a left adjoint, the so-called pro-reductive envelope. This left adjoint sends the additive group G_a to SL_2 (which happens to be semi-simple, as opposed to pro-reductive), thereby recovering the above form of Jacobson-Morozov. This generalized Jacobson- Morozov theorem was proven by by appealing to methods related to Tannakian categories and by by more geometric methods.
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That is, the image is the kernel of the cokernel, and the coimage is the cokernel of the kernel. Note that this notion of image may not correspond to the usual notion of image, or range, of a function, even assuming that the morphisms in the category are functions. For example, in the category of topological abelian groups, the image of a morphism actually corresponds to the inclusion of the closure of the range of the function. For this reason, people will often distinguish the meanings of the two terms in this context, using "image" for the abstract categorical concept and "range" for the elementary set-theoretic concept.
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In mathematics, the category of manifolds, often denoted Manp, is the category whose objects are manifolds of smoothness class Cp and whose morphisms are p-times continuously differentiable maps. This is a category because the composition of two Cp maps is again continuous and of class Cp. One is often interested only in Cp-manifolds modeled on spaces in a fixed category A, and the category of such manifolds is denoted Manp(A). Similarly, the category of Cp-manifolds modeled on a fixed space E is denoted Manp(E). One may also speak of the category of smooth manifolds, Man∞, or the category of analytic manifolds, Manω.
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One often considers continuous group actions: the group G is a topological group, X is a topological space, and the map is continuous with respect to the product topology of . The space X is also called a G-space in this case. This is indeed a generalization, since every group can be considered a topological group by using the discrete topology. All the concepts introduced above still work in this context, however we define morphisms between G-spaces to be continuous maps compatible with the action of G. The quotient X/G inherits the quotient topology from X, and is called the quotient space of the action.
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To wit, there is a category of connective spaces consisting of sets with collections of connected subsets satisfying connectivity axioms; their morphisms are those functions which map connected sets to connected sets . Topological spaces and graphs are special cases of connective spaces; indeed, the finite connective spaces are precisely the finite graphs. However, every graph can be canonically made into a topological space, by treating vertices as points and edges as copies of the unit interval (see topological graph theory#Graphs as topological spaces). Then one can show that the graph is connected (in the graph theoretical sense) if and only if it is connected as a topological space.
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The objects are the topological vector spaces over and the morphisms are the continuous -linear maps from one object to another. :Definition: A TVS homomorphism or topological homomorphism is a continuous linear map between topological vector spaces (TVSs) such that the induced map is an open mapping when , which is the range or image of , is given the subspace topology induced by Y. :Definition: A TVS embedding or a topological monomorphism is an injective topological homomorphism. Equivalently, a TVS-embedding is a linear map that is also a topological embedding. :Definition: A TVS isomorphism or an isomorphism in the category of TVSs is a bijective linear homeomorphism.
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In mathematics, the category of medial magmas, also known as the medial category, and denoted Med, is the category whose objects are medial magmas (that is, sets with a medial binary operation), and whose morphisms are magma homomorphisms (which are equivalent to homomorphisms in the sense of universal algebra). The category Med has direct products, so the concept of a medial magma object (internal binary operation) makes sense. As a result, Med has all its objects as medial objects, and this characterizes it. There is an inclusion functor from Set to Med as trivial magmas, with operations being the right projections : (x, y) -> y.
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That is, F is an étale sheaf if and only if the following condition is true. Suppose that is an object of Ét(X) and that is a jointly surjective family of étale morphisms over X. For each i, choose a section xi of F over Ui. The projection map , which is loosely speaking the inclusion of the intersection of Ui and Uj in Ui, induces a restriction map . If for all i and j the restrictions of xi and xj to are equal, then there must exist a unique section x of F over U which restricts to xi for all i. Suppose that X is a Noetherian scheme.
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Using the correspondence between n-simplices of a simplicial set X and morphisms \Delta^n \to X (a consequence of the Yoneda lemma), this definition can be written in terms of simplices. The image of the map fs: \Lambda_k^n \to Y can be thought of as a horn as described above. Asking that fs factors through yi corresponds to requiring that there is an n-simplex in Y whose faces make up the horn from fs (together with one other face). Then the required map x: \Delta^n\to X corresponds to a simplex in X whose faces include the horn from s.
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Here is an analogy: with the Taylor series method from calculus, you can approximate the shape of a smooth function f around a point x by using a sequence of increasingly accurate polynomial functions. In a similar way, with the calculus of functors method, you can approximate the behavior of certain kind of functor F at a particular object X by using a sequence of increasingly accurate polynomial functors. To be specific, let M be a smooth manifold and let O(M) be the category of open subspaces of M—i.e. the category where the objects are the open subspaces of M, and the morphisms are inclusion maps.
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In the mathematical theory of categories, a sketch is a category D, together with a set of cones intended to be limits and a set of cocones intended to be colimits. A model of the sketch in a category C is a functor :M:D\rightarrow C that takes each specified cone to a limit cone in C and each specified cocone to a colimit cocone in C. Morphisms of models are natural transformations. Sketches are a general way of specifying structures on the objects of a category, forming a category-theoretic analog to the logical concept of a theory and its models. They allow multisorted models and models in any category.
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Putting these together, the octahedral axiom asserts the "third isomorphism theorem": :(Z/X)/(Y/X)\cong Z/Y. If the triangulated category is the derived category D(A) of an abelian category A, and X, Y, Z are objects of A viewed as complexes concentrated in degree 0, and the maps X\to Y and Y\to Z are monomorphisms in A, then the cones of these morphisms in D(A) are actually isomorphic to the quotients above in A. Finally, formulates the octahedral axiom using a two-dimensional commutative diagram with 4 rows and 4 columns. also give generalizations of the octahedral axiom.
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Like polynomials, rational expressions can also be generalized to n indeterminates X1,..., Xn, by taking the field of fractions of F[X1,..., Xn], which is denoted by F(X1,..., Xn). An extended version of the abstract idea of rational function is used in algebraic geometry. There the function field of an algebraic variety V is formed as the field of fractions of the coordinate ring of V (more accurately said, of a Zariski-dense affine open set in V). Its elements f are considered as regular functions in the sense of algebraic geometry on non-empty open sets U, and also may be seen as morphisms to the projective line.
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A map between two representations (\rho, V_\rho),\, (\tau, V_\tau) of the same group G is a linear map T: V_\rho\to V_\tau, with the property that \tau(s)\circ T=T\circ\rho(s) holds for all s\in G. In other words, the following diagram commutes for all s\in G: :200px Such a map is also called G–linear, or an equivariant map. The kernel, the image and the cokernel of T are defined by default. The composition of equivariant maps is again an equivariant map. There is a category of representations with equivariant maps as its morphisms.
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At Carnegie Mellon University, Scott proposed the theory of equilogical spaces as a successor theory to domain theory; among its many advantages, the category of equilogical spaces is a cartesian closed category, whereas the category of domainsWhere here Dana Scott counts the category of domains to be the category whose objects are pointed directed-complete partial orders (DCPOs), and whose morphisms are the strict, Scott-continuous functions is not. In 1994, he was inducted as a Fellow of the Association for Computing Machinery. In 2012 he became a fellow of the American Mathematical Society.List of Fellows of the American Mathematical Society, retrieved 2013-07-14.
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Let , and be morphisms of a category containing objects A, B, C and D. By repeated composition, we can construct a morphism from A to D in two ways: :, and :. We have now the following coherence statement: :. In these two particular examples, the coherence statements are theorems for the case of an abstract category, since they follow directly from the axioms; in fact, they are axioms. For the case of a concrete mathematical structure, they can be viewed as conditions, namely as requirements for the mathematical structure under consideration to be a concrete category, requirements that such a structure may meet or fail to meet.
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In the mathematical field of group theory, a group is residually X (where X is some property of groups) if it "can be recovered from groups with property X". Formally, a group G is residually X if for every non-trivial element g there is a homomorphism h from G to a group with property X such that h(g) eq e. More categorically, a group is residually X if it embeds into its pro-X completion (see profinite group, pro-p group), that is, the inverse limit of the inverse system consisting of all morphisms \phi\colon G \to H from G to some group H with property X.
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The following outline is provided as an overview of and guide to category theory, the area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows (also called morphisms, although this term also has a specific, non category-theoretical sense), where these collections satisfy certain basic conditions. Many significant areas of mathematics can be formalised as categories, and the use of category theory allows many intricate and subtle mathematical results in these fields to be stated, and proved, in a much simpler way than without the use of categories.
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Stone duality provides a category theoretic duality between Boolean algebras and a class of topological spaces known as Boolean spaces. Building on nascent ideas of relational semantics (later formalized by Kripke) and a result of R. S. Pierce, Jónsson, Tarski and G. Hansoul extended Stone duality to Boolean algebras with operators by equipping Boolean spaces with relations that correspond to the operators via a power set construction. In the case of interior algebras the interior (or closure) operator corresponds to a pre-order on the Boolean space. Homomorphisms between interior algebras correspond to a class of continuous maps between the Boolean spaces known as pseudo-epimorphisms or p-morphisms for short.
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Every Grothendieck topos is an elementary topos, but the converse is not true (since every Grothendieck topos is cocomplete, which is not required from an elementary topos). The categories of finite sets, of finite G-sets (actions of a group G on a finite set), and of finite graphs are elementary topoi that are not Grothendieck topoi. If C is a small category, then the functor category SetC (consisting of all covariant functors from C to sets, with natural transformations as morphisms) is a topos. For instance, the category Grph of graphs of the kind permitting multiple directed edges between two vertices is a topos.
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For a given diagram F : J → C and functor G : C → D, if both F and GF have specified limits there is a unique canonical morphism :\tau_F : G \lim F \to \lim GF which respects the corresponding limit cones. The functor G preserves the limits of F if and only this map is an isomorphism. If the categories C and D have all limits of shape J then lim is a functor and the morphisms τF form the components of a natural transformation :\tau:G \lim \to \lim G^J. The functor G preserves all limits of shape J if and only if τ is a natural isomorphism.
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If F and G are functors between the categories C and D , then a natural transformation \eta from F to G is a family of morphisms that satisfies two requirements. # The natural transformation must associate, to every object X in C, a morphism \eta_X : F(X) \to G(X) between objects of D . The morphism \eta_X is called the component of \eta at X . # Components must be such that for every morphism f :X \to Y in C we have: :::\eta_Y \circ F(f) = G(f) \circ \eta_X The last equation can conveniently be expressed by the commutative diagram This is the commutative diagram which is part of the definition of a natural transformation between two functors.
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Like many categories, the category Manp is a concrete category, meaning its objects are sets with additional structure (i.e. a topology and an equivalence class of atlases of charts defining a Cp-differentiable structure) and its morphisms are functions preserving this structure. There is a natural forgetful functor :U : Manp -> Top to the category of topological spaces which assigns to each manifold the underlying topological space and to each p-times continuously differentiable function the underlying continuous function of topological spaces. Similarly, there is a natural forgetful functor :U′ : Manp -> Set to the category of sets which assigns to each manifold the underlying set and to each p-times continuously differentiable function the underlying function.
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Entailment as external implication between two terms expresses a metatruth outside the language of the logic, and is considered part of the metalanguage. Even when the logic under study is intuitionistic, entailment is ordinarily understood classically as two-valued: either the left side entails, or is less-or-equal to, the right side, or it is not. Similar but more complex translations to and from algebraic logics are possible for natural deduction systems as described above and for the sequent calculus. The entailments of the latter can be interpreted as two-valued, but a more insightful interpretation is as a set, the elements of which can be understood as abstract proofs organized as the morphisms of a category.
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The homotopy category of a model category C is the localization of C with respect to the class of weak equivalences. This definition of homotopy category does not depend on the choice of fibrations and cofibrations. However, the classes of fibrations and cofibrations are useful in describing the homotopy category in a different way and in particular avoiding set-theoretic issues arising in general localizations of categories. More precisely, the "fundamental theorem of model categories" states that the homotopy category of C is equivalent to the category whose objects are the objects of C which are both fibrant and cofibrant, and whose morphisms are left homotopy classes of maps (equivalently, right homotopy classes of maps) as defined above.
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In particular, it forms a category; the objects are taken to be the points of and the collection of morphisms from to is the collection of equivalence classes given above. The fact that this satisfies the definition of a category amounts to the standard fact that the equivalence class of the concatenation of two paths only depends on the equivalence classes of the individual paths.Spanier, section 1.7; Lemma 6 and Theorem 7. Likewise, the fact that this category is a groupoid, which asserts that every morphism is invertible, amounts to the standard fact that one can reverse the orientation of a path, and the equivalence class of the resulting concatenation contains the constant path.
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These terms come from the notion of covariant and contravariant functors in category theory. Consider the category C whose objects are types and whose morphisms represent the subtype relationship ≤. (This is an example of how any partially ordered set can be considered as a category.) Then for example the function type constructor takes two types p and r and creates a new type p → r; so it takes objects in C^2 to objects in C. By the subtyping rule for function types this operation reverses ≤ for the first parameter and preserves it for the second, so it is a contravariant functor in the first parameter and a covariant functor in the second.
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Grothendieck's relative point of view is a heuristic applied in certain abstract mathematical situations, with a rough meaning of taking for consideration families of 'objects' explicitly depending on parameters, as the basic field of study, rather than a single such object. It is named after Alexander Grothendieck, who made extensive use of it in treating foundational aspects of algebraic geometry. Outside that field, it has been influential particularly on category theory and categorical logic. In the usual formulation, the language of category theory is applied, to describe the point of view as treating, not objects X of a given category C as such, but morphisms :f: X -> S where S is a fixed object.
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In mathematics, Hurwitz's automorphisms theorem bounds the order of the group of automorphisms, via orientation-preserving conformal mappings, of a compact Riemann surface of genus g > 1, stating that the number of such automorphisms cannot exceed 84(g − 1). A group for which the maximum is achieved is called a Hurwitz group, and the corresponding Riemann surface a Hurwitz surface. Because compact Riemann surfaces are synonymous with non-singular complex projective algebraic curves, a Hurwitz surface can also be called a Hurwitz curve.Technically speaking, there is an equivalence of categories between the category of compact Riemann surfaces with the orientation-preserving conformal maps and the category of non-singular complex projective algebraic curves with the algebraic morphisms.
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It came to prominence when René Thom showed that cobordism groups could be computed by means of homotopy theory, via the Thom complex construction. Cobordism theory became part of the apparatus of extraordinary cohomology theory, alongside K-theory. It performed an important role, historically speaking, in developments in topology in the 1950s and early 1960s, in particular in the Hirzebruch–Riemann–Roch theorem, and in the first proofs of the Atiyah–Singer index theorem. In the 1980s the category with compact manifolds as objects and cobordisms between these as morphisms played a basic role in the Atiyah–Segal axioms for topological quantum field theory, which is an important part of quantum topology.
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Lawvere, Quantifiers and Sheaves ;Internal languages: This can be seen as a formalization and generalization of proof by diagram chasing. One defines a suitable internal language naming relevant constituents of a category, and then applies categorical semantics to turn assertions in a logic over the internal language into corresponding categorical statements. This has been most successful in the theory of toposes, where the internal language of a topos together with the semantics of intuitionistic higher-order logic in a topos enables one to reason about the objects and morphisms of a topos "as if they were sets and functions". This has been successful in dealing with toposes that have "sets" with properties incompatible with classical logic.
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In algebraic geometry and commutative algebra, a ring homomorphism f:A\to B is called formally smooth (from French: Formellement lisse) if it satisfies the following infinitesimal lifting property: Suppose B is given the structure of an A-algebra via the map f. Given a commutative A-algebra, C, and a nilpotent ideal N\subseteq C, any A-algebra homomorphism B\to C/N may be lifted to an A-algebra map B \to C. If moreover any such lifting is unique, then f is said to be formally étale. Formally smooth maps were defined by Alexander Grothendieck in Éléments de géométrie algébrique IV. For finitely presented morphisms, formal smoothness is equivalent to usual notion of smoothness.
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The visualization of orders with Hasse diagrams has a straightforward generalization: instead of displaying lesser elements below greater ones, the direction of the order can also be depicted by giving directions to the edges of a graph. In this way, each order is seen to be equivalent to a directed acyclic graph, where the nodes are the elements of the poset and there is a directed path from a to b if and only if a ≤ b. Dropping the requirement of being acyclic, one can also obtain all preorders. When equipped with all transitive edges, these graphs in turn are just special categories, where elements are objects and each set of morphisms between two elements is at most singleton.
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Many properties of morphisms can be restated in terms of points. For example, a map f is said to be a monomorphism if : For all maps g, h, f \circ g = f \circ h implies g = h. Suppose f \colon B \to C and g, h \colon A \to B in C. Then g and h are A-valued points of B, and therefore monomorphism is equivalent to the more familiar statement : f is a monomorphism if it is an injective function on points of B. Some care is necessary. f is an epimorphism if the dual condition holds: : For all maps g, h (of some suitable type), g \circ f = h \circ f implies g = h.
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In a category , a relation between objects and is a span of morphisms X\gets R\to Y that is jointly monic. Two such spans X\gets S\to Y and X\gets T\to Y are considered equivalent when there is an isomorphism between and that make everything commute; strictly speaking, relations are only defined up to equivalence (one may formalise this either by using equivalence classes or by using bicategories). If the category has products, a relation between and is the same thing as a monomorphism into (or an equivalence class of such). In the presence of pullbacks and a proper factorization system, one can define the composition of relations.
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Equivariance can be formalized using the concept of a -set for a group . This is a mathematical object consisting of a mathematical set and a group action (on the left) of on . If and are both -sets for the same group , then a function is said to be equivariant if : for all and all .. If one or both of the actions are right actions the equivariance condition may be suitably modified: :; (right-right) :; (right- left) :; (left-right) Equivariant maps are homomorphisms in the category of G-sets (for a fixed G).. Hence they are also known as G-morphisms, G-maps,. or G-homomorphisms.. Isomorphisms of G-sets are simply bijective equivariant maps.
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Thus any continuous function f from a space X to a space Y defines an inverse mapping f −1 from Ω(Y) to Ω(X). Furthermore, it is easy to check that f −1 (like any inverse image map) preserves finite intersections and arbitrary unions and therefore is a morphism of frames. If we define Ω(f) = f −1 then Ω becomes a contravariant functor from the category Top to the category Frm of frames and frame morphisms. Using the tools of category theory, the task of finding a characterization of topological spaces in terms of their open set lattices is equivalent to finding a functor from Frm to Top which is adjoint to Ω.
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Objects: all pairs (X,T) of set X together with a collection T of subsets of X satisfying: # The empty set and X are in T. # The intersection of any collection of sets in T is also in T. # The union of any pair of sets in T is also in T. :The sets in T are the closed sets. Morphisms: all functions such that the inverse image of every closed set is closed. Comments: This is the category that results by replacing each lattice of open sets in a topological space by its order-theoretic dual of closed sets, the lattice of complements of open sets. The relation between the two definitions is given by De Morgan's laws.
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In algebraic K-theory and homotopy theory there are several notions of categories equipped with some specified classes of morphisms. If C has a structure of an exact category, then by defining we(C) to be isomorphisms, co(C) to be admissible monomorphisms, one obtains a structure of a Waldhausen category on C. Both kinds of structure may be used to define K-theory of C, using the Q-construction for an exact structure and S-construction for a Waldhausen structure. An important fact is that the resulting K-theory spaces are homotopy equivalent. If C is a model category with a zero object, then the full subcategory of cofibrant objects in C may be given a Waldhausen structure.
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In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of abelian groups, Ab. The theory originated in an effort to unify several cohomology theories by Alexander Grothendieck and independently in the slightly earlier work of David Buchsbaum. Abelian categories are very stable categories; for example they are regular and they satisfy the snake lemma. The class of abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an abelian category, or the category of functors from a small category to an abelian category are abelian as well.
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While these matrices are rather degenerate, they do need to be included to get an additive category, since an additive category must have a zero object. Thinking about such matrices can be useful in one way, though: they highlight the fact that given any objects and in an additive category, there is exactly one morphism from to 0 (just as there is exactly one 0-by-1 matrix with entries in ) and exactly one morphism from 0 to (just as there is exactly one 1-by-0 matrix with entries in ) – this is just what it means to say that 0 is a zero object. Furthermore, the zero morphism from to is the composition of these morphisms, as can be calculated by multiplying the degenerate matrices.
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A coequalizer is a colimit of the diagram consisting of two objects X and Y and two parallel morphisms f, g : X → Y. More explicitly, a coequalizer can be defined as an object Q together with a morphism q : Y → Q such that q ∘ f = q ∘ g. Moreover, the pair (Q, q) must be universal in the sense that given any other such pair (Q′, q′) there exists a unique morphism u : Q → Q′ such that u ∘ q = q′. This information can be captured by the following commutative diagram: Image:Coequalizer-01.png As with all universal constructions, a coequalizer, if it exists, is unique up to a unique isomorphism (this is why, by abuse of language, one sometimes speaks of "the" coequalizer of two parallel arrows).
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Morphisms in monoidal categories can also be drawn as string diagrams since a strict monoidal category can be seen as a 2-category with only one object (there will therefore be only one type of planar region) and Mac Lane's strictification theorem states that any monoidal category is monoidally equivalent to a strict one. The graphical language of string diagrams for monoidal categories may be extended to represent expressions in categories with other structure, such as braided monoidal categories, dagger categories, etc. and is related to geometric presentations for braided monoidal categories and ribbon categories. In quantum computing, there are several diagrammatic languages based on string diagrams for reasoning about linear maps between qubits, the most well-known of which is the ZX-calculus.
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If R is a Noetherian local ring, the dimension of the tangent space is at least the dimension of R: :dim m/m2 ≧ dim R R is called regular if equality holds. In a more geometric parlance, when R is the local ring of a variety V in v, one also says that v is a regular point. Otherwise it is called a singular point. The tangent space has an interpretation in terms of homomorphisms to the dual numbers for K, :K[t]/t2: in the parlance of schemes, morphisms Spec K[t]/t2 to a scheme X over K correspond to a choice of a rational point x ∈ X(k) and an element of the tangent space at x.
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Unlike Grothendieck's definition, however, the Q-construction builds a category, not an abelian group, and unlike Segal's Γ-objects, the Q-construction works directly with short exact sequences. If C is an abelian category, then QC is a category with the same objects as C but whose morphisms are defined in terms of short exact sequences in C. The K-groups of the exact category are the homotopy groups of ΩBQC, the loop space of the geometric realization (taking the loop space corrects the indexing). Quillen additionally proved his " theorem" that his two definitions of K-theory agreed with each other. This yielded the correct K0 and led to simpler proofs, but still did not yield any negative K-groups.
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The category Ring is a concrete category meaning that the objects are sets with additional structure (addition and multiplication) and the morphisms are functions that preserve this structure. There is a natural forgetful functor :U : Ring → Set for the category of rings to the category of sets which sends each ring to its underlying set (thus "forgetting" the operations of addition and multiplication). This functor has a left adjoint :F : Set → Ring which assigns to each set X the free ring generated by X. One can also view the category of rings as a concrete category over Ab (the category of abelian groups) or over Mon (the category of monoids). Specifically, there are forgetful functors :A : Ring → Ab :M : Ring → Mon which "forget" multiplication and addition, respectively.
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It is not true in general, however, that all monomorphisms must be injective in other categories; that is, there are settings in which the morphisms are functions between sets, but one can have a function that is not injective and yet is a monomorphism in the categorical sense. For example, in the category Div of divisible (abelian) groups and group homomorphisms between them there are monomorphisms that are not injective: consider, for example, the quotient map , where Q is the rationals under addition, Z the integers (also considered a group under addition), and Q/Z is the corresponding quotient group. This is not an injective map, as for example every integer is mapped to 0. Nevertheless, it is a monomorphism in this category.
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In algebraic geometry, the term branched covering is used to describe morphisms f from an algebraic variety V to another one W, the two dimensions being the same, and the typical fibre of f being of dimension 0. In that case, there will be an open set W' of W (for the Zariski topology) that is dense in W, such that the restriction of f to W' (from V' = f^{-1}(W') to W', that is) is unramified. Depending on the context, we can take this as local homeomorphism for the strong topology, over the complex numbers, or as an étale morphism in general (under some slightly stronger hypotheses, on flatness and separability). Generically, then, such a morphism resembles a covering space in the topological sense.
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In category theory, the concept of an element, or a point, generalizes the more usual set theoretic concept of an element of a set to an object of any category. This idea often allows restating of definitions or properties of morphisms (such as monomorphism or product) given by a universal property in more familiar terms, by stating their relation to elements. Some very general theorems, such as Yoneda's lemma and the Mitchell embedding theorem, are of great utility for this, by allowing one to work in a context where these translations are valid. This approach to category theory, in particular the use of the Yoneda lemma in this way, is due to Grothendieck, and is often called the method of the functor of points.
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It is a simple exercise in topology to see that every three elements of a Puppe sequence are, up to a homotopy, of the form: : X\to Y\to C(f). By "up to a homotopy", we mean here that every 3 elements in a Puppe sequence are of the above form if regarded as objects and morphisms in the homotopy category. If one is now given a topological half-exact functor, the above property implies that, after acting with the functor in question on the Puppe sequence associated to A\to B, one obtains a long exact sequence. A result, due to John Milnor,John Milnor "Construction of Universal Bundles I" (1956) Annals of Mathematics, 63 pp. 272-284.
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The composition X\gets R\to Y\gets S\to Z is found by first pulling back the cospan R\to Y\gets S and then taking the jointly-monic image of the resulting span X\gets R\gets\bullet\to S\to Z. Composition of relations will be associative if the factorization system is appropriately stable. In this case, one can consider a category , with the same objects as , but where morphisms are relations between the objects. The identity relations are the diagonals X \to X\times X. A regular category (a category with finite limits and images in which covers are stable under pullback) has a stable regular epi/mono factorization system. The category of relations for a regular category is always an allegory.
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The traditional terminology also included differentials of the second kind and of the third kind. The idea behind this has been supported by modern theories of algebraic differential forms, both from the side of more Hodge theory, and through the use of morphisms to commutative algebraic groups. The Weierstrass zeta function was called an integral of the second kind in elliptic function theory; it is a logarithmic derivative of a theta function, and therefore has simple poles, with integer residues. The decomposition of a (meromorphic) elliptic function into pieces of 'three kinds' parallels the representation as (i) a constant, plus (ii) a linear combination of translates of the Weierstrass zeta function, plus (iii) a function with arbitrary poles but no residues at them.
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On the one hand σ-completeness is too weak to characterize inverse image maps (completeness is required), on the other hand it is too restrictive for a generalization. (Sikorski remarked on using non-σ- complete homomorphisms but included σ-completeness in his axioms for closure algebras.) Later J. Schmid defined a continuous homomorphism or continuous morphism for interior algebras as a Boolean homomorphism f between two interior algebras satisfying f(xC) ≤ f(x)C. This generalizes the forward image map of a continuous map - the image of a closure is contained in the closure of the image. This construction is covariant but not suitable for category theoretic applications as it only allows construction of continuous morphisms from continuous maps in the case of bijections. (C.
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Category theorists will often think of the ring R and the category R as two different representations of the same thing, so that a particularly perverse category theorist might define a ring as a preadditive category with exactly one object (in the same way that a monoid can be viewed as a category with only one object—and forgetting the additive structure of the ring gives us a monoid). In this way, preadditive categories can be seen as a generalisation of rings. Many concepts from ring theory, such as ideals, Jacobson radicals, and factor rings can be generalized in a straightforward manner to this setting. When attempting to write down these generalizations, one should think of the morphisms in the preadditive category as the "elements" of the "generalized ring".
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Equivariant maps can be generalized to arbitrary categories in a straightforward manner. Every group G can be viewed as a category with a single object (morphisms in this category are just the elements of G). Given an arbitrary category C, a representation of G in the category C is a functor from G to C. Such a functor selects an object of C and a subgroup of automorphisms of that object. For example, a G-set is equivalent to a functor from G to the category of sets, Set, and a linear representation is equivalent to a functor to the category of vector spaces over a field, VectK. Given two representations, ρ and σ, of G in C, an equivariant map between those representations is simply a natural transformation from ρ to σ.
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The congruence lattice Con A of an algebra A is an algebraic lattice. The (∨,0)-semilattice of compact elements of Con A is denoted by Conc A, and it is sometimes called the congruence semilattice of A. Then Con A is isomorphic to the ideal lattice of Conc A. By using the classical equivalence between the category of all (∨,0)-semilattices and the category of all algebraic lattices (with suitable definitions of morphisms), as it is outlined here, we obtain the following semilattice-theoretical formulation of CLP. \---- Semilattice- theoretical formulation of CLP: Is every distributive (∨,0)-semilattice isomorphic to the congruence semilattice of some lattice? \---- Say that a distributive (∨,0)-semilattice is representable, if it is isomorphic to Conc L, for some lattice L. So CLP asks whether every distributive (∨,0)-semilattice is representable.
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The Curtis–Hedlund–Lyndon theorem is a mathematical characterization of cellular automata in terms of their symbolic dynamics. It is named after Morton L. Curtis, Gustav A. Hedlund, and Roger Lyndon; in his 1969 paper stating the theorem, Hedlund credited Curtis and Lyndon as co-discoverers.. It has been called "one of the fundamental results in symbolic dynamics".. The theorem states that a function from a shift space to itself represents the transition function of a one-dimensional cellular automaton if and only if it is continuous (with respect to the Cantor topology) and equivariant (with respect to the shift map). More generally, it asserts that the morphisms between any two shift spaces (i.e., continuous mappings that commute with the shift) are exactly those mappings which can be defined uniformly by a local rule.
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In general, all constructions of algebraic topology are functorial; the notions of category, functor and natural transformation originated here. Fundamental groups and homology and cohomology groups are not only invariants of the underlying topological space, in the sense that two topological spaces which are homeomorphic have the same associated groups, but their associated morphisms also correspond — a continuous mapping of spaces induces a group homomorphism on the associated groups, and these homomorphisms can be used to show non-existence (or, much more deeply, existence) of mappings. One of the first mathematicians to work with different types of cohomology was Georges de Rham. One can use the differential structure of smooth manifolds via de Rham cohomology, or Čech or sheaf cohomology to investigate the solvability of differential equations defined on the manifold in question.
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This is an example of a branched covering of Riemann surfaces. This intuition also serves to define ramification in algebraic number theory. Given a (necessarily finite) extension of number fields F / E, a prime ideal p of OE generates the ideal pOF of OF. This ideal may or may not be a prime ideal, but, according to the Lasker–Noether theorem (see above), always is given by :pOF = q1e1 q2e2 ... qmem with uniquely determined prime ideals qi of OF and numbers (called ramification indices) ei. Whenever one ramification index is bigger than one, the prime p is said to ramify in F. The connection between this definition and the geometric situation is delivered by the map of spectra of rings Spec OF → Spec OE. In fact, unramified morphisms of schemes in algebraic geometry are a direct generalization of unramified extensions of number fields.
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Fix an arbitrary field k and let Fields/k denote the category of finitely generated field extensions of k with inclusions as morphisms. Consider a (covariant) functor F : Fields/k → Set. For a field extension K/k and an element a of F(K/k) a field of definition of a is an intermediate field K/L/k such that a is contained in the image of the map F(L/k) → F(K/k) induced by the inclusion of L in K. The essential dimension of a, denoted by ed(a), is the least transcendence degree (over k) of a field of definition for a. The essential dimension of the functor F, denoted by ed(F), is the supremum of ed(a) taken over all elements a of F(K/k) and objects K/k of Fields/k.
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If the composition fg of two morphisms is an epimorphism, then f must be an epimorphism. As some of the above examples show, the property of being an epimorphism is not determined by the morphism alone, but also by the category of context. If D is a subcategory of C, then every morphism in D that is an epimorphism when considered as a morphism in C is also an epimorphism in D. However the converse need not hold; the smaller category can (and often will) have more epimorphisms. As for most concepts in category theory, epimorphisms are preserved under equivalences of categories: given an equivalence F : C → D, a morphism f is an epimorphism in the category C if and only if F(f) is an epimorphism in D. A duality between two categories turns epimorphisms into monomorphisms, and vice versa.
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The Mayer–Vietoris sequence is such an approach, giving partial information about the (co)homology groups of any space by relating it to the (co)homology groups of two of its subspaces and their intersection. The most natural and convenient way to express the relation involves the algebraic concept of exact sequences: sequences of objects (in this case groups) and morphisms (in this case group homomorphisms) between them such that the image of one morphism equals the kernel of the next. In general, this does not allow (co)homology groups of a space to be completely computed. However, because many important spaces encountered in topology are topological manifolds, simplicial complexes, or CW complexes, which are constructed by piecing together very simple patches, a theorem such as that of Mayer and Vietoris is potentially of broad and deep applicability.
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A point of a topos X is defined as a geometric morphism from the topos of sets to X. If X is an ordinary space and x is a point of X, then the functor that takes a sheaf F to its stalk Fx has a right adjoint (the "skyscraper sheaf" functor), so an ordinary point of X also determines a topos-theoretic point. These may be constructed as the pullback-pushforward along the continuous map x: 1 → X. More precisely, those are the global points. They are not adequate in themselves for displaying the space-like aspect of a topos, because a non- trivial topos may fail to have any. Generalized points are geometric morphisms from a topos Y (the stage of definition) to X. There are enough of these to display the space-like aspect.
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In another group of dualities, the objects of one theory are translated into objects of another theory and the maps between objects in the first theory are translated into morphisms in the second theory, but with direction reversed. Using the parlance of category theory, this amounts to a contravariant functor between two categories and : which for any two objects X and Y of C gives a map That functor may or may not be an equivalence of categories. There are various situations, where such a functor is an equivalence between the opposite category of , and . Using a duality of this type, every statement in the first theory can be translated into a "dual" statement in the second theory, where the direction of all arrows has to be reversed.. Therefore, any duality between categories and is formally the same as an equivalence between and ( and ).
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Indeed, there is an equivalence of categories between the category of smooth projective algebraic curves over R (with regular maps as morphisms) and the category of compact connected Klein surfaces. This is akin to the correspondence between smooth projective algebraic curves over the complex numbers and compact connected Riemann surfaces. (Note that the algebraic curves considered here are abstract curves: integral, separated one-dimensional schemes of finite type over R. Such a curve need not have any R-rational points (like the curve X2+Y2+1=0 over R), in which case its Klein surface will have empty boundary.) There is also a one-to-one correspondence between compact connected Klein surfaces (up to equivalence) and algebraic function fields in one variable over R (up to R-isomorphism). This correspondence is akin to the one between compact connected Riemann surfaces and algebraic function fields over the complex numbers.
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The function field of an algebraic variety of dimension n over k is an algebraic function field of n variables over k. Two varieties are birationally equivalent if and only if their function fields are isomorphic. (But note that non-isomorphic varieties may have the same function field!) Assigning to each variety its function field yields a duality (contravariant equivalence) between the category of varieties over k (with dominant rational maps as morphisms) and the category of algebraic function fields over k. (The varieties considered here are to be taken in the scheme sense; they need not have any k-rational points, like the curve defined over the reals, that is with .) The case n = 1 (irreducible algebraic curves in the scheme sense) is especially important, since every function field of one variable over k arises as the function field of a uniquely defined regular (i.e.
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Cobordisms are objects of study in their own right, apart from cobordism classes. Cobordisms form a category whose objects are closed manifolds and whose morphisms are cobordisms. Roughly speaking, composition is given by gluing together cobordisms end-to-end: the composition of (W; M, N) and (W′; N, P) is defined by gluing the right end of the first to the left end of the second, yielding (W′ ∪N W; M, P). A cobordism is a kind of cospan:While every cobordism is a cospan, the category of cobordisms is not a "cospan category": it is not the category of all cospans in "the category of manifolds with inclusions on the boundary", but rather a subcategory thereof, as the requirement that M and N form a partition of the boundary of W is a global constraint. M → W ← N. The category is a dagger compact category.
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Objects: all pairs (X,N) of set X together with a neighbourhood function N : X → F(X), where F(X) denotes the set of all filters on X, satisfying for every x in X: #If U is in N(x), then x is in U. #If U is in N(x), then there exists V in N(x) such that U is in N(y) for all y in V. Morphisms: all neighbourhood- preserving functions, i.e., all functions f : (X, N) → (Y, N') such that if V is in N(f(x)), then there exists U in N(x) such that f(U) is contained in V. This is equivalent to asking that whenever V is in N(f(x)), then f−1(V) is in N(x). Comments: This definition axiomatizes the notion of neighbourhood. We say that U is a neighbourhood of x if U is in N(x).
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Let F : J → C be a diagram of shape J in a category C. A cone to F is an object N of C together with a family ψX : N → F(X) of morphisms indexed by the objects X of J, such that for every morphism f : X → Y in J, we have F(f) ∘ ψX = ψY. A limit of the diagram F : J → C is a cone (L, \varphi) to F such that for every other cone (N, ψ) to F there exists a unique morphism u : N → L such that \varphiX ∘ u = ψX for all X in J. A universal cone One says that the cone (N, ψ) factors through the cone (L, \varphi) with the unique factorization u. The morphism u is sometimes called the mediating morphism. Limits are also referred to as universal cones, since they are characterized by a universal property (see below for more information).
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Let G be a finite group (in fact everything will work verbatim for a profinite group). Then for any two finite G-sets X and Y we can define an equivalence relation among spans of G-sets of the form X\leftarrow U \rightarrow Y where two spans X\leftarrow U \rightarrow Y and X\leftarrow W \rightarrow Yare equivalent if and only if there is a G-equivariant bijection of U and W commuting with the projection maps to X and Y. This set of equivalence classes form naturally a monoid under disjoint union; we indicate with A(G)(X,Y) the group completion of that monoid. Taking pullbacks induces natural maps A(G)(X,Y)\times A(G)(Y,Z)\rightarrow A(G)(X,Z). Finally we can define the Burnside category A(G) of G as the category whose objects are finite G-sets and the morphisms spaces are the groups A(G)(X,Y).
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These generators and relations define K(X), and they imply that it is the universal way to assign invariants to vector bundles in a way compatible with exact sequences. Grothendieck took the perspective that the Riemann–Roch theorem is a statement about morphisms of varieties, not the varieties themselves. He proved that there is a homomorphism from K(X) to the Chow groups of X coming from the Chern character and Todd class of X. Additionally, he proved that a proper morphism to a smooth variety Y determines a homomorphism called the pushforward. This gives two ways of determining an element in the Chow group of Y from a vector bundle on X: Starting from X, one can first compute the pushforward in K-theory and then apply the Chern character and Todd class of Y, or one can first apply the Chern character and Todd class of X and then compute the pushforward for Chow groups.
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If the elements of B are viewed as "propositions about some object", then a 2-valued morphism on B can be interpreted as representing a particular "state of that object", namely the one where the propositions of B which are mapped to 1 are true, and the propositions mapped to 0 are false. Since the morphism conserves the Boolean operators (negation, conjunction, etc.), the set of true propositions will not be inconsistent but will correspond to a particular maximal conjunction of propositions, denoting the (atomic) state. (The true propositions form an ultrafilter, the false propositions form a maximal ideal, as mentioned above.) The transition between two states s1 and s2 of B, represented by 2-valued morphisms, can then be represented by an automorphism f from B to B, such that s2 o f = s1. The possible states of different objects defined in this way can be conceived as representing potential events.
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This is an initial morphism from Y to G, because any set map from Y to the underlying set GW of some group W will factor through \eta_Y:Y\to GFY via a unique group homomorphism from FY to W. This is precisely the universal property of the free group on Y. Terminal morphisms. For each group X, the group FGX is the free group generated freely by GX, the elements of X. Let \varepsilon_X:FGX\to X be the group homomorphism which sends the generators of FGX to the elements of X they correspond to, which exists by the universal property of free groups. Then each (GX,\varepsilon_X) is a terminal morphism from F to X, because any group homomorphism from a free group FZ to X will factor through \varepsilon_X:FGX\to X via a unique set map from Z to GX. This means that (F,G) is an adjoint pair. Hom-set adjunction.
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More generally, for a scheme X over a commutative ring R and any commutative R-algebra S, the set X(S) of S-points of X means the set of morphisms Spec(S) → X over Spec(R). The scheme X is determined up to isomorphism by the functor S ↦ X(S); this is the philosophy of identifying a scheme with its functor of points. Another formulation is that the scheme X over R determines a scheme XS over S by base change, and the S-points of X (over R) can be identified with the S-points of XS (over S). The theory of Diophantine equations traditionally meant the study of integral points, meaning solutions of polynomial equations in the integers Z rather than the rationals Q. For homogeneous polynomial equations such as x3 \+ y3 = z3, the two problems are essentially equivalent, since every rational point can be scaled to become an integral point.
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A presheaf F on a topological space is called a sheaf if it satisfies the sheaf condition: whenever an open subset is covered by open subsets Ui, and we are given elements of F(Ui) for all i whose restrictions to Ui ∩ Uj agree for all i, j, then they are images of a unique element of F(U). By analogy, an étale presheaf is called a sheaf if it satisfies the same condition (with intersections of open sets replaced by pullbacks of étale morphisms, and where a set of étale maps to U is said to cover U if the topological space underlying U is the union of their images). More generally, one can define a sheaf for any Grothendieck topology on a category in a similar way. The category of sheaves of abelian groups over a scheme has enough injective objects, so one can define right derived functors of left exact functors.
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In mathematics, specifically in category theory, a pre-abelian category is an additive category that has all kernels and cokernels. Spelled out in more detail, this means that a category C is pre-abelian if: # C is preadditive, that is enriched over the monoidal category of abelian groups (equivalently, all hom-sets in C are abelian groups and composition of morphisms is bilinear); # C has all finite products (equivalently, all finite coproducts); note that because C is also preadditive, finite products are the same as finite coproducts, making them biproducts; # given any morphism f: A → B in C, the equaliser of f and the zero morphism from A to B exists (this is by definition the kernel of f), as does the coequaliser (this is by definition the cokernel of f). Note that the zero morphism in item 3 can be identified as the identity element of the hom-set Hom(A,B), which is an abelian group by item 1; or as the unique morphism A → 0 → B, where 0 is a zero object, guaranteed to exist by item 2.
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A morphism of chain complexes induces a morphism H_\bullet(F) of their homology groups, consisting of the homomorphisms H_n(F) : H_n(C) \to H_n(D) for all n. A morphism F is called a quasi-isomorphism if it induces an isomorphism on the nth homology for all n. Many constructions of chain complexes arising in algebra and geometry, including singular homology, have the following functoriality property: if two objects X and Y are connected by a map f, then the associated chain complexes are connected by a morphism F=C(f) : C_\bullet(X) \to C_\bullet(Y), and moreover, the composition g\circ f of maps f: X -> Y and g: Y -> Z induces the morphism C(g\circ f): C_\bullet(X) \to C_\bullet(Z) that coincides with the composition C(g) \circ C(f). It follows that the homology groups H_\bullet(C) are functorial as well, so that morphisms between algebraic or topological objects give rise to compatible maps between their homology.
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If X is a Klein surface, a function f:X→Cu{∞} is called meromorphic if, on each coordinate patch, f or its complex conjugate is meromorphic in the ordinary sense, and if f takes only real values (or ∞) on the boundary of X. Given a connected Klein surface X, the set of meromorphic functions defined on X form a field M(X), an algebraic function field in one variable over R. M is a contravariant functor and yields a duality (contravariant equivalence) between the category of compact connected Klein surfaces (with non-constant morphisms) and the category of function fields in one variable over the reals. One can classify the compact connected Klein surfaces X up to homeomorphism (not up to equivalence!) by specifying three numbers (g, k, a): the genus g of the analytic double Σ, the number k of connected components of the boundary of X , and the number a, defined by a=0 if X is orientable and a=1 otherwise. We always have k ≤ g+1. The Euler characteristic of X equals 1-g.
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The octahedral axiom then asserts the existence of maps f and g forming an exact triangle, and so that f and g form commutative triangles in the other faces that contain them: :250px Two different pictures appear in ( also present the first one). The first presents the upper and lower pyramids of the above octahedron and asserts that given a lower pyramid, one can fill in an upper pyramid so that the two paths from Y to Y′, and from Y′ to Y, are equal (this condition is omitted, perhaps erroneously, from Hartshorne's presentation). The triangles marked + are commutative and those marked "d" are exact: :550px The second diagram is a more innovative presentation. Exact triangles are presented linearly, and the diagram emphasizes the fact that the four triangles in the "octahedron" are connected by a series of maps of triangles, where three triangles (namely, those completing the morphisms from X to Y, from Y to Z, and from X to Z) are given and the existence of the fourth is claimed.
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Given some category T of topological spaces (possibly with some additional structure) such as the category of all topological spaces Top or the category of pointed topological spaces, that is, topological spaces with a distinguished base point, and a functor F: T \to A from that category into some category A of algebraic structures such as the category of groups Grp or of abelian groups Ab which then associates such an algebraic structure to every topological space, then for every morphism f: X \to Y of T (which is usually a continuous map, possibly preserving some other structure such as the base point) this functor induces an induced morphism F(f): F(X) \to F(Y) in A (which is a group homomorphism if A is a category of groups) between the algebraic structures F(X) and F(Y) associated to X and Y, respectively. If F is not a functor but a contravariant functor then by definition it induces morphisms in the opposite direction: F(f): F(Y) \to F(X). Cohomology groups give an example.
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A gerbe on a topological space X is a stack G of groupoids over X which is locally non- empty (each point in X has an open neighbourhood U over which the section category G(U) of the gerbe is not empty) and transitive (for any two objects a and b of G(U) for any open set U, there is an open covering {Vi}i of U such that the restrictions of a and b to each Vi are connected by at least one morphism). A canonical example is the gerbe of principal bundles with a fixed structure group H: the section category over an open set U is the category of principal H-bundles on U with isomorphism as morphisms (thus the category is a groupoid). As principal bundles glue together (satisfy the descent condition), these groupoids form a stack. The trivial bundle X x H over X shows that the local non-emptiness condition is satisfied, and finally as principal bundles are locally trivial, they become isomorphic when restricted to sufficiently small open sets; thus the transitivity condition is satisfied as well.
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Building on the topological semantics introduced by Tang Tsao-Chen for Lewis's modal logic, McKinsey and Tarski showed that by generating a topology equivalent to using only the complexes that correspond to open elements as a basis, a representation of an interior algebra is obtained as a topological field of sets - a field of sets on a topological space that is closed with respect to taking interiors or closures. By equipping topological fields of sets with appropriate morphisms known as field maps C. Naturman showed that this approach can be formalized as a category theoretic Stone duality in which the usual Stone duality for Boolean algebras corresponds to the case of interior algebras having redundant interior operator (Boolean interior algebras). The pre-order obtained in the Jónsson–Tarski approach corresponds to the accessibility relation in the Kripke semantics for an S4 theory, while the intermediate field of sets corresponds to a representation of the Lindenbaum–Tarski algebra for the theory using the sets of possible worlds in the Kripke semantics in which sentences of the theory hold. Moving from the field of sets to a Boolean space somewhat obfuscates this connection.
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