The composition of two open maps is again open; the composition of two closed maps is again closed. The categorical sum of two open maps is open, or of two closed maps is closed. The categorical product of two open maps is open, however, the categorical product of two closed maps need not be closed. A bijective map is open if and only if it is closed.
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Named after (1596–1650), French philosopher, mathematician, and scientist, whose formulation of analytic geometry gave rise to the concept of Cartesian product, which was later generalized to the notion of categorical product.
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Functions between orders become functors between categories. Many ideas of order theory are just concepts of category theory in small. For example, an infimum is just a categorical product. More generally, one can capture infima and suprema under the abstract notion of a categorical limit (or colimit, respectively).
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Like Set, FinSet and FinOrd are topoi. As in Set, in FinSet the categorical product of two objects A and B is given by the cartesian product , the categorical sum is given by the disjoint union , and the exponential object BA is given by the set of all functions with domain A and codomain B. In FinOrd, the categorical product of two objects n and m is given by the ordinal product , the categorical sum is given by the ordinal sum , and the exponential object is given by the ordinal exponentiation nm. The subobject classifier in FinSet and FinOrd is the same as in Set. FinOrd is an example of a PRO.
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M.W. Shields "Concurrent Machines", Computer Journal, (1985) 28 pp. 449–465. History monoids are isomorphic to trace monoids (free partially commutative monoids) and to the monoid of dependency graphs. As such, they are free objects and are universal. The history monoid is a type of semi-abelian categorical product in the category of monoids.
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There are thus no zero objects in Ord. The categorical product in Ord is given by the product order on the cartesian product. We have a forgetful functor Ord → Set that assigns to each preordered set the underlying set, and to each order-preserving function the underlying function. This functor is faithful, and therefore Ord is a concrete category.
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For small categories, this is the same as the action on objects of the categorical product in the category Cat. A functor whose domain is a product category is known as a bifunctor. An important example is the Hom functor, which has the product of the opposite of some category with the original category as domain: :Hom : Cop × C → Set.
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Because the Segre map is to the categorical product of projective spaces, it is a natural mapping for describing non-entangled states in quantum mechanics and quantum information theory. More precisely, the Segre map describes how to take products of projective Hilbert spaces. In algebraic statistics, Segre varieties correspond to independence models. The Segre embedding of P2×P2 in P8 is the only Severi variety of dimension 4.
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In graph theory, the Cartesian product of two graphs G and H is the graph denoted by , whose vertex set is the (ordinary) Cartesian product and such that two vertices (u,v) and (u′,v′) are adjacent in , if and only if and v is adjacent with v′ in H, or and u is adjacent with u′ in G. The Cartesian product of graphs is not a product in the sense of category theory. Instead, the categorical product is known as the tensor product of graphs.
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Quantum mechanics may be modelled on a projective Hilbert space, and the categorical product of two such spaces is the Segre embedding. In the bipartite case, a quantum state is separable if and only if it lies in the image of the Segre embedding. Jon Magne Leinaas, Jan Myrheim and Eirik Ovrum in their paper "Geometrical aspects of entanglement""Geometrical aspects of entanglement", Physical Review A 74, 012313 (2006) describe the problem and study the geometry of the separable states as a subset of the general state matrices. This subset have some intersection with the subset of states holding Peres-Horodecki criterion.
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In mathematics, specifically in the field known as category theory, a monoidal category where the monoidal ("tensor") product is the categorical product is called a cartesian monoidal category. Any category with finite products (a "finite product category") can be thought of as a cartesian monoidal category. In any cartesian monoidal category, the terminal object is the tensor unit. Dually, a monoidal finite coproduct category with the monoidal structure given by the coproduct and unit the initial object is called a cocartesian monoidal category, and any finite coproduct category can be thought of as a cocartesian monoidal category.
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In topology and related areas of mathematics, a product space is the Cartesian product of a family of topological spaces equipped with a natural topology called the product topology. This topology differs from another, perhaps more obvious, topology called the box topology, which can also be given to a product space and which agrees with the product topology when the product is over only finitely many spaces. However, the product topology is "correct" in that it makes the product space a categorical product of its factors, whereas the box topology is too fine; in that sense the product topology is the natural topology on the Cartesian product.
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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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