“category”的概念、定义、翻译、参考文献-科学参考

category

Mathematics

A category is a class of objects, together with morphisms between objects, such that composition of morphisms is associative, and an identity morphism 1_X exists for each object X. Category theory is an alternative approach to set theory when studying the foundations of mathematics. As an example, the class of sets as the objects, with morphisms being functions, makes a category; note there is no ‘set of all sets’.

Computer

A collection of objects A, together with a related set of morphisms M. An object is a generalization of a set and a morphism is a generalization of a function that maps between sets.
The set M is the disjoint union of sets of the form [A,B], where A and B are elements of A; if α‎ is a member of [A,B], A is the domain of α‎, B is the codomain of α‎, and α‎ is said to be a morphism from A to B. For each triple (A,B,C) of elements of A there is a dyadic operation ° from the Cartesian product
$[B, C] \times [A, B]$
to [A,C]. The image β‎°α‎ of the ordered pair (β‎,α‎) is the composition of β‎ with α‎; the composition operation is associative. In addition, when the composition is defined there is an identity morphism for each A in A.
Examples of categories include the set of groups and homomorphisms on groups, and the set of rings and homomorphisms on rings. See functor.

Biology

See rank.

单词	category
释义	category Mathematics A category is a class of objects, together with morphisms between objects, such that composition of morphisms is associative, and an identity morphism 1_X exists for each object X. Category theory is an alternative approach to set theory when studying the foundations of mathematics. As an example, the class of sets as the objects, with morphisms being functions, makes a category; note there is no ‘set of all sets’. Computer A collection of objects A, together with a related set of morphisms M. An object is a generalization of a set and a morphism is a generalization of a function that maps between sets. The set M is the disjoint union of sets of the form [A,B], where A and B are elements of A; if α‎ is a member of [A,B], A is the domain of α‎, B is the codomain of α‎, and α‎ is said to be a morphism from A to B. For each triple (A,B,C) of elements of A there is a dyadic operation ° from the Cartesian product $[B, C] \times [A, B]$ to [A,C]. The image β‎°α‎ of the ordered pair (β‎,α‎) is the composition of β‎ with α‎; the composition operation is associative. In addition, when the composition is defined there is an identity morphism for each A in A. Examples of categories include the set of groups and homomorphisms on groups, and the set of rings and homomorphisms on rings. See functor. Biology See rank.
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