“Product category”的意思、由来-开放百科全书

词条

Product category

释义

Definition
Relation to other categorical concepts
Generalization to several arguments
References

{{Redirect|Product of categories|the operation on objects of a category|Product (category theory)}}

In the mathematical field of category theory, the product of two categories C and D, denoted {{nowrap| C × D}} and called a product category, is an extension of the concept of the Cartesian product of two sets. Product categories are used to define bifunctors and multifunctors.{{sfn|Mac Lane|1978}}

Definition

The product category {{nowrap| C × D}} has:

as objects:

pairs of objects {{nowrap| (A, B)}}, where A is an object of C and B of D;

as arrows from {{nowrap| (A₁, B₁)}} to {{nowrap| (A₂, B₂)}}:

pairs of arrows {{nowrap| (f, g)}}, where {{nowrap| f : A₁ → A₂}} is an arrow of C and {{nowrap| g : B₁ → B₂}} is an arrow of D;

as composition, component-wise composition from the contributing categories:

{{nowrap|1= (f₂, g₂) o (f₁, g₁) = (f₂ o f₁, g₂ o g₁)}};

as identities, pairs of identities from the contributing categories:

1_{(A, B)} = (1_A, 1_B).

Relation to other categorical concepts

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 : C^op × C → Set.

Generalization to several arguments

Just as the binary Cartesian product is readily generalized to an n-ary Cartesian product, binary product of two categories can be generalized, completely analogously, to a product of n categories. The product operation on categories is commutative and associative, up to isomorphism, and so this generalization brings nothing new from a theoretical point of view.

References

Definition 1.6.5 in {{Cite book| publisher = Cambridge University Press| isbn = 0-521-44178-1| volume = Volume 1| last = Borceux| first = Francis| title = Handbook of categorical algebra| series = Encyclopedia of mathematics and its applications 50-51, 53 [i.e. 52]| date = 1994 | page=22 }}
{{nlab|id=product+category|title=Product category}}
{{Cite book|url=https://www.worldcat.org/oclc/851741862|title=Categories for the Working Mathematician|last=Mac Lane|first=Saunders|date=1978|publisher=Springer New York|year=|isbn=1441931236|edition=Second|location=New York, NY|pages=49–51|oclc=851741862}}

1 : Category theory

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