To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
Languages
Recent
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

Product category

From Wikipedia, the free encyclopedia

In the mathematical field of category theory, the product of two categories C and D, denoted 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.[1]

Definition

The product category C × D has:

  • as objects:
    pairs of objects (A, B), where A is an object of C and B of D;
  • as arrows from (A1, B1) to (A2, B2):
    pairs of arrows (f, g), where f : A1A2 is an arrow of C and g : B1B2 is an arrow of D;
  • as composition, component-wise composition from the contributing categories:
    (f2, g2) o (f1, g1) = (f2 o f1, g2 o g1);
  • as identities, pairs of identities from the contributing categories:
    1(A, B) = (1A, 1B).

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 : Cop × CSet.

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 Borceux, Francis (1994). Handbook of categorical algebra. Encyclopedia of mathematics and its applications 50-51, 53 [i.e. 52]. Volume 1. Cambridge University Press. p. 22. ISBN 0-521-44178-1. |volume= has extra text (help)
  • Product category in nLab
  • Mac Lane, Saunders (1978). Categories for the Working Mathematician (Second ed.). New York, NY: Springer New York. pp. 49–51. ISBN 1441931236. OCLC 851741862.
This page was last edited on 1 April 2020, at 05:36
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.