# Product (Category Theory)

This simultaneously captures the concept of a product of sets, posets, groups, topological spaces etc. In addition, like any universal construction, this characterization does not differentiate between isomorphic versions of the product, thus allowing one to abstract away from an arbitrary, specific construction.

## Definition

Given a pair of objects \(X\) and \(Y\) in a category \(\mathbb{C}\), the **product** of \(X\) and \(Y\) is an object \(P\) *along with a pair of morphisms* \(f: P \rightarrow X\) and \(g: P \rightarrow Y\) satisfying the following universal condition:

Given any other object \(W\) and morphisms \(u: W \rightarrow X\) and \(v:W \rightarrow Y\) there is a *unique* morphism \(h: W \rightarrow P\) such that \(fh = u\) and \(gh = v\).

Children:

- Universal property of the product
The product can be defined in a very general way, applicable to the natural numbers, to sets, to algebraic structures, and so on.

Parents:

- Category theory
How mathematical objects are related to others in the same category.

Would Product (mathematics) be an appropriate name, or does category theory’s use of the term point to only a subset of the things product can mean?

@1yq Whether Product (mathematics) is appropriate really depends if you’re asking a category theorist (who would say yes) or not . ;-)

In seriousness, specific kinds of products include cartesian products, products of algebraic structures, products of topological spaces and the most well known: product of numbers. All of these are special cases of the categorical product (if you pick your category right), but I can imagine someone wanting to look up ‘product’ as in multiplication and getting hit with category theory.

I don’t know. It’s a matter of taste I suppose. I get the idea that category theory is not yet quite widely-known enough for this to be considered “the” definition by most mathematicians, but if other contributors feel it should be given that status I certainly won’t complain. I just thought this was the safer approach.

See, for example product on Wikipedia.

Yeah, I think keeping it as it is now is probably the best way of following the “one idea per page” methodology. The page on Products (mathematics) can have this page as child.