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 $C$, the product of $X$ and $Y$ is an object $P$ along with a pair of morphisms $f:P\to X$ and $g:P\to Y$ satisfying the following universal condition:

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