Product (Category Theory)

This simultaneously captures the concept of a product of , , , etc. In addition, like any , this characterization does not differentiate between versions of the product, thus allowing one to abstract away from an arbitrary, .

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 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 $gh=v$.