Product (category theory)

A binary categorical product of objects of a category $A,B$ is an object $P$ with categorical morphisms $p_A:P\to A$ and $p_B:P\to B$. Its universal property is that, for every object $X$ and pair of arrows $f:X\to A$, $g:X\to B$, there is exactly one arrow $h:X\to P$ satisfyingThe arrows $p_A,p_B$ are called the product projections. Existence lets us combine two maps; uniqueness means there is no extra unspecified information in the combined map.

In the Category of sets, take $A=\{0,1\}$ and $B=\{a,b,c\}$. Their Cartesian product contains six ordered pairs, and the projection maps select the first and second coordinates. Let $X=\{u,v,w\}$, $f(u)=0$, $f(v)=1$, $f(w)=1$, and $g(u)=a$, $g(v)=b$, $g(w)=a$. The unique required map is $h(u)=(0,a)$, $h(v)=(1,b)$, $h(w)=(1,a)$. Reading either coordinate recovers the corresponding input map.

In a partially ordered set viewed as a category, the projection arrows mean $P\le A$ and $P\le B$. The universal property says that every common lower bound $X$ satisfies $X\le P$, so $P$ is the greatest lower bound. For subsets ordered by inclusion this is $A\cap B$, the intersection; it is a different construction from the Cartesian product in the Category of sets because the allowed arrows have changed.

The universal property also compares any two proposed products: each supplies a unique map to the other compatible with the projections. The two composites have the same projections as the corresponding identity morphisms, so uniqueness makes them identities. Thus a categorical product is determined up to a unique projection-preserving isomorphism, which explains why we can use whichever concrete construction is convenient.