Categorical product

An object A×B equipped with projections, universal among cones to A and B.

Categorical product

Let $\mathcal C$ be a category and let $A,B$ be objects of $\mathcal C$ .

Definition

A (binary) categorical product of $A$ and $B$ is a triple $(A\times B,\pi_1,\pi_2)$ consisting of an object $A\times B$ and morphisms

\pi_1:A\times B\to A,\qquad \pi_2:A\times B\to B

such that for every object $X$ and every pair of morphisms $f:X\to A$ , $g:X\to B$ , there exists a unique morphism

\langle f,g\rangle : X\to A\times B

making the equations

\pi_1\circ \langle f,g\rangle = f,\qquad \pi_2\circ \langle f,g\rangle = g

hold (see composition ).

In diagram form:

\begin{CD} & X @>\langle f,g\rangle>> A\times B \\ @V f VV @VV \pi_1 V \\ A && \end{CD} \qquad \begin{CD} & X @>\langle f,g\rangle>> A\times B \\ @V g VV @VV \pi_2 V \\ B && \end{CD}

A product is unique up to unique isomorphism : if $(P,\pi_1,\pi_2)$ and $(P',\pi_1',\pi_2')$ are both products of $A,B$ , there is a unique isomorphism $P\cong P'$ compatible with projections.

This is a special case of a limit (the limit of the discrete diagram $A\;\;B$ ).

Examples

$\mathbf{Set}$ .
The categorical product is the usual cartesian product $A\times B$ of sets, with projections $\pi_1(a,b)=a$ , $\pi_2(a,b)=b$ .
$\mathbf{Grp}$ .
For groups $G,H$ , the product is the direct product $G\times H$ with coordinate projections, characterized by: giving a homomorphism $X\to G\times H$ is equivalent to giving a pair of homomorphisms $X\to G$ and $X\to H$ .
$\mathbf{Top}$ (and similarly $\mathbf{Ab}$ , $R$ -Mod).
The product of spaces $X,Y$ is the set-theoretic product $X\times Y$ equipped with the product topology; the projections are continuous and satisfy the same universal property.