Definition:Product (Category Theory)/Binary Product

Definition

Let $\mathbf C$ be a metacategory.

Let $A$ and $B$ be objects of $\mathbf C$.

A (binary) product diagram for $A$ and $B$ comprises an object $P$ and morphisms $p_1: P \to A$, $p_2: P \to B$:

$\begin{xy}\xymatrix@+1em@L+3px{

&

P
 \ar[l]_*+{p_1}
 \ar[r]^*+{p_2}

&

}\end{xy}$

subjected to the following universal mapping property:

For any object $X$ and morphisms $x_1, x_2$ like so:

$\begin{xy}\xymatrix@+1em@L+3px{

&

X
 \ar[l]_*+{x_1}
 \ar[r]^*+{x_2}

&

}\end{xy}$

there is a unique morphism $u: X \to P$ such that:

$\begin{xy}\xymatrix@+1em@L+3px{

&

X
 \ar[ld]_*+{x_1}
 \ar@{-->}[d]^*+{u}
 \ar[rd]^*+{x_2}

\\

&

P
 \ar[l]^*+{p_1}
 \ar[r]_*+{p_2}

&

}\end{xy}$

is a commutative diagram, i.e., $x_1 = p_1 \circ u$ and $x_2 = p_2 \circ u$.

In this situation, $P$ is called a (binary) product of $A$ and $B$ and may be denoted $A \times B$.

Generally, one writes $\left\langle{x_1, x_2}\right\rangle$ for the unique morphism $u$ determined by above diagram.

The morphisms $p_1$ and $p_2$ are often taken to be implicit.

They are called projections; if necessary, $p_1$ can be called the first projection and $p_2$ the second projection.

This article is complete as far as it goes, but it could do with expansion.
In particular: the projection definition may merit its own, separate page
You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information.
To discuss this page in more detail, feel free to use the talk page.
When this work has been completed, you may remove this instance of {{Expand}} from the code.
If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.