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{

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

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

\\ A & P \ar[l]^*+{p_1} \ar[r]_*+{p_2} & B }\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.

Also see

 * Definition:Set Product, an archetypal example in the category of sets $\mathbf{Set}$
 * Binary Product is Finite Product
 * Definition:Binary Coproduct (Category Theory)