# Definition:Product (Category Theory)

## Definition

### Binary Product

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 $p_1: P \to A$, $p_2: P \to B$:

$\begin{xy}\xymatrix@[email protected]+3px{ A & P \ar[l]_*+{p_1} \ar[r]^*+{p_2} & B }\end{xy}$

subjected to the following universal mapping property:

For any object $X$ and morphisms $x_1, x_2$ like so:
$\begin{xy}\xymatrix@[email protected]+3px{ A & X \ar[l]_*+{x_1} \ar[r]^*+{x_2} & B }\end{xy}$
there is a unique morphism $u: X \to P$ such that:
$\begin{xy}\xymatrix@[email protected]+3px{ & X \ar[ld]_*+{x_1} \ar@{-->}[d]^*+{u} \ar[rd]^*+{x_2} \\ A & P \ar[l]^*+{p_1} \ar[r]_*+{p_2} & B }\end{xy}$
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.

### General Definition

Let $\mathbf C$ be a metacategory.

Let $\CC$ be any collection of objects of $\mathbf C$.

Let $\map {\mathbf {Dis} } \CC$ be the discrete category on $\CC$, considered as a subcategory of $\mathbf C$.

A product for $\CC$, denoted $\ds \prod \CC$, is a limit for the inclusion functor $D: \map {\mathbf {Dis} } \CC \to \mathbf C$, considered as a diagram.

For an object $C$ in $\CC$, the associated morphism $\ds \prod \CC \to C$ is denoted $\pr_C$ and called the projection on $C$.

The whole construction is pictured in the following commutative diagram:

$\begin{xy}\[email protected]@L+3px{ & & A \ar@{-->}[dd] \ar[dddl]_*+{a_C} \ar[dddr]^*+{a_C'} \\ \\ & & \ds \prod \CC \ar[dl]^*{\pr_C} \ar[dr]_*{\pr_{C'}} \\ \map {\mathbf {Dis} } \CC & C & \dots \quad \dots & C' }\end{xy}$

### Finite Product

Let $\ds \prod \CC$ be a product for a finite set $\CC$ of objects of $\mathbf C$.

Then $\ds \prod \CC$ is called a finite product.