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

- $\begin{xy}\xymatrix@+1em@L+3px{ A & P \ar[l]_*+{p_1} \ar[r]^*+{p_2} & B }\end{xy}$

subjected to the following universal mapping property:

- $\begin{xy}\xymatrix@+1em@L+3px{ A & X \ar[l]_*+{x_1} \ar[r]^*+{x_2} & B }\end{xy}$

- $\begin{xy}\xymatrix@+1em@L+3px{ & X \ar[ld]_*+{x_1} \ar@{-->}[d]^*+{u} \ar[rd]^*+{x_2} \\ A & P \ar[l]^*+{p_1} \ar[r]_*+{p_2} & B }\end{xy}$

- 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**.

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### 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**.