Definition:Product (Category Theory)

Definition
Let $\mathbf C$ be a metacategory.

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

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


 * $\begin{xy}

<-4em,0em>*+{A} = "A", <0em,0em>*+{P} = "P", <4em,0em>*+{B} = "B",

"P";"A" **@{-} ?>*@{>} ?*!/^.8em/{p_1}, "P";"B" **@{-} ?>*@{>} ?*!/_.8em/{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}

<-4em,0em>*+{A} = "A", <0em,0em>*+{X} = "X", <4em,0em>*+{B} = "B",

"X";"A" **@{-} ?>*@{>} ?*!/^.8em/{x_1}, "X";"B" **@{-} ?>*@{>} ?*!/_.8em/{x_2}, \end{xy}$


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


 * $\begin{xy}

<0em,5em>*+{X} = "X", <-5em,0em>*+{A} = "A", <0em,0em>*+{P} = "P", <5em,0em>*+{B} = "B",

"X";"A" **@{-} ?>*@{>} ?*!/^.8em/{x_1}, "X";"B" **@{-} ?>*@{>} ?*!/_.8em/{x_2}, "X";"P" **@{--} ?>*@{>} ?*!/_.6em/{u}, "P";"A" **@{-} ?>*@{>} ?*!/_.8em/{p_1}, "P";"B" **@{-} ?>*@{>} ?*!/^.8em/{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 product of $A$ and $B$ and may be denoted $A \times B$.

We generally write $\left\langle{x_1, x_2}\right\rangle$ 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

 * Coproduct, the dual notion
 * Product (Category Theory) is Unique
 * Set Product, an archetypal example in the category of sets $\mathbf{Set}$