Definition:Product (Category Theory)/General Definition

Definition
Let $\mathbf C$ be a metacategory.

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

Let $\mathbf{Dis} \left({\mathcal C}\right)$ be the discrete category on $\mathcal C$, considered as a subcategory of $\mathbf C$.

A product for $\mathcal C$, denoted $\displaystyle \prod \mathcal C$, is a limit for the inclusion functor $D: \mathbf{Dis} \left({\mathcal C}\right) \to \mathbf C$, considered as a diagram.

For an object $C$ in $\mathcal C$, the associated morphism $\displaystyle \prod \mathcal C \to C$ is denoted $\operatorname{pr}_C$ and called the projection on $C$.

The whole construction is pictured in the following commutative diagram:


 * $\begin{xy}\xymatrix@R-.5em@L+3px{

& & A \ar@{-->}[dd] \ar[dddl]_*+{a_C} \ar[dddr]^*+{a_C'}

\\ \\ & & \displaystyle \prod \mathcal C \ar[dl]^*{\operatorname{pr}_C} \ar[dr]_*{\operatorname{pr}_{C'}}

\\ \mathbf{Dis} \left({\mathcal C}\right) & C & \dots \quad \dots & C' }\end{xy}$

Also known as
If $\mathcal C = \left({C_i}\right)_{i \in I}$ is a set, indexed by some indexing set $I$, the notations $\displaystyle \prod_{i \mathop \in I} C_i$ and $\displaystyle \prod_i C_i$ are often seen.

In this situation, one writes $\operatorname{pr}_i$ for $\operatorname{pr}_{C_i}$ and calls it the $i$th projection.

Also see

 * Empty Product is Terminal Object
 * Unary Product for Object is Itself