Product (Category Theory)

Definition

In category theory, a product is a limit of a discrete diagram.

$$\prod _{i:I}{X}_i:\mathcal{C}\forall (I,X):\mathrm{Fam}(\mathcal{C})$$

Like with coproducts, we say:

• $\mathcal{C}$ has binary products whenever $I\cong 2$.
• $\mathcal{C}$ has finite products whenever $|I|<{\aleph }_0$.
• $\mathcal{C}$ has small products whenever $I$ fits in Set.