A *finite product* is a product (Cartesian product) of a finite number of factors.

Finite products are generated from the empty product (the terminal object) and binary products (those with two factors, often – but not always – understood by default under “product”.)

Similarly a *finite coproduct* is a coproduct of a finite number of summands. This is generated from the empty coproduct (the initial object) and binary coproducts.

**(categories with finite products are cosifted)**

Let $\mathcal{C}$ be a small category which has finite products. Then $\mathcal{C}$ is a *cosifted category*, equivalently its opposite category $\mathcal{C}^{op}$ is a *sifted category*, equivalently colimits over $\mathcal{C}^{op}$ with values in Set are *sifted colimits*, equivalently colimits over $\mathcal{C}^{op}$ with values in Set *commute with finite products*, as follows:

For $\mathbf{X}, \mathbf{Y} \in [\mathcal{C}^{op}, Set]$ to functors on the opposite category of $\mathcal{C}$ (hence two presheaves on $\mathcal{C}$) we have a natural isomorphism

$\underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim}
\left(
\mathbf{X} \times \mathbf{Y}
\right)
\;\simeq\;
\left(
\underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim}
\mathbf{X}
\right)
\times
\left(
\underset{\underset{\mathcal{C}^{op}}{\longrightarrow}}{\lim}
\mathbf{Y}
\right)
\,.$

- Saunders MacLane, §III.5 of:
*Categories for the Working Mathematician*, Graduate Texts in Mathematics**5**Springer (1971, second ed. 1997) [doi:10.1007/978-1-4757-4721-8]

Last revised on May 20, 2023 at 08:40:46. See the history of this page for a list of all contributions to it.