# nLab finite product

Contents

### Context

#### Limits and colimits

limits and colimits

# Contents

## Definition

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.

## Properties

###### Example

(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

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

Last revised on September 28, 2021 at 05:19:03. See the history of this page for a list of all contributions to it.