# Definition:Category with Products

Jump to navigation
Jump to search

## Definition

Let $\mathbf C$ be a metacategory.

Then $\mathbf C$ is said to **have products** or to be a **(meta)category with products** if and only if:

### Category with Binary Products

Let $\mathbf C$ be a metacategory.

Then $\mathbf C$ is said to **have binary products** or to be a **(meta)category with binary products** if and only if:

- For all objects $C, D \in \mathbf C_0$, there is a binary product $C \times D$ for $C$ and $D$.

### Category with Finite Products

Let $\mathbf C$ be a metacategory.

Then $\mathbf C$ is said to **have products** or to be a **(meta)category with products** if and only if:

- For all finite sets of objects $\CC \subseteq \mathbf C_0$, there is a product $\ds \prod \CC$ for $\CC$.

## Also known as

Some authors, sensibly perhaps, say $\mathbf C$ as above **has small products**.

As it is rare that products are taken over collections too large to be sets, the standard terminology has deemed this abuse of language admissible.