Definition:Category with Products

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:

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

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.

Also see

• Complete Category