---
tag: ZY6R4H65P8
kind: example
name: Category of small categories
msc: [18A05, 18N10]
instance_of: [A2DE66937D]
uses: [A2DE66937D, 28NQ0KRHD3]
sources:
  - url: https://en.wikipedia.org/wiki/Category_of_small_categories
same_as:
  wikidata: Q5051852
  wikipedia: Category of small categories
---

## Example

A [[category|A2DE66937D]] is *small* when its objects and its morphisms form sets.
The *category of small categories* $\mathbf{Cat}$ has the small categories as objects and the [[functors|28NQ0KRHD3]] between them as morphisms.
The composite of $F \colon \mathcal{A} \to \mathcal{B}$ and $G \colon \mathcal{B} \to \mathcal{C}$ is the functor $G \circ F$, given on objects and on morphisms by composing the maps.
The identity functor of $\mathcal{A}$ is the identity morphism.
Composition of functors is associative because it is composition of maps on objects and on morphisms, so $\mathbf{Cat}$ is a category.

## Remarks

The empty category is an [[initial object|EK85VSSRMR]], and the category with one object and one morphism is a terminal object.
The [[product|S5744QZV4N]] of $\mathcal{A}$ and $\mathcal{B}$ is the [[product category|AFNE9C4NH6]] $\mathcal{A} \times \mathcal{B}$.
$\mathbf{Cat}$ is [[Cartesian closed|BFBNQ36QS8]]: the exponential object of $\mathcal{A}$ and $\mathcal{B}$ is the [[functor category|1X0WKFQX8K]] $\mathcal{B}^{\mathcal{A}}$.
With [[natural transformations|KYCPCC57FT]] as morphisms between functors, $\mathbf{Cat}$ becomes a [[2-category|CQACXG10JZ]].
The smallness condition avoids a paradox: $\mathbf{Cat}$ itself is not small, so it is not one of its own objects.

## Explanations

### L1 human

Categories can themselves be the objects of a category, with functors as the arrows between them.
To avoid "the set of all sets" style problems, one only takes categories whose objects and arrows form sets.
For example, a [[monoid|0N6R902FG1]] is a category with one object, and a monoid homomorphism is then exactly a functor, so monoids sit inside $\mathbf{Cat}$.