# Category of small categories

In mathematics, specifically in category theory, the **category of small categories**, denoted by **Cat**, is the category whose objects are all small categories and whose morphisms are functors between categories. **Cat** may actually be regarded as a 2-category with natural transformations serving as 2-morphisms.

The initial object of **Cat** is the *empty category* **0**, which is the category of no objects and no morphisms.[1] The terminal object is the *terminal category* or *trivial category* **1** with a single object and morphism.[2]

The category **Cat** is itself a large category, and therefore not an object of itself. In order to avoid problems analogous to Russell's paradox one cannot form the “category of all categories”. But it is possible to form a quasicategory (meaning objects and morphisms merely form a conglomerate) of all categories.

## Free category

The category **Cat** has a forgetful functor *U* into the quiver category **Quiv**:

*U*:**Cat**→**Quiv**

This functor forgets the identity morphisms of a given category, and it forgets morphism compositions. The left adjoint of this functor is a functor *F* taking **Quiv** to the corresponding free categories:

*F*:**Quiv**→**Cat**

## 1-Categorical properties

**Cat**has all small limits and colimits.**Cat**is a Cartesian closed category, with exponential given by the functor category .**Cat**is*not*locally Cartesian closed.**Cat**is locally finitely presentable.

## See also

## References

- Kashiwara, Masaki; Schapira, Pierre (2006).
*Categories and sheaves*.