Definition:Category of Categories

Definition
The category of categories, denoted $\mathbf{Cat}$, is the metacategory with:


 * objects: All small categories;
 * morphisms: All functors between them.

This forms a metacategory, as shown on Category of Categories is Category.

Also known as
Some authors use a Germanic font to denote categories and write $\mathfrak{Cat}$ instead.

Others use a calligraphic font, yielding $\mathscr{Cat}$.

As these are hard to read (when we don't know that it says "Cat") we discourage these.

Note
The reason to call $\mathbf{Cat}$ a metacategory is foundational; allowing it to be a category would bring us to axiomatic troubles.

Also see

 * Category of Finite Sets