Definition:Category of Categories

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

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.

These issues also make it preferable to let $\mathbf{Cat}$ comprise only the small categories.

Also see

 * Category of Categories is Category


 * Definition:Category of Finite Sets