Definition:Category of Categories
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Definition
The category of categories, denoted $\mathbf{Cat}$, is the metacategory with:
| Objects: | small categories | |
| Morphisms: | functors | |
| Composition: | composition of functors | |
| Identity morphisms: | identity functors |
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
- Results about the category of categories can be found here.
- Definition:Category of Finite Sets
Sources
- 2010: Steve Awodey: Category Theory (2nd ed.) ... (previous) ... (next): $\S 1.4.6$