The Stacks project

Remark 4.29.3. Big $2$-categories. In many texts a $2$-category is allowed to have a class of objects (but hopefully a “class of classes” is not allowed). We will allow these “big” $2$-categories as well, but only in the following list of cases (to be updated as we go along):

1. The $2$-category of categories $\textit{Cat}$.

2. The $(2, 1)$-category of categories $\textit{Cat}$.

3. The $2$-category of groupoids $\textit{Groupoids}$; this is a $(2, 1)$-category.

4. The $2$-category of fibred categories over a fixed category.

5. The $(2, 1)$-category of fibred categories over a fixed category.

6. The $2$-category of categories fibred in groupoids over a fixed category; this is a $(2, 1)$-category.

7. The $2$-category of stacks over a fixed site.

8. The $(2, 1)$-category of stacks over a fixed site.

9. The $2$-category of stacks in groupoids over a fixed site; this is a $(2, 1)$-category.

10. The $2$-category of stacks in setoids over a fixed site; this is a $(2, 1)$-category.

11. The $2$-category of algebraic stacks over a fixed scheme; this is a $(2, 1)$-category.

See Definition 4.30.1. Note that in each case the class of objects of the $2$-category $\mathcal{C}$ is a proper class, but for all objects $x, y \in \mathop{\mathrm{Ob}}\nolimits (C)$ the category $\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(x, y)$ is “small” (according to our conventions).