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):
The 2-category of categories \textit{Cat}.
The (2, 1)-category of categories \textit{Cat}.
The 2-category of groupoids \textit{Groupoids}; this is a (2, 1)-category.
The 2-category of fibred categories over a fixed category.
The (2, 1)-category of fibred categories over a fixed category.
The 2-category of categories fibred in groupoids over a fixed category; this is a (2, 1)-category.
The 2-category of stacks over a fixed site.
The (2, 1)-category of stacks over a fixed site.
The 2-category of stacks in groupoids over a fixed site; this is a (2, 1)-category.
The 2-category of stacks in setoids over a fixed site; this is a (2, 1)-category.
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).
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