The Stacks project

4.30 (2, 1)-categories

Some $2$-categories have the property that all $2$-morphisms are isomorphisms. These will play an important role in the following, and they are easier to work with.

Definition 4.30.1. A (strict) $(2, 1)$-category is a $2$-category in which all $2$-morphisms are isomorphisms.

Example 4.30.2. The $2$-category $\textit{Cat}$, see Remark 4.29.3, can be turned into a $(2, 1)$-category by only allowing isomorphisms of functors as $2$-morphisms.

In fact, more generally any $2$-category $\mathcal{C}$ produces a $(2, 1)$-category by considering the sub $2$-category $\mathcal{C}'$ with the same objects and $1$-morphisms but whose $2$-morphisms are the invertible $2$-morphisms of $\mathcal{C}$. In this situation we will say “let $\mathcal{C}'$ be the $(2, 1)$-category associated to $\mathcal{C}$” or similar. For example, the $(2, 1)$-category of groupoids means the $2$-category whose objects are groupoids, whose $1$-morphisms are functors and whose $2$-morphisms are isomorphisms of functors. Except that this is a bad example as a transformation between functors between groupoids is automatically an isomorphism!

Remark 4.30.3. Thus there are variants of the construction of Example 4.30.2 above where we look at the $2$-category of groupoids, or categories fibred in groupoids over a fixed category, or stacks. And so on.