Definition 4.29.1. A (strict) *$(2, 1)$-category* is a $2$-category in which all $2$-morphisms are isomorphisms.

## 4.29 (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.

Example 4.29.2. The $2$-category $\textit{Cat}$, see Remark 4.28.3, 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 4.29.3. Thus there are variants of the construction of Example 4.29.2 above where we look at the $2$-category of groupoids, or categories fibred in groupoids over a fixed category, or stacks. And so on.

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (0)