The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

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.

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

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.


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