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*}

18.8 2-morphisms of ringed topoi

This is a brief section concerning the notion of a $2$-morphism of ringed topoi.

Definition 18.8.1. Let $f, g : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D})$ be two morphisms of ringed topoi. A 2-morphism from $f$ to $g$ is given by a transformation of functors $t : f_* \to g_*$ such that

\[ \xymatrix{ & \mathcal{O}_\mathcal {D} \ar[ld]_{f^\sharp } \ar[rd]^{g^\sharp } \\ f_*\mathcal{O}_\mathcal {C} \ar[rr]^ t & & g_*\mathcal{O}_\mathcal {C} } \]

is commutative.

Pictorially we sometimes represent $t$ as follows:

\[ \xymatrix{ (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}_\mathcal {C}) \rrtwocell ^ f_ g{t} & & (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}_\mathcal {D}) } \]

As in Sites, Section 7.36 giving a 2-morphism $t : f_* \to g_*$ is equivalent to giving $t : g^{-1} \to f^{-1}$ (usually denoted by the same symbol) such that the diagram

\[ \xymatrix{ f^{-1}\mathcal{O}_\mathcal {D} \ar[rd]_{f^\sharp } & & g^{-1}\mathcal{O}_\mathcal {D} \ar[ll]^ t \ar[ld]^{g^\sharp } \\ & \mathcal{O}_\mathcal {C} } \]

is commutative. As in Sites, Section 7.36 the axioms of a strict 2-category hold with horizontal and vertical compositions defined as explained in loc. cit.


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