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*}

7.36 2-morphisms of topoi

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

Definition 7.36.1. Let $f, g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be two morphisms of topoi. A 2-morphism from $f$ to $g$ is given by a transformation of functors $t : f_* \to g_*$.

Pictorially we sometimes represent $t$ as follows:

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

Note that since $f^{-1}$ is adjoint to $f_*$ and $g^{-1}$ is adjoint to $g_*$ we see that $t$ induces also a transformation of functors $t : g^{-1} \to f^{-1}$ (usually denoted by the same symbol) uniquely characterized by the condition that the diagram

\[ \xymatrix{ \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{G}, f_*\mathcal{F}) \ar[d]_{t \circ -} \ar@{=}[r] & \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(f^{-1}\mathcal{G}, \mathcal{F}) \ar[d]^{- \circ t} \\ \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(\mathcal{G}, g_*\mathcal{F}) \ar@{=}[r] & \mathop{Mor}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(g^{-1}\mathcal{G}, \mathcal{F}) } \]

commutes. Because of set theoretic difficulties (see Remark 7.15.4) we do not obtain a 2-category of topoi. But we can still define horizontal and vertical composition and show that the axioms of a strict 2-category listed in Categories, Section 4.28 hold. Namely, vertical composition of 2-morphisms is clear (just compose transformations of functors), composition of 1-morphisms has been defined in Definition 7.15.1, and horizontal composition of

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \rtwocell ^ f_ g{t} & \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \rtwocell ^{f'}_{g'}{s} & \mathop{\mathit{Sh}}\nolimits (\mathcal{E}) } \]

is defined by the transformation of functors $s \star t$ introduced in Categories, Definition 4.27.1. Explicitly, $s \star t$ is given by

\[ \xymatrix{ f'_*f_*\mathcal{F} \ar[r]^{f'_*t} & f'_*g_*\mathcal{F} \ar[r]^ s & g'_*g_*\mathcal{F} } \quad \text{or}\quad \xymatrix{ f'_*f_*\mathcal{F} \ar[r]^ s & g'_*f_*\mathcal{F} \ar[r]^{g'_*t} & g'_*g_*\mathcal{F} } \]

(these maps are equal). Since these definitions agree with the ones in Categories, Section 4.27 it follows from Categories, Lemma 4.27.2 that the axioms of a strict 2-category hold with these definitions.


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