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 4.29 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 4.28.1. 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 4.28 it follows from Categories, Lemma 4.28.2 that the axioms of a strict 2-category hold with these definitions.

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