## 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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