The Stacks project

Definition 18.7.1. Ringed topoi.

1. A ringed topos is a pair $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ where $\mathcal{C}$ is a site and $\mathcal{O}$ is a sheaf of rings on $\mathcal{C}$. The sheaf $\mathcal{O}$ is called the structure sheaf of the ringed topos.

2. Let $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$, $(\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}')$ be ringed topoi. A morphism of ringed topoi

   \[ (f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O}) \longrightarrow (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}'), \mathcal{O}') \]

   is given by a morphism of topoi $f : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}')$ (see Sites, Definition 7.15.1) together with a map of sheaves of rings $f^\sharp : f^{-1}\mathcal{O}' \to \mathcal{O}$, which by adjunction is the same thing as a map of sheaves of rings $f^\sharp : \mathcal{O}' \to f_*\mathcal{O}$.

3. Let $(f, f^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_1), \mathcal{O}_1) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_2), \mathcal{O}_2)$ and $(g, g^\sharp ) : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_2), \mathcal{O}_2) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_3), \mathcal{O}_3)$ be morphisms of ringed topoi. Then we define the composition of morphisms of ringed topoi by the rule

   \[ (g, g^\sharp ) \circ (f, f^\sharp ) = (g \circ f, f^\sharp \circ g^\sharp ). \]

   Here we use composition of morphisms of topoi defined in Sites, Definition 7.15.1 and $f^\sharp \circ g^\sharp $ indicates the morphism of sheaves of rings

   \[ \mathcal{O}_3 \xrightarrow {g^\sharp } g_*\mathcal{O}_2 \xrightarrow {g_*f^\sharp } g_*f_*\mathcal{O}_1 = (g \circ f)_*\mathcal{O}_1 \]