The Stacks project

Lemma 85.5.2. Let $\mathcal{C}_ n, f_\varphi , u_\varphi $ and $\mathcal{C}'_ n, f'_\varphi , u'_\varphi $ be as in Situation 85.3.3. Let $h$ be a morphism between simplicial sites as in Remark 85.5.1. Then we obtain a morphism of topoi

\[ h_{total} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}'_{total}) \]

and commutative diagrams

\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n) \ar[d]_{g_ n} \ar[r]_{h_ n} & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}'_ n) \ar[d]^{g'_ n} \\ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total}) \ar[r]^{h_{total}} & \mathop{\mathit{Sh}}\nolimits (\mathcal{C}'_{total}) } \]

Moreover, we have $(g'_ n)^{-1} \circ h_{total, *} = h_{n, *} \circ g_ n^{-1}$.

Proof. Case A. Say $h_ n$ corresponds to the continuous functor $v_ n : \mathcal{C}'_ n \to \mathcal{C}_ n$. Then we can define a functor $v_{total} : \mathcal{C}'_{total} \to \mathcal{C}_{total}$ by using $v_ n$ in degree $n$. This is clearly a continuous functor (see definition of coverings in Lemma 85.3.1). Let $h_{total}^{-1} = v_{total, s} : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}'_{total}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total})$ and $h_{total, *} = v_{total}^ s = v_{total}^ p : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}'_{total})$ be the adjoint pair of functors constructed and studied in Sites, Sections 7.13 and 7.14. To see that $h_{total}$ is a morphism of topoi we still have to verify that $h_{total}^{-1}$ is exact. We first observe that $(g'_ n)^{-1} \circ h_{total, *} = h_{n, *} \circ g_ n^{-1}$; this is immediate by computing sections over an object $U$ of $\mathcal{C}'_ n$. Thus, if we think of a sheaf $\mathcal{F}$ on $\mathcal{C}_{total}$ as a system $(\mathcal{F}_ n, \mathcal{F}(\varphi ))$ as in Lemma 85.3.4, then $h_{total, *}\mathcal{F}$ corresponds to the system $(h_{n, *}\mathcal{F}_ n, h_{n, *}\mathcal{F}(\varphi ))$. Clearly, the functor $(\mathcal{F}'_ n, \mathcal{F}'(\varphi )) \to (h_ n^{-1}\mathcal{F}'_ n, h_ n^{-1}\mathcal{F}'(\varphi ))$ is its left adjoint. By uniqueness of adjoints, we conclude that $h_{total}^{-1}$ is given by this rule on systems. In particular, $h_{total}^{-1}$ is exact (by the description of sheaves on $\mathcal{C}_{total}$ given in the lemma and the exactness of the functors $h_ n^{-1}$) and we have our morphism of topoi. Finally, we obtain $g_ n^{-1} \circ h_{total}^{-1} = h_ n^{-1} \circ (g'_ n)^{-1}$ as well, which proves that the displayed diagram of the lemma commutes.

Case B. Here we have a functor $v_{total} : \mathcal{C}_{total} \to \mathcal{C}'_{total}$ by using $v_ n$ in degree $n$. This is clearly a cocontinuous functor (see definition of coverings in Lemma 85.3.2). Let $h_{total}$ be the morphism of topoi associated to $v_{total}$. The commutativity of the displayed diagram of the lemma follows immediately from Sites, Lemma 7.21.2. Taking left adjoints the final equality of the lemma becomes

\[ h_{total}^{-1} \circ (g'_ n)^{Sh}_! = g^{Sh}_{n!} \circ h_ n^{-1} \]

This follows immediately from the explicit description of the functors $(g'_ n)^{Sh}_!$ and $g^{Sh}_{n!}$ in Lemma 85.3.5, the fact that $h_ n^{-1} \circ (f'_\varphi )^{-1} = f_\varphi ^{-1} \circ h_ m^{-1}$ for $\varphi : [m] \to [n]$, and the fact that we already know $h_{total}^{-1}$ commutes with restrictions to the degree $n$ parts of the simplicial sites. $\square$

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