The Stacks project

Lemma 85.2.8. Let $Y$ be a simplicial space and let $a : Y \to X$ be an augmentation (Simplicial, Definition 14.20.1). Let $a_ n : Y_ n \to X$ be the corresponding morphisms of topological spaces. There is a canonical morphism of topoi

\[ a : \mathop{\mathit{Sh}}\nolimits (Y_{Zar}) \to \mathop{\mathit{Sh}}\nolimits (X) \]

with the following properties:

  1. $a^{-1}\mathcal{F}$ is the sheaf restricting to $a_ n^{-1}\mathcal{F}$ on $Y_ n$,

  2. $a_ m \circ Y(\varphi ) = a_ n$ for all $\varphi : [m] \to [n]$,

  3. $a \circ g_ n = a_ n$ as morphisms of topoi with $g_ n$ as in Lemma 85.2.5,

  4. $a_*\mathcal{G}$ for $\mathcal{G} \in \mathop{\mathit{Sh}}\nolimits (Y_{Zar})$ is the equalizer of the two maps $a_{0, *}\mathcal{G}_0 \to a_{1, *}\mathcal{G}_1$.

Proof. Part (2) holds for augmentations of simplicial objects in any category. Thus $Y(\varphi )^{-1} a_ m^{-1} \mathcal{F} = a_ n^{-1}\mathcal{F}$ which defines an $Y(\varphi )$-map from $a_ m^{-1}\mathcal{F}$ to $a_ n^{-1}\mathcal{F}$. Thus we can use (1) as the definition of $a^{-1}\mathcal{F}$ (using Lemma 85.2.2) and (4) as the definition of $a_*$. If this defines a morphism of topoi then part (3) follows because we'll have $g_ n^{-1} \circ a^{-1} = a_ n^{-1}$ by construction. To check $a$ is a morphism of topoi we have to show that $a^{-1}$ is left adjoint to $a_*$ and we have to show that $a^{-1}$ is exact. The last fact is immediate from the exactness of the functors $a_ n^{-1}$.

Let $\mathcal{F}$ be an object of $\mathop{\mathit{Sh}}\nolimits (X)$ and let $\mathcal{G}$ be an object of $\mathop{\mathit{Sh}}\nolimits (Y_{Zar})$. Given $\beta : a^{-1}\mathcal{F} \to \mathcal{G}$ we can look at the components $\beta _ n : a_ n^{-1}\mathcal{F} \to \mathcal{G}_ n$. These maps are adjoint to maps $\beta _ n : \mathcal{F} \to a_{n, *}\mathcal{G}_ n$. Compatibility with the simplicial structure shows that $\beta _0$ maps into $a_*\mathcal{G}$. Conversely, suppose given a map $\alpha : \mathcal{F} \to a_*\mathcal{G}$. For any $n$ choose a $\varphi : [0] \to [n]$. Then we can look at the composition

\[ \mathcal{F} \xrightarrow {\alpha } a_*\mathcal{G} \to a_{0, *}\mathcal{G}_0 \xrightarrow {\mathcal{G}(\varphi )} a_{n, *}\mathcal{G}_ n \]

These are adjoint to maps $a_ n^{-1}\mathcal{F} \to \mathcal{G}_ n$ which define a morphism of sheaves $a^{-1}\mathcal{F} \to \mathcal{G}$. We omit the proof that the constructions given above define mutually inverse bijections

\[ \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (Y_{Zar})}(a^{-1}\mathcal{F}, \mathcal{G}) = \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (X)}(\mathcal{F}, a_*\mathcal{G}) \]

This finishes the proof. An interesting observation is here that this morphism of topoi does not correspond to any obvious geometric functor between the sites defining the topoi. $\square$

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