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85.4 Augmentations of simplicial sites

We continue in the fashion described in Section 85.3 working out the meaning of augmentations in cases A and B treated in that section.

Remark 85.4.1. In Situation 85.3.3 an augmentation $a_0$ towards a site $\mathcal{D}$ will mean

  1. $a_0 : \mathcal{C}_0 \to \mathcal{D}$ is a morphism of sites given by a continuous functor $u_0 : \mathcal{D} \to \mathcal{C}_0$ such that for all $\varphi , \psi : [0] \to [n]$ we have $u_\varphi \circ u_0 = u_\psi \circ u_0$.

  2. $a_0 : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_0) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ is a morphism of topoi given by a cocontinuous functor $u_0 : \mathcal{C}_0 \to \mathcal{D}$ such that for all $\varphi , \psi : [0] \to [n]$ we have $u_0 \circ u_\varphi = u_0 \circ u_\psi $.

Lemma 85.4.2. In Situation 85.3.3 let $a_0$ be an augmentation towards a site $\mathcal{D}$ as in Remark 85.4.1. Then $a_0$ induces

  1. a morphism of topoi $a_ n : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ for all $n \geq 0$,

  2. a morphism of topoi $a : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$

such that

  1. for all $\varphi : [m] \to [n]$ we have $a_ m \circ f_\varphi = a_ n$,

  2. if $g_ n : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total})$ is as in Lemma 85.3.5, then $a \circ g_ n = a_ n$, and

  3. $a_*\mathcal{F}$ for $\mathcal{F} \in \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total})$ is the equalizer of the two maps $a_{0, *}\mathcal{F}_0 \to a_{1, *}\mathcal{F}_1$.

Proof. Case A. Let $u_ n : \mathcal{D} \to \mathcal{C}_ n$ be the common value of the functors $u_\varphi \circ u_0$ for $\varphi : [0] \to [n]$. Then $u_ n$ corresponds to a morphism of sites $a_ n : \mathcal{C}_ n \to \mathcal{D}$, see Sites, Lemma 7.14.4. The same lemma shows that for all $\varphi : [m] \to [n]$ we have $a_ m \circ f_\varphi = a_ n$.

Case B. Let $u_ n : \mathcal{C}_ n \to \mathcal{D}$ be the common value of the functors $u_0 \circ u_\varphi $ for $\varphi : [0] \to [n]$. Then $u_ n$ is cocontinuous and hence defines a morphism of topoi $a_ n : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_ n) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D)}$, see Sites, Lemma 7.21.2. The same lemma shows that for all $\varphi : [m] \to [n]$ we have $a_ m \circ f_\varphi = a_ n$.

Consider the functor $a^{-1} : \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total})$ which to a sheaf of sets $\mathcal{G}$ associates the sheaf $\mathcal{F} = a^{-1}\mathcal{G}$ whose components are $a_ n^{-1}\mathcal{G}$ and whose transition maps $\mathcal{F}(\varphi )$ are the identifications

\[ f_\varphi ^{-1}\mathcal{F}_ m = f_\varphi ^{-1} a_ m^{-1}\mathcal{G} = a_ n^{-1}\mathcal{G} = \mathcal{F}_ n \]

for $\varphi : [m] \to [n]$, see the description of $\mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total})$ in Lemma 85.3.4. Since the functors $a_ n^{-1}$ are exact, $a^{-1}$ is an exact functor. Finally, for $a_* : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}_{total}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ we take the functor which to a sheaf $\mathcal{F}$ on $\mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ associates

\[ \xymatrix{ a_*\mathcal{F} \ar@{=}[r] & \text{Equalizer}(a_{0, *}\mathcal{F}_0 \ar@<1ex>[r] \ar@<-1ex>[r] & a_{1, *}\mathcal{F}_1) } \]

Here the two maps come from the two maps $\varphi : [0] \to [1]$ via

\[ a_{0, *}\mathcal{F}_0 \to a_{0, *}f_{\varphi , *} f_\varphi ^{-1}\mathcal{F}_0 \xrightarrow {\mathcal{F}(\varphi )} a_{0, *}f_{\varphi , *} \mathcal{F}_1 = a_{1, *}\mathcal{F}_1 \]

where the first arrow comes from $1 \to f_{\varphi , *} f_\varphi ^{-1}$. Let $\mathcal{G}_\bullet $ denote the constant simplicial sheaf with value $\mathcal{G}$ and let $a_{\bullet , *}\mathcal{F}$ denote the simplicial sheaf having $a_{n, *}\mathcal{F}_ n$ in degree $n$. By the usual adjuntion for the morphisms of topoi $a_ n$ we see that a map $a^{-1}\mathcal{G} \to \mathcal{F}$ is the same thing as a map

\[ \mathcal{G}_\bullet \longrightarrow a_{\bullet , *}\mathcal{F} \]

of simplicial sheaves. By Simplicial, Lemma 14.20.2 this is the same thing as a map $\mathcal{G} \to a_*\mathcal{F}$. Thus $a^{-1}$ and $a_*$ are adjoint functors and we obtain our morphism of topoi $a$1. The equalities $a \circ g_ n = f_ n$ follow immediately from the definitions. $\square$

[1] In case B the morphism $a$ corresponds to the cocontinuous functor $\mathcal{C}_{total} \to \mathcal{D}$ sending $U$ in $\mathcal{C}_ n$ to $u_ n(U)$.

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