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\input{preamble}
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\begin{document}
\title{Simplicial Spaces}
\maketitle
\phantomsection
\label{section-phantom}
\tableofcontents
\section{Introduction}
\label{section-introduction}
\noindent
This chapter develops some theory concerning simplicial topological spaces,
simplicial ringed spaces, simplicial schemes, and simplicial algebraic spaces.
The theory of simplicial spaces sometimes allows one to prove local to global
principles which appear difficult to prove in other ways.
Some example applications can be found in the papers
\cite{faltings_finiteness}, \cite{Kiehl}, and \cite{HodgeIII}.
\medskip\noindent
We assume throughout that the reader is familiar with the basic concepts
and results of the chapter Simplicial Methods, see
Simplicial, Section \ref{simplicial-section-introduction}.
In particular, we continue to write $X$ and not $X_\bullet$
for a simplicial object.
\section{Simplicial topological spaces}
\label{section-simplicial-top}
\noindent
A {\it simplicial space} is a simplicial object in the category of
topological spaces where morphisms are continuous maps of topological
spaces. (We will use ``simplicial algebraic space'' to refer to simplicial
objects in the category of algebraic spaces.)
We may picture a simplicial space $X$ as follows
$$
\xymatrix{
X_2
\ar@<2ex>[r]
\ar@<0ex>[r]
\ar@<-2ex>[r]
&
X_1
\ar@<1ex>[r]
\ar@<-1ex>[r]
\ar@<1ex>[l]
\ar@<-1ex>[l]
&
X_0
\ar@<0ex>[l]
}
$$
Here there are two morphisms $d^1_0, d^1_1 : X_1 \to X_0$
and a single morphism $s^0_0 : X_0 \to X_1$, etc.
It is important to keep in mind that $d^n_i : X_n \to X_{n - 1}$
should be thought of as a ``projection forgetting the
$i$th coordinate'' and $s^n_j : X_n \to X_{n + 1}$ as the diagonal
map repeating the $j$th coordinate.
\medskip\noindent
Let $X$ be a simplicial space. We associate a site
$X_{Zar}$\footnote{This notation is similar to the notation in
Sites, Example \ref{sites-example-site-topological}
and
Topologies, Definition \ref{topologies-definition-big-small-Zariski}.}
to $X$ as follows.
\begin{enumerate}
\item An object of $X_{Zar}$ is an open $U$ of $X_n$ for some $n$,
\item a morphism $U \to V$ of $X_{Zar}$ is given by a
$\varphi : [m] \to [n]$ where $n, m$ are such that
$U \subset X_n$, $V \subset X_m$ and $\varphi$ is such that
$X(\varphi)(U) \subset V$, and
\item a covering $\{U_i \to U\}$ in $X_{Zar}$ means
that $U, U_i \subset X_n$ are open, the maps $U_i \to U$ are
given by $\text{id} : [n] \to [n]$, and $U = \bigcup U_i$.
\end{enumerate}
Note that in particular, if $U \to V$ is a morphism of $X_{Zar}$
given by $\varphi$, then $X(\varphi) : X_n \to X_m$ does in fact
induce a continuous map $U \to V$ of topological spaces.
\noindent
It is clear that the above is a special case of a construction that
associates to any diagram of topological spaces a site. We formulate
the obligatory lemma.
\begin{lemma}
\label{lemma-simplicial-site}
Let $X$ be a simplicial space. Then $X_{Zar}$
as defined above is a site.
\end{lemma}
\begin{proof}
Omitted.
\end{proof}
\noindent
Let $X$ be a simplicial space. Let $\mathcal{F}$ be a sheaf on $X_{Zar}$.
It is clear from the definition of coverings, that the restriction
of $\mathcal{F}$ to the opens of $X_n$ defines a sheaf $\mathcal{F}_n$
on the topological space $X_n$. For every $\varphi : [m] \to [n]$ the
restriction maps of $\mathcal{F}$ for pairs $U \subset X_n$, $V \subset X_m$
with $X(\varphi)(U) \subset V$, define an $X(\varphi)$-map
$\mathcal{F}(\varphi) : \mathcal{F}_m \to \mathcal{F}_n$, see
Sheaves, Definition \ref{sheaves-definition-f-map}.
Moreover, given $\varphi : [m] \to [n]$ and $\psi : [l] \to [m]$
we have
$$
\mathcal{F}(\varphi) \circ \mathcal{F}(\psi) =
\mathcal{F}(\varphi \circ \psi)
$$
(LHS uses composition of $f$-maps, see
Sheaves, Definition \ref{sheaves-definition-composition-f-maps}).
Clearly, the converse is true as well: if we have a system
$(\{\mathcal{F}_n\}_{n \geq 0},
\{\mathcal{F}(\varphi)\}_{\varphi \in \text{Arrows}(\Delta)})$
as above, satisfying the displayed equalities,
then we obtain a sheaf on $X_{Zar}$.
\begin{lemma}
\label{lemma-describe-sheaves-simplicial-site}
Let $X$ be a simplicial space. There is an equivalence of
categories between
\begin{enumerate}
\item $\Sh(X_{Zar})$, and
\item category of systems $(\mathcal{F}_n, \mathcal{F}(\varphi))$
described above.
\end{enumerate}
\end{lemma}
\begin{proof}
See discussion above.
\end{proof}
\begin{lemma}
\label{lemma-simplicial-space-site-functorial}
Let $f : Y \to X$ be a morphism of simplicial spaces.
Then the functor $u : X_{Zar} \to Y_{Zar}$
which associates to the open $U \subset X_n$ the open
$f_n^{-1}(U) \subset Y_n$ defines a morphism of sites
$f_{Zar} : Y_{Zar} \to X_{Zar}$.
\end{lemma}
\begin{proof}
It is clear that $u$ is a continuous functor. Hence we obtain functors
$f_{Zar, *} = u^s$ and $f_{Zar}^{-1} = u_s$, see
Sites, Section \ref{sites-section-morphism-sites}.
To see that we obtain a morphism of sites we have to show
that $u_s$ is exact. We will use
Sites, Lemma \ref{sites-lemma-directed-morphism} to see this.
Let $V \subset Y_n$ be an open subset. The category
$\mathcal{I}_V^u$ (see Sites, Section \ref{sites-section-functoriality-PSh})
consists of pairs $(U, \varphi)$ where
$\varphi : [m] \to [n]$ and $U \subset X_m$ open such that
$Y(\varphi)(V) \subset f_m^{-1}(U)$. Moreover, a morphism
$(U, \varphi) \to (U', \varphi')$ is given by a
$\psi : [m'] \to [m]$ such that $X(\psi)(U) \subset U'$
and $\varphi \circ \psi = \varphi'$.
It is our task to show that $\mathcal{I}_V^u$ is cofiltered.
\medskip\noindent
We verify the conditions of
Categories, Definition \ref{categories-definition-codirected}.
Condition (1) holds because $(X_n, \text{id}_{[n]})$ is an object.
Let $(U, \varphi)$ be an object. The condition
$Y(\varphi)(V) \subset f_m^{-1}(U)$ is equivalent to
$V \subset f_n^{-1}(X(\varphi)^{-1}(U))$. Hence we obtain a morphism
$(X(\varphi)^{-1}(U), \text{id}_{[n]}) \to (U, \varphi)$ given
by setting $\psi = \varphi$. Moreover, given a pair of objects
of the form $(U, \text{id}_{[n]})$ and $(U', \text{id}_{[n]})$
we see there exists an object, namely $(U \cap U', \text{id}_{[n]})$,
which maps to both of them. Thus condition (2) holds.
To verify condition (3) suppose given two morphisms
$a, a': (U, \varphi) \to (U', \varphi')$ given by $\psi, \psi' : [m'] \to [m]$.
Then precomposing with the morphism
$(X(\varphi)^{-1}(U), \text{id}_{[n]}) \to (U, \varphi)$ given
by $\varphi$ equalizes $a, a'$ because
$\varphi \circ \psi = \varphi' = \varphi \circ \psi'$.
This finishes the proof.
\end{proof}
\begin{lemma}
\label{lemma-describe-functoriality}
Let $f : Y \to X$ be a morphism of simplicial spaces. In terms of the
description of sheaves in
Lemma \ref{lemma-describe-sheaves-simplicial-site} the
morphism $f_{Zar}$ of Lemma \ref{lemma-simplicial-space-site-functorial}
can be described as follows.
\begin{enumerate}
\item If $\mathcal{G}$ is a sheaf on $Y$, then
$(f_{Zar, *}\mathcal{G})_n = f_{n, *}\mathcal{G}_n$.
\item If $\mathcal{F}$ is a sheaf on $X$, then
$(f_{Zar}^{-1}\mathcal{F})_n = f_n^{-1}\mathcal{F}_n$.
\end{enumerate}
\end{lemma}
\begin{proof}
The first part is immediate from the definitions. For the second part, note
that in the proof of
Lemma \ref{lemma-simplicial-space-site-functorial}
we have shown that for a $V \subset Y_n$ open the category
$(\mathcal{I}_V^u)^{opp}$ contains as a cofinal subcategory
the category of opens $U \subset X_n$ with $f_n^{-1}(U) \supset V$
and morphisms given by inclusions. Hence we see that the restriction
of $u_p\mathcal{F}$ to opens of $Y_n$ is the presheaf
$f_{n, p}\mathcal{F}_n$ as defined in
Sheaves, Lemma \ref{sheaves-lemma-pullback-presheaves}.
Since $f_{Zar}^{-1}\mathcal{F} = u_s\mathcal{F}$ is the sheafification
of $u_p\mathcal{F}$ and since sheafification uses only coverings and
since coverings in $Y_{Zar}$ use only inclusions between opens on the
same $Y_n$, the result follows from the fact that $f_n^{-1}\mathcal{F}_n$
is (correspondingly) the sheafification of $f_{n, p}\mathcal{F}_n$, see
Sheaves, Section \ref{sheaves-section-presheaves-functorial}.
\end{proof}
\noindent
Let $X$ be a topological space. In
Sites, Example \ref{sites-example-site-topological}
we denoted $X_{Zar}$ the site consisting of opens of $X$
with inclusions as morphisms and coverings given by open coverings.
We identify the topos $\Sh(X_{Zar})$ with the category
of sheaves on $X$.
\begin{lemma}
\label{lemma-restriction-to-components}
Let $X$ be a simplicial space. The functor
$X_{n, Zar} \to X_{Zar}$, $U \mapsto U$ is continuous
and cocontinuous. The associated morphism of
topoi $g_n : \Sh(X_n) \to \Sh(X_{Zar})$ satisfies
\begin{enumerate}
\item $g_n^{-1}$ associates to the sheaf $\mathcal{F}$ on $X$
the sheaf $\mathcal{F}_n$ on $X_n$,
\item $g_n^{-1} : \Sh(X_{Zar}) \to \Sh(X_n)$ has a left adjoint $g^{Sh}_{n!}$,
\item $g^{Sh}_{n!}$ commutes with finite connected limits,
\item $g_n^{-1} : \textit{Ab}(X_{Zar}) \to \textit{Ab}(X_n)$
has a left adjoint $g_{n!}$, and
\item $g_{n!}$ is exact.
\end{enumerate}
\end{lemma}
\begin{proof}
Besides the properties of our functor mentioned in the statement,
the category $X_{n, Zar}$ has fibre products and equalizers
and the functor commutes with them (beware that $X_{Zar}$ does not
have all fibre products). Hence the lemma follows from the discussion in
Sites, Sections \ref{sites-section-cocontinuous-functors} and
\ref{sites-section-cocontinuous-morphism-topoi}
and
Modules on Sites, Section \ref{sites-modules-section-exactness-lower-shriek}.
More precisely,
Sites, Lemmas \ref{sites-lemma-cocontinuous-morphism-topoi},
\ref{sites-lemma-when-shriek}, and
\ref{sites-lemma-preserve-equalizers}
and
Modules on Sites, Lemmas
\ref{sites-modules-lemma-g-shriek-adjoint} and
\ref{sites-modules-lemma-exactness-lower-shriek}.
\end{proof}
\begin{lemma}
\label{lemma-restriction-injective-to-component}
Let $X$ be a simplicial space. If $\mathcal{I}$ is an injective abelian
sheaf on $X_{Zar}$, then $\mathcal{I}_n$ is an injective abelian sheaf
on $X_n$.
\end{lemma}
\begin{proof}
This follows from
Homology, Lemma \ref{homology-lemma-adjoint-preserve-injectives}
and
Lemma \ref{lemma-restriction-to-components}.
\end{proof}
\begin{lemma}
\label{lemma-restriction-to-components-functorial}
Let $f : Y \to X$ be a morphism of simplicial spaces. Then
$$
\xymatrix{
\Sh(Y_n) \ar[d] \ar[r]_{f_n} & \Sh(X_n) \ar[d] \\
\Sh(Y_{Zar}) \ar[r]^{f_{Zar}} & \Sh(X_{Zar})
}
$$
is a commutative diagram of topoi.
\end{lemma}
\begin{proof}
Direct from the description of pullback functors in
Lemmas \ref{lemma-describe-functoriality} and
\ref{lemma-restriction-to-components}.
\end{proof}
\begin{lemma}
\label{lemma-augmentation}
Let $Y$ be a simplicial space and let $a : Y \to X$ be an augmentation
(Simplicial, Definition \ref{simplicial-definition-augmentation}).
Let $a_n : Y_n \to X$ be the corresponding morphisms of topological spaces.
There is a canonical morphism of topoi
$$
a : \Sh(Y_{Zar}) \to \Sh(X)
$$
with the following properties:
\begin{enumerate}
\item $a^{-1}\mathcal{F}$ is the sheaf restricting to $a_n^{-1}\mathcal{F}$
on $Y_n$,
\item $a_m \circ Y(\varphi) = a_n$ for all $\varphi : [m] \to [n]$,
\item $a \circ g_n = a_n$ as morphisms of topoi with
$g_n$ as in Lemma \ref{lemma-restriction-to-components-site},
\item $a_*\mathcal{G}$ for $\mathcal{G} \in \Sh(Y_{Zar})$
is the equalizer of the two maps
$a_{0, *}\mathcal{G}_0 \to a_{1, *}\mathcal{G}_1$.
\end{enumerate}
\end{lemma}
\begin{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 \ref{lemma-describe-sheaves-simplicial-site}) 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 $a_n^{-1}$.
\medskip\noindent
Let $\mathcal{F}$ be an object of $\Sh(X)$ and let $\mathcal{G}$
be an object of $\Sh(Y_{Zar})$. 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
$$
\Mor_{\Sh(Y_{Zar})}(a^{-1}\mathcal{F}, \mathcal{G}) =
\Mor_{\Sh(X)}(\mathcal{F}, a_*\mathcal{G})
$$
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.
\end{proof}
\begin{lemma}
\label{lemma-simplicial-resolution-Z}
Let $X$ be a simplicial topological space. The complex of
abelian presheaves on $X_{Zar}$
$$
\ldots \to \mathbf{Z}_{X_2} \to \mathbf{Z}_{X_1} \to \mathbf{Z}_{X_0}
$$
with boundary $\sum (-1)^i d^n_i$ is a resolution
of the constant presheaf $\mathbf{Z}$.
\end{lemma}
\begin{proof}
Let $U \subset X_m$ be an object of $X_{Zar}$. Then the value of
the complex above on $U$ is the complex of abelian groups
$$
\ldots \to
\mathbf{Z}[\Mor_\Delta([2], [m])] \to
\mathbf{Z}[\Mor_\Delta([1], [m])] \to
\mathbf{Z}[\Mor_\Delta([0], [m])]
$$
In other words, this is the complex associated to the
free abelian group on the simplicial set $\Delta[m]$, see
Simplicial, Example \ref{simplicial-example-simplex-simplicial-set}.
Since $\Delta[m]$ is homotopy equivalent to $\Delta[0]$, see
Simplicial, Example \ref{simplicial-example-simplex-contractible},
and since ``taking free abelian groups'' is a functor,
we see that the complex above is homotopy equivalent to
the free abelian group on $\Delta[0]$
(Simplicial, Remark \ref{simplicial-remark-homotopy-better} and
Lemma \ref{simplicial-lemma-homotopy-equivalence-s-N}).
This complex is acyclic in positive degrees
and equal to $\mathbf{Z}$ in degree $0$.
\end{proof}
\begin{lemma}
\label{lemma-simplicial-sheaf-cohomology}
Let $X$ be a simplicial topological space. Let $\mathcal{F}$ be an abelian
sheaf on $X$. There is a spectral sequence $(E_r, d_r)_{r \geq 0}$ with
$$
E_1^{p, q} = H^q(X_p, \mathcal{F}_p)
$$
converging to $H^{p + q}(X_{Zar}, \mathcal{F})$.
This spectral sequence is functorial in $\mathcal{F}$.
\end{lemma}
\begin{proof}
Let $\mathcal{F} \to \mathcal{I}^\bullet$ be an injective resolution.
Consider the double complex with terms
$$
A^{p, q} = \mathcal{I}^q(X_p)
$$
and first differential given by the alternating sum along the maps
$d^{p + 1}_i$-maps $\mathcal{I}_p^q \to \mathcal{I}_{p + 1}^q$, see
Lemma \ref{lemma-describe-sheaves-simplicial-site}. Note that
$$
A^{p, q} = \Gamma(X_p, \mathcal{I}_p^q) =
\Mor_{\textit{PSh}}(h_{X_p}, \mathcal{I}^q) =
\Mor_{\textit{PAb}}(\mathbf{Z}_{X_p}, \mathcal{I}^q)
$$
Hence it follows from Lemma \ref{lemma-simplicial-resolution-Z} and
Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-injective-abelian-sheaf-injective-presheaf}
that the rows of the double complex are exact in positive degrees and
evaluate to $\Gamma(X_{Zar}, \mathcal{I}^q)$ in degree $0$.
On the other hand, since restriction is exact
(Lemma \ref{lemma-restriction-to-components})
the map
$$
\mathcal{F}_p \to \mathcal{I}_p^\bullet
$$
is a resolution. The sheaves $\mathcal{I}_p^q$ are injective
abelian sheaves on $X_p$
(Lemma \ref{lemma-restriction-injective-to-component}).
Hence the cohomology of the columns computes the groups
$H^q(X_p, \mathcal{F}_p)$. We conclude by applying
Homology, Lemmas \ref{homology-lemma-first-quadrant-ss} and
\ref{homology-lemma-double-complex-gives-resolution}.
\end{proof}
\begin{lemma}
\label{lemma-augmentation-pushforward-higher-direct-image}
Let $X$ be a simplicial space and let $a : X \to Y$
be an augmentation. Let $\mathcal{F}$ be an abelian sheaf
on $X_{Zar}$. Then $R^na_*\mathcal{F}$ is the sheaf associated
to the presheaf
$$
V \longmapsto H^n((X \times_Y V)_{Zar}, \mathcal{F}|_{(X \times_Y V)_{Zar}})
$$
\end{lemma}
\begin{proof}
This is the analogue of
Cohomology, Lemma \ref{cohomology-lemma-describe-higher-direct-images} or of
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-higher-direct-images}
and we strongly encourge the reader to skip the proof.
Choosing an injective resolution of $\mathcal{F}$ on
$X_{Zar}$ and using the definitions we see that it suffices to show:
(1) the restriction of an injective abelian
sheaf on $X_{Zar}$ to $(X \times_Y V)_{Zar}$ is an injective abelian sheaf and
(2) $a_*\mathcal{F}$ is equal to the rule
$$
V \longmapsto H^0((X \times_Y V)_{Zar}, \mathcal{F}|_{(X \times_Y V)_{Zar}})
$$
Part (2) follows from the following facts
\begin{enumerate}
\item[(2a)] $a_*\mathcal{F}$ is the equalizer of the two maps
$a_{0, *}\mathcal{F}_0 \to a_{1, *}\mathcal{F}_1$
by Lemma \ref{lemma-augmentation},
\item[(2b)] $a_{0, *}\mathcal{F}_0(V) =
H^0(a_0^{-1}(V), \mathcal{F}_0)$ and
$a_{1, *}\mathcal{F}_1(V) = H^0(a_1^{-1}(V), \mathcal{F}_1)$,
\item[(2c)] $X_0 \times_Y V = a_0^{-1}(V)$ and $X_1 \times_Y V = a_1^{-1}(V)$,
\item[(2d)] $H^0((X \times_Y V)_{Zar}, \mathcal{F}|_{(X \times_Y V)_{Zar}})$
is the equalizer of the two maps
$H^0(X_0 \times_Y V, \mathcal{F}_0) \to H^0(X_1 \times_Y V, \mathcal{F}_1)$
for example by Lemma \ref{lemma-simplicial-sheaf-cohomology}.
\end{enumerate}
Part (1) follows after one defines an exact left adjoint
$j_! : \textit{Ab}((X \times_Y V)_{Zar}) \to \textit{Ab}(X_{Zar})$
(extension by zero) to restriction
$\textit{Ab}(X_{Zar}) \to \textit{Ab}((X \times_Y V)_{Zar})$
and using Homology, Lemma \ref{homology-lemma-adjoint-preserve-injectives}.
\end{proof}
\noindent
Let $X$ be a topological space. Denote $X_\bullet$ the constant simplicial
topological space with value $X$. By
Lemma \ref{lemma-describe-sheaves-simplicial-site}
a sheaf on $X_{\bullet, Zar}$ is the same
thing as a cosimplicial object in the category of sheaves on $X$.
\begin{lemma}
\label{lemma-constant-simplicial-space}
Let $X$ be a topological space. Let $X_\bullet$ be the constant
simplicial topological space with value $X$. The functor
$$
X_{\bullet, Zar} \longrightarrow X_{Zar},\quad
U \longmapsto U
$$
is continuous and cocontinuous and defines a morphism of
topoi $g : \Sh(X_{\bullet, Zar}) \to \Sh(X)$ as well as a left adjoint
$g_!$ to $g^{-1}$. We have
\begin{enumerate}
\item $g^{-1}$ associates to a sheaf on $X$ the constant cosimplicial
sheaf on $X$,
\item $g_!$ associates to a sheaf $\mathcal{F}$ on $X_{\bullet, Zar}$ the
sheaf $\mathcal{F}_0$, and
\item $g_*$ associates to a sheaf $\mathcal{F}$ on $X_{\bullet, Zar}$ the
equalizer of the two maps $\mathcal{F}_0 \to \mathcal{F}_1$.
\end{enumerate}
\end{lemma}
\begin{proof}
The statements about the functor are straightforward to verify.
The existence of $g$ and $g_!$ follow from
Sites, Lemmas \ref{sites-lemma-cocontinuous-morphism-topoi} and
\ref{sites-lemma-when-shriek}. The description of
$g^{-1}$ is immediate from Sites, Lemma \ref{sites-lemma-when-shriek}.
The description of $g_*$ and $g_!$ follows as the functors given are
right and left adjoint to $g^{-1}$.
\end{proof}
\section{Simplicial sites and topoi}
\label{section-simplicial-sites}
\noindent
It seems natural to define a {\it simplicial site} as a simplicial
object in the (big) category whose objects are sites
and whose morphisms are morphisms of sites.
See Sites, Definitions \ref{sites-definition-site} and
\ref{sites-definition-morphism-sites}
with composition of morphisms as in
Sites, Lemma \ref{sites-lemma-composition-morphisms-sites}.
But here are some variants one might want to consider:
(a) we could work with cocontinuous functors
(see Sites, Sections \ref{sites-section-cocontinuous-functors} and
\ref{sites-section-cocontinuous-morphism-topoi}) between sites instead,
(b) we could work in a suitable $2$-category of sites where one introduces
the notion of a $2$-morphism between morphisms of sites,
(c) we could work in a $2$-category constructed out of cocontinuous
functors. Instead of picking one of these variants as a definition
we will simply develop theory as needed.
\medskip\noindent
Certainly a {\it simplicial topos} should probably be defined as a
pseudo-functor from $\Delta^{opp}$ into the $2$-category of topoi.
See Categories, Definition \ref{categories-definition-functor-into-2-category}
and Sites, Section \ref{sites-section-topoi} and
\ref{sites-section-2-category}. We will try to avoid working with such
a beast if possible.
\medskip\noindent
{\bf Case A.}
Let $\mathcal{C}$ be a simplicial object in the category whose objects
are sites and whose morphisms are morphisms of sites. This means that
for every morphism $\varphi : [m] \to [n]$ of $\Delta$ we have a morphism
of sites $f_\varphi : \mathcal{C}_n \to \mathcal{C}_m$. This morphism is
given by a continuous functor in the opposite direction which we will denote
$u_\varphi : \mathcal{C}_m \to \mathcal{C}_n$.
\begin{lemma}
\label{lemma-simplicial-site-site}
Let $\mathcal{C}$ be a simplicial object in the category of sites.
With notation as above we construct a site $\mathcal{C}_{total}$ as follows.
\begin{enumerate}
\item An object of $\mathcal{C}_{total}$ is an object $U$ of
$\mathcal{C}_n$ for some $n$,
\item a morphism $(\varphi, f) : U \to V$ of $\mathcal{C}_{total}$
is given by a map $\varphi : [m] \to [n]$ with
$U \in \Ob(\mathcal{C}_n)$, $V \in \Ob(\mathcal{C}_m)$
and a morphism $f : U \to u_\varphi(V)$ of $\mathcal{C}_n$, and
\item a covering $\{(\text{id}, f_i) : U_i \to U\}$ in $\mathcal{C}_{total}$
is given by an $n$ and a covering $\{f_i : U_i \to U\}$
of $\mathcal{C}_n$.
\end{enumerate}
\end{lemma}
\begin{proof}
Composition of $(\varphi, f) : U \to V$ with $(\psi, g) : V \to W$
is given by $(\varphi \circ \psi, u_\varphi(g) \circ f)$.
This uses that $u_\varphi \circ u_\psi = u_{\varphi \circ \psi}$.
\medskip\noindent
Let $\{(\text{id}, f_i) : U_i \to U\}$ be a covering as in (3)
and let $(\varphi, g) : W \to U$ be a morphism with
$W \in \Ob(\mathcal{C}_m)$. We claim that
$$
W \times_{(\varphi, g), U, (\text{id}, f_i)} U_i =
W \times_{g, u_\varphi(U), u_\varphi(f_i)} u_\varphi(U_i)
$$
in the category $\mathcal{C}_{total}$. This makes sense as by our
definition of morphisms of sites, the required fibre products
in $\mathcal{C}_m$ exist since $u_\varphi$ transforms coverings into
coverings. The same reasoning implies the claim (details omitted).
Thus we see that the collection of coverings is stable under base
change. The other axioms of a site are immediate.
\end{proof}
\noindent
{\bf Case B.}
Let $\mathcal{C}$ be a simplicial object in the category whose objects are
sites and whose morphisms are cocontinuous functors. This means that for
every morphism $\varphi : [m] \to [n]$ of $\Delta$ we have a cocontinuous
functor denoted $u_\varphi : \mathcal{C}_n \to \mathcal{C}_m$. The associated
morphism of topoi is denoted
$f_\varphi : \Sh(\mathcal{C}_n) \to \Sh(\mathcal{C}_m)$.
\begin{lemma}
\label{lemma-simplicial-cocontinuous-site}
Let $\mathcal{C}$ be a simplicial object in the category whose objects are
sites and whose morphisms are cocontinuous functors. With notation as above,
assume the functors $u_\varphi : \mathcal{C}_n \to \mathcal{C}_m$
have property $P$ of Sites, Remark \ref{sites-remark-cartesian-cocontinuous}.
Then we can construct a site $\mathcal{C}_{total}$ as follows.
\begin{enumerate}
\item An object of $\mathcal{C}_{total}$ is an object $U$ of
$\mathcal{C}_n$ for some $n$,
\item a morphism $(\varphi, f) : U \to V$ of $\mathcal{C}_{total}$
is given by a map $\varphi : [m] \to [n]$ with
$U \in \Ob(\mathcal{C}_n)$, $V \in \Ob(\mathcal{C}_m)$
and a morphism $f : u_\varphi(U) \to V$ of $\mathcal{C}_m$, and
\item a covering $\{(\text{id}, f_i) : U_i \to U\}$ in $\mathcal{C}_{total}$
is given by an $n$ and a covering $\{f_i : U_i \to U\}$
of $\mathcal{C}_n$.
\end{enumerate}
\end{lemma}
\begin{proof}
Composition of $(\varphi, f) : U \to V$ with $(\psi, g) : V \to W$
is given by $(\varphi \circ \psi, g \circ u_\psi(f))$.
This uses that $u_\psi \circ u_\varphi = u_{\varphi \circ \psi}$.
\medskip\noindent
Let $\{(\text{id}, f_i) : U_i \to U\}$ be a covering as in (3)
and let $(\varphi, g) : W \to U$ be a morphism with
$W \in \Ob(\mathcal{C}_m)$. We claim that
$$
W \times_{(\varphi, g), U, (\text{id}, f_i)} U_i =
W \times_{g, U, f_i} U_i
$$
in the category $\mathcal{C}_{total}$ where the right hand side
is the object of $\mathcal{C}_m$ defined in
Sites, Remark \ref{sites-remark-cartesian-cocontinuous}
which exists by property $P$. Compatibility of this type of fibre product
with compositions of functors implies the claim (details omitted).
Since the family $\{W \times_{g, U, f_i} U_i \to W\}$ is a
covering of $\mathcal{C}_m$ by property $P$ we see that
the collection of coverings is stable under base
change. The other axioms of a site are immediate.
\end{proof}
\begin{situation}
\label{situation-simplicial-site}
Here we have one of the following two cases:
\begin{enumerate}
\item[(A)] $\mathcal{C}$ is a simplicial object in the category whose
objects are sites and whose morphisms are morphisms of sites. For every
morphism $\varphi : [m] \to [n]$ of $\Delta$ we have a morphism of sites
$f_\varphi : \mathcal{C}_n \to \mathcal{C}_m$ given by a continuous
functor $u_\varphi : \mathcal{C}_m \to \mathcal{C}_n$.
\item[(B)] $\mathcal{C}$ is a simplicial object in the category whose
objects are sites and whose morphisms are cocontinuous functors having
property $P$ of Sites, Remark \ref{sites-remark-cartesian-cocontinuous}.
For every morphism $\varphi : [m] \to [n]$ of $\Delta$ we have a cocontinuous
functor $u_\varphi : \mathcal{C}_n \to \mathcal{C}_m$ which induces a
morphism of topoi $f_\varphi : \Sh(\mathcal{C}_n) \to \Sh(\mathcal{C}_m)$.
\end{enumerate}
As usual we will denote $f_\varphi^{-1}$ and $f_{\varphi, *}$ the
pullback and pushforward. We let $\mathcal{C}_{total}$ denote the
site defined in
Lemma \ref{lemma-simplicial-site-site} (case A) or
Lemma \ref{lemma-simplicial-cocontinuous-site} (case B).
\end{situation}
\noindent
Let $\mathcal{C}$ be as in Situation \ref{situation-simplicial-site}.
Let $\mathcal{F}$ be a sheaf on $\mathcal{C}_{total}$.
It is clear from the definition of coverings, that the restriction
of $\mathcal{F}$ to the objects of $\mathcal{C}_n$ defines a sheaf
$\mathcal{F}_n$ on the site $\mathcal{C}_n$. For every
$\varphi : [m] \to [n]$ the restriction maps of $\mathcal{F}$
along the morphisms $(\varphi, f) : U \to V$ with
$U \in \Ob(\mathcal{C}_n)$ and $V \in \Ob(\mathcal{C}_m)$
define an element $\mathcal{F}(\varphi)$ of
$$
\Mor_{\Sh(\mathcal{C}_m)}(\mathcal{F}_m, f_{\varphi, *}\mathcal{F}_n) =
\Mor_{\Sh(\mathcal{C}_n)}(f_\varphi^{-1}\mathcal{F}_m, \mathcal{F}_n)
$$
Moreover, given $\varphi : [m] \to [n]$ and $\psi : [l] \to [m]$
the diagrams
$$
\vcenter{
\xymatrix{
\mathcal{F}_l \ar[rr]_{\mathcal{F}(\varphi \circ \psi)}
\ar[rd]_{\mathcal{F}(\psi)}
& & f_{\varphi \circ \psi, *} \mathcal{F}_n \\
& f_{\psi, *}\mathcal{F}_m \ar[ur]_{f_{\psi, *}\mathcal{F}(\varphi)}
}
}
\quad\text{and}\quad
\vcenter{
\xymatrix{
f_{\varphi \circ \psi}^{-1}\mathcal{F}_l
\ar[rr]_{\mathcal{F}(\varphi \circ \psi)}
\ar[rd]_{f_\varphi^{-1}\mathcal{F}(\psi)}
& & \mathcal{F}_n \\
& f_\varphi^{-1}\mathcal{F}_m \ar[ur]_{\mathcal{F}(\varphi)}
}
}
$$
commute. Clearly, the converse statement is true as well: if we have a system
$(\{\mathcal{F}_n\}_{n \geq 0},
\{\mathcal{F}(\varphi)\}_{\varphi \in \text{Arrows}(\Delta)})$
satisfying the commutativity constraints above,
then we obtain a sheaf on $\mathcal{C}_{total}$.
\begin{lemma}
\label{lemma-describe-sheaves-simplicial-site-site}
In Situation \ref{situation-simplicial-site} there is an equivalence of
categories between
\begin{enumerate}
\item $\Sh(\mathcal{C}_{total})$, and
\item the category of systems $(\mathcal{F}_n, \mathcal{F}(\varphi))$
described above.
\end{enumerate}
In particular, the topos $\Sh(\mathcal{C}_{total})$ only depends on
the topoi $\Sh(\mathcal{C}_n)$ and the morphisms of topoi $f_\varphi$.
\end{lemma}
\begin{proof}
See discussion above.
\end{proof}
\begin{lemma}
\label{lemma-restriction-to-components-site}
In Situation \ref{situation-simplicial-site} the functor
$\mathcal{C}_n \to \mathcal{C}_{total}$, $U \mapsto U$ is continuous
and cocontinuous. The associated morphism of
topoi $g_n : \Sh(\mathcal{C}_n) \to \Sh(\mathcal{C}_{total})$ satisfies
\begin{enumerate}
\item $g_n^{-1}$ associates to the sheaf $\mathcal{F}$ on $\mathcal{C}_{total}$
the sheaf $\mathcal{F}_n$ on $\mathcal{C}_n$,
\item $g_n^{-1} : \Sh(\mathcal{C}_{total}) \to \Sh(\mathcal{C}_n)$
has a left adjoint $g^{Sh}_{n!}$,
\item for $\mathcal{G}$ in $\Sh(\mathcal{C}_n)$ the restriction of
$g_{n!}^{Sh}\mathcal{G}$ to $\mathcal{C}_m$ is
$\coprod\nolimits_{\varphi : [n] \to [m]} f_\varphi^{-1}\mathcal{G}$,
\item $g_{n!}^{Sh}$ commutes with finite connected limits,
\item $g_n^{-1} : \textit{Ab}(\mathcal{C}_{total}) \to
\textit{Ab}(\mathcal{C}_n)$ has a left adjoint $g_{n!}$,
\item for $\mathcal{G}$ in $\textit{Ab}(\mathcal{C}_n)$ the restriction of
$g_{n!}\mathcal{G}$ to $\mathcal{C}_m$ is
$\bigoplus\nolimits_{\varphi : [n] \to [m]} f_\varphi^{-1}\mathcal{G}$, and
\item $g_{n!}$ is exact.
\end{enumerate}
\end{lemma}
\begin{proof}
Case A. If $\{U_i \to U\}_{i \in I}$ is a covering in $\mathcal{C}_n$
then the image $\{U_i \to U\}_{i \in I}$ is a covering in $\mathcal{C}_{total}$
by definition (Lemma \ref{lemma-simplicial-site-site}). For a morphism
$V \to U$ of $\mathcal{C}_n$, the fibre product
$V \times_U U_i$ in $\mathcal{C}_n$ is also the
the fibre product in $\mathcal{C}_{total}$ (by the claim in the
proof of Lemma \ref{lemma-simplicial-site-site}).
Therefore our functor is continuous. On the other hand, our functor
defines a bijection between coverings of $U$ in $\mathcal{C}_n$
and coverings of $U$ in $\mathcal{C}_{total}$. Therefore it is
certainly the case that our functor is cocontinuous.
\medskip\noindent
Case B. If $\{U_i \to U\}_{i \in I}$ is a covering in $\mathcal{C}_n$
then the image $\{U_i \to U\}_{i \in I}$ is a covering in $\mathcal{C}_{total}$
by definition (Lemma \ref{lemma-simplicial-cocontinuous-site}). For a morphism
$V \to U$ of $\mathcal{C}_n$, the fibre product
$V \times_U U_i$ in $\mathcal{C}_n$ is also the
the fibre product in $\mathcal{C}_{total}$ (by the claim in the
proof of Lemma \ref{lemma-simplicial-cocontinuous-site}).
Therefore our functor is continuous. On the other hand, our functor
defines a bijection between coverings of $U$ in $\mathcal{C}_n$
and coverings of $U$ in $\mathcal{C}_{total}$. Therefore it is
certainly the case that our functor is cocontinuous.
\medskip\noindent
At this point part (1) and the existence of $g^{Sh}_{n!}$ and $g_{n!}$
in cases A and B follows from
Sites, Lemmas \ref{sites-lemma-cocontinuous-morphism-topoi} and
\ref{sites-lemma-when-shriek}
and
Modules on Sites, Lemmas \ref{sites-modules-lemma-g-shriek-adjoint} and
\ref{sites-modules-lemma-back-and-forth}.
\medskip\noindent
Proof of (3). Let $\mathcal{G}$ be a sheaf on $\mathcal{C}_n$.
Consider the sheaf $\mathcal{H}$ on $\mathcal{C}_{total}$
whose degree $m$ part is the sheaf
$$
\mathcal{H}_m = \coprod\nolimits_{\varphi : [n] \to [m]}
f_\varphi^{-1}\mathcal{G}
$$
given in part (3) of the statement of the lemma.
Given a map $\psi : [m] \to [m']$ the map
$\mathcal{H}(\psi) : f_\psi^{-1}\mathcal{H}_m \to \mathcal{H}_{m'}$
is given on components by the identifications
$$
f_\psi^{-1} f_\varphi^{-1} \mathcal{G} \to
f_{\psi \circ \varphi}^{-1}\mathcal{G}
$$
Observe that given a map $\alpha : \mathcal{H} \to \mathcal{F}$
of sheaves on $\mathcal{C}_{total}$ we obtain a map
$\mathcal{G} \to \mathcal{F}_n$
corresponding to the restriction of $\alpha_n$ to the component
$\mathcal{G}$ in $\mathcal{H}_n$. Conversely, given a map
$\beta : \mathcal{G} \to \mathcal{F}_n$ of sheaves on $\mathcal{C}_n$
we can define
$\alpha : \mathcal{H} \to \mathcal{F}$ by letting $\alpha_m$
be the map which on components
$$
f_\varphi^{-1}\mathcal{G} \to \mathcal{F}_m
$$
uses the maps adjoint to $\mathcal{F}(\varphi) \circ f_\varphi^{-1}\beta$.
We omit the arguments showing these two constructions give
mutually inverse maps
$$
\Mor_{\Sh(\mathcal{C}_n)}(\mathcal{G}, \mathcal{F}_n) =
\Mor_{\Sh(\mathcal{C}_{total})}(\mathcal{H}, \mathcal{F})
$$
Thus $\mathcal{H} = g^{Sh}_{n!}\mathcal{G}$ as desired.
\medskip\noindent
Proof of (4). If $\mathcal{G}$ is an abelian sheaf on $\mathcal{C}_n$,
then we proceed in exactly the same ammner as above, except that
we define $\mathcal{H}$ is the abelian sheaf on $\mathcal{C}_{total}$
whose degree $m$ part is the sheaf
$$
\bigoplus\nolimits_{\varphi : [n] \to [m]} f_\varphi^{-1}\mathcal{G}
$$
with transition maps defined exactly as above. The bijection
$$
\Mor_{\textit{Ab}(\mathcal{C}_n)}(\mathcal{G}, \mathcal{F}_n) =
\Mor_{\textit{Ab}(\mathcal{C}_{total})}(\mathcal{H}, \mathcal{F})
$$
is proved exactly as above.
Thus $\mathcal{H} = g_{n!}\mathcal{G}$ as desired.
\medskip\noindent
The exactness properties of $g^{Sh}_{n!}$ and $g_{n!}$ follow
from formulas given for these functors.
\end{proof}
\begin{lemma}
\label{lemma-restriction-injective-to-component-site}
\begin{slogan}
An injective abelian sheaf on a simplicial site is injective on each component
\end{slogan}
In Situation \ref{situation-simplicial-site}.
If $\mathcal{I}$ is injective in $\textit{Ab}(\mathcal{C}_{total})$,
then $\mathcal{I}_n$ is injective in $\textit{Ab}(\mathcal{C}_n)$.
If $\mathcal{I}^\bullet$ is a K-injective complex in
$\textit{Ab}(\mathcal{C}_{total})$,
then $\mathcal{I}_n^\bullet$ is K-injective in $\textit{Ab}(\mathcal{C}_n)$.
\end{lemma}
\begin{proof}
The first statement follows from
Homology, Lemma \ref{homology-lemma-adjoint-preserve-injectives}
and
Lemma \ref{lemma-restriction-to-components-site}.
The second statement from
Derived Categories, Lemma \ref{derived-lemma-adjoint-preserve-K-injectives}
and
Lemma \ref{lemma-restriction-to-components-site}.
\end{proof}
\section{Augmentations of simplicial sites}
\label{section-augmentation-simplicial-sites}
\noindent
We continue in the fashion described in
Section \ref{section-simplicial-sites}
working out the meaning of augmentations in cases A and B
treated in that section.
\begin{remark}
\label{remark-augmentation-site}
In Situation \ref{situation-simplicial-site} an
{\it augmentation $a_0$ towards a site $\mathcal{D}$} will mean
\begin{enumerate}
\item[(A)] $a_0 : \mathcal{C}_0 \to \mathcal{D}$ 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
$u_\varphi \circ u_0 = u_\psi \circ u_0$.
\item[(B)] $a_0 : \Sh(\mathcal{C}_0) \to \Sh(\mathcal{D})$ 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
$u_0 \circ u_\varphi = u_0 \circ u_\psi$.
\end{enumerate}
\end{remark}
\begin{lemma}
\label{lemma-augmentation-site}
In Situation \ref{situation-simplicial-site} let $a_0$ be an
augmentation towards a site $\mathcal{D}$ as in
Remark \ref{remark-augmentation-site}. Then $a_0$ induces
\begin{enumerate}
\item a morphism of topoi $a_n : \Sh(\mathcal{C}_n) \to \Sh(\mathcal{D})$
for all $n \geq 0$,
\item a morphism of topoi $a : \Sh(\mathcal{C}_{total}) \to \Sh(\mathcal{D})$
\end{enumerate}
such that
\begin{enumerate}
\item for all $\varphi : [m] \to [n]$ we have $a_m \circ f_\varphi = a_n$,
\item if $g_n : \Sh(\mathcal{C}_n) \to \Sh(\mathcal{C}_{total})$
is as in Lemma \ref{lemma-restriction-to-components-site}, then
$a \circ g_n = a_n$, and
\item $a_*\mathcal{F}$ for $\mathcal{F} \in \Sh(\mathcal{C}_{total})$
is the equalizer of the two maps
$a_{0, *}\mathcal{F}_0 \to a_{1, *}\mathcal{F}_1$.
\end{enumerate}
\end{lemma}
\begin{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 \ref{sites-lemma-composition-morphisms-sites}.
The same lemma shows that for all $\varphi : [m] \to [n]$ we have
$a_m \circ f_\varphi = a_n$.
\medskip\noindent
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
$a_n : \Sh(\mathcal{C}_n) \to \Sh(\mathcal{D)}$, see
Sites, Lemma \ref{sites-lemma-composition-cocontinuous}.
The same lemma shows that for all $\varphi : [m] \to [n]$ we have
$a_m \circ f_\varphi = a_n$.
\medskip\noindent
Consider the functor $a^{-1} : \Sh(\mathcal{D}) \to \Sh(\mathcal{C}_{total})$
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
$\Sh(\mathcal{C}_{total})$ in
Lemma \ref{lemma-describe-sheaves-simplicial-site-site}.
Since the functors $a_n^{-1}$ are exact, $a^{-1}$ is an exact functor.
Finally, for $a_* : \Sh(\mathcal{C}_{total}) \to \Sh(\mathcal{D})$
we take the functor which to a sheaf $\mathcal{F}$ on $\Sh(\mathcal{D})$
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
$$
a_{0, *}\mathcal{F}_0 \to
a_{0, *}f_{\varphi, *} f_\varphi^{-1}\mathcal{F}_0
\xrightarrow{\mathcal{F}(\varphi)}
a_{0, *}f_{\varphi, *} \mathcal{F}_0 = a_{1, *}\mathcal{F}_1
$$
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 \ref{simplicial-lemma-augmentation-howto}
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 $a$\footnote{In case B the morphism $a$
corresponds to the cocontinuous functor
$\mathcal{C}_{total} \to \mathcal{D}$ sending
$U$ in $\mathcal{C}_n$ to $u_n(U)$.}. The equalities
$a \circ g_n = f_n$ follow immediately from the definitions.
\end{proof}
\section{Morphisms of simplicial sites}
\label{section-morphism-simplicial-sites}
\noindent
We continue in the fashion described in
Section \ref{section-simplicial-sites}
working out the meaning of morphisms of simplicial sites
in cases A and B treated in that section.
\begin{remark}
\label{remark-morphism-simplicial-sites}
Let $\mathcal{C}_n, f_\varphi, u_\varphi$ and
$\mathcal{C}'_n, f'_\varphi, u'_\varphi$ be as in
Situation \ref{situation-simplicial-site}. A
{\it morphism $h$ between simplicial sites} will mean
\begin{enumerate}
\item[(A)] Morphisms of sites
$h_n : \mathcal{C}_n \to \mathcal{C}'_n$
such that $f'_\varphi \circ h_n = h_m \circ f_\varphi$
as morphisms of sites for all $\varphi : [m] \to [n]$.
\item[(B)] Cocontinuous functors
$v_n : \mathcal{C}_n \to \mathcal{C}'_n$
inducing morphisms of topoi $h_n : \Sh(\mathcal{C}_n) \to \Sh(\mathcal{C}'_n)$
such that $u'_\varphi \circ v_n = v_m \circ u_\varphi$
as functors for all $\varphi : [m] \to [n]$.
\end{enumerate}
In both cases we have
$f'_\varphi \circ h_n = h_m \circ f_\varphi$
as morphisms of topoi, see
Sites, Lemma \ref{sites-lemma-composition-cocontinuous}
for case B and Sites,
Definition \ref{sites-definition-composition-morphisms-sites}
for case A.
\end{remark}
\begin{lemma}
\label{lemma-morphism-simplicial-sites}
Let $\mathcal{C}_n, f_\varphi, u_\varphi$ and
$\mathcal{C}'_n, f'_\varphi, u'_\varphi$ be as in
Situation \ref{situation-simplicial-site}.
Let $h$ be a morphism between simplicial sites as in
Remark \ref{remark-morphism-simplicial-sites}.
Then we obtain a morphism of topoi
$$
h_{total} : \Sh(\mathcal{C}_{total}) \to \Sh(\mathcal{C}'_{total})
$$
and commutative diagrams
$$
\xymatrix{
\Sh(\mathcal{C}_n) \ar[d]_{g_n} \ar[r]_{h_n} &
\Sh(\mathcal{C}'_n) \ar[d]^{g'_n} \\
\Sh(\mathcal{C}_{total}) \ar[r]^{h_{total}} &
\Sh(\mathcal{C}'_{total})
}
$$
Moreover, we have $(g'_n)^{-1} \circ h_{total, *} = h_{n, *} \circ g_n^{-1}$.
\end{lemma}
\begin{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 \ref{lemma-simplicial-site-site}).
Let
$h_{total}^{-1} = v_{total, s} :
\Sh(\mathcal{C}'_{total}) \to \Sh(\mathcal{C}_{total})$ and
$h_{total, *} = v_{total}^s = v_{total}^p :
\Sh(\mathcal{C}_{total}) \to \Sh(\mathcal{C}'_{total})$
be the adjoint pair of functors constructed and studied in
Sites, Sections \ref{sites-section-continuous-functors} and
\ref{sites-section-morphism-sites}.
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 \ref{lemma-describe-sheaves-simplicial-site-site}, 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.
\medskip\noindent
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 \ref{lemma-simplicial-cocontinuous-site}).
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 \ref{sites-lemma-composition-cocontinuous}.
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 \ref{lemma-restriction-to-components-site},
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.
\end{proof}
\begin{lemma}
\label{lemma-direct-image-morphism-simplicial-sites}
With notation and hypotheses as in Lemma \ref{lemma-morphism-simplicial-sites}.
For $K \in D(\mathcal{C}_{total})$ we have
$(g'_n)^{-1}Rh_{total, *}K = Rh_{n, *}g_n^{-1}K$.
\end{lemma}
\begin{proof}
Let $\mathcal{I}^\bullet$ be a K-injective complex on $\mathcal{C}_{total}$
representing $K$. Then $g_n^{-1}K$ is represented by
$g_n^{-1}\mathcal{I}^\bullet = \mathcal{I}_n^\bullet$
which is K-injective by
Lemma \ref{lemma-restriction-injective-to-component-site}.
We have $(g'_n)^{-1}h_{total, *}\mathcal{I}^\bullet =
h_{n, *}g_n^{-1}\mathcal{I}_n^\bullet$ by
Lemma \ref{lemma-morphism-simplicial-sites}
which gives the desired equality.
\end{proof}
\begin{remark}
\label{remark-morphism-augmentation-simplicial-sites}
Let $\mathcal{C}_n, f_\varphi, u_\varphi$ and
$\mathcal{C}'_n, f'_\varphi, u'_\varphi$ be as in
Situation \ref{situation-simplicial-site}.
Let $a_0$, resp.\ $a'_0$ be an augmentation
towards a site $\mathcal{D}$, resp.\ $\mathcal{D}'$
as in Remark \ref{remark-augmentation-site}.
Let $h$ be a morphism between simplicial sites as in
Remark \ref{remark-morphism-simplicial-sites}.
We say a morphism of topoi $h_{-1} : \Sh(\mathcal{D}) \to \Sh(\mathcal{D}')$
is {\it compatible with $h$, $a_0$, $a'_0$} if
\begin{enumerate}
\item[(A)] $h_{-1}$ comes from a morphism of sites
$h_{-1} : \mathcal{D} \to \mathcal{D}'$
such that $a'_0 \circ h_0 = h_{-1} \circ a_0$
as morphisms of sites.
\item[(B)] $h_{-1}$ comes from a cocontinuous functor
$v_{-1} : \mathcal{D} \to \mathcal{D}'$
such that $u'_0 \circ v_0 = v_{-1} \circ u_0$
as functors.
\end{enumerate}
In both cases we have $a'_0 \circ h_0 = h_{-1} \circ a_0$
as morphisms of topoi, see
Sites, Lemma \ref{sites-lemma-composition-cocontinuous}
for case B and Sites,
Definition \ref{sites-definition-composition-morphisms-sites}
for case A.
\end{remark}
\begin{lemma}
\label{lemma-morphism-augmentation-simplicial-sites}
Let $\mathcal{C}_n, f_\varphi, u_\varphi, \mathcal{D}, a_0$,
$\mathcal{C}'_n, f'_\varphi, u'_\varphi, \mathcal{D}', a'_0$, and
$h_n$, $n \geq -1$ be as in
Remark \ref{remark-morphism-augmentation-simplicial-sites}.
Then we obtain a commutative diagram
$$
\xymatrix{
\Sh(\mathcal{C}_{total}) \ar[d]_a \ar[r]_{h_{total}} &
\Sh(\mathcal{C}'_{total}) \ar[d]^{a'} \\
\Sh(\mathcal{D}) \ar[r]^{h_{-1}} &
\Sh(\mathcal{D}')
}
$$
\end{lemma}
\begin{proof}
The morphism $h$ is defined in Lemma \ref{lemma-morphism-simplicial-sites}.
The morphisms $a$ and $a'$ are defined in Lemma \ref{lemma-augmentation-site}.
Thus the only thing is to prove the commutativity of the diagram.
To do this, we prove that
$a^{-1} \circ h_{-1}^{-1} = h_{total}^{-1} \circ (a')^{-1}$.
By the commutative diagrams of
Lemma \ref{lemma-morphism-simplicial-sites}
and the description of $\Sh(\mathcal{C}_{total})$
and $\Sh(\mathcal{C}'_{total})$ in terms of components
in Lemma \ref{lemma-describe-sheaves-simplicial-site-site},
it suffices to show that
$$
\xymatrix{
\Sh(\mathcal{C}_n) \ar[d]_{a_n} \ar[r]_{h_n} &
\Sh(\mathcal{C}'_n) \ar[d]^{a'_n} \\
\Sh(\mathcal{D}) \ar[r]^{h_{-1}} &
\Sh(\mathcal{D}')
}
$$
commutes for all $n$. This follows from the case for $n = 0$
(which is an assumption in
Remark \ref{remark-morphism-augmentation-simplicial-sites})
and for $n > 0$ we pick $\varphi : [0] \to [n]$
and then the required commutativity follows from the case $n = 0$
and the relations $a_n = a_0 \circ f_\varphi$
and $a'_n = a'_0 \circ f'_\varphi$
as well as the commutation relations
$f'_\varphi \circ h_n = h_0 \circ f_\varphi$.
\end{proof}
\section{Ringed simplicial sites}
\label{section-simplicial-sites-modules}
\noindent
Let us endow our simplicial topos with a sheaf of rings.
\begin{lemma}
\label{lemma-restriction-module-to-components-site}
In Situation \ref{situation-simplicial-site}. Let $\mathcal{O}$
be a sheaf of rings on $\mathcal{C}_{total}$.
There is a canonical morphism of ringed topoi
$g_n : (\Sh(\mathcal{C}_n), \mathcal{O}_n) \to
(\Sh(\mathcal{C}_{total}), \mathcal{O})$
agreeing with the morphism $g_n$ of
Lemma \ref{lemma-restriction-to-components-site} on underlying topoi.
The functor
$g_n^* : \textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_n)$
has a left adjoint $g_{n!}$.
For $\mathcal{G}$ in $\textit{Mod}(\mathcal{O}_n)$-modules the
restriction of $g_{n!}\mathcal{G}$ to $\mathcal{C}_m$ is
$$
\bigoplus\nolimits_{\varphi : [n] \to [m]} f_\varphi^*\mathcal{G}
$$
where $f_\varphi : (\Sh(\mathcal{C}_m), \mathcal{O}_m) \to
(\Sh(\mathcal{C}_n), \mathcal{O}_n)$ is the morphism of ringed topoi
agreeing with the previously defined $f_\varphi$ on topoi and
using the map
$\mathcal{O}(\varphi) : f_\varphi^{-1}\mathcal{O}_n \to \mathcal{O}_m$
on sheaves of rings.
\end{lemma}
\begin{proof}
By Lemma \ref{lemma-restriction-to-components-site} we have
$g_n^{-1}\mathcal{O} = \mathcal{O}_n$ and hence we obtain our
morphism of ringed topoi. By Modules on Sites, Lemma
\ref{sites-modules-lemma-lower-shriek-modules}
we obtain the adjoint $g_{n!}$. To prove the formula for $g_{n!}$
we first define a sheaf of $\mathcal{O}$-modules $\mathcal{H}$
on $\mathcal{C}_{total}$ with degree $m$ component
the $\mathcal{O}_m$-module
$$
\mathcal{H}_m =
\bigoplus\nolimits_{\varphi : [n] \to [m]} f_\varphi^*\mathcal{G}
$$
Given a map $\psi : [m] \to [m']$ the map
$\mathcal{H}(\psi) : f_\psi^{-1}\mathcal{H}_m \to \mathcal{H}_{m'}$
is given on components by
$$
f_\psi^{-1} f_\varphi^*\mathcal{G} \to
f_\psi^* f_\varphi^*\mathcal{G} \to
f_{\psi \circ \varphi}^*\mathcal{G}
$$
Since this map $f_\psi^{-1}\mathcal{H}_m \to \mathcal{H}_{m'}$ is
$\mathcal{O}(\psi) : f_\psi^{-1}\mathcal{O}_m \to \mathcal{O}_{m'}$-semi-linear,
this indeed does define an $\mathcal{O}$-module
(use Lemma \ref{lemma-describe-sheaves-simplicial-site-site}).
Then one proves directly that
$$
\Mor_{\mathcal{O}_n}(\mathcal{G}, \mathcal{F}_n) =
\Mor_{\mathcal{O}}(\mathcal{H}, \mathcal{F})
$$
proceeding as in the proof of Lemma \ref{lemma-restriction-to-components-site}.
Thus $\mathcal{H} = g_{n!}\mathcal{G}$ as desired.
\end{proof}
\begin{lemma}
\label{lemma-restriction-injective-to-component-limp}
In Situation \ref{situation-simplicial-site}.
Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$.
If $\mathcal{I}$ is injective in $\textit{Mod}(\mathcal{O})$, then
$\mathcal{I}_n$ is a limp sheaf on $\mathcal{C}_n$.
\end{lemma}
\begin{proof}
This follows from
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-pullback-injective-limp}
applied to the inclusion functor $\mathcal{C}_n \to \mathcal{C}_{total}$
and its properties proven in Lemma \ref{lemma-restriction-to-components-site}.
\end{proof}
\begin{lemma}
\label{lemma-exactness-g-shriek-modules}
With assumptions as in
Lemma \ref{lemma-restriction-module-to-components-site} the functor
$g_{n!} : \textit{Mod}(\mathcal{O}_n) \to \textit{Mod}(\mathcal{O})$
is exact if the maps $f_\varphi^{-1}\mathcal{O}_n \to \mathcal{O}_m$
are flat for all $\varphi : [n] \to [m]$.
\end{lemma}
\begin{proof}
Recall that $g_{n!}\mathcal{G}$ is the $\mathcal{O}$-module
whose degree $m$ part is the $\mathcal{O}_m$-module
$$
\bigoplus\nolimits_{\varphi : [n] \to [m]} f_\varphi^*\mathcal{G}
$$
Here the morphism of ringed topoi
$f_\varphi : (\Sh(\mathcal{C}_m), \mathcal{O}_m) \to
(\Sh(\mathcal{C}_n), \mathcal{O}_n)$ uses the map
$f_\varphi^{-1}\mathcal{O}_n \to \mathcal{O}_m$ of the
statement of the lemma. If these maps are flat, then
$f_\varphi^*$ is exact
(Modules on Sites, Lemma \ref{sites-modules-lemma-flat-pullback-exact}).
By definition of the site $\mathcal{C}_{total}$ we see that these
functors have the desired exactness properties and we conclude.
\end{proof}
\begin{lemma}
\label{lemma-restriction-injective-to-component-site-module}
In Situation \ref{situation-simplicial-site}.
Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$
such that $f_\varphi^{-1}\mathcal{O}_n \to \mathcal{O}_m$
is flat for all $\varphi : [n] \to [m]$.
If $\mathcal{I}$ is injective in $\textit{Mod}(\mathcal{O})$, then
$\mathcal{I}_n$ is injective in $\textit{Mod}(\mathcal{O}_n)$.
\end{lemma}
\begin{proof}
This follows from
Homology, Lemma \ref{homology-lemma-adjoint-preserve-injectives}
and
Lemma \ref{lemma-exactness-g-shriek-modules}.
\end{proof}
\section{Cohomology on simplicial sites}
\label{section-cohomology-simplicial-sites}
\noindent
Let $\mathcal{C}$ be as in Situation \ref{situation-simplicial-site}.
In statement of the following lemmas we will let
$g_n : \Sh(\mathcal{C}_n) \to \Sh(\mathcal{C}_{total})$ be the
morphism of topoi of
Lemma \ref{lemma-restriction-to-components-site}. If $\varphi : [m] \to [n]$
is a morphism of $\Delta$, then the diagram of topoi
$$
\xymatrix{
\Sh(\mathcal{C}_n) \ar[rd]_{g_n} \ar[rr]_{f_\varphi} & &
\Sh(\mathcal{C}_m) \ar[ld]^{g_m} \\
& \Sh(\mathcal{C}_{total})
}
$$
is not commutative, but there is a $2$-morphism $g_n \to g_m \circ f_\varphi$
coming from the maps
$\mathcal{F}(\varphi) : f_\varphi^{-1}\mathcal{F}_m \to \mathcal{F}_n$.
See Sites, Section \ref{sites-section-2-category}.
\begin{lemma}
\label{lemma-simplicial-resolution-Z-site}
In Situation \ref{situation-simplicial-site} and with notation as above
there is a complex
$$
\ldots \to g_{2!}\mathbf{Z} \to g_{1!}\mathbf{Z} \to g_{0!}\mathbf{Z}
$$
of abelian sheaves on $\mathcal{C}_{total}$ which forms a resolution of
the constant sheaf with value $\mathbf{Z}$ on $\mathcal{C}_{total}$.
\end{lemma}
\begin{proof}
We will use the description of the functors $g_{n!}$ in
Lemma \ref{lemma-restriction-to-components-site} without further mention.
As maps of the complex we take $\sum (-1)^i d^n_i$ where
$d^n_i : g_{n!}\mathbf{Z} \to g_{n - 1!}\mathbf{Z}$ is the
adjoint to the map $\mathbf{Z} \to
\bigoplus_{[n - 1] \to [n]} \mathbf{Z} = g_n^{-1}g_{n - 1!}\mathbf{Z}$
corresponding to the factor labeled with $\delta^n_i : [n - 1] \to [n]$.
Then $g_m^{-1}$ applied to the complex gives the complex
$$
\ldots \to
\bigoplus\nolimits_{\alpha \in \Mor_\Delta([2], [m])]} \mathbf{Z} \to
\bigoplus\nolimits_{\alpha \in \Mor_\Delta([1], [m])]} \mathbf{Z} \to
\bigoplus\nolimits_{\alpha \in \Mor_\Delta([0], [m])]} \mathbf{Z}
$$
on $\mathcal{C}_m$.
In other words, this is the complex associated to the
free abelian sheaf on the simplicial set $\Delta[m]$, see
Simplicial, Example \ref{simplicial-example-simplex-simplicial-set}.
Since $\Delta[m]$ is homotopy equivalent to $\Delta[0]$, see
Simplicial, Example \ref{simplicial-example-simplex-contractible},
and since ``taking free abelian sheaf on'' is a functor,
we see that the complex above is homotopy equivalent to
the free abelian sheaf on $\Delta[0]$
(Simplicial, Remark \ref{simplicial-remark-homotopy-better} and
Lemma \ref{simplicial-lemma-homotopy-equivalence-s-N}).
This complex is acyclic in positive degrees
and equal to $\mathbf{Z}$ in degree $0$.
\end{proof}
\begin{lemma}
\label{lemma-cech-complex}
In Situation \ref{situation-simplicial-site}. Let $\mathcal{F}$ be an abelian
sheaf on $\mathcal{C}_{total}$ there is a canonical complex
$$
0 \to \Gamma(\mathcal{C}_{total}, \mathcal{F}) \to
\Gamma(\mathcal{C}_0, \mathcal{F}_0) \to
\Gamma(\mathcal{C}_1, \mathcal{F}_1) \to
\Gamma(\mathcal{C}_2, \mathcal{F}_2) \to \ldots
$$
which is exact in degrees $-1, 0$ and exact everywhere
if $\mathcal{F}$ is injective.
\end{lemma}
\begin{proof}
Observe that
$\Hom(\mathbf{Z}, \mathcal{F}) = \Gamma(\mathcal{C}_{total}, \mathcal{F})$
and
$\Hom(g_{n!}\mathbf{Z}, \mathcal{F}) = \Gamma(\mathcal{C}_n, \mathcal{F}_n)$.
Hence this lemma is an immediate consequence of
Lemma \ref{lemma-simplicial-resolution-Z-site}
and the fact that $\Hom(-, \mathcal{F})$ is exact if
$\mathcal{F}$ is injective.
\end{proof}
\begin{lemma}
\label{lemma-simplicial-sheaf-cohomology-site}
In Situation \ref{situation-simplicial-site}. For $K$ in
$D^+(\mathcal{C}_{total})$ there is a spectral sequence
$(E_r, d_r)_{r \geq 0}$ with
$$
E_1^{p, q} = H^q(\mathcal{C}_p, K_p),\quad
d_1^{p, q} : E_1^{p, q} \to E_1^{p + 1, q}
$$
converging to $H^{p + q}(\mathcal{C}_{total}, K)$.
This spectral sequence is functorial in $K$.
\end{lemma}
\begin{proof}
Let $\mathcal{I}^\bullet$ be a bounded below complex of injectives
representing $K$. Consider the double complex with terms
$$
A^{p, q} = \Gamma(\mathcal{C}_p, \mathcal{I}^q_p)
$$
where the horizontal arrows come from Lemma \ref{lemma-cech-complex}
and the vertical arrows from the differentials of the
complex $\mathcal{I}^\bullet$. The rows of the double complex are exact
in positive degrees and evaluate to
$\Gamma(\mathcal{C}_{total}, \mathcal{I}^q)$ in degree $0$.
On the other hand, since restriction to $\mathcal{C}_p$ is exact
(Lemma \ref{lemma-restriction-to-components-site})
the complex $\mathcal{I}_p^\bullet$ represents $K_p$ in
$D(\mathcal{C}_p)$. The sheaves $\mathcal{I}_p^q$ are injective
abelian sheaves on $\mathcal{C}_p$
(Lemma \ref{lemma-restriction-injective-to-component-site}).
Hence the cohomology of the columns computes the groups
$H^q(\mathcal{C}_p, K_p)$. We conclude by applying
Homology, Lemmas \ref{homology-lemma-first-quadrant-ss} and
\ref{homology-lemma-double-complex-gives-resolution}.
\end{proof}
\section{Cohomology and augmentations of simplicial sites}
\label{section-cohomology-augmentation-simplicial-sites}
\noindent
Consider a simplicial site $\mathcal{C}$ as in
Situation \ref{situation-simplicial-site}.
Let $a_0$ be an augmentation towards a site $\mathcal{D}$ as in
Remark \ref{remark-augmentation-site}.
By Lemma \ref{lemma-augmentation-site} we obtain a morphism of topoi
$$
a : \Sh(\mathcal{C}_{total} \longrightarrow \Sh(\mathcal{D})
$$
and morphisms of topoi
$g_n : \Sh(\mathcal{C}_n) \to \Sh(\mathcal{C}_{total})$
as in Lemma \ref{lemma-restriction-to-components-site}.
The compositions $a \circ g_n$ are denoted
$a_n : \Sh(\mathcal{C}_n) \to \Sh(\mathcal{D})$.
Furthermore, the simplicial structure gives
morphisms of topoi
$f_\varphi : \Sh(\mathcal{C}_n) \to \Sh(\mathcal{C}_m)$
such that $a_n \circ f_\varphi = a_m$ for all $\varphi : [m] \to [n]$.
\begin{lemma}
\label{lemma-simplicial-resolution-augmentation}
In Situation \ref{situation-simplicial-site} let
$a_0$ be an augmentation towards a site $\mathcal{D}$
as in Remark \ref{remark-augmentation-site}.
For any abelian sheaf $\mathcal{G}$ on $\mathcal{D}$
there is an exact complex
$$
\ldots \to
g_{2!}(a_2^{-1}\mathcal{G}) \to
g_{1!}(a_1^{-1}\mathcal{G}) \to
g_{0!}(a_0^{-1}\mathcal{G}) \to
a^{-1}\mathcal{G} \to 0
$$
of abelian sheaves on $\mathcal{C}_{total}$.
\end{lemma}
\begin{proof}
We encourage the reader to read the proof of
Lemma \ref{lemma-simplicial-resolution-Z-site} first.
We will use Lemma \ref{lemma-augmentation-site} and
the description of the functors $g_{n!}$ in
Lemma \ref{lemma-restriction-to-components-site} without further mention.
In particular $g_{n!}(a_n^{-1}\mathcal{G})$ is the
sheaf on $\mathcal{C}_{total}$ whose restriction to $\mathcal{C}_m$
is the sheaf
$$
\bigoplus\nolimits_{\varphi : [n] \to [m]} f_\varphi^{-1}a_n^{-1}\mathcal{G} =
\bigoplus\nolimits_{\varphi : [n] \to [m]} a_m^{-1}\mathcal{G}
$$
As maps of the complex we take $\sum (-1)^i d^n_i$ where
$d^n_i : g_{n!}(a_n^{-1}\mathcal{G}) \to g_{n - 1!}(a_{n - 1}^{-1}\mathcal{G})$
is the adjoint to the map
$a_n^{-1}\mathcal{G} \to \bigoplus_{[n - 1] \to [n]} a_n^{-1}\mathcal{G} =
g_n^{-1}g_{n - 1!}(a_{n - 1}^{-1}\mathcal{G})$
corresponding to the factor labeled with $\delta^n_i : [n - 1] \to [n]$.
The map $g_{0!}(a_0^{-1}\mathcal{G}) \to a^{-1}\mathcal{G}$ is adjoint
to the identity map of $a_0^{-1}\mathcal{G}$.
Then $g_m^{-1}$ applied to the chain complex in degrees
$\ldots, 2, 1, 0$ gives the complex
$$
\ldots \to
\bigoplus\nolimits_{\alpha \in \Mor_\Delta([2], [m])]} a_m^{-1}\mathcal{G} \to
\bigoplus\nolimits_{\alpha \in \Mor_\Delta([1], [m])]} a_m^{-1}\mathcal{G} \to
\bigoplus\nolimits_{\alpha \in \Mor_\Delta([0], [m])]} a_m^{-1}\mathcal{G}
$$
on $\mathcal{C}_m$. This is equal to $a_m^{-1}\mathcal{G}$
tensored over the constant sheaf $\mathbf{Z}$ with the complex
$$
\ldots \to
\bigoplus\nolimits_{\alpha \in \Mor_\Delta([2], [m])]} \mathbf{Z} \to
\bigoplus\nolimits_{\alpha \in \Mor_\Delta([1], [m])]} \mathbf{Z} \to
\bigoplus\nolimits_{\alpha \in \Mor_\Delta([0], [m])]} \mathbf{Z}
$$
discussed in the proof of Lemma \ref{lemma-simplicial-resolution-Z-site}.
There we have seen that this complex is homotopy equivalent to
$\mathbf{Z}$ placed in degree $0$ which finishes the proof.
\end{proof}
\begin{lemma}
\label{lemma-augmentation-cech-complex}
In Situation \ref{situation-simplicial-site} let
$a_0$ be an augmentation towards a site $\mathcal{D}$
as in Remark \ref{remark-augmentation-site}.
For an abelian sheaf $\mathcal{F}$ on $\mathcal{C}_{total}$
there is a canonical complex
$$
0 \to a_*\mathcal{F} \to a_{0, *}\mathcal{F}_0 \to a_{1, *}\mathcal{F}_1 \to
a_{2, *}\mathcal{F}_2 \to \ldots
$$
on $\mathcal{D}$ which is exact in degrees $-1, 0$ and
exact everywhere if $\mathcal{F}$ is injective.
\end{lemma}
\begin{proof}
To construct the complex, by the Yoneda lemma, it suffices for any
abelian sheaf $\mathcal{G}$ on $\mathcal{D}$ to construct a complex
$$
0 \to \Hom(\mathcal{G}, a_*\mathcal{F}) \to
\Hom(\mathcal{G}, a_{0, *}\mathcal{F}_0) \to
\Hom(\mathcal{G}, a_{1, *}\mathcal{F}_1) \to \ldots
$$
functorially in $\mathcal{G}$. To do this apply $\Hom(-, \mathcal{F})$
to the exact complex of Lemma \ref{lemma-simplicial-resolution-augmentation}
and use adjointness of pullback and pushforward.
The exactness properties in degrees $-1, 0$ follow from
the construction as $\Hom(-, \mathcal{F})$ is left exact.
If $\mathcal{F}$ is an injective abelian sheaf, then the
complex is exact because $\Hom(-, \mathcal{F})$ is exact.
\end{proof}
\begin{lemma}
\label{lemma-augmentation-spectral-sequence}
In Situation \ref{situation-simplicial-site} let
$a_0$ be an augmentation towards a site $\mathcal{D}$
as in Remark \ref{remark-augmentation-site}.
For any $K$ in $D^+(\mathcal{C}_{total})$ there is a spectral
sequence
$(E_r, d_r)_{r \geq 0}$ with
$$
E_1^{p, q} = R^qa_{p, *} K_p,\quad
d_1^{p, q} : E_1^{p, q} \to E_1^{p + 1, q}
$$
converging to $R^{p + q}a_*K$. This spectral sequence is functorial in $K$.
\end{lemma}
\begin{proof}
Let $\mathcal{I}^\bullet$ be a bounded below complex of injectives
representing $K$. Consider the double complex with terms
$$
A^{p, q} = a_{p, *}\mathcal{I}^q_p
$$
where the horizontal arrows come from
Lemma \ref{lemma-augmentation-cech-complex}
and the vertical arrows from the differentials of the
complex $\mathcal{I}^\bullet$. The rows of the double complex are exact
in positive degrees and evaluate to $a_*\mathcal{I}^q$ in degree $0$.
On the other hand, since restriction to $\mathcal{C}_p$ is exact
(Lemma \ref{lemma-restriction-to-components-site})
the complex $\mathcal{I}_p^\bullet$ represents $K_p$ in
$D(\mathcal{C}_p)$. The sheaves $\mathcal{I}_p^q$ are injective
abelian sheaves on $\mathcal{C}_p$
(Lemma \ref{lemma-restriction-injective-to-component-site}).
Hence the cohomology of the columns computes $R^qa_{p, *}K_p$.
We conclude by applying
Homology, Lemmas \ref{homology-lemma-first-quadrant-ss} and
\ref{homology-lemma-double-complex-gives-resolution}.
\end{proof}
\section{Cohomology on ringed simplicial sites}
\label{section-cohomology-simplicial-sites-modules}
\noindent
This section is the analogue of
Section \ref{section-cohomology-simplicial-sites}
for sheaves of modules.
\medskip\noindent
In Situation \ref{situation-simplicial-site} let $\mathcal{O}$
be a sheaf of rings on $\mathcal{C}_{total}$.
In statement of the following lemmas we will let
$g_n : (\Sh(\mathcal{C}_n), \mathcal{O}_n) \to
(\Sh(\mathcal{C}_{total}), \mathcal{O})$
be the morphism of ringed topoi of
Lemma \ref{lemma-restriction-module-to-components-site}.
If $\varphi : [m] \to [n]$ is a morphism of $\Delta$, then the diagram
of ringed topoi
$$
\xymatrix{
(\Sh(\mathcal{C}_n), \mathcal{O}_n) \ar[rd]_{g_n} \ar[rr]_{f_\varphi} & &
(\Sh(\mathcal{C}_m), \mathcal{O}_m) \ar[ld]^{g_m} \\
& (\Sh(\mathcal{C}_{total}), \mathcal{O})
}
$$
is not commutative, but there is a $2$-morphism $g_n \to g_m \circ f_\varphi$
coming from the maps
$\mathcal{F}(\varphi) : f_\varphi^{-1}\mathcal{F}_m \to \mathcal{F}_n$.
See Sites, Section \ref{sites-section-2-category}.
\begin{lemma}
\label{lemma-simplicial-resolution-ringed}
In Situation \ref{situation-simplicial-site} let $\mathcal{O}$
be a sheaf of rings on $\mathcal{C}_{total}$. There is a complex
$$
\ldots \to g_{2!}\mathcal{O}_2 \to g_{1!}\mathcal{O}_1 \to g_{0!}\mathcal{O}_0
$$
of $\mathcal{O}$-modules which forms a resolution of
$\mathcal{O}$.
Here $g_{n!}$ is as in Lemma \ref{lemma-restriction-module-to-components-site}.
\end{lemma}
\begin{proof}
We will use the description of $g_{n!}$ given in
Lemma \ref{lemma-restriction-to-components-site}.
As maps of the complex we take $\sum (-1)^i d^n_i$ where
$d^n_i : g_{n!}\mathcal{O}_n \to g_{n - 1!}\mathcal{O}_{n - 1}$
is the adjoint to the map
$\mathcal{O}_n \to \bigoplus_{[n - 1] \to [n]} \mathcal{O}_n =
g_n^*g_{n - 1!}\mathcal{O}_{n - 1}$
corresponding to the factor labeled with $\delta^n_i : [n - 1] \to [n]$.
Then $g_m^{-1}$ applied to the complex gives the complex
$$
\ldots \to
\bigoplus\nolimits_{\alpha \in \Mor_\Delta([2], [m])]} \mathcal{O}_m \to
\bigoplus\nolimits_{\alpha \in \Mor_\Delta([1], [m])]} \mathcal{O}_m \to
\bigoplus\nolimits_{\alpha \in \Mor_\Delta([0], [m])]} \mathcal{O}_m
$$
on $\mathcal{C}_m$.
In other words, this is the complex associated to the
free $\mathcal{O}_m$-module on the simplicial set $\Delta[m]$, see
Simplicial, Example \ref{simplicial-example-simplex-simplicial-set}.
Since $\Delta[m]$ is homotopy equivalent to $\Delta[0]$, see
Simplicial, Example \ref{simplicial-example-simplex-contractible},
and since ``taking free abelian sheaf on'' is a functor,
we see that the complex above is homotopy equivalent to
the free abelian sheaf on $\Delta[0]$
(Simplicial, Remark \ref{simplicial-remark-homotopy-better} and
Lemma \ref{simplicial-lemma-homotopy-equivalence-s-N}).
This complex is acyclic in positive degrees
and equal to $\mathcal{O}_m$ in degree $0$.
\end{proof}
\begin{lemma}
\label{lemma-cech-complex-modules}
In Situation \ref{situation-simplicial-site} let $\mathcal{O}$
be a sheaf of rings. Let $\mathcal{F}$ be a
sheaf of $\mathcal{O}$-modules. There is a canonical complex
$$
0 \to \Gamma(\mathcal{C}_{total}, \mathcal{F}) \to
\Gamma(\mathcal{C}_0, \mathcal{F}_0) \to
\Gamma(\mathcal{C}_1, \mathcal{F}_1) \to
\Gamma(\mathcal{C}_2, \mathcal{F}_2) \to \ldots
$$
which is exact in degrees $-1, 0$ and exact everywhere
if $\mathcal{F}$ is an injective $\mathcal{O}$-module.
\end{lemma}
\begin{proof}
Observe that
$\Hom(\mathcal{O}, \mathcal{F}) = \Gamma(\mathcal{C}_{total}, \mathcal{F})$
and
$\Hom(g_{n!}\mathcal{O}_n, \mathcal{F}) = \Gamma(\mathcal{C}_n, \mathcal{F}_n)$.
Hence this lemma is an immediate consequence of
Lemma \ref{lemma-simplicial-resolution-ringed}
and the fact that $\Hom(-, \mathcal{F})$ is exact if
$\mathcal{F}$ is injective.
\end{proof}
\begin{lemma}
\label{lemma-simplicial-module-cohomology-site}
In Situation \ref{situation-simplicial-site} let $\mathcal{O}$
be a sheaf of rings. For $K$ in $D^+(\mathcal{O})$
there is a spectral sequence $(E_r, d_r)_{r \geq 0}$ with
$$
E_1^{p, q} = H^q(\mathcal{C}_p, K_p),\quad
d_1^{p, q} : E_1^{p, q} \to E_1^{p + 1, q}
$$
converging to $H^{p + q}(\mathcal{C}_{total}, K)$.
This spectral sequence is functorial in $K$.
\end{lemma}
\begin{proof}
Let $\mathcal{I}^\bullet$ be a bounded below complex of injective
$\mathcal{O}$-modules representing $K$. Consider the double complex with terms
$$
A^{p, q} = \Gamma(\mathcal{C}_p, \mathcal{I}^q_p)
$$
where the horizontal arrows come from
Lemma \ref{lemma-cech-complex-modules}
and the vertical arrows from the differentials of the
complex $\mathcal{I}^\bullet$. Observe that
$\Gamma(\mathcal{D}, -) =
\Hom_{\mathcal{O}_\mathcal{D}}(\mathcal{O}_\mathcal{D}, -)$
on $\textit{Mod}(\mathcal{O}_\mathcal{D})$. Hence the lemma
says rows of the double complex are exact
in positive degrees and evaluate to
$\Gamma(\mathcal{C}_{total}, \mathcal{I}^q)$ in degree $0$.
Thus the total complex associated to the double complex
computes $R\Gamma(\mathcal{C}_{total}, K)$ by
Homology, Lemma \ref{homology-lemma-double-complex-gives-resolution}.
On the other hand, since restriction to $\mathcal{C}_p$ is exact
(Lemma \ref{lemma-restriction-to-components-site})
the complex $\mathcal{I}_p^\bullet$ represents $K_p$ in
$D(\mathcal{C}_p)$. The sheaves $\mathcal{I}_p^q$ are
are limp on $\mathcal{C}_p$
(Lemma \ref{lemma-restriction-injective-to-component-limp}).
Hence the cohomology of the columns computes the groups
$H^q(\mathcal{C}_p, K_p)$ by Leray's acyclicity lemma
(Derived Categories, Lemma \ref{derived-lemma-leray-acyclicity})
and
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-limp-acyclic}.
We conclude by applying
Homology, Lemma \ref{homology-lemma-first-quadrant-ss}.
\end{proof}
\section{Cohomology and augmentations of ringed simplicial sites}
\label{section-cohomology-augmentation-ringed-simplicial-sites}
\noindent
This section is the analogue of
Section \ref{section-cohomology-augmentation-simplicial-sites}
for sheaves of modules.
\medskip\noindent
Consider a simplicial site $\mathcal{C}$ as in
Situation \ref{situation-simplicial-site}.
Let $a_0$ be an augmentation towards a site $\mathcal{D}$ as in
Remark \ref{remark-augmentation-site}.
Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$.
Let $\mathcal{O}_\mathcal{D}$ be a sheaf of rings on $\mathcal{D}$.
Suppose we are given a morphism
$$
a^\sharp : \mathcal{O}_\mathcal{D} \longrightarrow a_*\mathcal{O}
$$
where $a$ is as in Lemma \ref{lemma-augmentation-site}.
Consequently, we obtain a morphism of ringed topoi
$$
a :
(\Sh(\mathcal{C}_{total}), \mathcal{O})
\longrightarrow
(\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})
$$
We will think of $g_n : (\Sh(\mathcal{C}_n), \mathcal{O}_n) \to
(\Sh(\mathcal{C}_{total}), \mathcal{O})$ as a morphism of ringed topoi
as in
Lemma \ref{lemma-restriction-module-to-components-site}, then
taking the composition $a_n = a \circ g_n$
(Lemma \ref{lemma-augmentation-site})
as morphisms of ringed topoi we obtain
$$
a_n :
(\Sh(\mathcal{C}_n), \mathcal{O}_n)
\longrightarrow
(\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})
$$
Using the transition maps $f_\varphi^{-1}\mathcal{O}_m \to \mathcal{O}_n$
we obtain morphisms of ringed topoi
$$
f_\varphi : (\Sh(\mathcal{C}_n), \mathcal{O}_n) \to
(\Sh(\mathcal{C}_m), \mathcal{O}_m)
$$
such that $a_n \circ f_\varphi = a_m$ as morphisms of
ringed topoi for all $\varphi : [m] \to [n]$.
\begin{lemma}
\label{lemma-simplicial-resolution-augmentation-modules}
With notation as above. For a $\mathcal{O}_\mathcal{D}$-module $\mathcal{G}$
there is an exact complex
$$
\ldots \to
g_{2!}(a_2^*\mathcal{G}) \to
g_{1!}(a_1^*\mathcal{G}) \to
g_{0!}(a_0^*\mathcal{G}) \to
a^*\mathcal{G} \to 0
$$
of sheaves of $\mathcal{O}$-modules on $\mathcal{C}_{total}$.
Here $g_{n!}$ is as in Lemma \ref{lemma-restriction-module-to-components-site}.
\end{lemma}
\begin{proof}
Observe that $a^*\mathcal{G}$ is the $\mathcal{O}$-module on
$\mathcal{C}_{total}$ whose restriction to $\mathcal{C}_m$
is the $\mathcal{O}_m$-module $a_m^*\mathcal{G}$.
The description of the functors $g_{n!}$ on modules
in Lemma \ref{lemma-restriction-module-to-components-site}
shows that $g_{n!}(a_n^*\mathcal{G})$ is the
$\mathcal{O}$-module on $\mathcal{C}_{total}$
whose restriction to $\mathcal{C}_m$ is the $\mathcal{O}_m$-module
$$
\bigoplus\nolimits_{\varphi : [n] \to [m]} f_\varphi^*a_n^*\mathcal{G} =
\bigoplus\nolimits_{\varphi : [n] \to [m]} a_m^*\mathcal{G}
$$
The rest of the proof is exactly the same as the proof of
Lemma \ref{lemma-simplicial-resolution-augmentation},
replacing $a_m^{-1}\mathcal{G}$ by $a_m^*\mathcal{G}$.
\end{proof}
\begin{lemma}
\label{lemma-augmentation-cech-complex-modules}
With notation as above.
For an $\mathcal{O}$-module $\mathcal{F}$ on $\mathcal{C}_{total}$
there is a canonical complex
$$
0 \to a_*\mathcal{F} \to a_{0, *}\mathcal{F}_0 \to a_{1, *}\mathcal{F}_1 \to
a_{2, *}\mathcal{F}_2 \to \ldots
$$
of $\mathcal{O}_\mathcal{D}$-modules which is exact in degrees $-1, 0$.
If $\mathcal{F}$ is an injective $\mathcal{O}$-module, then the complex
is exact in all degrees and remains exact on applying the functor
$\Hom_{\mathcal{O}_\mathcal{D}}(\mathcal{G}, -)$ for any
$\mathcal{O}_\mathcal{D}$-module $\mathcal{G}$.
\end{lemma}
\begin{proof}
To construct the complex, by the Yoneda lemma, it suffices for any
$\mathcal{O}_\mathcal{D}$-modules $\mathcal{G}$ on $\mathcal{D}$
to construct a complex
$$
0 \to \Hom_{\mathcal{O}_\mathcal{D}}(\mathcal{G}, a_*\mathcal{F}) \to
\Hom_{\mathcal{O}_\mathcal{D}}(\mathcal{G}, a_{0, *}\mathcal{F}_0) \to
\Hom_{\mathcal{O}_\mathcal{D}}(\mathcal{G}, a_{1, *}\mathcal{F}_1) \to \ldots
$$
functorially in $\mathcal{G}$. To do this apply
$\Hom_\mathcal{O}(-, \mathcal{F})$
to the exact complex of
Lemma \ref{lemma-simplicial-resolution-augmentation-modules}
and use adjointness of pullback and pushforward.
The exactness properties in degrees $-1, 0$ follow from
the construction as $\Hom_\mathcal{O}(-, \mathcal{F})$ is left exact.
If $\mathcal{F}$ is an injective $\mathcal{O}$-module, then the
complex is exact because $\Hom_\mathcal{O}(-, \mathcal{F})$ is exact.
\end{proof}
\begin{lemma}
\label{lemma-augmentation-spectral-sequence-modules}
With notation as above for any $K$ in $D^+(\mathcal{O})$ there is a spectral
sequence $(E_r, d_r)_{r \geq 0}$ in $\textit{Mod}(\mathcal{O}_\mathcal{D})$
with
$$
E_1^{p, q} = R^qa_{p, *} K_p
$$
converging to $R^{p + q}a_*K$. This spectral sequence is functorial in $K$.
\end{lemma}
\begin{proof}
Let $\mathcal{I}^\bullet$ be a bounded below complex of injective
$\mathcal{O}$-modules representing $K$. Consider the double complex with terms
$$
A^{p, q} = a_{p, *}\mathcal{I}^q_p
$$
where the horizontal arrows come from
Lemma \ref{lemma-augmentation-cech-complex-modules}
and the vertical arrows from the differentials of the
complex $\mathcal{I}^\bullet$. The lemma
says rows of the double complex are exact
in positive degrees and evaluate to
$a_*\mathcal{I}^q$ in degree $0$.
Thus the total complex associated to the double complex
computes $Ra_*K$ by
Homology, Lemma \ref{homology-lemma-double-complex-gives-resolution}.
On the other hand, since restriction to $\mathcal{C}_p$ is exact
(Lemma \ref{lemma-restriction-to-components-site})
the complex $\mathcal{I}_p^\bullet$ represents $K_p$ in
$D(\mathcal{C}_p)$. The sheaves $\mathcal{I}_p^q$ are
are limp on $\mathcal{C}_p$
(Lemma \ref{lemma-restriction-injective-to-component-limp}).
Hence the cohomology of the columns are the sheaves
$R^qa_{p, *}K_p$ by Leray's acyclicity lemma
(Derived Categories, Lemma \ref{derived-lemma-leray-acyclicity})
and
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-limp-acyclic}.
We conclude by applying
Homology, Lemma \ref{homology-lemma-first-quadrant-ss}.
\end{proof}
\section{Cartesian sheaves and modules}
\label{section-cartesian}
\noindent
Here is the definition.
\begin{definition}
\label{definition-cartesian-sheaf}
In Situation \ref{situation-simplicial-site}.
\begin{enumerate}
\item A sheaf $\mathcal{F}$ of sets or of abelian groups on
$\mathcal{C}$ is {\it cartesian} if the maps
$\mathcal{F}(\varphi) : f_\varphi^{-1}\mathcal{F}_m \to \mathcal{F}_n$
are isomorphisms for all $\varphi : [m] \to [n]$.
\item If $\mathcal{O}$ is a sheaf of rings on $\mathcal{C}_{total}$,
then a sheaf $\mathcal{F}$ of $\mathcal{O}$-modules is
{\it cartesian} if the maps $f_\varphi^*\mathcal{F}_m \to \mathcal{F}_n$
are isomorphisms for all $\varphi : [m] \to [n]$.
\item An object $K$ of $D(\mathcal{C}_{total})$ is {\it cartesian} if the maps
$f_\varphi^{-1}K_m \to K_n$
are isomorphisms for all $\varphi : [m] \to [n]$.
\item If $\mathcal{O}$ is a sheaf of rings on $\mathcal{C}_{total}$, then
an object $K$ of $D(\mathcal{O})$ is {\it cartesian} if the maps
$Lf_\varphi^*K_m \to K_n$
are isomorphisms for all $\varphi : [m] \to [n]$.
\end{enumerate}
\end{definition}
\noindent
Of course there is a general notion of a cartesian section of a
fibred category and the above are merely examples of this.
The property on pullbacks needs only be checked for the degeneracies.
\begin{lemma}
\label{lemma-check-cartesian-module}
In Situation \ref{situation-simplicial-site}.
\begin{enumerate}
\item A sheaf $\mathcal{F}$ of sets or abelian groups is cartesian
if and only if the maps
$(f_{\delta^n_j})^{-1}\mathcal{F}_{n - 1} \to \mathcal{F}_n$
are isomorphisms.
\item An object $K$ of $D(\mathcal{C}_{total})$ is cartesian
if and only if the maps
$(f_{\delta^n_j})^{-1}K_{n - 1} \to K_n$
are isomorphisms.
\item If $\mathcal{O}$ is a sheaf of rings on $\mathcal{C}_{total}$
a sheaf $\mathcal{F}$ of $\mathcal{O}$-modules is cartesian
if and only if the maps
$(f_{\delta^n_j})^*\mathcal{F}_{n - 1} \to \mathcal{F}_n$
are isomorphisms.
\item If $\mathcal{O}$ is a sheaf of rings on $\mathcal{C}_{total}$
an object $K$ of $D(\mathcal{O})$ is cartesian
if and only if the maps
$L(f_{\delta^n_j})^*K_{n - 1} \to K_n$
are isomorphisms.
\item Add more here.
\end{enumerate}
\end{lemma}
\begin{proof}
In each case the key is that the pullback functors
compose to pullback functor; for part (4) see
Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-derived-pullback-composition}.
We show how the argument works in case (1) and omit the proof
in the other cases.
The category $\Delta$ is generated by the morphisms
the morphisms $\delta^n_j$ and $\sigma^n_j$, see
Simplicial, Lemma \ref{simplicial-lemma-face-degeneracy}.
Hence we only need to check the maps
$(f_{\delta^n_j})^{-1}\mathcal{F}_{n - 1} \to \mathcal{F}_n$
and $(f_{\sigma^n_j})^{-1}\mathcal{F}_{n + 1} \to \mathcal{F}_n$ are
isomorphisms, see
Simplicial, Lemma \ref{simplicial-lemma-characterize-simplicial-object}
for notation. Since $\sigma^n_j \circ \delta_j^{n + 1} = \text{id}_{[n]}$
the composition
$$
\mathcal{F}_n =
(f_{\sigma^n_j})^{-1}
(f_{\delta_j^{n + 1}})^{-1}
\mathcal{F}_n \to
(f_{\sigma^n_j})^{-1}
\mathcal{F}_{n + 1} \to
\mathcal{F}_n
$$
is the identity. Thus the result for $\delta^{n + 1}_j$ implies the result
for $\sigma^n_j$.
\end{proof}
\begin{lemma}
\label{lemma-augmentation-cartesian-module}
In Situation \ref{situation-simplicial-site} let
$a_0$ be an augmentation towards a site $\mathcal{D}$ as in
Remark \ref{remark-augmentation-site}.
\begin{enumerate}
\item The pullback $a^{-1}\mathcal{G}$ of a sheaf of sets or abelian groups
on $\mathcal{D}$ is cartesian.
\item The pullback $a^{-1}K$ of an object $K$ of $D(\mathcal{D})$
is cartesian.
\end{enumerate}
Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$ and
$\mathcal{O}_\mathcal{D}$ a sheaf of rings on $\mathcal{D}$
and $a^\sharp : \mathcal{O}_\mathcal{D} \to a_*\mathcal{O}$ a
morphism as in
Section \ref{section-cohomology-augmentation-ringed-simplicial-sites}.
\begin{enumerate}
\item[(3)] The pullback $a^*\mathcal{F}$ of a sheaf of
$\mathcal{O}_\mathcal{D}$-modules is cartesian.
\item[(4)] The derived pullback $La^*K$ of an object
$K$ of $D(\mathcal{O}_\mathcal{D})$ is cartesian.
\end{enumerate}
\end{lemma}
\begin{proof}
This follows immediately from the identities
$a_m \circ f_\varphi = a_n$ for all $\varphi : [m] \to [n]$.
See Lemma \ref{lemma-augmentation-site} and the discussion in
Section \ref{section-cohomology-augmentation-ringed-simplicial-sites}.
\end{proof}
\begin{lemma}
\label{lemma-characterize-cartesian}
In Situation \ref{situation-simplicial-site}.
The category of cartesian sheaves of sets (resp.\ abelian groups)
is equivalent to the category of pairs $(\mathcal{F}, \alpha)$
where $\mathcal{F}$ is a a sheaf of sets (resp.\ abelian groups)
on $\mathcal{C}_0$ and
$$
\alpha :
(f_{\delta_1^1})^{-1}\mathcal{F}
\longrightarrow (f_{\delta_0^1})^{-1}\mathcal{F}
$$
is an isomorphism of sheaves of sets (resp.\ abelian groups)
on $\mathcal{C}_1$ such that
$(f_{\delta^2_1})^{-1}\alpha =
(f_{\delta^2_0})^{-1}\alpha \circ (f_{\delta^2_2})^{-1}\alpha$
as maps of sheaves on $\mathcal{C}_2$.
\end{lemma}
\begin{proof}
We abbreviate
$d^n_j = f_{\delta^n_j} : \Sh(\mathcal{C}_n) \to \Sh(\mathcal{C}_{n - 1})$.
The condition on $\alpha$ in the statement of the lemma makes sense because
$$
d^1_1 \circ d^2_2 = d^1_1 \circ d^2_1, \quad
d^1_1 \circ d^2_0 = d^1_0 \circ d^2_2, \quad
d^1_0 \circ d^2_0 = d^1_0 \circ d^2_1
$$
as morphisms of topoi $\Sh(\mathcal{C}_2) \to \Sh(\mathcal{C}_0)$, see
Simplicial, Remark \ref{simplicial-remark-relations}. Hence we
can picture these maps as follows
$$
\xymatrix{
& (d^2_0)^{-1}(d^1_1)^{-1}\mathcal{F} \ar[r]_-{(d^2_0)^{-1}\alpha} &
(d^2_0)^{-1}(d^1_0)^{-1}\mathcal{F} \ar@{=}[rd] & \\
(d^2_2)^{-1}(d^1_0)^{-1}\mathcal{F} \ar@{=}[ru] & & &
(d^2_1)^{-1}(d^1_0)^{-1}\mathcal{F} \\
& (d^2_2)^{-1}(d^1_1)^{-1}\mathcal{F} \ar[lu]^{(d^2_2)^{-1}\alpha} \ar@{=}[r] &
(d^2_1)^{-1}(d^1_1)^{-1}\mathcal{F} \ar[ru]_{(d^2_1)^{-1}\alpha}
}
$$
and the condition signifies the diagram is commutative. It is clear that
given a cartesian sheaf $\mathcal{G}$ of sets (resp.\ abelian groups)
on $\mathcal{C}_{total}$
we can set $\mathcal{F} = \mathcal{G}_0$ and $\alpha$ equal to the composition
$$
(d_1^1)^{-1}\mathcal{G}_0 \to \mathcal{G}_1
\leftarrow (d_1^0)^{-1}\mathcal{G}_0
$$
where the arrows are invertible as $\mathcal{G}$ is cartesian.
To prove this functor
is an equivalence we construct a quasi-inverse. The construction of
the quasi-inverse is analogous to the construction discussed in
Descent, Section \ref{descent-section-descent-modules} from which we borrow
the notation $\tau^n_i : [0] \to [n]$, $0 \mapsto i$ and
$\tau^n_{ij} : [1] \to [n]$, $0 \mapsto i$, $1 \mapsto j$.
Namely, given a pair $(\mathcal{F}, \alpha)$
as in the lemma we set $\mathcal{G}_n = (f_{\tau^n_n})^{-1}\mathcal{F}$.
Given $\varphi : [n] \to [m]$ we define
$\mathcal{G}(\varphi) : (f_\varphi)^{-1}\mathcal{G}_n \to \mathcal{G}_m$
using
$$
\xymatrix{
(f_\varphi)^{-1}\mathcal{G}_n \ar@{=}[r] &
(f_\varphi)^{-1}(f_{\tau^n_n})^{-1}\mathcal{F} \ar@{=}[r] &
(f_{\tau^m_{\varphi(n)}})^{-1}\mathcal{F} \ar@{=}[r] &
(f_{\tau^m_{\varphi(n)m}})^{-1}(d^1_1)^{-1}\mathcal{F}
\ar[d]^{(f_{\tau^m_{\varphi(n)m}})^{-1}\alpha} \\
&
\mathcal{G}_m \ar@{=}[r] &
(f_{\tau^m_m})^{-1}\mathcal{F} \ar@{=}[r] &
(f_{\tau^m_{\varphi(n)m}})^{-1}(d^1_0)^{-1}\mathcal{F}
}
$$
We omit the verification that the commutativity of the displayed diagram
above implies the maps compose correctly and hence give rise to a
sheaf on $\mathcal{C}_{total}$, see
Lemma \ref{lemma-describe-sheaves-simplicial-site-site}.
We also omit the verification
that the two functors are quasi-inverse to each other.
\end{proof}
\begin{lemma}
\label{lemma-characterize-cartesian-modules}
In Situation \ref{situation-simplicial-site}
let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$.
The category of cartesian $\mathcal{O}$-modules
is equivalent to the category of pairs $(\mathcal{F}, \alpha)$
where $\mathcal{F}$ is a $\mathcal{O}_0$-module
and
$$
\alpha :
(f_{\delta_1^1})^*\mathcal{F}
\longrightarrow (f_{\delta_0^1})^*\mathcal{F}
$$
is an isomorphism of $\mathcal{O}_1$-modules such that
$(f_{\delta^2_1})^*\alpha =
(f_{\delta^2_0})^*\alpha \circ (f_{\delta^2_2})^*\alpha$
as $\mathcal{O}_2$-module maps.
\end{lemma}
\begin{proof}
The proof is identical to the proof of
Lemma \ref{lemma-characterize-cartesian}
with pullback of sheaves of abelian groups replaced
by pullback of modules.
\end{proof}
\begin{lemma}
\label{lemma-Serre-subcat-cartesian-modules}
In Situation \ref{situation-simplicial-site}.
\begin{enumerate}
\item The full subcategory of cartesian abelian sheaves forms a
weak Serre subcategory of $\textit{Ab}(\mathcal{C}_{total})$.
Colimits of systems of cartesian abelian sheaves are cartesian.
\item Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$
such that the morphisms
$$
f_{\delta^n_j} : (\Sh(\mathcal{C}_n), \mathcal{O}_n)
\to (\Sh(\mathcal{C}_{n - 1}), \mathcal{O}_{n - 1})
$$
are flat. The full subcategory of cartesian $\mathcal{O}$-modules forms a
weak Serre subcategory of $\textit{Mod}(\mathcal{O})$.
Colimits of systems of cartesian $\mathcal{O}$-modules are cartesian.
\end{enumerate}
\end{lemma}
\begin{proof}
To see we obtain a weak Serre subcategory in (1)
we check the conditions listed in
Homology, Lemma \ref{homology-lemma-characterize-weak-serre-subcategory}.
First, if $\varphi : \mathcal{F} \to \mathcal{G}$ is a map
between cartesian abelian sheaves, then
$\Ker(\varphi)$ and $\Coker(\varphi)$ are cartesian too
because the restriction functors
$\Sh(\mathcal{C}_{total}) \to \Sh(\mathcal{C}_n)$
and the functors $f_\varphi^{-1}$ are exact.
Similarly, if
$$
0 \to \mathcal{F} \to \mathcal{H} \to \mathcal{G} \to 0
$$
is a short exact sequence of abelian sheaves on $\mathcal{C}_{total}$
with $\mathcal{F}$ and $\mathcal{G}$ cartesian, then it follows that
$\mathcal{H}$ is cartesian from the 5-lemma. To see the property of
colimits, use that colimits commute with pullback as pullback is a
left adjoint. In the case of modules
we argue in the same manner, using the exactness of flat pullback
(Modules on Sites, Lemma \ref{sites-modules-lemma-flat-pullback-exact})
and the fact that it suffices to check the condition
for $f_{\delta^n_j}$, see Lemma \ref{lemma-check-cartesian-module}.
\end{proof}
\begin{remark}[Warning]
\label{remark-warning-cartesian-modules}
Lemma \ref{lemma-Serre-subcat-cartesian-modules} notwithstanding, it
can happen that the category of cartesian $\mathcal{O}$-modules is
abelian without being a Serre subcategory of $\textit{Mod}(\mathcal{O})$.
Namely, suppose that we only know that
$f_{\delta_1^1}$ and $f_{\delta_0^1}$ are flat.
Then it follows easily from
Lemma \ref{lemma-characterize-cartesian-modules}
that the category of cartesian $\mathcal{O}$-modules is abelian.
But if $f_{\delta_0^2}$ is not flat (for example),
there is no reason for the inclusion functor
from the category of cartesian $\mathcal{O}$-modules
to all $\mathcal{O}$-modules to be exact.
\end{remark}
\begin{lemma}
\label{lemma-derived-cartesian-modules}
In Situation \ref{situation-simplicial-site}.
\begin{enumerate}
\item An object $K$ of $D(\mathcal{C}_{total})$ is cartesian if and only
if $H^q(K)$ is a cartesian abelian sheaf for all $q$.
\item Let $\mathcal{O}$ be a sheaf
of rings on $\mathcal{C}_{total}$ such that the morphisms
$f_{\delta^n_j} : (\Sh(\mathcal{C}_n), \mathcal{O}_n)
\to (\Sh(\mathcal{C}_{n - 1}), \mathcal{O}_{n - 1})$ are flat.
Then an object $K$ of $D(\mathcal{O})$ is cartesian if and only
if $H^q(K)$ is a cartesian $\mathcal{O}$-module for all $q$.
\end{enumerate}
\end{lemma}
\begin{proof}
Part (1) is true because the pullback functors $(f_\varphi)^{-1}$
are exact. Part (2) follows from the characterization in
Lemma \ref{lemma-check-cartesian-module}
and the fact that $L(f_{\delta^n_j})^* = (f_{\delta^n_j})^*$
by flatness.
\end{proof}
\begin{lemma}
\label{lemma-derived-cartesian-shriek}
In Situation \ref{situation-simplicial-site}.
\begin{enumerate}
\item An object $K$ of $D(\mathcal{C}_{total})$ is cartesian if and only
the canonical map
$$
g_{n!}K_n \longrightarrow
g_{n!}\mathbf{Z} \otimes^\mathbf{L}_\mathbf{Z} K
$$
is an isomorphism for all $n$.
\item Let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$
such that the morphisms $f_\varphi^{-1}\mathcal{O}_n \to \mathcal{O}_m$
are flat for all $\varphi : [n] \to [m]$. Then an object $K$ of
$D(\mathcal{O})$ is cartesian if and only the canonical map
$$
g_{n!}K_n \longrightarrow
g_{n!}\mathcal{O}_n \otimes^\mathbf{L}_\mathcal{O} K
$$
is an isomorphism for all $n$.
\end{enumerate}
\end{lemma}
\begin{proof}
Proof of (1). Since $g_{n!}$ is exact, it induces a functor
on derived categories adjoint to $g_n^{-1}$.
The map is the adjoint of the map
$K_n \to (g_n^{-1}g_{n!}\mathbf{Z}) \otimes^\mathbf{L}_\mathbf{Z} K_n$
corresponding to $\mathbf{Z} \to g_n^{-1}g_{n!}\mathbf{Z}$
which in turn is adjoint to
$\text{id} : g_{n!}\mathbf{Z} \to g_{n!}\mathbf{Z}$.
Using the description of $g_{n!}$
given in Lemma \ref{lemma-restriction-to-components-site}
we see that the restriction to $\mathcal{C}_m$ of this map
is
$$
\bigoplus\nolimits_{\varphi : [n] \to [m]} f_\varphi^{-1}K_n
\longrightarrow
\bigoplus\nolimits_{\varphi : [n] \to [m]} K_m
$$
Thus the statement is clear.
\medskip\noindent
Proof of (2). Since $g_{n!}$ is exact
(Lemma \ref{lemma-exactness-g-shriek-modules}), it induces a functor
on derived categories adjoint to $g_n^*$ (also exact).
The map is the adjoint of the map
$K_n \to (g_n^*g_{n!}\mathcal{O}_n) \otimes^\mathbf{L}_{\mathcal{O}_n} K_n$
corresponding to $\mathcal{O}_n \to g_n^*g_{n!}\mathcal{O}_n$
which in turn is adjoint to
$\text{id} : g_{n!}\mathcal{O}_n \to g_{n!}\mathcal{O}_n$.
Using the description of $g_{n!}$
given in Lemma \ref{lemma-restriction-module-to-components-site}
we see that the restriction to $\mathcal{C}_m$ of this map
is
$$
\bigoplus\nolimits_{\varphi : [n] \to [m]} f_\varphi^*K_n
\longrightarrow
\bigoplus\nolimits_{\varphi : [n] \to [m]}
f_\varphi^*\mathcal{O}_n \otimes_{\mathcal{O}_m} K_m =
\bigoplus\nolimits_{\varphi : [n] \to [m]} K_m
$$
Thus the statement is clear.
\end{proof}
\begin{lemma}
\label{lemma-quasi-coherent-sheaf}
In Situation \ref{situation-simplicial-site}
let $\mathcal{O}$ be a sheaf of rings on $\mathcal{C}_{total}$.
Let $\mathcal{F}$ be a sheaf of $\mathcal{O}$-modules.
Then $\mathcal{F}$ is quasi-coherent in the sense of
Modules on Sites, Definition \ref{sites-modules-definition-site-local}
if and only if $\mathcal{F}$ is cartesian
and $\mathcal{F}_n$ is a quasi-coherent $\mathcal{O}_n$-module for all $n$.
\end{lemma}
\begin{proof}
Assume $\mathcal{F}$ is quasi-coherent. Since pullbacks of
quasi-coherent modules are quasi-coherent
(Modules on Sites, Lemma \ref{sites-modules-lemma-local-pullback})
we see that $\mathcal{F}_n$ is a quasi-coherent $\mathcal{O}_n$-module
for all $n$. To show that $\mathcal{F}$ is cartesian, let $U$
be an object of $\mathcal{C}_n$ for some $n$. Let us view $U$
as an object of $\mathcal{C}_{total}$. Because $\mathcal{F}$
is quasi-coherent there exists a covering $\{U_i \to U\}$
and for each $i$ a presentation
$$
\bigoplus\nolimits_{j \in J_i} \mathcal{O}_{\mathcal{C}_{total}/U_i} \to
\bigoplus\nolimits_{k \in K_i} \mathcal{O}_{\mathcal{C}_{total}/U_i} \to
\mathcal{F}|_{\mathcal{C}_{total}/U_i} \to 0
$$
Observe that $\{U_i \to U\}$ is a covering of $\mathcal{C}_n$ by
the construction of the site $\mathcal{C}_{total}$.
Next, let $V$ be an object of $\mathcal{C}_m$ for some $m$ and let
$V \to U$ be a morphism of $\mathcal{C}_{total}$ lying over
$\varphi : [n] \to [m]$. The fibre products $V_i = V \times_U U_i$
exist and we get an induced covering $\{V_i \to V\}$ in $\mathcal{C}_m$.
Restricting the presentation above to the sites
$\mathcal{C}_n/U_i$ and $\mathcal{C}_m/V_i$ we obtain
presentations
$$
\bigoplus\nolimits_{j \in J_i} \mathcal{O}_{\mathcal{C}_m/U_i} \to
\bigoplus\nolimits_{k \in K_i} \mathcal{O}_{\mathcal{C}_m/U_i} \to
\mathcal{F}_n|_{\mathcal{C}_n/U_i} \to 0
$$
and
$$
\bigoplus\nolimits_{j \in J_i} \mathcal{O}_{\mathcal{C}_m/V_i} \to
\bigoplus\nolimits_{k \in K_i} \mathcal{O}_{\mathcal{C}_m/V_i} \to
\mathcal{F}_m|_{\mathcal{C}_m/V_i} \to 0
$$
These presentations are compatible with the map
$\mathcal{F}(\varphi) : f_\varphi^*\mathcal{F}_n \to \mathcal{F}_m$
(as this map is defined using the restriction maps of $\mathcal{F}$
along morphisms of $\mathcal{C}_{total}$ lying over $\varphi$).
We conclude that $\mathcal{F}(\varphi)|_{\mathcal{C}_m/V_i}$
is an isomorphism. As $\{V_i \to V\}$ is a covering we conclude
$\mathcal{F}(\varphi)|_{\mathcal{C}_m/V}$ is an isomorphism.
Since $V$ and $U$ were arbitrary this proves that $\mathcal{F}$ is cartesian.
(In case A use Sites, Lemma \ref{sites-lemma-morphism-of-sites-covering}.)
\medskip\noindent
Conversely, assume $\mathcal{F}_n$ is quasi-coherent
for all $n$ and that $\mathcal{F}$ is cartesian.
Then for any $n$ and object $U$ of $\mathcal{C}_n$ we
can choose a covering $\{U_i \to U\}$ of $\mathcal{C}_n$
and for each $i$ a presentation
$$
\bigoplus\nolimits_{j \in J_i} \mathcal{O}_{\mathcal{C}_m/U_i} \to
\bigoplus\nolimits_{k \in K_i} \mathcal{O}_{\mathcal{C}_m/U_i} \to
\mathcal{F}_n|_{\mathcal{C}_n/U_i} \to 0
$$
Pulling back to $\mathcal{C}_{total}/U_i$ we obtain complexes
$$
\bigoplus\nolimits_{j \in J_i} \mathcal{O}_{\mathcal{C}_{total}/U_i} \to
\bigoplus\nolimits_{k \in K_i} \mathcal{O}_{\mathcal{C}_{total}/U_i} \to
\mathcal{F}|_{\mathcal{C}_{total}/U_i} \to 0
$$
of modules on $\mathcal{C}_{total}/U_i$. Then the property that
$\mathcal{F}$ is cartesian implies that this is exact.
We omit the details.
\end{proof}
\section{Formalities on cohomological descent}
\label{section-formal-cohomological-descent}
\noindent
Here is a typical result.
\begin{lemma}
\label{lemma-trivialities-cohomological-descent-abelian}
In Situation \ref{situation-simplicial-site} let $a_0$ be an augmentation
towards a site $\mathcal{D}$ as in Remark \ref{remark-augmentation-site}.
Suppose given strictly full weak Serre subcategories
$$
\mathcal{A} \subset \textit{Ab}(\mathcal{D}),\quad
\mathcal{A}_n \subset \textit{Ab}(\mathcal{C}_n)
$$
Then
\begin{enumerate}
\item[(1)]
the collection of abelian sheaves $\mathcal{F}$ on $\mathcal{C}_{total}$
whose restriction to $\mathcal{C}_n$ is in $\mathcal{A}_n$ for all $n$
is a strictly full weak Serre subcategory
$\mathcal{A}_{total} \subset \textit{Ab}(\mathcal{C}_{total})$.
\end{enumerate}
If $a_n^{-1}$ sends $\mathcal{A}$ into $\mathcal{A}_n$
for all $n$, then
\begin{enumerate}
\item[(2)] $a^{-1}$ sends $\mathcal{A}$ into $\mathcal{A}_{total}$ and
\item[(3)] $a^{-1}$ sends $D_\mathcal{A}(\mathcal{D})$ into
$D_{\mathcal{A}_{total}}(\mathcal{C}_{total})$.
\end{enumerate}
If $R^qa_{n, *}$ sends $\mathcal{A}_n$ into $\mathcal{A}$
for all $n, q$, then
\begin{enumerate}
\item[(4)] $R^qa_*$ sends $\mathcal{A}_{total}$ into $\mathcal{A}$ for all $q$,
and
\item[(5)] $Ra_*$ sends $D_{\mathcal{A}_{total}}^+(\mathcal{C}_{total})$
into $D_\mathcal{A}^+(\mathcal{D})$.
\end{enumerate}
\end{lemma}
\begin{proof}
The only interesting assertions are (4) and (5).
Part (4) follows from the spectral sequence in
Lemma \ref{lemma-augmentation-spectral-sequence}
and Homology, Lemma \ref{homology-lemma-biregular-ss-converges}.
Then part (5) follows by considering the spectral sequence
associated to the canonical filtration on an object
$K$ of $D_{\mathcal{A}_{total}}^+(\mathcal{C}_{total})$ given by truncations.
We omit the details.
\end{proof}
\begin{lemma}
\label{lemma-downstairs}
Let $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to
(\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi.
Consider the full subcategory $D' \subset D(\mathcal{O}_\mathcal{D})$
consisting of objects $K$ such that
$$
K \longrightarrow Rf_*Lf^*K
$$
is an isomorphism. Then $D'$ is a saturated triangulated strictly full
subcategory of $D(\mathcal{O}_\mathcal{D})$ and the functor
$Lf^* : D' \to D(\mathcal{O}_\mathcal{C})$ is fully faithful.
\end{lemma}
\begin{proof}
See Derived Categories, Definition \ref{derived-definition-saturated}
for the definition of saturated in this setting. See
Derived Categories, Lemma \ref{derived-lemma-triangulated-subcategory}
for a discussion of triangulated subcategories.
The canonical map of the lemma is the unit of the adjoint
pair of functors $(Lf^*, Rf_*)$, see
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-adjoint}.
Having said this the proof that $D'$ is a saturated triangulated subcategory
is omitted; it follows formally from the fact that
$Lf^*$ and $Rf_*$ are exact functors of triangulated categories.
The final part follows formally from
fact that $Lf^*$ and $Rf_*$ are adjoint; compare with
Categories, Lemma \ref{categories-lemma-adjoint-fully-faithful}.
\end{proof}
\begin{lemma}
\label{lemma-upstairs}
Let $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to
(\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi.
Consider the full subcategory $D' \subset D(\mathcal{O}_\mathcal{C})$
consisting of objects $K$ such that
$$
Lf^*Rf_*K \longrightarrow K
$$
is an isomorphism. Then $D'$ is a saturated triangulated strictly full
subcategory of $D(\mathcal{O}_\mathcal{C})$ and the functor
$Rf_* : D' \to D(\mathcal{O}_\mathcal{D})$ is fully faithful.
\end{lemma}
\begin{proof}
See Derived Categories, Definition \ref{derived-definition-saturated}
for the definition of saturated in this setting. See
Derived Categories, Lemma \ref{derived-lemma-triangulated-subcategory}
for a discussion of triangulated subcategories.
The canonical map of the lemma is the counit of the adjoint
pair of functors $(Lf^*, Rf_*)$, see
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-adjoint}.
Having said this the proof that $D'$ is a saturated triangulated subcategory
is omitted; it follows formally from the fact that
$Lf^*$ and $Rf_*$ are exact functors of triangulated categories.
The final part follows formally from
fact that $Lf^*$ and $Rf_*$ are adjoint; compare with
Categories, Lemma \ref{categories-lemma-adjoint-fully-faithful}.
\end{proof}
\begin{lemma}
\label{lemma-bounded-in-image-upstairs}
Let $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to
(\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi.
Let $K$ be an object of $D(\mathcal{O}_\mathcal{C})$. Assume
\begin{enumerate}
\item $f$ is flat,
\item $K$ is bounded below,
\item $f^*Rf_*H^q(K) \to H^q(K)$ is an isomorphism.
\end{enumerate}
Then $f^*Rf_*K \to K$ is an isomorphism.
\end{lemma}
\begin{proof}
Observe that $f^*Rf_*K \to K$ is an isomorphism if and only
if it is an isomorphism on cohomology sheaves $H^j$. Observe that
$H^j(f^*Rf_*K) = f^*H^j(Rf_*K) = f^*H^j(Rf_*\tau_{\leq j}K) =
H^j(f^*Rf_*\tau_{\leq j}K)$.
Hence we may assume that $K$ is bounded. Then property (3)
tells us the cohomology sheaves are in the triangulated
subcategory $D' \subset D(\mathcal{O}_\mathcal{C})$ of
Lemma \ref{lemma-upstairs}. Hence $K$ is in it too.
\end{proof}
\begin{lemma}
\label{lemma-bounded-in-image-downstairs}
Let $f : (\Sh(\mathcal{C}), \mathcal{O}_\mathcal{C}) \to
(\Sh(\mathcal{D}), \mathcal{O}_\mathcal{D})$ be a morphism of ringed topoi.
Let $K$ be an object of $D(\mathcal{O}_\mathcal{D})$. Assume
\begin{enumerate}
\item $f$ is flat,
\item $K$ is bounded below,
\item $H^q(K) \to Rf_*f^*H^q(K)$ is an isomorphism.
\end{enumerate}
Then $K \to Rf_*f^*K$ is an isomorphism.
\end{lemma}
\begin{proof}
Observe that $K \to Rf_*f^*K$ is an isomorphism if and only
if it is an isomorphism on cohomology sheaves $H^j$. Observe that
$H^j(Rf_*f^*K) = H^j(Rf_*\tau_{\leq j}f^*K) = H^j(Rf_*f^*\tau_{\leq j}K)$.
Hence we may assume that $K$ is bounded. Then property (3)
tells us the cohomology sheaves are in the triangulated
subcategory $D' \subset D(\mathcal{O}_\mathcal{D})$ of
Lemma \ref{lemma-downstairs}. Hence $K$ is in it too.
\end{proof}
\begin{lemma}
\label{lemma-equivalence-bounded}
Let $f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{C}'), \mathcal{O}')$
be a morphism of ringed topoi.
Let $\mathcal{A} \subset \textit{Mod}(\mathcal{O})$
and $\mathcal{A}' \subset \textit{Mod}(\mathcal{O}')$
be weak Serre subcategories. Assume
\begin{enumerate}
\item $f$ is flat,
\item $f^*$ induces an equivalence of categories
$\mathcal{A}' \to \mathcal{A}$,
\item $\mathcal{F}' \to Rf_*f^*\mathcal{F}'$ is an isomorphism
for $\mathcal{F}' \in \Ob(\mathcal{A}')$.
\end{enumerate}
Then
$f^* : D_{\mathcal{A}'}^+(\mathcal{O}') \to D_\mathcal{A}^+(\mathcal{O})$
is an equivalence of categories with quasi-inverse given by
$Rf_* : D_\mathcal{A}^+(\mathcal{O}) \to D_{\mathcal{A}'}^+(\mathcal{O}')$.
\end{lemma}
\begin{proof}
By assumptions (2) and (3) and
Lemmas \ref{lemma-bounded-in-image-upstairs} and \ref{lemma-downstairs}
we see that
$f^* : D_{\mathcal{A}'}^+(\mathcal{O}') \to D_\mathcal{A}^+(\mathcal{O})$
is fully faithful.
Let $\mathcal{F} \in \Ob(\mathcal{A})$. Then we can write
$\mathcal{F} = f^*\mathcal{F}'$. Then
$Rf_*\mathcal{F} = Rf_* f^*\mathcal{F}' = \mathcal{F}'$.
In particular, we have $R^pf_*\mathcal{F} = 0$ for $p > 0$
and $f_*\mathcal{F} \in \Ob(\mathcal{A}')$.
Thus for any $K \in D^+_\mathcal{A}(\mathcal{O})$ we see,
using the spectral sequence $E_2^{p, q} = R^pf_*H^q(K)$
converging to $R^{p + q}f_*K$,
that $Rf_*K$ is in $D^+_{\mathcal{A}'}(\mathcal{O}')$.
Of course, it also follows from
Lemmas \ref{lemma-bounded-in-image-downstairs} and \ref{lemma-upstairs}
that $Rf_* : D_\mathcal{A}^+(\mathcal{O}) \to D_{\mathcal{A}'}^+(\mathcal{O}')$
is fully faithful. Since $f^*$ and $Rf_*$ are adjoint
we then get the result of the lemma, for example by
Categories, Lemma \ref{categories-lemma-adjoint-fully-faithful}.
\end{proof}
\begin{lemma}
\label{lemma-equivalence-unbounded-one}
\begin{reference}
This is analogous to \cite[Theorem 2.2.3]{six-I}.
\end{reference}
Let $f : (\Sh(\mathcal{C}), \mathcal{O}) \to (\Sh(\mathcal{C}'), \mathcal{O}')$
be a morphism of ringed topoi.
Let $\mathcal{A} \subset \textit{Mod}(\mathcal{O})$
and $\mathcal{A}' \subset \textit{Mod}(\mathcal{O}')$
be weak Serre subcategories. Assume
\begin{enumerate}
\item $f$ is flat,
\item $f^*$ induces an equivalence of categories
$\mathcal{A}' \to \mathcal{A}$,
\item $\mathcal{F}' \to Rf_*f^*\mathcal{F}'$ is an isomorphism
for $\mathcal{F}' \in \Ob(\mathcal{A}')$,
\item $\mathcal{C}, \mathcal{O}, \mathcal{A}$ satisfy the
assumption of
Cohomology on Sites, Situation \ref{sites-cohomology-situation-olsson-laszlo},
\item $\mathcal{C}', \mathcal{O}', \mathcal{A}'$ satisfy the
assumption of
Cohomology on Sites, Situation \ref{sites-cohomology-situation-olsson-laszlo}.
\end{enumerate}
Then $f^* : D_{\mathcal{A}'}(\mathcal{O}') \to D_\mathcal{A}(\mathcal{O})$
is an equivalence of categories with quasi-inverse given by
$Rf_* : D_\mathcal{A}(\mathcal{O}) \to D_{\mathcal{A}'}(\mathcal{O}')$.
\end{lemma}
\begin{proof}
Since $f^*$ is exact, it is clear that $f^*$ defines a functor
$f^* : D_{\mathcal{A}'}(\mathcal{O}') \to D_\mathcal{A}(\mathcal{O})$
as in the statement of the lemma and that moreover this
functor commutes with the truncation functors $\tau_{\geq -n}$.
We already know that $f^*$ and $Rf_*$ are quasi-inverse
equivalence on the corresponding bounded below categories,
see Lemma \ref{lemma-equivalence-bounded}.
By Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-olsson-laszlo-map-version-one}
with $N = 0$ we see that $Rf_*$ indeed defines a functor
$Rf_* : D_\mathcal{A}(\mathcal{O}) \to D_{\mathcal{A}'}(\mathcal{O}')$
and that moreover this functor commutes with
the truncation functors $\tau_{\geq -n}$.
Thus for $K$ in $D_\mathcal{A}(\mathcal{O})$ the map
$f^*Rf_*K \to K$ is an isomorphism as this is true
on trunctions.
Similarly, for $K'$ in $D_{\mathcal{A}'}(\mathcal{O}')$ the map
$K' \to Rf_*f^*K'$ is an isomorphism as this is true
on trunctions.
This finishes the proof.
\end{proof}
\begin{lemma}
\label{lemma-equivalence-unbounded-two}
\begin{reference}
This is analogous to \cite[Theorem 2.2.3]{six-I}.
\end{reference}
Let $f : (\mathcal{C}, \mathcal{O}) \to (\mathcal{C}', \mathcal{O}')$
be a morphism of ringed sites.
Let $\mathcal{A} \subset \textit{Mod}(\mathcal{O})$
and $\mathcal{A}' \subset \textit{Mod}(\mathcal{O}')$
be weak Serre subcategories. Assume
\begin{enumerate}
\item $f$ is flat,
\item $f^*$ induces an equivalence of categories
$\mathcal{A}' \to \mathcal{A}$,
\item $\mathcal{F}' \to Rf_*f^*\mathcal{F}'$ is an isomorphism
for $\mathcal{F}' \in \Ob(\mathcal{A}')$,
\item $\mathcal{C}, \mathcal{O}, \mathcal{A}$ satisfy the
assumption of
Cohomology on Sites, Situation \ref{sites-cohomology-situation-olsson-laszlo},
\item $f : (\mathcal{C}, \mathcal{O}) \to (\mathcal{C}', \mathcal{O}')$
and $\mathcal{A}$ satisfy the assumption of
Cohomology on Sites, Situation
\ref{sites-cohomology-situation-olsson-laszlo-prime}.
\end{enumerate}
Then $f^* : D_{\mathcal{A}'}(\mathcal{O}') \to D_\mathcal{A}(\mathcal{O})$
is an equivalence of categories with quasi-inverse given by
$Rf_* : D_\mathcal{A}(\mathcal{O}) \to D_{\mathcal{A}'}(\mathcal{O}')$.
\end{lemma}
\begin{proof}
The proof of this lemma is exactly the same as the proof
of Lemma \ref{lemma-equivalence-unbounded-one}
except the reference to
Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-olsson-laszlo-map-version-one}
is replaced by a reference to
Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-olsson-laszlo-map-version-two}.
\end{proof}
\noindent
Let $\mathcal{C}$ be a category. Let
$\text{Cov}(\mathcal{C}) \supset \text{Cov}'(\mathcal{C})$
be two ways to endow $\mathcal{C}$ with the structure of a site.
Denote $\tau$ the topology corresponding to $\text{Cov}(\mathcal{C})$
and $\tau'$ the topology corresponding to $\text{Cov}'(\mathcal{C})$.
Then the identity functor on $\mathcal{C}$ defines a morphism
of sites
$$
\epsilon : \mathcal{C}_\tau \longrightarrow \mathcal{C}_{\tau'}
$$
where $\epsilon_*$ is the identity functor on underlying presheaves and
where $\epsilon^{-1}$ is the $\tau$-sheafification of a $\tau'$-sheaf
(hence clearly exact). Let $\mathcal{O}$ be a sheaf of rings for the
$\tau$-topology. Then $\mathcal{O}$ is also a sheaf for the $\tau'$-topology
and $\epsilon$ becomes a morphism of ringed sites
$$
\epsilon :
(\mathcal{C}_\tau, \mathcal{O}_\tau)
\longrightarrow
(\mathcal{C}_{\tau'}, \mathcal{O}_{\tau'})
$$
\begin{lemma}
\label{lemma-compare-topologies-derived-adequate-modules}
With $\epsilon : (\mathcal{C}_\tau, \mathcal{O}_\tau) \to
(\mathcal{C}_{\tau'}, \mathcal{O}_{\tau'})$ as above.
Let $\mathcal{B} \subset \Ob(\mathcal{C})$ be a subset.
Let $\mathcal{A} \subset \textit{PMod}(\mathcal{O})$
be a full subcategory. Assume
\begin{enumerate}
\item every object of $\mathcal{A}$ is a sheaf for the $\tau$-topology,
\item $\mathcal{A}$ is a weak Serre subcategory of
$\textit{Mod}(\mathcal{O}_\tau)$,
\item every object of $\mathcal{C}$ has a $\tau'$-covering whose
members are elements of $\mathcal{B}$, and
\item for every $U \in \mathcal{B}$ we have $H^p_\tau(U, \mathcal{F}) = 0$,
$p > 0$ for all $\mathcal{F} \in \mathcal{A}$.
\end{enumerate}
Then $\mathcal{A}$ is a weak Serre subcategory of
$\textit{Mod}(\mathcal{O}_{\tau'})$ and there is an equivalence
of triangulated categories
$D_\mathcal{A}(\mathcal{O}_\tau) = D_\mathcal{A}(\mathcal{O}_{\tau'})$
given by $\epsilon^*$ and $R\epsilon_*$.
\end{lemma}
\begin{proof}
Since $\epsilon^{-1}\mathcal{O}_{\tau'} = \mathcal{O}_\tau$
we see that $\epsilon$ is a flat morphism of ringed sites
and that in fact $\epsilon^{-1} = \epsilon^*$ on sheaves
of modules. By property (1) we can think of every object of
$\mathcal{A}$ as a sheaf of $\mathcal{O}_\tau$-modules
and as a sheaf of $\mathcal{O}_{\tau'}$-modules.
In other words, we have fully faithful inclusion functors
$$
\mathcal{A} \to \textit{Mod}(\mathcal{O}_\tau) \to
\textit{Mod}(\mathcal{O}_{\tau'})
$$
To avoid confusion we will denote
$\mathcal{A}' \subset \textit{Mod}(\mathcal{O}_{\tau'})$
the image of $\mathcal{A}$. Then it is clear that
$\epsilon_* : \mathcal{A} \to \mathcal{A}'$ and
$\epsilon^* : \mathcal{A}' \to \mathcal{A}$ are
quasi-inverse equivalences (see discussion preceding
the lemma and use that objects of $\mathcal{A}'$ are
sheaves in the $\tau$ topology).
\medskip\noindent
Conditions (3) and (4) imply that $R^p\epsilon_*\mathcal{F} = 0$
for $p > 0$ and $\mathcal{F} \in \Ob(\mathcal{A})$.
This is true because $R^p\epsilon_*$ is the sheaf associated
to the presheave $U \mapsto H^p_\tau(U, \mathcal{F})$, see
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-higher-direct-images}.
Thus any exact complex in $\mathcal{A}$ (which is the same thing
as an exact complex in $\textit{Mod}(\mathcal{O}_\tau)$
whose terms are in $\mathcal{A}$, see
Homology, Lemma \ref{homology-lemma-characterize-weak-serre-subcategory})
remains exact upon applying the functor $\epsilon_*$.
\medskip\noindent
Consider an exact sequence
$$
\mathcal{F}'_0 \to \mathcal{F}'_1 \to
\mathcal{F}'_2 \to \mathcal{F}'_3 \to
\mathcal{F}'_4
$$
in $\textit{Mod}(\mathcal{O}_{\tau'})$ with
$\mathcal{F}'_0, \mathcal{F}'_1, \mathcal{F}'_3, \mathcal{F}'_4$ in
$\mathcal{A}'$. Apply the exact functor $\epsilon^*$ to get
an exact sequence
$$
\epsilon^*\mathcal{F}'_0 \to \epsilon^*\mathcal{F}'_1 \to
\epsilon^*\mathcal{F}'_2 \to \epsilon^*\mathcal{F}'_3 \to
\epsilon^*\mathcal{F}'_4
$$
in $\textit{Mod}(\mathcal{O}_\tau)$. Since $\mathcal{A}$ is
a weak Serre subcategory and since
$\epsilon^*\mathcal{F}'_0, \epsilon^*\mathcal{F}'_1,
\epsilon^*\mathcal{F}'_3, \epsilon^*\mathcal{F}'_4$ are in
$\mathcal{A}$, we conclude that $\epsilon^*\mathcal{F}_2$
is in $\mathcal{A}$ by
Homology, Definition \ref{homology-definition-serre-subcategory}.
Consider the map of sequences
$$
\xymatrix{
\mathcal{F}'_0 \ar[r] \ar[d] &
\mathcal{F}'_1 \ar[r] \ar[d] &
\mathcal{F}'_2 \ar[r] \ar[d] &
\mathcal{F}'_3 \ar[r] \ar[d] &
\mathcal{F}'_4 \ar[d] \\
\epsilon_*\epsilon^*\mathcal{F}'_0 \ar[r] &
\epsilon_*\epsilon^*\mathcal{F}'_1 \ar[r] &
\epsilon_*\epsilon^*\mathcal{F}'_2 \ar[r] &
\epsilon_*\epsilon^*\mathcal{F}'_3 \ar[r] &
\epsilon_*\epsilon^*\mathcal{F}'_4
}
$$
The lower row is exact by the discussion in the preceding
paragraph. The vertical arrows with index $0$, $1$, $3$, $4$
are isomorphisms by the discussion in the first paragraph.
By the $5$ lemma (Homology, Lemma \ref{homology-lemma-five-lemma})
we find that $\mathcal{F}'_2 \cong \epsilon_*\epsilon^*\mathcal{F}'_2$
and hence $\mathcal{F}'_2$ is in $\mathcal{A}'$.
In this way we see that $\mathcal{A}'$ is a weak Serre subcategory
of $\textit{Mod}(\mathcal{O}_{\tau'})$, see
Homology, Definition \ref{homology-definition-serre-subcategory}.
\medskip\noindent
At this point it makes sense to talk about the
derived categories $D_\mathcal{A}(\mathcal{O}_\tau)$ and
$D_{\mathcal{A}'}(\mathcal{O}_{\tau'})$, see
Derived Categories, Section \ref{derived-section-triangulated-sub}.
To finish the proof we show that conditions
(1) -- (5) of Lemma \ref{lemma-equivalence-unbounded-two} apply.
We have already seen (1), (2), (3) above.
Note that since every object has a $\tau'$-covering
by objects of $\mathcal{B}$, a fortiori every object has
a $\tau$-covering by objects of $\mathcal{B}$. Hence
condition (4) of Lemma \ref{lemma-equivalence-unbounded-two} is satisfied.
Similarly, condition (5) is satisfied as well.
\end{proof}
\section{Simplicial systems of the derived category}
\label{section-glueing}
\noindent
In this section we are going to prove a special case of
\cite[Proposition 3.2.9]{BBD} in the setting of derived
categories of abelian sheaves. The case of modules
is discussed in Section \ref{section-glueing-modules}.
\begin{definition}
\label{definition-cartesian-derived}
In Situation \ref{situation-simplicial-site}. A
{\it simplicial system of the derived category}
consists of the following data
\begin{enumerate}
\item for every $n$ an object $K_n$ of $D(\mathcal{C}_n)$,
\item for every $\varphi : [m] \to [n]$ a map
$K_\varphi : f_\varphi^{-1}K_m \to K_n$ in $D(\mathcal{C}_n)$
\end{enumerate}
subject to the condition that
$$
K_{\varphi \circ \psi} = K_\varphi \circ f_\varphi^{-1}K_\psi :
f_{\varphi \circ \psi}^{-1}K_l = f_\varphi^{-1} f_\psi^{-1}K_l
\longrightarrow
K_n
$$
for any morphisms $\varphi : [m] \to [n]$ and $\psi : [l] \to [m]$ of $\Delta$.
We say the simplicial system is {\it cartesian} if the maps $K_\varphi$
are isomorphisms for all $\varphi$.
Given two simplicial systems of the derived category
there is an obvious notion of a
{\it morphism of simplicial systems of the derived category}.
\end{definition}
\noindent
We have given this notion a ridiculously long name intentionally.
The goal is to show that a simplicial system of the derived category
comes from an object of $D(\mathcal{C}_{total})$ under certain
hypotheses.
\begin{lemma}
\label{lemma-cartesian-objects-derived}
In Situation \ref{situation-simplicial-site}.
If $K \in D(\mathcal{C}_{total})$ is an object,
then $(K_n, K(\varphi))$ is a simplicial system of the derived category.
If $K$ is cartesian, so is the system.
\end{lemma}
\begin{proof}
This is obvious.
\end{proof}
\begin{lemma}
\label{lemma-abelian-postnikov}
In Situation \ref{situation-simplicial-site}. Let $K$ be
an object of $D(\mathcal{C}_{total})$. Set
$$
X_n = (g_{n!}\mathbf{Z})
\otimes^\mathbf{L}_\mathbf{Z} K
\quad\text{and}\quad
Y_n =
(g_{n!}\mathbf{Z} \to \ldots \to g_{0!}\mathbf{Z})[-n]
\otimes^\mathbf{L}_\mathbf{Z} K
$$
as objects of $D(\mathcal{C}_{total})$ where the maps are
as in Lemma \ref{lemma-simplicial-resolution-Z-site}.
With the evident canonical maps $Y_n \to X_n$ and
$Y_0 \to Y_1[1] \to Y_2[2] \to \ldots$ we have
\begin{enumerate}
\item the distinguished triangles $Y_n \to X_n \to Y_{n - 1} \to Y_n[1]$
define a Postnikov system
(Derived Categories, Definition \ref{derived-definition-postnikov-system})
for $\ldots \to X_2 \to X_1 \to X_0$,
\item $K = \text{hocolim} Y_n[n]$ in $D(\mathcal{C}_{total})$.
\end{enumerate}
\end{lemma}
\begin{proof}
First, if $K = \mathbf{Z}$, then this is the construction of
Derived Categories, Example \ref{derived-example-key-postnikov}
applied to the complex
$$
\ldots \to
g_{2!}\mathbf{Z} \to
g_{1!}\mathbf{Z} \to
g_{0!}\mathbf{Z}
$$
in $\textit{Ab}(\mathcal{C}_{total})$ combined with the fact that
this complex represents $K = \mathbf{Z}$ in $D(\mathcal{C}_{total})$
by Lemma \ref{lemma-simplicial-resolution-Z-site}.
The general case follows from this, the fact that the exact functor
$- \otimes^\mathbf{L}_\mathbf{Z} K$ sends Postnikov systems to
Postnikov systems, and
that $- \otimes^\mathbf{L}_\mathbf{Z} K$ commutes with homotopy colimits.
\end{proof}
\begin{lemma}
\label{lemma-nullity-cartesian-objects-derived}
In Situation \ref{situation-simplicial-site}.
If $K, K' \in D(\mathcal{C}_{total})$.
Assume
\begin{enumerate}
\item $K$ is cartesian,
\item $\Hom(K_i[i], K'_i) = 0$ for $i > 0$, and
\item $\Hom(K_i[i + 1], K'_i) = 0$ for $i \geq 0$.
\end{enumerate}
Then any map $K \to K'$ which induces the zero map $K_0 \to K'_0$ is zero.
\end{lemma}
\begin{proof}
Consider the objects $X_n$ and the Postnikov system $Y_n$
associated to $K$ in Lemma \ref{lemma-abelian-postnikov}.
As $K = \text{hocolim} Y_n[n]$ the map $K \to K'$ induces
a compatible family of morphisms $Y_n[n] \to K'$.
By (1) and Lemma \ref{lemma-derived-cartesian-shriek} we have
$X_n = g_{n!}K_n$. Since $Y_0 = X_0$ we find that
$K_0 \to K'_0$ being zero implies $Y_0 \to K'$ is zero.
Suppose we've shown that the map $Y_n[n] \to K'$ is zero
for some $n \geq 0$. From the distinguished triangle
$$
Y_n[n] \to Y_{n + 1}[n + 1] \to X_{n + 1}[n + 1] \to Y_n[n + 1]
$$
we get an exact sequence
$$
\Hom(X_{n + 1}[n + 1], K') \to
\Hom(Y_{n + 1}[n + 1], K') \to
\Hom(Y_n[n], K')
$$
As $X_{n + 1}[n + 1] = g_{n + 1!}K_{n + 1}[n + 1]$ the first group is equal to
$$
\Hom(K_{n + 1}[n + 1], K'_{n + 1})
$$
which is zero by assumption (2). By induction we conclude all the maps
$Y_n[n] \to K'$ are zero. Consider the defining distinguished triangle
$$
\bigoplus Y_n[n] \to
\bigoplus Y_n[n] \to
K \to
(\bigoplus Y_n[n])[1]
$$
for the homotopy colimit. Arguing as above, we find that it suffices
to show that
$$
\Hom((\bigoplus Y_n[n])[1], K') = \prod \Hom(Y_n[n + 1], K')
$$
is zero for all $n \geq 0$. To see this, arguing as above,
it suffices to show that
$$
\Hom(K_n[n + 1], K'_n) = 0
$$
for all $n \geq 0$ which follows from condition (3).
\end{proof}
\begin{lemma}
\label{lemma-hom-cartesian-objects-derived}
In Situation \ref{situation-simplicial-site}.
If $K, K' \in D(\mathcal{C}_{total})$.
Assume
\begin{enumerate}
\item $K$ is cartesian,
\item $\Hom(K_i[i - 1], K'_i) = 0$ for $i > 1$.
\end{enumerate}
Then any map $\{K_n \to K'_n\}$ between the associated simplicial systems
of $K$ and $K'$ comes from a map $K \to K'$ in $D(\mathcal{C}_{total})$.
\end{lemma}
\begin{proof}
Let $\{K_n \to K'_n\}_{n \geq 0}$
be a morphism of simplicial systems of the derived category.
Consider the objects $X_n$ and Postnikov system $Y_n$
associated to $K$ of Lemma \ref{lemma-abelian-postnikov}.
By (1) and Lemma \ref{lemma-derived-cartesian-shriek} we have
$X_n = g_{n!}K_n$. In particular, the map $K_0 \to K'_0$
induces a morphism $X_0 \to K'$. Since $\{K_n \to K'_n\}$
is a morphism of systems, a computation (omitted) shows that
the composition
$$
X_1 \to X_0 \to K'
$$
is zero. As $Y_0 = X_0$ and as $Y_1$ fits into a distinguished
triangle
$$
Y_1 \to X_1 \to Y_0 \to Y_1[1]
$$
we conclude that there exists a morphism $Y_1[1] \to K'$ whose
composition with $X_0 = Y_0 \to Y_1[1]$ is the morphism $X_0 \to K'$
given above. Suppose given a map $Y_n[n] \to K'$ for $n \geq 1$.
From the distinguished triangle
$$
X_{n + 1}[n] \to Y_n[n] \to Y_{n + 1}[n + 1] \to X_{n + 1}[n + 1]
$$
we get an exact sequence
$$
\Hom(Y_{n + 1}[n + 1], K') \to
\Hom(Y_n[n], K') \to
\Hom(X_{n + 1}[n], K')
$$
As $X_{n + 1}[n] = g_{n + 1!}K_{n + 1}[n]$ the last group is equal to
$$
\Hom(K_{n + 1}[n], K'_{n + 1})
$$
which is zero by assumption (2). By induction we get a system of
maps $Y_n[n] \to K'$ compatible with transition maps and reducing
to the given map on $Y_0$. This produces a map
$$
\gamma :
K = \text{hocolim} Y_n[n]
\longrightarrow
K'
$$
This map in any case has the property that the diagram
$$
\xymatrix{
X_0 \ar[rd] \ar[r] &
K \ar[d]^\gamma \\
& K'
}
$$
is commutative. Restricting to
$\mathcal{C}_0$ we deduce that the map $\gamma_0 : K_0 \to K'_0$
is the same as the first map $K_0 \to K'_0$ of the morphism
of simplicial systems. Since $K$ is cartesian, this easily gives that
$\{\gamma_n\}$ is the map of simplicial systems we started out with.
\end{proof}
\begin{lemma}
\label{lemma-cartesian-object-derived-from-simplicial}
In Situation \ref{situation-simplicial-site}. Let
$(K_n, K_\varphi)$ be a simplicial system of the derived category.
Assume
\begin{enumerate}
\item $(K_n, K_\varphi)$ is cartesian,
\item $\Hom(K_i[t], K_i) = 0$ for $i \geq 0$ and $t > 0$.
\end{enumerate}
Then there exists a cartesian object $K$ of $D(\mathcal{C}_{total})$
whose associated simplicial system is isomorphic to $(K_n, K_\varphi)$.
\end{lemma}
\begin{proof}
Set $X_n = g_{n!}K_n$ in $D(\mathcal{C}_{total})$. For each $n \geq 1$
we have
$$
\Hom(X_n, X_{n - 1}) =
\Hom(K_n, g_n^{-1}g_{n - 1!}K_{n - 1}) =
\bigoplus\nolimits_{\varphi : [n - 1] \to [n]}
\Hom(K_n, f_\varphi^{-1}K_{n - 1})
$$
Thus we get a map $X_n \to X_{n - 1}$ corresponding to the
alternating sum of the maps
$K_\varphi^{-1} : K_n \to f_\varphi^{-1}K_{n - 1}$
where $\varphi$ runs over $\delta^n_0, \ldots, \delta^n_n$.
We can do this because $K_\varphi$ is invertible by assumption (1).
Please observe the similarity with the definition of the maps
in the proof of Lemma \ref{lemma-simplicial-resolution-Z-site}.
We obtain a complex
$$
\ldots \to X_2 \to X_1 \to X_0
$$
in $D(\mathcal{C}_{total})$. We omit the computation which shows
that the compositions are zero. By
Derived Categories, Lemma \ref{derived-lemma-existence-postnikov-system}
if we have
$$
\Hom(X_i[i - j - 2], X_j) = 0\text{ for }i > j + 2
$$
then we can extend this complex to a Postnikov system.
The group is equal to
$$
\Hom(K_i[i - j - 2], g_i^{-1}g_{j!}K_j)
$$
Again using that $(K_n, K_\varphi)$ is cartesian we see that
$g_i^{-1}g_{j!}K_j$ is isomorphic to a finite direct sum of copies of
$K_i$. Hence the group vanishes by assumption (2).
Let the Postnikov system be given by $Y_0 = X_0$ and distinguished
sequences $Y_n \to X_n \to Y_{n - 1} \to Y_n[1]$ for $n \geq 1$.
We set
$$
K = \text{hocolim} Y_n[n]
$$
To finish the proof we have to show that $g_m^{-1}K$ is isomorphic
to $K_m$ for all $m$ compatible with the maps $K_\varphi$. Observe that
$$
g_m^{-1} K = \text{hocolim} g_m^{-1}Y_n[n]
$$
and that $g_m^{-1}Y_n[n]$ is a Postnikov system for $g_m^{-1}X_n$.
Consider the isomorphisms
$$
g_m^{-1}X_n =
\bigoplus\nolimits_{\varphi : [n] \to [m]} f_\varphi^{-1}K_n
\xrightarrow{\bigoplus K_\varphi}
\bigoplus\nolimits_{\varphi : [n] \to [m]} K_m
$$
These maps define an isomorphism of complexes
$$
\xymatrix{
\ldots \ar[r] &
g_m^{-1}X_2 \ar[r] \ar[d] &
g_m^{-1}X_1 \ar[r] \ar[d] &
g_m^{-1}X_0 \ar[d] \\
\ldots \ar[r] &
\bigoplus\nolimits_{\varphi : [2] \to [m]} K_m \ar[r] &
\bigoplus\nolimits_{\varphi : [1] \to [m]} K_m \ar[r] &
\bigoplus\nolimits_{\varphi : [0] \to [m]} K_m
}
$$
in $D(\mathcal{C}_m)$ where the arrows in the bottom row are as
in the proof of Lemma \ref{lemma-simplicial-resolution-Z-site}.
The squares commute by our choice of the arrows of the complex
$\ldots \to X_2 \to X_1 \to X_0$; we omit the computation.
The bottom row complex has a postnikov tower given by
$$
Y'_{m, n} =
\left(\bigoplus\nolimits_{\varphi : [n] \to [m]} \mathbf{Z} \to
\ldots \to
\bigoplus\nolimits_{\varphi : [0] \to [m]} \mathbf{Z}\right)[-n]
\otimes^\mathbf{L}_\mathbf{Z} K_m
$$
and $\text{hocolim} Y'_{m, n} = K_m$
(please compare with the proof of Lemma \ref{lemma-abelian-postnikov}
and Derived Categories, Example \ref{derived-example-key-postnikov}).
Applying the second part of
Derived Categories, Lemma \ref{derived-lemma-existence-postnikov-system}
the vertical maps in the big diagram extend to an isomorphism
of Postnikov systems provided we have
$$
\Hom(g_m^{-1}X_i[i - j - 1], \bigoplus\nolimits_{\varphi : [j] \to [m]} K_m)
= 0\text{ for }i > j + 1
$$
The is true if $\Hom(K_m[i - j - 1], K_m) = 0$ for $i > j + 1$
which holds by assumption (2). Choose an isomorphism given
by $\gamma_{m, n} : g_m^{-1}Y_n \to Y'_{m, n}$ of Postnikov systems
in $D(\mathcal{C}_m)$. By uniqueness of homotopy colimits,
we can find an isomorphism
$$
g_m^{-1} K = \text{hocolim} g_m^{-1}Y_n[n]
\xrightarrow{\gamma_m}
\text{hocolim} Y'_{m, n} = K_m
$$
compatible with $\gamma_{m, n}$.
\medskip\noindent
We still have to prove that the maps $\gamma_m$ fit into commutative diagrams
$$
\xymatrix{
f_\varphi^{-1}g_m^{-1}K \ar[d]_{f_\varphi^{-1}\gamma_m} \ar[r]_{K(\varphi)} &
g_n^{-1}K \ar[d]^{\gamma_n} \\
f_\varphi^{-1}K_m \ar[r]^{K_\varphi} &
K_n
}
$$
for every $\varphi : [m] \to [n]$. Consider the diagram
$$
\xymatrix{
f_\varphi^{-1}(\bigoplus_{\psi : [0] \to [m]} f_\psi^{-1}K_0)
\ar@{=}[r] \ar[d]_{f_\varphi^{-1}(\bigoplus K_\psi)} &
f_\varphi^{-1}g_m^{-1}X_0 \ar[d] \ar[r]_{X_0(\varphi)} &
g_n^{-1}X_0 \ar[d] &
\bigoplus_{\chi : [0] \to [n]} f_\chi^{-1}K_0
\ar@{=}[l] \ar[d]^{\bigoplus K_\chi} \\
f_\varphi^{-1}(\bigoplus_{\psi : [0] \to [m]} K_m) \ar@{=}[d] &
f_\varphi^{-1}g_m^{-1}K \ar[d]_{f_\varphi^{-1}\gamma_m} \ar[r]_{K(\varphi)} &
g_n^{-1}K \ar[d]^{\gamma_n} &
\bigoplus_{\chi : [0] \to [n]} K_n \ar@{=}[d] \\
f_\varphi^{-1}Y'_{0, m} \ar[r] &
f_\varphi^{-1}K_m \ar[r]^{K_\varphi} &
K_n &
Y'_{0, n} \ar[l]
}
$$
The top middle square is commutative as $X_0 \to K$ is a morphism
of simplicial objects. The left, resp.\ the right rectangles are
commutative as $\gamma_m$, resp.\ $\gamma_n$ is compatible with
$\gamma_{0, m}$, resp.\ $\gamma_{0, n}$ which are the arrows
$\bigoplus K_\psi$ and $\bigoplus K_\chi$ in the diagram.
Going around the outer rectangle of the diagram
is commutative as $(K_n, K_\varphi)$ is a simplical system
and the map $X_0(\varphi)$ is given by the obvious identifications
$f_\varphi^{-1}f_\psi^{-1}K_0 = f_{\varphi \circ \psi}^{-1}K_0$.
Note that the arrow $\bigoplus_\psi K_m \to Y'_{0, m} \to K_m$
induces an isomorphism on any of the direct summands
(because of our explicit construction of the Postnikov
systems $Y'_{i, j}$ above).
Hence, if we take a direct summand summand of
the upper left and corner, then this maps isomorphically to
$f_\varphi^{-1}g_m^{-1}K$ as $\gamma_m$ is an isomorphism.
Working out what the above says,
but looking only at this direct summand we conclude the lower
middle square commutes as we well. This concludes the proof.
\end{proof}
\section{Simplicial systems of the derived category: modules}
\label{section-glueing-modules}
\noindent
In this section we are going to prove a special case of
\cite[Proposition 3.2.9]{BBD} in the setting of derived
categories of $\mathcal{O}$-modules. The (slightly) easier
case of abelian sheaves is discussed in Section \ref{section-glueing}.
\begin{definition}
\label{definition-cartesian-derived-modules}
In Situation \ref{situation-simplicial-site}. Let $\mathcal{O}$
be a sheaf of rings on $\mathcal{C}_{total}$. A
{\it simplicial system of the derived category of modules}
consists of the following data
\begin{enumerate}
\item for every $n$ an object $K_n$ of $D(\mathcal{O}_n)$,
\item for every $\varphi : [m] \to [n]$ a map
$K_\varphi : Lf_\varphi^*K_m \to K_n$ in $D(\mathcal{O}_n)$
\end{enumerate}
subject to the condition that
$$
K_{\varphi \circ \psi} = K_\varphi \circ Lf_\varphi^*K_\psi :
Lf_{\varphi \circ \psi}^*K_l = Lf_\varphi^* Lf_\psi^*K_l
\longrightarrow
K_n
$$
for any morphisms $\varphi : [m] \to [n]$ and $\psi : [l] \to [m]$ of $\Delta$.
We say the simplicial system is {\it cartesian} if the maps $K_\varphi$
are isomorphisms for all $\varphi$.
Given two simplicial systems of the derived category
there is an obvious notion of a
{\it morphism of simplicial systems of the derived category of modules}.
\end{definition}
\noindent
We have given this notion a ridiculously long name intentionally.
The goal is to show that a simplicial system of the derived category
of modules comes from an object of $D(\mathcal{O})$ under certain
hypotheses.
\begin{lemma}
\label{lemma-cartesian-objects-derived-modules}
In Situation \ref{situation-simplicial-site} let $\mathcal{O}$ be a
sheaf of rings on $\mathcal{C}_{total}$.
If $K \in D(\mathcal{O})$ is an object, then $(K_n, K(\varphi))$
is a simplicial system of the derived category of modules.
If $K$ is cartesian, so is the system.
\end{lemma}
\begin{proof}
This is immediate from the definitions.
\end{proof}
\begin{lemma}
\label{lemma-modules-postnikov}
In Situation \ref{situation-simplicial-site} let $\mathcal{O}$
be a sheaf of rings on $\mathcal{C}_{total}$. Let $K$ be
an object of $D(\mathcal{C}_{total})$. Set
$$
X_n = (g_{n!}\mathcal{O}_n)
\otimes^\mathbf{L}_\mathcal{O} K
\quad\text{and}\quad
Y_n =
(g_{n!}\mathcal{O}_n \to \ldots \to g_{0!}\mathcal{O}_0)[-n]
\otimes^\mathbf{L}_\mathcal{O} K
$$
as objects of $D(\mathcal{O})$ where the maps are
as in Lemma \ref{lemma-simplicial-resolution-Z-site}.
With the evident canonical maps $Y_n \to X_n$ and
$Y_0 \to Y_1[1] \to Y_2[2] \to \ldots$ we have
\begin{enumerate}
\item the distinguished triangles $Y_n \to X_n \to Y_{n - 1} \to Y_n[1]$
define a Postnikov system
(Derived Categories, Definition \ref{derived-definition-postnikov-system})
for $\ldots \to X_2 \to X_1 \to X_0$,
\item $K = \text{hocolim} Y_n[n]$ in $D(\mathcal{O})$.
\end{enumerate}
\end{lemma}
\begin{proof}
First, if $K = \mathcal{O}$, then this is the construction of
Derived Categories, Example \ref{derived-example-key-postnikov}
applied to the complex
$$
\ldots \to
g_{2!}\mathcal{O}_2 \to
g_{1!}\mathcal{O}_1 \to
g_{0!}\mathcal{O}_0
$$
in $\textit{Ab}(\mathcal{C}_{total})$ combined with the fact that
this complex represents $K = \mathcal{O}$ in $D(\mathcal{C}_{total})$
by Lemma \ref{lemma-simplicial-resolution-ringed}.
The general case follows from this, the fact that the exact functor
$- \otimes^\mathbf{L}_\mathcal{O} K$ sends Postnikov systems to