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 \input{preamble} % OK, start here. % \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