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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{abelian sheaf cohomology} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{cohomology}{}\paragraph*{{Cohomology}}\label{cohomology} [[!include cohomology - contents]] \hypertarget{topos_theory}{}\paragraph*{{$(\infty,1)$-Topos Theory}}\label{topos_theory} [[!include (infinity,1)-topos - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak \noindent\hyperlink{more_details_on_this_idea}{More details on this idea}\dotfill \pageref*{more_details_on_this_idea} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{AsIntrinsicInfinityToposCohomology}{As intrinsic (∞,1)-topos cohomology}\dotfill \pageref*{AsIntrinsicInfinityToposCohomology} \linebreak \noindent\hyperlink{relation_to_derived_direct_images}{Relation to derived direct images}\dotfill \pageref*{relation_to_derived_direct_images} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} \hypertarget{general}{}\subsubsection*{{General}}\label{general} The [[cohomology]] $H^n(X,F)$ of a [[topological space]] $X$ with values in a [[sheaf of abelian groups]] / [[abelian sheaf]] $F$ was originally defined as the value of the right [[derived functor]] of the [[global section]] functor, the [[derived direct image]] functor. But by embedding sheaves with values in abelian groups as special cases of [[simplicial presheaf|simplicial sheaves]] into the more general context of [[(∞,1)-sheaves|∞-groupoid-valued sheaves]] via the [[Dold-Kan correspondence]] and thus the abelian sheaf cohomology into the more general context of the intrinsic [[nonabelian cohomology]] of an [[(∞,1)-topos]] $\mathbf{H} = Sh_{(\infty,1)}(C)$, this definition becomes equivalent to a special case of the general notion of [[nonabelian cohomology]] defined simply as the set of homotopy classes of maps \begin{displaymath} H^n(X,F) = \pi_0 \mathbf{H}(X,\mathbf{B}^n F) \end{displaymath} from the space $X$ regarded a (``nonabelian''!) sheaf, to the [[Eilenberg-MacLane object]] in degree $n$, defined by $F$. The relation of this more conceptual and more general point of view on abelian sheaf cohomology to the original definition was originally clarified in \begin{itemize}% \item [[Kenneth Brown]], [[BrownAHT|Abstract Homotopy Theory and Generalized Sheaf Cohomology]] \end{itemize} (whose proof is reproduced below). Brown constructed effectively the [[homotopy category of an (∞,1)-category|homotopy category]] of $\mathbf{H}$ using a model of a [[category of fibrant objects]] paralleling the [[model structure on simplicial presheaves]] as a [[presentable (∞,1)-category|presentation]] of the [[(∞,1)-category of (∞,1)-sheaves]]. This says that ordinary abelian sheaf cohomology in fact computes the equivalence classes of the [[∞-stackification]] of a sheaf with values in [[chain complexes]] of [[abelian groups]]. The general [[(∞,1)-topos]]-theoreric perspective on cohomology is described in more detail at [[cohomology]]. The details on how to realize abelian sheaf cohomology as an example of this are discussed below. \hypertarget{more_details_on_this_idea}{}\subsubsection*{{More details on this idea}}\label{more_details_on_this_idea} Using the [[Dold-Kan correspondence]] in [[higher topos theory]], [[complex]]es of [[abelian sheaf|abelian sheaves]] can be understood as a generalization of [[topological space]]s, in a precise sense recalled below. This induces a notion of cohomology of [[complex]]es of [[abelian sheaf|abelian sheaves]] from the familiar notion of cohomology of spaces. Which is a simple one: recall that the [[cohomology]] of one [[topological space]] $X$ with coefficients in another space $A$ is nothing but the space of morphisms (continuous maps) $H(X,A) := [X,A]$ from $X$ to $A$ -- or, in a more restrictive sense traditionally adopted, the set $\Pi_0[X,A]$ of connected path components of this space. Similarly, when considering [[chain complexes]] of [[abelian sheaves]] in their natural [[higher topos theory|higher topos theoretic]] home, the cohomology of a complex of sheaves $A$ on a space $X$ is nothing but the hom-space $H(X,A) = [X,A]$ -- where the space $X$ itself is regarded as a special case of a sheaf. One reason this conceptually simple picture is not usually presented is that the space $X$ is typically not represented by an \emph{abelian} complex of sheaves, so that the full simplicity of the story becomes manifest only in general [[nonabelian cohomology]]. More precisely, via the [[Dold-Kan correspondence]] (non-negatively graded) [[complex]]es of [[abelian sheaf|abelian sheaves]] -- which are equivalently [[sheaf|sheaves]] with values in (non-negatively graded) [[category of chain complexes|categories of chain complexes]] -- can be regarded as special cases of [[simplicial presheaf|simplicial sheaves]]. But thanks to the [[model structure on simplicial presheaves|model category structure]] on the category of [[simplicial presheaf|simplicial sheaves]], this in turn is a model for the [[(infinity,1)-topos]] of [[space and quantity|generalized spaces]] called [[infinity-stack]]s. The very point of $(\infty,1)$-[[(infinity,1)-topos|topoi]] is that they are [[(infinity,1)-category|(infintiy,1)-categories]] which behave in all structural aspects relevant for [[homotopy theory]] as the archetypical example [[Top]] does. In particular, as in [[Top]], the notion of [[cohomology]] in any [[(infinity,1)-topos]] simply coincides with that of [[hom-space]]s. In the 1-categorical [[model category|model theoretic models]] these hom-spaces are computed technically by [[derived functor]]s. More precisely, the Hom-space $[X,A]$ for $X$ an ordinary space computes the [[global section]]s $\Gamma(X,A)$ of the complex of [[abelian sheaf|abelian sheaves]] $A$ which is computed by the right [[derived functor]] of the [[global section]] $R \Gamma(X,-)$ of the [[global section functor]] $\Gamma(X,-)$, which does exist entirely within the abelian context. This, then, is the definition of sheaf cohomology as usually presented: the cohomology of the complex $R \Gamma(X,A)$. \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{AsIntrinsicInfinityToposCohomology}{}\subsubsection*{{As intrinsic (∞,1)-topos cohomology}}\label{AsIntrinsicInfinityToposCohomology} Under the ([[stable Dold-Kan correspondence|stable]]) [[Dold-Kan correspondence]] we have the following identification of sheaves taking values in [[chain complexes]] with sheaves taking values in [[infinity-groupoids]] and [[spectrum|spectra]], crucial for a conceptual understanding of abelian sheaf cohomology: let $X$ be a [[site]] \begin{itemize}% \item the category $Sh(X, Ch_+(Ab))$ of non-negatively graded [[chain complex]]es of [[abelian sheaf|abelian sheaves]] is [[homotopical functor|homotopically]] [[equivalence|equivalent]] to the category $Sh(X, sAb)$ of [[sheaf|sheaves]] with values in simplicial abelian groups (i.e. [[Kan complex]]es with strict abelian group structure); \item the category $Sh(X, Ch(Ab))$ of unbounded [[chain complex]]es of [[abelian sheaf|abelian sheaves]] is [[equivalence of categories|equivalent]] to $Sh(X, Sp(Ab))$, the category of sheaves with values in [[combinatorial spectrum|combinatorial spectra]] internal to abelian groups. \end{itemize} Let how $F \in Sh(X,Ab)$ be a [[sheaf]] on a [[site]] $X$ with values in the category [[Ab]] of abelian groups. For $n \in \mathbb{N}$ write $B^n F \in Sh(X, Ch_+(Ab))$ for the [[complex]] of [[sheaf|sheaves]] with values in abelian groups which is trivial everywhere except in degree $n$, where it is given by $F$. By the [[Dold-Kan correspondence]] we can regard $B^n F$ equivalently as a complex of sheaves of abelian groups as well as sheaf with values in [[infinity-groupoid]]s. Write $H$ for the [[(infinity,1)-category]] of [[simplicial presheaf|simplicial sheaves]] on $X$ and $H_{ab}$ for the [[(infinity,1)-category]] of complexes of [[abelian sheaf|abelian sheaves]] on $X$. Write $X$ for the terminal sheaf of $X$, i.e. for the [[sheaf]] that corresponds to the space $X$ itself. Then \begin{displaymath} H^n(X,A) := \pi_0 H(X,\mathbf{B}^n F) \end{displaymath} is the degree $n$ [[cohomology]] class of $X$ with values in $F$, regarded as computed in [[nonabelian cohomology]]. Now write $\mathbb{Z}[X]$ for the free abelianization of the sheaf $X$. This is the sheaf constant on the abelian group $\mathbb{Z}$ of integers. Then the above cohomology set, which of course happens to be a cohomology group here, due to the abelianness of $F$, is canonically isomorphic to the cohomology set \begin{displaymath} \cdots \simeq \pi_0 H_{ab}(\mathbb{Z}[X], \mathbf{B}^n F) \end{displaymath} which can be regarded as the [[hom-set]] in the [[derived category]] of [[complex]]es of [[abelian sheaf|abelian sheaves]]. This, in turn, is the same as the traditional expression \begin{displaymath} \cdots \simeq R^n \Gamma(X,F) \end{displaymath} giving the $n$th [[derived functor]] of the [[global section functor]] of the [[abelian sheaf]] $F$. This, finally, is the same group as obtained by choosing any [[complex]] $I_F$ of [[abelian sheaf|abelian sheaves]] that is [[injective complex|injective]] and [[quasi-isomorphism|quasi-isomorphic]] to $F$ regarded as a complex concentrated in degree 0 and then computing the $n$ [[homology]] group of the complex $\Gamma(X,I_F)$ of global sections of $F$: \begin{displaymath} \cdots \simeq H_n(\Gamma(X,I_F)) \,. \end{displaymath} Historically the development of abelian sheaf cohomology was precisely in reverse order to this derivation from the general $(\infty,1)$-[[(infinity,1)-category|categorical]] [[cohomology]]. \begin{theorem} \label{}\hypertarget{}{} Let $X$ be a [[topological space]], $F$ a [[sheaf]] on (the [[category of open subsets]] of) $X$ with values in abelian groups, and $\mathbf{B}^n F = K(F,n)$ the image of the complex of abelian sheaves $F[n]$ ($F$ in degree $n$, trivial elsewhere) under the [[Dold-Kan correspondence]] in sheaves with values in [[Kan complex]]es \begin{displaymath} \Gamma \circ (-) : Sh(X,Ch_+(Ab)) \to Sh(X, AbSimpGrp) \end{displaymath} \begin{displaymath} F[n] \mapsto K(F,n) =: \mathbf{B}^n F \,. \end{displaymath} Then we have the following natural isomorphism of cohomologies: \begin{displaymath} H^n(X,F) \simeq H(X, \mathbf{B}^n F) \end{displaymath} where \begin{itemize}% \item on the left we have ordinary abelian sheaf cohomology defined as the right [[derived functor]] of the global sections functor \begin{displaymath} H^n(X,F) := (R \Gamma_X)(F) \,; \end{displaymath} \item on the right we have [[nonabelian cohomology]], namely the hom-set in the [[homotopy category]] of Kan complex valued [[model structure on simplicial presheaves|simplicial sheaves]] \begin{displaymath} H(X, \mathbf{B}^n F) := Ho_{Sh(X,\infty Grpd)}(X,\mathbf{B}^n X) \,. \end{displaymath} \end{itemize} \end{theorem} \begin{proof} This is the first four steps in the proof of theorem 2 in [[BrownAHT]]. The proof proceeds along the following four steps, which we describe in more detail below: \begin{displaymath} \begin{aligned} H^n(X,F) & \simeq Ho_{Sh(X,Ch(Ab))}(\mathbb{Z}, F[n]) \\ & \simeq Ho_{Sh(X,Ch_+(Ab))}(\mathbb{Z}, F[n]) \\ & \simeq Ho_{Sh(X,AbSimpGrp)}(\mathbb{Z}X, K(F,n)) \\ & \simeq Ho_{Sh(X,\infty Grpd)}(X, K(F,n)) \end{aligned} \end{displaymath} \begin{enumerate}% \item By the [[derived functor]] definition of sheaf cohomology, $H^n(X,F)$ is the cohomology of any complex of sheaves $I^\bullet \in Sh(X,Ch(Ab))$ that is [[injective object|injective]] and weakly equivalent to $F[n]$, $F[n] \stackrel{\simeq}{\to} I^\bullet$: \begin{displaymath} H^n(X,F) \simeq H^0(I^\bullet(X)) \,. \end{displaymath} On the other hand, by the general formula for hom-sets in [[homotopy category|homtotopy categories]] obtained by localizing at the [[calculus of fractions|multiplicative system]] given by [[quasi-isomorphism]]s of complexes (e.g. def. 13.1.2 in [[Categories and Sheaves|CaS]]) we have \begin{displaymath} Ho_{Sh(X,Ch(Ab))}(\mathbb{Z}, F[n]) \simeq colim_{Y^\bullet \stackrel{\simeq}{\to} \mathbb{Z}} Hom_{K(Sh(X,Ab))}(Y, I^\bullet) \,. \end{displaymath} But due to the injectiveness of $I^\bullet$, the integrand on the right is constant (lemma 14.1.5 in [[Categories and Sheaves|CaS]]) and hence the colimit is isomorphic to $\cdots \simeq Hom_{K(Sh(X,Ab))}(\mathbb{Z}, I^\bullet) \simeq H^0(I^\bullet(X))$, as desired. \item The second step uses that the inclusion functor \begin{displaymath} Ho_{Sh(X,Ch_+(Ab))} \hookrightarrow Ho_{Sh(X,Ch(Ab))} \end{displaymath} is [[full and faithful functor|full and faithful]]. This in turn follows from \begin{itemize}% \item first observing that the inclusion $S : Sh(X,Ch_+(Ab)) \hookrightarrow Sh(X, Ch(Ab))$ of chain complexes concentrated in non-negative degree into all complexes of sheaves is [[full and faithful functor|full and faithful]] and has the obvious [[right adjoint]] $T : Sh(X,Ch(Ab)) \to Sh(X, Ch_+(Ab))$ obtained by \textbf{t}runcating a complex. \item By inspection, or else by the general properties of [[adjoint functor]]s (see the list of properties given there) this implies that $Id \to T \circ S$ is an [[isomorphism]]. This implies that also $Id \to Ho T \circ Ho S$ is an [[isomorphism]]. \item But by the adjoint functor lemma for homotopical categories, $Ho S$ is also left adjoint to $Ho T$ (since both preserve weak equivalences). So that once again with the general properties of [[adjoint functor]]s it follows that $Ho S$ is [[full and faithful functor|full and faithful]]. \end{itemize} \item The third step uses that the [[normalized chain complex]] functor $Sh(X,AbSimpGrp) \to Sh(X, Ch_+(Ab))$ is an [[equivalence of categories]] that preserves the respective weak equivalences and homotopies. \item The fourth step finally uses that the [[stuff, structure, property|forgetful functor]] $Sh(X, SimpAbGrp) \to Sh(X, \infty Grpd)$ that only remembers the [[Kan complex]] underlying a [[simplicial group]] has a [[left adjoint]], the free abelian group functor $\mathbb{Z} : Sh(X,\infty Grpd) \to Sh(X, AbSimpGrp)$ (see [[Dold-Kan correspondence]] for details), and that preserves weak equivalences (see the discussion at [[simplicial group]] for more on that). \end{enumerate} \end{proof} \hypertarget{relation_to_derived_direct_images}{}\subsubsection*{{Relation to derived direct images}}\label{relation_to_derived_direct_images} \begin{prop} \label{}\hypertarget{}{} Let $f^{-1} \colon Y \to X$ be a [[morphism of sites]]. Then the $q$th [[derived functor]] $R^q f_\ast$ of the induced [[direct image]] functor sends $\mathcal{F} \in Ab(Sh(X_{et}))$ to the [[sheafification]] of the [[presheaf]] \begin{displaymath} U_Y \mapsto H^q(f^{-1}(U_Y), \mathcal{F}) \,, \end{displaymath} where on the right we have the degree $q$ [[abelian sheaf cohomology]] [[cohomology group|group]] with [[coefficients]] in the given $\mathcal{F}$. \end{prop} (e.g. \hyperlink{Tamme}{Tamme, I (3.7.1), II (1.3.4)}, \hyperlink{Milne}{Milne, 12.1}). \begin{proof} We have a [[commuting diagram]] \begin{displaymath} \itexarray{ Ab(PSh(X)) &\stackrel{(-)\circ f^{-1}}{\longrightarrow}& Ab(PSh(Y)) \\ \uparrow^{\mathrlap{inc}} && \downarrow^{L} \\ Ab(Sh(X)) &\stackrel{f_\ast}{\longrightarrow}& Ab(Sh(Y)) } \,, \end{displaymath} where the right vertical morphism is [[sheafification]]. Because $(-) \circ f^{-1}$ and $L$ are both [[exact functors]] it follows that for $I^\bullet \to \mathcal{F}$ an [[injective resolution]] that \begin{displaymath} \begin{aligned} R^p f_\ast(\mathcal{F}) & :\simeq H^p( f_\ast I) \\ & = H^p(L I^\bullet(f^{-1}(-))) \\ & = L (H^p(I^\bullet)(f^{-1}(-))) \end{aligned} \end{displaymath} \end{proof} \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \begin{itemize}% \item [[de Rham cohomology]] \item [[Dolbeault cohomology]] \item [[Spencer cohomology]] \item [[Deligne cohomology]] \item [[etale cohomology]] \item [[crystalline cohomology]] \item [[syntomic cohomology]] \end{itemize} \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[sheaf]], [[cohomology]] \item \textbf{abelian sheaf cohomology} \begin{itemize}% \item [[model structure on chain complexes]] \item [[resolutions]]: [[soft sheaf]], [[fine sheaf]], [[flabby sheaf]] \item [[triangulated category of sheaves]] \item [[Verdier duality]] \end{itemize} \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} The traditional definition of sheaf cohomology in terms of the right [[derived functor]] of the [[global sections]] functor can be found recalled for instance in these notes: \begin{itemize}% \item Ch\^e{}nevert, Kassaei, \emph{Sheaf Cohomology} (\href{http://www.math.mcgill.ca/goren/SeminarOnCohomology/Sheaf_Cohomology.pdf}{pdf}) \item Cibotaru, \emph{Sheaf cohomology} (\href{http://www.nd.edu/~lnicolae/sheaves_coh.pdf}{pdf}) \item [[Patrick Morandi]], \emph{Sheaf cohomology} (\href{http://sierra.nmsu.edu/morandi/notes/SheafCohomology.pdf}{pdf}) \end{itemize} Its discussion in the more general [[nonabelian cohomology]] and [[infinity-stack]] context emphasized above is due to \begin{itemize}% \item [[Kenneth Brown]], \emph{[[BrownAHT|Abstract Homotopy Theory and Generalized Sheaf Cohomology]]} \end{itemize} This uses homotopical structures of a [[category of fibrant objects]] on complexes of abelian sheaves. Discussion of actual [[model structure on chain complexes]] of abelian sheaves is in \begin{itemize}% \item [[Mark Hovey]], \emph{Model category structures on chain complexes of sheaves}, Trans. Amer. Math. Soc. 353, 6 (\href{http://www.mathaware.org/tran/2001-353-06/S0002-9947-01-02721-0/S0002-9947-01-02721-0.pdf}{pdf}) \end{itemize} A discussion of the [[?ech cohomology]] description of sheaf cohomology along the above lines is in \begin{itemize}% \item [[Tibor Beke]], \emph{Higher ech Theory} (\href{http://www.math.uiuc.edu/K-theory/0646/}{web}, \href{http://www.math.uiuc.edu/K-theory/0646/cech.pdf}{pdf}) \end{itemize} See also \begin{itemize}% \item [[Günter Tamme]], section II 1 of \emph{[[Introduction to Étale Cohomology]]} \end{itemize} \begin{itemize}% \item [[James Milne]], section 7 of \emph{[[Lectures on Étale Cohomology]]} \end{itemize} [[!redirects chain complex of sheaves]] [[!redirects chain complexes of sheaves]] [[!redirects sheaf of chain complexes]] [[!redirects sheaves of chain complexes]] [[!redirects abelian presheaf]] [[!redirects abelian presheaves]] \end{document}