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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*{hypercover} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{locality_and_descent}{}\paragraph*{{Locality and descent}}\label{locality_and_descent} [[!include descent and locality - contents]] \hypertarget{model_category_theory}{}\paragraph*{{Model category theory}}\label{model_category_theory} [[!include model category theory - 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{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{existence_and_refinements}{Existence and refinements}\dotfill \pageref*{existence_and_refinements} \linebreak \noindent\hyperlink{cech_nerves}{Cech nerves}\dotfill \pageref*{cech_nerves} \linebreak \noindent\hyperlink{OverGeneralSites}{Over general sites}\dotfill \pageref*{OverGeneralSites} \linebreak \noindent\hyperlink{OverVerdierSites}{Over Verdier sites}\dotfill \pageref*{OverVerdierSites} \linebreak \noindent\hyperlink{HypercoverHomology}{Hypercover homology}\dotfill \pageref*{HypercoverHomology} \linebreak \noindent\hyperlink{DescentAndCohomology}{Descent and cohomology}\dotfill \pageref*{DescentAndCohomology} \linebreak \noindent\hyperlink{reference}{Reference}\dotfill \pageref*{reference} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} A \emph{hypercover} is the generalization of a [[Cech nerve]] of a [[cover]]: it is a simplicial [[resolution]] of an object obtained by iteratively applying covering families. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} Let \begin{displaymath} (L \dashv i) : Sh(C) \stackrel{\overset{L}{\leftarrow}}{\hookrightarrow} PSh(C) \end{displaymath} be the [[geometric embedding]] defining a [[sheaf topos]] $Sh(C)$ into a [[presheaf topos]] $PSh(C)$. \begin{defn} \label{CoskeletonDefinition}\hypertarget{CoskeletonDefinition}{} A morphism \begin{displaymath} (Y \stackrel{f}{\to} X) \in PSh(C)^{\Delta^{op}} \end{displaymath} in the [[category of simplicial objects]] in $PSh(C)$, hence the category of [[simplicial presheaves]], is called a \textbf{hypercover} if for all $n \in \mathbb{N}$ the canonical morphism \begin{displaymath} Y_{n} \to (\mathbf{cosk}_{n-1} Y)_n \times_{(\mathbf{cosk}_{n-1} X)_n} X_n \end{displaymath} in $PSh(C)$ are [[local epimorphisms]] (in other words, $f$ is a ``[[Reedy model structure|Reedy]] local-epimorphism''). Here this morphism into the [[fiber product]] is that induced from the [[naturality square]] \begin{displaymath} \itexarray{ Y &\longrightarrow& X \\ \downarrow && \downarrow \\ \mathbf{cosk}_{n-1} Y &\longrightarrow& \mathbf{cosk}_{n-1} X } \end{displaymath} of the [[unit of a monad|unit]] of the [[coskeleton]] [[functor]] $\mathbf{cosk}_n : PSh(C)^{\Delta^{op}} \to PSh(C)^{\Delta^{op}}$. A hypercover is called \textbf{bounded} by $n \in \mathbb{N}$ if for all $k \geq n$ the morphisms $Y_{k} \to (\mathbf{cosk}_{k-1} Y)_k \times_{(\mathbf{cosk}_{k-1} X)_k} X_k$ are [[isomorphism]]s. The smallest $n$ for which this holds is called the \textbf{height} of the hypercover. \end{defn} \begin{defn} \label{SplitHypercover}\hypertarget{SplitHypercover}{} A hypercover that also satisfies the \href{model+structure+on+simplicial+presheaves#CofibrantObjects}{cofibrancy condition} in the projective [[local model structure on simplicial presheaves]] in that \begin{enumerate}% \item it is simplicial-degree wise a [[coproduct]] of [[representable functor|representables]]; \item degenerate cells split off as a [[coproduct|direct summand]]) \end{enumerate} is called a \textbf{[[split hypercover]]}. \end{defn} (\hyperlink{DuggerHollanderIsaksen02}{Dugger-Hollander-Isaksen 02, def. 4.13}) see also (\hyperlink{Low}{Low 14-05-26, 8.2.15}) \begin{remark} \label{}\hypertarget{}{} Definition \ref{CoskeletonDefinition} is equivalent to saying that $f : Y \to X$ is a \emph{local} acyclic fibration: for all $U \in C$ and $n \in \mathbb{N}$ every lifting problem \begin{displaymath} \left( \itexarray{ \partial \Delta[n] \cdot U &\to& Y \\ \downarrow && \downarrow^{\mathrlap{f}} \\ \Delta[n]\cdot U &\to& X } \right) \;\;\simeq \;\; \left( \itexarray{ \partial \Delta[n] &\to& Y(U) \\ \downarrow && \downarrow^{\mathrlap{f(U)}} \\ \Delta[n] &\to& X(U) } \right) \end{displaymath} has a solution $(\sigma_i)$ after refining to some [[covering]] family $\{U_i \to U\}$ of $U$ \begin{displaymath} \forall i : \left( \itexarray{ \partial \Delta[n] &\to& Y(U_i) \\ \downarrow &{}^{\mathllap{\exists \sigma_i}}\nearrow& \downarrow^{\mathrlap{f(U_i)}} \\ \Delta[n] &\to& X(U_i) } \right) \,, \end{displaymath} \end{remark} (\hyperlink{DuggerHollanderIsaksen02}{Dugger-Hollander-Isaksen 02, prop. 7.2} \begin{remark} \label{}\hypertarget{}{} If the [[topos]] $Sh(C)$ has [[point of a topos|enough points]] a morphism $f : Y \to X$ in $Sh(C)^{\Delta^{op}}$ is a hypercover if all its [[stalk]]s are acyclic [[Kan fibration]]s. \end{remark} In this form the notion of hypercover appears for instance in (\hyperlink{Brown73}{Brown 73}). In some situations, we may be interested primarily in hypercovers that are built out of data entirely in the site $C$. We obtain such hypercovers by restricting $X$ to be a discrete simplicial object which is representable, and each $Y_n$ to be a coproduct of representables. This notion can equivalently be formulated in terms of diagrams $(\Delta/A) \to C$, where $A$ is some simplicial set and $(\Delta/A)$ is its [[category of simplices]]. \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \begin{example} \label{}\hypertarget{}{} Consider the case that $X = const X_0$ is simplicially constant. Then the conditions on a morphism $Y \to X$ to be a hypercover is as follows. \begin{itemize}% \item in degree 0: $Y_0 \to X_0$ is a local epimorphism. \item in degree 1: The commuting diagram in question is \begin{displaymath} \itexarray{ Y_1 &\to& X_0 \\ \downarrow && \downarrow^{\mathrlap{diag}} \\ Y_0 \times Y_0 &\to& X_0 \times X_0 } \,. \end{displaymath} Its [[pullback]] is $(Y_0 \times Y_0)_{X_0 \times X_0} X_0 \simeq Y_0 \times_{X_0} Y_0$, Hence the condition is that $Y_1 \to Y_0 \times_{X_0} Y_0$ is a local epimorphism. \item in degree 2: The commuting diagram in question is \begin{displaymath} \itexarray{ Y_2 &\to& X_0 \\ \downarrow && \downarrow^{Id} \\ (Y_1 \times_{Y_0} Y_1 \times_{Y_0}Y_1)_{\times_{Y_0 \times Y_0}} Y_0 &\to& X_0 } \,. \end{displaymath} So the condition is that the vertical morphism is a local epi. \item Similarly, in any degree $n \geq 2$ the condition is that \begin{displaymath} Y_n \to (\mathbf{cosk}_{n-1} Y)_n \end{displaymath} is a local epimorphism. \end{itemize} \end{example} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{existence_and_refinements}{}\subsubsection*{{Existence and refinements}}\label{existence_and_refinements} \hypertarget{cech_nerves}{}\paragraph*{{Cech nerves}}\label{cech_nerves} \begin{prop} \label{}\hypertarget{}{} For $U = \{U_i \to X\}$ a [[cover]], the [[Cech nerve]] projection $C(U) \to X$ is a hypercover of height 0. \end{prop} \hypertarget{OverGeneralSites}{}\paragraph*{{Over general sites}}\label{OverGeneralSites} \begin{prop} \label{ExistenceOfSplitRefinements}\hypertarget{ExistenceOfSplitRefinements}{} Given any [[site]] $(\mathcal{C},J)$ and given a diagram of simplicial presheaves \begin{displaymath} \itexarray{ && Y \\ && \downarrow^{\mathrlap{hcov}} \\ X' &\longrightarrow& X } \end{displaymath} where the vertical morphism is a hypercover, then there exists a completion to a [[commuting diagram]] \begin{displaymath} \itexarray{ Y' &\longrightarrow& Y \\ \downarrow^{\mathrlap{hcov}} && \downarrow^{\mathrlap{hcov}} \\ X' &\longrightarrow& X } \end{displaymath} where the left vertical morphism is a split hypercover, def. \ref{SplitHypercover}. Moreover, if $(\mathcal{D}, K)\to (\mathcal{C},J)$ is a [[dense subsite]] then $Y'$ as above exists such that it is simplicial-degree wise a [[coproduct]] of ([[representable functor|representables]] by) objects of $\mathcal{D}$. \end{prop} (e.g. \hyperlink{Low}{Low 14-05-26, lemma 8.2.20}) \begin{remark} \label{}\hypertarget{}{} In particular taking $X'\to X$ in prop. \ref{ExistenceOfSplitRefinements} to be an identity, the proposition says that every hypercover may be refined by a split hypercover. \end{remark} (see also \hyperlink{Low}{Low 14-05-26, lemma 8.2.23}) \hypertarget{OverVerdierSites}{}\paragraph*{{Over Verdier sites}}\label{OverVerdierSites} \begin{defn} \label{}\hypertarget{}{} A \textbf{[[Verdier site]]} is a [[small category]] with finite [[pullbacks]] equipped with a [[basis for a Grothendieck topology]] such that the generating [[covering]] maps $U_i \to X$ all have the property that their [[diagonal]] \begin{displaymath} U_i \to U_i \times_X U_i \end{displaymath} is also a generating covering. We say that $U_i \to X$ is \textbf{basal}. \end{defn} \begin{example} \label{}\hypertarget{}{} It is sufficient that all the $U_i \to X$ are [[monomorphism]]s. Examples include the standard [[open cover]]-topology on [[Top]]. \end{example} \begin{defn} \label{BasalHypercover}\hypertarget{BasalHypercover}{} A \textbf{basal hypercover} over a [[Verdier site]] is a hypercover $U \to X$ such that for all $n \in \mathbb{N}$ the components of the maps into the matching object $U_n \to M U_n$ are basal maps, as above. \end{defn} \begin{theorem} \label{}\hypertarget{}{} Over a Verdier site, every hypercover may be refined by a split (def. \ref{SplitHypercover}) and basal hypercover (def. \ref{BasalHypercover}). \end{theorem} This is (\hyperlink{DuggerHollanderIsaksen02}{Dugger-Hollander-Isaksen 02, theorem 8.6}). \hypertarget{HypercoverHomology}{}\subsubsection*{{Hypercover homology}}\label{HypercoverHomology} Let $f : Y \to X$ be a hypercover. We may regard this as an object in the [[overcategory]] $Sh(C)/X$. By the discussion this is equivalently $Sh(C/X)$. Write $Ab(Sh(C/X))$ for the category of abelian [[group object]]s in the [[sheaf topos]] $Sh(C/X)$. This is an [[abelian category]]. Forming in the sheaf topos the [[free functor|free]] abelian group on $f_n$ for each $n \in \mathbb{N}$, we obtain a simplicial abelian group object $\bar f \in Ab(Sh(C/X))^{\Delta}$. As such this has a [[Moore complex|normalized chain complex]] $N_\bullet(\bar f)$. \begin{prop} \label{}\hypertarget{}{} For $f : Y \to X$ a hypercover, the [[chain homology]] of $N(\bar f)$ vanishes in positive degree and is the group of [[integer]]s in degree 0, as an object in $Ab(Sh(C)(X)$: \begin{displaymath} H_p(N(f)) \simeq \left\{ \itexarray{ 0 & for \; p \geq 1 \\ \mathbb{Z} & for \; p = 0 } \right. \,. \end{displaymath} \end{prop} \hypertarget{DescentAndCohomology}{}\subsubsection*{{Descent and cohomology}}\label{DescentAndCohomology} The following theorem characterizes the [[∞-stack]]/[[(∞,1)-sheaf]]-condition for the [[presentable (∞,1)-category|presentation]] of an [[(∞,1)-topos]] by a [[local model structure on simplicial presheaves]] in terms of descent along hypercovers. \begin{theorem} \label{DescentFromDescentAlongHypercovers}\hypertarget{DescentFromDescentAlongHypercovers}{} In the [[local model structure on simplicial presheaves]] $PSh(C)^{\Delta^{op}}$ an object is fibrant precisely if it is fibrant in the global [[model structure on simplicial presheaves]] and in addition satisfies [[descent]] along all hypercovers over representables that are degreewise [[coproduct]]s of representables. \end{theorem} This is the central theorem in (\hyperlink{DuggerHollanderIsaksen02}{Dugger-Hollander-Isaksen 02}). The following theorem is a corollary of this theorem, using the discussion at [[abelian sheaf cohomology]]. But historically it predates the above- theorem. \begin{theorem} \label{}\hypertarget{}{} \textbf{(Verdier's hypercovering theorem)} For $X$ a [[topological space]] and $F$ a [[sheaf]] of [[abelian group]]s on $X$, we have that the [[abelian sheaf cohomology]] of $X$ with coefficients in $F$ is given \begin{displaymath} H^q(X, F) \simeq {\lim_{\to}}_{Y \to X} H^q(Hom_{Sh}(Y^\bullet,F)) \end{displaymath} by computing for each hypercover $Y \to X$ the [[cochain cohomology]] of the [[Moore complex]] of the cosimplicial abelian group obtained by evaluating $F$ degreewise on the hypercover, and then taking the [[colimit]] of the result over the [[poset]] of all hypercovers over $X$. \end{theorem} A proof of this result in terms of the structure of a [[category of fibrant objects]] on the category of simplicial presheaves appears in (\hyperlink{Brown73}{Brown 73, section 3}). \hypertarget{reference}{}\subsection*{{Reference}}\label{reference} The concept of hypercovers was introduced for [[abelian sheaf cohomology]] in \begin{itemize}% \item [[Jean-Louis Verdier]], Expos\'e{} V, sect. 7 of [[SGA4]], \end{itemize} An early standard reference founding [[étale homotopy theory]] is \begin{itemize}% \item [[Michael Artin]], [[Barry Mazur]], \emph{\'E{}tale Homotopy} , Lecture Notes in Mathematics 100, Springer- Verlag, Berline-Heidelberg-New York (1972). \end{itemize} The modern reformulation of their notion of hypercover in terms of simplicial presheaves is mentioned for instance at the end of section 2, on \href{http://www.math.uwo.ca/~jardine/papers/Fields-01.pdf#page=6}{p. 6} of \begin{itemize}% \item [[John Frederick Jardine]], \emph{Fields Lectures: Simplicial presheaves} (\href{http://www.math.uwo.ca/~jardine/papers/Fields-01.pdf}{pdf}) \end{itemize} A discussion of hypercovers of [[topological spaces]] and relation to [[étale homotopy type]] of [[smooth schemes]] and [[A1-homotopy theory]] is in \begin{itemize}% \item [[Daniel Isaksen]], \emph{\'E{}tale realization of the $\mathbb{A}^1$-homotopy theory of schemes}, 2001 (\href{http://www.math.uiuc.edu/K-theory/0495/}{K-theory archive}) \item [[Daniel Dugger]], [[Daniel Isaksen]], \emph{Hypercovers in topology}, 2005 (\href{http://www.math.uiuc.edu/K-theory/0528/hypercover.pdf}{pdf}, \href{http://www.math.uiuc.edu/K-theory/0528/}{K-Theory archive}) \end{itemize} A discussion in a topos with enough points in in \begin{itemize}% \item [[Kenneth Brown]], \emph{[[BrownAHT|Abstract homotopy theory and generalized sheaf cohomology]]}, 1973 \end{itemize} A thorough discussion of hypercovers over representables and their role in [[descent]] for simplicial presheaves is in \begin{itemize}% \item [[Daniel Dugger]], [[Sharon Hollander]], [[Daniel Isaksen]], \emph{Hypercovers and simplicial presheaves}, Mathematical Proceedings of the Cambridge Philosophical Society. Vol. 136. No. 1., 2004. (\href{http://arxiv.org/abs/math/0205027}{arXiv:0205027}, \href{http://www.math.uiuc.edu/K-theory/0563/}{K-theory archive}) \end{itemize} On the Verdier hypercovering theorem see \begin{itemize}% \item [[Kenneth Brown]], \emph{[[BrownAHT|Abstract Homotopy Theory and Generalized Sheaf Cohomology]]} \item [[John Frederick Jardine]], \emph{The Verdier hypercovering theorem} (\href{http://www.math.uwo.ca/~jardine/papers/preprints/Verdier4.pdf}{pdf}) \item [[Zhen Lin Low]], \emph{Cocycles in categories of fibrant objects}, (\href{http://arxiv.org/abs/1502.03925v3}{pdf}) \end{itemize} Split hypercover refinement over general sites is discussed in \begin{itemize}% \item [[Zhen Lin Low]], \emph{[[Notes on homotopical algebra]]} \end{itemize} [[!redirects hypercovers]] [[!redirects Verdier hypercover theorem]] [[!redirects Verdier hypercovering theorem]] \end{document}