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\title{Hypercoverings}

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\tableofcontents

\section{Introduction}
\label{section-introduction}

\noindent
Hypercoverings can be used to compute cohomology of abelian sheaves on sites
without recourse to injective resolutions. See \cite[Expose V, Sec. 7]{SGA4}.
A nice manuscript on cohomological descent is the text by Brian Conrad,
see \url{http://www.math.lsa.umich.edu/~bdconrad/papers/hypercover.pdf}.
Probably it is useless to try to improve on Brian's article, so we look
at the question a little differently (more naively).

\section{Definitions}
\label{section-definitions}

\noindent
Let $\mathcal{C}$ be a category. Let $\Delta$ be the category of finite 
ordered sets with objects $[0]=\{0\}, [1]=\{0,1\}, [2]=\{0,1,2\},\ldots$ 
and order preserving maps. A simplicial object $U_\bullet$ of $\mathcal{C}$ 
is a contravariant functor $U_\bullet : \Delta \to \mathcal{C}$. This means 
there are objects $U_0,U_1,U_2,\ldots$ and morphisms $U_\bullet(\varphi) : 
U_n \to U_m$, where $\varphi$ is any order preserving map 
$\varphi : [m] \to [n]$.

\smallskip\noindent
In particular there is a unique morphism $U_0 \to U_n$ and there are
exactly $n+1$ morphisms $U_n \to U_0$ corresponding to the $n+1$ maps
$[0] \to [n]$. Obviously we need some more notation to be able to talk 
intelligently about these simplicial objects.

\begin{definition}
\label{definition-face-degeneracy}
For any integer $n\geq 1$, and any $0\leq j \leq n$ we let $d^n_j : [n-1]
\to [n]$ denote the injective order preserving map skipping $j$. For any
integer $n\geq 0$, and any $0\leq j \leq n$ we denote $s^n_j : [n+1] 
\to [n]$ the surjective order preserving map with 
$(s^n_j)^{-1}(\{j\}) = \{j, j+1\}$.
\end{definition}

\noindent
We get a unique morphism $U_\bullet(s^0_0) : U_0 \to U_1$ and
two morphisms $U_\bullet(d^1_0), U_\bullet(d^1_1) : U_1 \to U_0$.
There are two morphisms $U_\bullet(s^1_0), U_\bullet(s^1_1) :
U_1 \to U_2$ and three morphisms $U_\bullet(d^2_0), 
U_\bullet(d^2_1), U_\bullet(d^2_2) : U_3 \to U_2$. And so on.
FIXME: This notation...

\smallskip\noindent
FIXME: Much more.

\begin{example}
\label{example-simplicial-products}
(1) The simplest example is the {\it constant} simplicial object with
value $X \in \text{Ob}(\mathcal{C})$. In other words, $U_n=X$ and
all maps are $\text{id}_X$. \\
(2) Suppose that $Y\to X$ is a morphism of $C$ such that all
the fibred products $Y_{/X}^nY = \times_X Y \times_X \ldots Y$ exist.
Then we set $U_n = Y^{n+1}_{/X}$, and we let $s: [n] \to [m]$
correspond to the map (on ``coordinates'') $(y_0,\ldots, y_m) 
\mapsto (y_{s(0)},\ldots, y_{s(n)})$.
\end{example}

\subsection{Goals}

\noindent
Assume that $\mathcal{C}$ is a site with the property
that the set of coverings consisting of $1$ morphism is cofinal.
Let $\mathcal{F}$ be a sheaf of abelian groups on
the site $\mathcal{C}$ which is assumed to have the property
that the set of coverings consisting of $1$ morphism is cofinal.
Choose an injective resolution $\mathcal{F} \to \mathcal{J}^\bullet$
(for example a canonical one, see 
Injectives, \autoref{injectives-subsection-injectives-sheaves}).
Let $X$ be an object of $\mathcal{C}$. We want to compute 
$R\Gamma(X, \mathcal{F}) = \Gamma(X, \mathcal{J}^\bullet)$
or at least the cohomology groups $H^j(X, \mathcal{F})$.
The idea is to construct simplicial objects $U_\bullet$ 
augmented towards $X$, so $U_\bullet \to X$, such that 
$$
R\Gamma(X, \mathcal{F}) 
= \text{Tot}(R\Gamma(U_\bullet, \mathcal{J}^\bullet))
\leqno{(*)}
$$
is a quasi-isomorphism (for any $\mathcal{F}$). On the right hand 
side this is the total complex associated to the double complex. 
(The maps are always canonical since we have the resolution over 
all of $\mathcal{C}$.)
The complex $\Gamma(U_\bullet, \mathcal{F})$ maps into the
complex on the right. We will show that for any
element $\eta \in H^j(X, \mathcal{F})$ there exists a choice
of $U_\bullet \to X$ such that $\eta$ comes from an element
in $H^j(U_\bullet, \mathcal{F})$. This is a first step and
it already allows us to define cup products for example.
The starting point is the following.

\begin{lemma}
\label{lemma-product-hypercovering}
Suppose that $\{Y \to X\}$ is a covering in the topology of
$\mathcal{C}$. Let $U_n = Y^n_{/X}$ be the simplicial
object defined in Example \ref{example-simplicial-products}.
The augmentation $U_\bullet \to X$ has the property
that $(*)$ is a quasi-isomorphism for all $\mathcal{F}$.
\end{lemma}

\begin{proof}
FIXME.
\end{proof}

\subsection{Making simplicial objects}
\label{subsection-making-simplicial}

\noindent
Suppose that $U_\bullet$ is a simplicial object of $\mathcal{C}$. Now let
$n\geq 0$ and let $V \to U_n$ be a representable morphism of 
$\mathcal{C}$. This means that the fibre products $V \times_{U_n} W$ 
exist for all morphisms $W \to U_n$.

\smallskip\noindent
For any $m$ consider the fibre product (over $U_m$)
$$
U'_m = \prod\nolimits_{\varphi \in \text{Mor}_\Delta([n],[m])}
V\times_{U_n, U_\bullet(\varphi)} U_m.
$$
By our assumption on the morphism $V \to U_n$ this fibre product
exists. For any $\psi : [m1] \to [m2]$ there is a canonical morphism
$U'_{m2} \to U'_{m1}$ coming from the map $\text{Mor}_\Delta([n],[m1])
\to \text{Mor}_\Delta([n],[m2]), \varphi \mapsto \varphi \circ \psi$,
the identity map on $V$ and the canonical map $U_\bullet(\psi) : 
U_{m2} \to U_{m1}$

\smallskip\noindent
Clearly, these data give rise to a simplicial object $U'_\bullet$ in
$\mathcal{C}$. The natural morphisms $U'_m \to U_m$ give rise to a
morphism of simplicial objects $U'_\bullet \to U_\bullet$. Note that
the morphism $U'_n \to U_n$ factors throught the morphism $V \to U_n$
by projection onto the factor corresponding to $\varphi=\text{id}_{[n]}$.
Also, note that if $\mathcal{C}$ is a site and if 
$\{V \to U_n\}$ is a covering in the site then for any $m$ it is true
that $\{U'_m \to U_m\}$ is a covering. This proves the following lemma.

\begin{lemma}
\label{lemma-construct-new-covers}
Suppose that $U_\bullet$ and $V\to U_n$ are as above such that
$\{V \to U_n\}$ is a covering for the topology on the site
$\mathcal{C}$. The morphism of simplicial objects 
$U'_\bullet \to U_\bullet$ constructed above has the following 
properties:
(1) The morphism $U'_n \to U_n$ factors trough $V \to U_n$.
(2) For any $m$ the set $\{U'_m \to U_m\}$ is a covering
in the topology of $\mathcal{C}$.
\end{lemma}

\subsection{Doubly simplicial stuff}
\label{subsection-doubly-simplicial}

\noindent
A doubly simplicial object of $\mathcal{C}$ is a functor
$U_{\bullet,\bullet} : (\Delta\times\Delta)^\circ \to \mathcal{C}$.
By subdividing we can make this into a simplicial object 
$W(U_{\bullet,\bullet})$ with the same cohomology. FIXME: Explain this.

\noindent
Suppose that $U'_\bullet \to U_\bullet$ is a morphism of simplicial
objects of $\mathcal{C}$ such that each of the morphisms $U'_n \to
U_n$ is representable. Then we can construct a doubly-simplicial
object $U'_{\bullet,\bullet}$ by setting $U'_{n,0}= U'_n$,
$$
U'_{n,1} = U'_n \times_{U_n} U'_n,
$$
etc. Compare Example \ref{example-simplicial-products}. 
Out of this object we can construct a single simplicial object
$W(U'_{\bullet,\bullet})$ as explained above. Construct the 
natural morphism of simplicial objects 
$W(U'_{\bullet,\bullet}) \to U_\bullet$.

\begin{lemma}
Suppose that every $\{U'_n \to U_n\}$ is a covering for the topology
of $\mathcal{C}$. Suppose that $\mathcal{F}$ is a sheaf on 
$\mathcal{C}$. Then there is a natural
morphism of complexes
$$
R\Gamma(U_\bullet, \mathcal{F}) \to 
R\Gamma(W(U_{\bullet,\bullet}), \mathcal{F})
$$
which is a quasi-isomorphism. FIXME: Something like this in any case.
\end{lemma}

\section{The general case}

\noindent
Mention how things work more generally, for example if $\mathcal{C}$
does not have the property that coverings consisting of a single map
are cofinal. State the theorem in the correct generality.


\smallskip\noindent
To continue reading,
\begin{enumerate}

\item visit the next section: Stacks,
\autoref{stacks-section-introduction}, or 

\item go back to the
table of contents: \url{index.html#contents}.

\end{enumerate}

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