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\begin{document}

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\section*{Čech nerve}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory}

[[!include homotopy - contents]]

\hypertarget{ech_nerve}{}\section*{{ech nerve}}\label{ech_nerve}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{in_category_theory}{In category theory}\dotfill \pageref*{in_category_theory} \linebreak
\noindent\hyperlink{in_category_theory_2}{In $(\infty,1)$-category theory}\dotfill \pageref*{in_category_theory_2} \linebreak
\noindent\hyperlink{applications_and_occurences}{Applications and occurences}\dotfill \pageref*{applications_and_occurences} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{in_category_theory}{}\subsubsection*{{In category theory}}\label{in_category_theory}

\begin{defn}
\label{}\hypertarget{}{}
In a [[category]] $C$ with [[pullbacks]] (possibly [[homotopy pullbacks]]), given a [[morphism]] $U \to X$ in $C$ its corresponding \textbf{ech nerve} $C(U)$ is the [[simplicial object]] in $C$ that in degree $k$ is given by the $(k+1)$-fold [[fiber product]] of $U$ over $X$ with itself :

\begin{displaymath}
C(U) \coloneqq
  \left(
    \cdots
    U \times_X U \times_X U 
   \overset{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} U \times_X U \stackrel{\longrightarrow}{\longrightarrow} U
  \right)
  \,.
\end{displaymath}
\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
This is the [[internal nerve]] of the [[internal groupoid]] corresponding to the [[kernel pair]] of the morphism $U \to X$.

\end{remark}
\hypertarget{in_category_theory_2}{}\subsubsection*{{In $(\infty,1)$-category theory}}\label{in_category_theory_2}

The notion of Cech nerve makes sense in any [[(∞,1)-category]] with [[limit in a quasi-category|(∞,1)-pullbacks]].

See [[groupoid object in an (∞,1)-category]].

\hypertarget{applications_and_occurences}{}\subsection*{{Applications and occurences}}\label{applications_and_occurences}

\begin{itemize}%
\item The [[cohomology]] theory obtained by mapping out of [[?ech covers]] instead of general [[hypercovers]] is [[?ech cohomology]].


\item A [[groupoid object in an (infinity,1)-category]] that is a ech nerve $U \to X$ exhibits $X$ as a [[delooping]].

\begin{itemize}%
\item In an [[infinity-stack]] [[(infinity,1)-topos]] every [[groupoid object in an (infinity,1)-category]] is a ech nerve.

\end{itemize}


\end{itemize}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{example}
\label{FromACover}\hypertarget{FromACover}{}
For $U = \coprod_i U_i$ the [[disjoint union]] of a [[covering]] [[sieve]] $\{U_i \to X\}$ with respect to a [[coverage]], the objectwise [[simplicial homotopy group|connected components]] of the ech nerve is the [[subfunctor]] corresponding to the [[sieve]]

\begin{displaymath}
\Pi_0 C(U) = \bigcup_i hom(-,U_i)
  \,.
\end{displaymath}
This is described in more detail in the section ``Interpretation in terms of higher descent and codescent'' at [[sieve]]. This example is important in understanding the construction of the [[etale homotopy type]] of a scheme or more generally of objects in certain types of topos.

\end{example}
\begin{remark}
\label{}\hypertarget{}{}
This example is more or less the way that [[Eduard ?ech]] gave the original form of the construction that now carries his name. More on this can be found in the entry on [[ech methods]], and the discussion there of the \textbf{nerve of an open cover}. For the case of triangulable spaces (polyhedra), for the cover by open stars of vertices of a triangulation, one retrieves the simplicial complex used to triangulate the space. This is one of the strong roots of the modern theory of cohomology as nerves of open covers can be seen as analogues of triangulations, and then [[?ech cohomology]] is seen to extend simplicial cohomology to spaces that are locally nice. That is one of the first steps on the long route to [[Grothendieck]]`s definition of topos as a generalisation of space, expressly so as to define a cohomology further extending ech cohomology to the geometric objects studied in algebraic geometry.

\end{remark}
\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[?ech groupoid]]


\item [[effective epimorphism]]


\item [[nerve theorem]]



\end{itemize}
[[!redirects Cech nerve]] [[!redirects Cech-nerve]] [[!redirects ?ech nerve]] [[!redirects ?ech-nerve]] [[!redirects ?ech nerve]] [[!redirects ?ech-nerve]] [[!redirects Cech complex]]

[[!redirects ?ech nerves]] [[!redirects Cech nerves]]



\end{document}