homotopy theory, (∞,1)-category theory, homotopy type theory
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In a category $C$ with pullbacks (possibly homotopy pullbacks), given a morphism $U \to X$ in $C$ its corresponding Čech 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 :
This is the internal nerve of the internal groupoid corresponding to the kernel pair of the morphism $U \to X$.
The notion of Cech nerve makes sense in any (∞,1)-category with (∞,1)-pullbacks.
See groupoid object in an (∞,1)-category.
The cohomology theory obtained by mapping out of Čech covers instead of general hypercovers is Čech cohomology.
A groupoid object in an (infinity,1)-category that is a Čech nerve $U \to X$ exhibits $X$ as a delooping.
For $U = \coprod_i U_i$ the disjoint union of a covering sieve $\{U_i \to X\}$ with respect to a coverage, the objectwise connected components of the Čech nerve is the subfunctor corresponding to the sieve
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.
This example is more or less the way that Eduard Čech gave the original form of the construction that now carries his name. More on this can be found in the entry on Čech methods, and the discussion there of the 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 Čech 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 Čech cohomology to the geometric objects studied in algebraic geometry.