Čech nerve in nLab
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Homotopy theory
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Čech nerve
Definition
In category theory
Definition
In a category CC with pullbacks (possibly homotopy pullbacks), given a morphism U→XU \to X in CC its corresponding Čech nerve C(U)C(U) is the simplicial object in CC that in degree kk is given by the (k+1)(k+1)-fold fiber product of UU over XX with itself :
C(U)≔(⋯U× XU× XU⟶⟶⟶U× XU⟶⟶U). 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) \,.
In (∞,1)(\infty,1)-category theory
The notion of Čech nerve makes sense in any (∞,1)-category with (∞,1)-pullbacks.
Consider a morphism f:A→Bf: A \rightarrow B in an (∞,1)(\infty,1)-category MM. The Cech nerve of that morphism is, by definition, the augmented simplicial object in MM obtained by the right Kan extension of that morphism (regarded as a functor from the obvious category into MM) along the inclusion of the first two objects into the augmented simplicial category. See Lurie HTT 6.1.2.11. One can show that this gives a groupoid object in MM. The Cech nerve is essentially unique by uniqueness of adjoint ∞−\infty- functors.
See groupoid object in an (∞,1)-category.
Applications and occurences
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The cohomology theory obtained by mapping out of Čech covers instead of general hypercovers is Čech cohomology.
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A groupoid object in an (infinity,1)-category that is a Čech nerve U→XU \to X exhibits XX as a delooping.
- In an infinity-stack (infinity,1)-topos every groupoid object in an (infinity,1)-category is a Čech nerve.
Examples
Example
For U=∐ iU iU = \coprod_i U_i the disjoint union of a covering sieve {U i→X}\{U_i \to X\} with respect to a coverage, the objectwise connected components of the Čech nerve is the subfunctor corresponding to the sieve
Π 0C(U)=⋃ ihom(−,U i). \Pi_0 C(U) = \bigcup_i hom(-,U_i) \,.
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.
References
An early appearance (in the form of the 0-coskeleton) is in Enlightenment 8.5(b) of
- Michael Artin, Barry Mazur, Etale Homotopy, Lecture Notes in Mathematics 100 (1969), doi.
An early appearance of the term “Čech nerve” is in
- Eric M. Friedlander, Etale Homotopy of Simplicial Schemes, Annals of Mathematics Studies 104 (1983), doi.
Last revised on May 22, 2022 at 14:42:46. See the history of this page for a list of all contributions to it.