nLab
Čech nerve

Definition

Quite generally, 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 $k$-fold fiber product of $U$ over $X$ with itself :

$$C(U):=(\cdots U{\times }_{X}U{\times }_{X}U\overrightarrow{\rightrightarrows }U{\times }_{X}U\rightrightarrows U).$$C(U) := \left( \cdots U \times_X U \times_X U \overset{\to}\rightrightarrows U \times_X U \rightrightarrows U \right) \,.

This mainly plays a role in the general context of the model structure on simplicial presheaves, where Čech covers are special cases of hypercovers.

Applications and occurences

• 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.

  • In an infinity-stack (infinity,1)-topos every groupoid object in an (infinity,1)-category is a Čech nerve.

Examples

• For $U={\coprod }_i{U}_i$ the disjoint union of over 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

  $${\Pi }_{0}C(U)=\bigcup _i\mathrm{hom}(-,U_i).$$\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.