nLab Čech nerve

Definition

In category theory

Definition

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 :

$$C(U) \coloneqq \left( \cdots U \times_X U \times_X U \overset{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} U \times_X U \stackrel{\longrightarrow}{\longrightarrow} U \right) \,.$$

Remark

This is the internal nerve of the internal groupoid corresponding to the kernel pair of the morphism $U \to X$.

In $(\infty,1)$-category theory

The notion of Čech nerve makes sense in any (∞,1)-category with (∞,1)-pullbacks.

Consider a morphism $f: A \rightarrow B$ in an $(\infty,1)$-category $M$. The Cech nerve of that morphism is, by definition, the augmented simplicial object in $M$ obtained by the right Kan extension of that morphism (regarded as a functor from the obvious category into $M$) 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 $M$. The Cech nerve is essentially unique by uniqueness of adjoint $\infty-$ functors.

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

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

Example

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

$$\Pi_0 C(U) = \bigcup_i hom(-,U_i) \,.$$

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.

Remark (Historical and pedagogic note)

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.

Related entries

• Čech groupoid

• effective epimorphism

• nerve theorem

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.