nLab Čech cohomology

Idea

General idea

Čech cohomology is a tool, or an algorithm, which, when it applies, computes abelian sheaf cohomology (of some $X$ with coefficients in some $A$) by use of coverings and systems of coefficients on the covering and all its non-empty finite intersections. More generally, it applies also to nonabelian cohomology and may hence for instance (in degree 1) be used to compute classes of principal bundles and generally (in higher degree) those of principal ∞-bundles. Quite generally Čech cohomology is the way to express the intrinsic cohomology of (∞,1)-sheaf (∞,1)-toposes by mapping out of Čech nerves of covers.

Hence Čech cohomology is more an algorithm for computing cohomology (see also at Čech methods) than a cohomology theory in itself. In particular it applies (when it applies) to all sorts of coefficient sheaves/stacks/∞-stacks $A$.

When no further qualification of the coefficients $A$ is given then one usually has in mind by default either that $A = \mathbb{Z}$ is the group of integers (or a delooping thereof, in which case one computes ordinary cohomology by Čech methods) or, if working on a ringed space/ringed topos/structured (∞,1)-topos that $A = \mathcal{O}^\times$ is the group of units in the structure sheaf.

Notice that there are technical conditions on a site and a cover to ensure that the result of the Čech cohomology algorithm really coincides with the more abstractly defined abelian sheaf cohomology – and generally with the hom-spaces in the given (∞,1)-topos. (Technically the condition is that the Čech nerve is a cofibrant resolution in a model structure on simplicial presheaves for which the coefficient sheaf is a fibrant object). If these conditions do not apply, then the definition of Čech cohomology groups still works as such, but typically no longer qualifies as a “cohomology theory”, axiomatically. They are just some groups. Strictly speaking one should maybe not use the term “Čech cohomology” in such cases, but beware that some authors do.

More technical idea

From the discussion at abelian sheaf cohomology we know that the right derived functor definition computes the hom-set in the homotopy category of an (infinity,1)-topos $\mathbf{H}$ that may alternatively be computed as a colimit over resolutions of the domain object

$$H(X,A) \coloneqq \pi_0 \mathbf{H}(X,A) = colim_{Y \stackrel{\in W \cap F}{\longrightarrow} X} C_H(Y,A)/_{homotopy}$$

where the colimit is over all acyclic fibrations $Y \stackrel{\in W \cap F}{\longrightarrow} X$ in an appropriate model category $C_H$ that presents $\mathbf{H}$. For $\mathbf{H}$ an infinity-stack (infinity,1)-topos this $C_H$ is a model structure on simplicial presheaves and the acyclic fibrations $Y \stackrel{\in W \cap F}{\to} X$ for $X$ an ordinary space are the hypercovers.

Now, for some coefficient objects $A$ it is sufficient to take the colimit here not over all hypercovers, but just over Čech covers. The resulting formula

$$H(X,A) = colim_{Y \stackrel{\mathop{Čech} cover}{\to} X} C_H(Y,A)/_{homotopy}$$

is then called the formula for Čech cohomology on $X$ with values in $A$.

Here a Čech cover is a simplicial presheaf that arises from an ordinary covering map $U \to X$ of $X$ by another space $U$ as the corresponding Čech nerve simplicial presheaf

$$\begin{aligned} Y := \left( \cdots U \times_X U \times_X U \stackrel{\longrightarrow}{\stackrel{\longrightarrow}{\longrightarrow}} U \times_X U \stackrel{\longrightarrow}{\longrightarrow} U \right) \simeq hocolim_{[k] \in \Delta} U^{\times_X k} \end{aligned} \,.$$

See descent for simplicial presheaves for more on the manipulations involved here.

To amplify, a general hypercover would start in degree 0 with a $U$ as above, but then in the next degree would have a cover $V \to U \times_X U$ of the fiber product, and so on, each fiber product in turn being covered by another space.

If $Y$ is not simply a Čech cover but also not the most general hypercover in that this iterative choice of further covering stops in degree $n$, then one also speaks of Čech cover of level $n$ and of the corresponding cohomology formula as higher Čech cohomology. See for instance the reference by Tibor Beke below.

General (“nonabelian”) Čech cohomology

We start with describing the general “ nonabelian ” Čech cohomology (compare the terminology and remarks at cohomology and nonabelian cohomology), i.e. the plain unwrapping of the above definition, before assuming that our coefficient object is abelian and before applying the Moore complex functor that sends the following simplicial computation to the maybe more familiar one in chain complexes.

The reasoning parallels that described at descent to some extent, but is maybe still worthwhile repeating here.

So let $C$ be some site and let $SSh(C)$ the category of simplicial (pre)sheaves on $C$. Let $C(U)$ be the Čech nerve for a cover $\pi : U \to X$ of some simplicial presheaf $X$ and let $A$ be any other simplicial presheaf that serves as the coefficient object.

Using end and coend notation we compute the required hom

$$\begin{aligned} SSh(C(U), A) & \simeq SSh( \int^{[k] \in \Delta} \Delta^k \cdot C(U)_k, A ) \\ & \simeq \int_{[k] \in \Delta} SSh(\Delta^k \cdot C(U)_k, A ) \\ & \simeq \int_{[k] \in \Delta} SSet(\Delta^k , SSh(C(U)_k,A) ) \\ & \simeq \int_{[k] \in \Delta} SSet(\Delta^k , A(U \times_X \cdots \times_X U) ) \\ & \simeq \int_{[k] \in \Delta} SSet(\Delta^k , \prod_{i_0, \cdots, i_k} A(U_{i_0 i_1, \cdots ,i_k}) ) \end{aligned} \,.$$

Here in the last line we have assumed that $U = \sqcup_i U_i$ for $\{U_i \to X\}$ an open cover of some space $X$ and we abbreviate as usual with $U_{i_0, i_1, \cdots, i_k}$ the $(k+1)$-fold intersections $\cap_{0 \leq r \leq k} U_{i_r}$.

An object in this last expression – an $A$-valued cocycle on $X$ relative to $U$ – is a collection of morphisms in SSet of the form

$$\left( \array{ \vdots && \vdots \\ \Delta^2 &\stackrel{f}{\longrightarrow}& \prod_{i j k} A(U_{i j k}) \\ \Delta^1 &\stackrel{g}{\longrightarrow}& \prod_{i j}A(U_{ i j}) \\ \Delta^0 &\stackrel{a}{\longrightarrow}& \prod_i A(U_i) } \right)$$

that make all diagrams of the form

$$\array{ \vdots && \vdots \\ \uparrow && \uparrow \\ \Delta^2 &\stackrel{f}{\longrightarrow}& \prod_{i j k} A(U_{i j k}) \\ \uparrow && \uparrow \\ \Delta^1 &\stackrel{g}{\to}& \prod_{i j}A(U_{ i j}) \\ \uparrow^{\delta_{\cdot}} && \uparrow^{A(d_{C(U)})_\cdot} \\ \Delta^0 &\stackrel{a}{\to}& \prod_i A(U_i) }$$

commute. Here the vertical morphism on the right, when one traces their origin through the derivation above, are the various inclusion and restriction maps of elements of $A$ evaluated on a $k$-fold intersection to some other intersection.

For instance the three possible vertical maps at the bottom have components on the cartesian factors given by

$$A(d_1)|_{U_i} := (-)|_{U_{i j}} : A(U_i) \to A(U_{i j})$$

and

$$A(d_0)|_{U_i} := (-)|_{U_{i j}} : A(U_j) \to A(U_{i j})$$

and

$$A(s_0)|_{U_{i i}} : A(U_{i i}) \to A(U_{i }) \,.$$

This picks

• an object $a \in A(U)$

• a morphism $g : \pi_1 ^* a \to \pi_2^* a in A(U \times_X U)$

• a 2-morphism

  $$\array{ && \pi_{2}^* a \\ & {}^{\pi_{12}^* g}\nearrow &\Downarrow^{f}& \searrow^{\pi_{23}^* g} \\ \pi_{1}^* a &&\stackrel{\pi_{13}^* g}{\to}&& \pi_3^* a }$$

  in $A(U \times_X U \times_X U)$

• etc.

If we follow tradition and write

• $U = \sqcup_i U_i$;

• and $U \times_X U = \sqcup_{i j} U_{i j} := \sqcup_{i j} U_i \cap U_j$

• etc

this is

• an collection of objects $(a_i \in A(U_i))$

• a collection of morphisms $(g_{i j} : a_i|_{U_{ij}} \to a_{j}|_{U_{i j}} in A(U_{i j}))$

• a collection of 2-morphisms

  $$\left( \array{ && a_{j}|_{U_{i j k}} \\ & {}^{g_{ij}|_{U_{i j k}}}\nearrow &\Downarrow^{f_{i j k}}& \searrow^{g_{j k}|_{U_{i j k}}} \\ a_i|_{U_{i j k}} &&\stackrel{g_{i k}|_{U_{i j k}}}{\to}&& a_k|_{U_{i j k}} } \right)$$

  in $A(U_{i j k})$

• etc.

This is a Čech-cocycle on $X$ with values in $A$ relative to $U$.

A transformation/homotopy/coboundary between two such cocycles is a cylinder over these diagrams, i.e.

• a collection of morphism $h_i : a_i \to a'_i in A(U_i)$;

• a collection of 2-morphism

  $$\array{ a_i|_{U_{i j}} &\stackrel{g_{i j}}{\to}& a_j|_{U_{i j}} \\ \;\;\;\downarrow^{h_i|_{U_{i j}}} &\Downarrow^{a_{i j}}& \;\;\;\downarrow^{h_j|_{U_{i j}}} \\ a'_i|_{U_{i j}} &\stackrel{g_{i j}}{\to}& a'_j|_{U_{i j}} }$$

  on $U_{i j}$

• a collection of 3-morphisms

  $$\array{ && a_j|_{U_{i j k}} \\ & {}^{g_{ij}|_{U_{i j k}}}\nearrow &\Downarrow^{f_{i j k}}& \searrow^{g_{j k}|_{U_{i j k}}} \\ a_i|_{U_{i j k}} &&\stackrel{g_{i k}}{\to}&& a_k|_{U_{i j k}} \\ \;\;\;\downarrow^{h_i|_{U_{i j}}} &&\Downarrow^{a_{i k}}&& \;\;\;\downarrow^{h_j|_{U_{i j}}} \\ a'_i|_{U_{i j k}} &&\stackrel{g_{i k}}{\to}&& a'_k|_{U_{i j k}} } \;\; \;\; \;\; \;\; \;\; \stackrel{\rho_{i j k}}{\Rightarrow} \;\; \;\; \;\; \;\; \;\; \array{ && a_j|_{U_{i j k}} \\ & {}^{g_{ij}|_{U_{i j k}}}\nearrow &\Downarrow^{a_{i j}\cdot a_{j k}|_{U_{i j k}}}& \searrow^{g_{j k}|_{U_{i j k}}} \\ a_i|_{U_{i j k}} && a'_j|_{U_{i j k}} && a_k|_{U_{i j k}} \\ \downarrow^{h_{i}|_{U_{i j k}}} & {}^{g'_{i j}|_{U_{i j k}}}\nearrow & \Downarrow^{f'_{ijk}} & \searrow^{g'_{j k}|_{U_{i j k}}} & \downarrow^{h_{k}|_{U_{i j k}}} \\ a'_i|_{U_{i j k}} &&\stackrel{g'_{i k}|_{U_{i j k}}}{\to}&& a'_k|_{U_{i j k}} }$$

We now plug in some concrete coefficient object $n$-types for low $n$ and reproduce some concrete formulas from this.

Nonabelian 1-cocycles: principal bundles

For $G$ a group let $\mathbf{B} G$ by abuse of notation denote

• first of all the corresponding

deloopedgroupoid,

• then the corresponding nerve

• then finally the corresponding simplicial sheaf that for each object $U$ assigns the nerve for the group $Hom(U,G)$:

  $$\mathbf{B}G : U \mapsto \left\{ \array{ \mathbf{B}G(U)_0 = Hom(U,{*}) = {*} \\ \mathbf{B}G(U)_1 = Hom(U,G) \\ \mathbf{B}G(U)_2 = Hom(U,G)\times Hom(U,G) \\ \vdots } \right.$$

The above then says that

• a $\mathbf{B}G$-Čech cocycle is

  • a collection of $G$-valued functions

    $$(g_{ij} \in Hom(U_{i j}, G))$$

  • a collection of identities between $G$-valued functions

    $$g_{ij}|_{U_{i j k }} \cdot g_{j k}|_{U_{i j k }} = g_{i k}|_{U_{i j k }}$$

• a $\mathbf{B}G$-Čech coboundary is

  • a collection of $G$-valued functions

    $$(h_i \in Hom(U_i , G))$$

  • a collection of identites between $G$-valued functions

    $$g_{ij} \cdot h_j|_{U_{i j}} = h_i|_{U_{i j}} \cdot g'_{j k} \,.$$

Remembering that the Čech cohomology is the colimit over refinement of covers over cohomology classes defined this way, one sees the standard

Theorem

Čech cohomology with coefficients in $\mathbf{B}G$ (as above) classies $G$-principal bundles

$$colim_{U \stackrel{\mathop{Čech} cover}{\to} X} SSh(C(U), \mathbf{B}G)/_{homtopy} \simeq G Bund(X)/_\sim$$

Nonabelian 2-cocycles: gerbes and principal 2-bundles

In one degree higher the general homotopy 2-type coefficient object is modeled using a strict 2-group $H$ coming from a crossed module $G_2 \stackrel{\delta}{\to} G_1$.

The nerve of the corresponding delooped 2-groupoid, regarded immediately as a simplicial sheaf starts out like

$$\mathbf{B}(G_2 \to G_1) : U \mapsto \left\{ \array{ \mathbf{B}(G_2 \to G_1)(U)_{0} = Hom(U, G_1) \\ \mathbf{B}(G_2 \to G_1)(U)_{1} = Hom(U, G_1) \times Hom(U, G_1) \times Hom(U, G_2) \\ \vdots } \right. \,.$$

One finds that a Čech-cocycle now is (see also the diagrams at group cohomology for this)

• a collection of functions

  $$(g_{i j} \in Hom(U_{i j}, G_1))$$

• a collection of functions

  $$(f_{i j k }\in Hom(U_{i j k}, G_2) )$$

• a collection of identities

  $$\delta(f_{i j k}|_{U_{i j k}}) g_{i j}|_{U_{i j k}} g_{j k}|_{U_{i j k}} = g_{i k}|_{U_{i j k}}$$

• and a collection of identities

  $$f_{i k l}|_{U_{i j k l}} \cdot f_{i j k}|_{U_{i j k l}} = f_{i j l}|_{U_{i j k l}} \cdot \alpha(g_{i j}|_{U_{i j k l}})(f_{j k l}|_{U_{i j k l}}),$$

where $\alpha:G_1\to Aut(G_2)$ is the homomorphism associated with the crossed-module description of the 2-group.

This is a nonabelian Čech 2-cocycle.

Reading off the formulas for the coboundaries is left as an excercise for the reader.

For the specical case the

$$(G_2 \to G_1) = (G \to Aut(G))$$

this is the nonabelian cohomology classifying

gerbes.

Abelian Čech cohomology

In much of the literature Čech cohomology denotes exclusively the abelian case, which we now describe.

In the special case that the coefficient simplicial presheaf $A$ is in the image of the nerve operation

$$N : Ch_+ \to sAb \to SSet$$

on chain complexes with values in simplicial sets that happen to be abelian simplicial groups, the computation of Čech cocycles may be entirely pulled back to the world of homological algebra by making use of the Dold-Kan correspondence that provides an adjoint equivalence between simplicial sets with values in abelian groups and non-negatively graded chain complexes.

Definition

We discuss the double complex which gives the traditional definition of Čech cohomology with coefficients in sheaves of abelian groups.

Let

• $X$ be a topological space;

• $A_\bullet = (\cdots \stackrel{\partial}{\to} A_2 \stackrel{\partial}{\to} A_1 \stackrel{\partial}{\to} A_0)$ be a chain complex of abelian sheaves on $X$ (i.e. on the category of open subsets $S \colon Op(X)$ of $X$, regarded as a site with the usal covering Grothendieck topology);

• $\{U_i \to X\}$ be an open cover of $X$.

We write

$$U_{i_0,\cdots i_k} \coloneqq U_{i_0} \underset{X}{\times} U_{i_1} \underset{X}{\times} \cdots \underset{X}{\times} U_{i_k}$$

for the $k$-fold intersections of these open subsets. Given an inclusion of open subsets $U_1 \to U_2$ and given any group element $a\in A_\bullet(U_2)$ we write $a|_{U_1}$ for its restriction along this map.

More generally, we may allow $S$ to be any site. For simplicity of the following formulas assume that $S$ has finite products (which in the case that $S$ is a category of open subsets are the fiber products above.) Then $A_\bullet$ is a chain complex of abelian sheaves on that site.

Definition

(Čech complex)

The Čech cochain complex $C^\bullet((X,\{U_i\}),A_\bullet)$ of $X$ with respect to the cover $\{U_i \to X\}$ and with coefficients in $A_\bullet$ is in degree $k \in \mathbb{N}$ given by the abelian group

$$C^k((X,\{U_i\}),A_\bullet) \coloneqq \oplus_{{l,n} \atop {k = n-l}} \oplus_{i_0, i_1, \cdots, i_n} A_l(U_{i_0, \cdots, i_n})$$

which is the direct sum of the values of $A_\bullet$ on the given intersections as indicated; and whose differential

$$d \colon C^{k}((X,\{U_i\}),A_\bullet) \longrightarrow C^{k+1}((X,\{U_i\}),A_\bullet)$$

is defined componentwise (see at matrix calculus for conventions on maps between direct sums) by

$$\begin{aligned} (d a)_{i_0, \cdots, i_{k+1}} & \coloneqq (\partial_A a + (-1)^k \delta a)_{i_0, \cdots, i_{k+1}} \\ & \coloneqq \partial_A a_{i_0, \cdots, i_{k+1}} + (-1)^k \sum_{0 \leq j \leq k+1} (-1)^{j} a_{i_0, \cdots, i_{j-1}, i_{j+1}, \cdots, i_{k+1}} |_{U_{i_0, \cdots, i_{k+1}}} \end{aligned}$$

where on the right the sum is over all components of $a$ obtained via the canonical restrictions obtained by discarding one of the original $(k+1)$ subscripts.

The Čech cohomology groups of $X$ with coefficients in $A_\bullet$ relative to the given cover are the chain homology groups of the Čech complex

$$H_{Cech}^k((X,\{U_i\}), A_\bullet) \coloneqq H^k(C^\bullet((X,\{U_i\}),A_\bullet)) \,.$$

The Čech cohomology groups as such are the colimit (“directed limit”) of these groups over refinements of covers

$$H^k_{Cech}(X, A_\bullet) \coloneqq \underset{\longrightarrow}{\lim}_{\{U_i \to X\}} H_{Cech}^k((X,\{U_i\}), A_\bullet) \,.$$

Remark

Often Čech cohomology is considered for the case that $A_\bullet$ is concentrated in a single degree, in which case the first term in the sum defining the differential in def. disappears. When $A_\bullet$ is not concentrated in a single degree, then for emphasis and following terminology of hypercohomology one may speak of the Čech hypercomplex computing Čech hypercohomology.

Remark

The Čech chain complex in def. is the total complex of the double complex whose vertical differential is that of $A_\bullet$ and whose horizontal differential is the Čech differential $\delta$ given by alternating sums over restrictions along patch inclusions

$$\array{ \vdots && \vdots \\ \downarrow^{\mathrlap{\partial_A}} && \downarrow^{\mathrlap{\partial_A}} \\ \oplus_i A_1(U_i) &\stackrel{\delta}{\longrightarrow}& \oplus_{i_1, i_2} A_1(U_{i_1, i_2}) &\stackrel{\delta}{\longrightarrow}& \cdots \\ \downarrow^{\mathrlap{\partial_A}} && \downarrow^{\mathrlap{\partial_A}} \\ \oplus_i A_0(U_i) &\stackrel{\delta}{\longrightarrow}& \oplus_{i_1, i_2} A_0(U_{i_1, i_2}) &\stackrel{\delta}{\longrightarrow}& \cdots }$$

Relation to nonabelian Čech cohomology

We discuss how the abelian Čech complex of def. arises as a special case of the simplicial nonabelian cocycle complex of general nonabelian Čech cohomology above, under the Dold-Kan correspondence.

Remark

It may be helpful to keep the following pictures in mind to match the signs to the orientations of simplices.

In dimension one we have:

$$(0 \to 1) \;\;\; \mapsto \;\;\; (a_0 \stackrel{a_{0 1}}{\to} a_1) \;\;\; \leftrightarrow \;\;\; (a_1 = a_0 + d_A a_{01})$$

In dimension 2 we have:

$$\array{ && 1 \\ & \nearrow &\Downarrow& \searrow \\ 0 &&\to&& 2 } \;\;\; \mapsto \array{ && a_1 \\ & {}^{a_{0 1}}\nearrow &\Downarrow^{a_{0 1 2}} & \searrow^{a_{1 2}} \\ a_0 &&\stackrel{a_{0 2}}{\to}&& a_2 } \;\;\; \leftrightarrow (a_{0 2} = a_{0 1} + a_{1 2} + d a_{0 1 2}) \,.$$

Here the relation on the right is the Dold-Kan correspondence relating coboundaries in the complex $A_\bullet$ to simplices .

Remark

The following derivation of abelian Čech cohomology from nonabelian Čech cohomology restricted to nerves of chain complexes needs of the Dold-Kan correspondence only that it is an adjunction, not that it is an adjoint equivalence.

Proposition (recovering abelian Čech cohomology)

Let $A_\bullet$ be a sheaf with values in $Ch_+$ and write $N(A_\bullet)$ for the corresponding simplicial sheaf under the nerve operation of the Dold-Kan correspondence.

Then the general (nonabelian) Čech cohomology of $N(A_\bullet)$ as defined above coincides with the cohomology of the Čech complex of $A_\bullet$:

$$H(X, N(A_\bullet)) \simeq colim_U H_0(C(U,A_\bullet)) \,.$$

Proof

The underlying idea is to use the adjunction of the Dold-Kan correspondence to move the nerve operation $N(A_\bullet)$ on the right to the free abelian chain complex operation $C_bullet(F(C(U)))$ on the Čech cover simplicial sheaf $C(U)$ and then use the Yoneda lemma to evaluate $A_\bullet$ on $N_\bullet(F(C(U)))$. The result is the Čech complex of $A_\bullet$. Cocycles and homotopies/coboundaries are in bijection on both sides.

Spelled out in full detail this looks a bit more lengthy, but is nothing but this simple idea.

Before starting the computation notice the following observation on the image of the Čech cover $C(U)$ under the Dold-Kan correspondence:

For $Y = C(U)$ a Čech cover on a sieve $\{U_i \to X\}_{i \in I}$, and for $W$ any test object, the non-degenerate $n$-simplices in $C(U)(W)$ are the

$$U_{i_0, i_1, \cdots, i_n}(W)$$

with all indices pairwise different, $i_k \neq i_l$.

Accordingly the free abelian simplicial group-valued sheaf $F(C(U))$ is for each $W$ and $n$ the free abelian group generated from these $U_{i_0, i_1, \cdots, i_n}(W)$ with pairwise distinct elements, i.e. that given by formal interger combinations of these element.

In turn, the normalized Moore complex sheaf

$$N_\bullet(F(C(U)))$$

assigns to a test domain $W$ the complex that in each degree is the free abelian group on these elements.

With this in hand, we compute the set of $N(A_\bullet)$-valued cocycles on $U$ as follows:

$$\begin{aligned} Hom_{SPSh}(C(U), N(A_\bullet)) &\simeq \int_W Hom_{SSet}( C(U)(W) , N(A_\bullet)(W) ) \\ & \simeq \int_W Hom_{SSet}( C(U)(W) , N(A(W)_\bullet) ) \\ & \simeq \int_W Hom_{SAb}( F(C(U)(W)) , N(A(W)_\bullet) ) \\ & \simeq \int_W Hom_{Ch_+}( N_\bullet(F(C(U)(W))) , A(W)_\bullet ) \\ & \simeq \int_W \int_{n} Hom_{Ab}( N_n(F(C(U)(W)), A(W)_n ) \\ & \simeq \int_{n} \int_W Hom_{Ab}( N_n(F(C(U)(W)), A(W)_n ) \\ & \simeq \int_{n} \int_W Hom_{Set}( \coprod_{i_0, i_1, \cdots, i_k} U_{i_0,i_1, \dots, i_n}(W) , A(W)_n ) ) \\ & \simeq \int_{n} \prod_{i_0, i_1, \cdots, i_n} A(U_{i_0,i_1, \dots, i_n})_n \\ & = ker d_{C(U, A_\bullet)} \subset C(U,A_\bullet)_0 \end{aligned} \,.$$

Here

• The first step is the definition of morphism of presheaves using the end notation;

• the second step is the definition of the nerve of of chain complexes applied to chain complex valued sheaves;

• the third step uses that the free simplicial abelian group functor is left adjoint to the forgetful one that remembers the underlying simplicial set;

• the fourth step then uses the Dold-Kan correspondence, or actually just that the normalized Moore complex functor is left adjoint to the nerve of chain complexes;

• the fifth step expresses the set of morphisms of chain complexes as an end (being itself a natural transformation);

• the sixth step uses the Fubini theorem of enriched category theory to commute the two ends;

• the seventh step uses that the chain complex in the left argument is generated freely on elements of a set to rewrite the hom of abelian groups into one of sets;

• this finally allows to apply the Yoneda lemma in step eight

• and in step nine, by inspection, one notices that the result thus obtained is the set of 0-cycles in the Čech complex $C(U,A_\bullet)$ as previously defined.

An entirely analogous argument shows that dividing out homotopies is respected.

One starts by observing that the cohomology coboundaries are given by homotopies in the hom-simplicial set, i.e. by

$$Hom_{SPSh}(C(U) \times \Delta^1, N(A_\bullet)) \,.$$

With this one goes in the above computation. After applying the Dold-Kan adjunction we now have on the left in the integrand the term

$$N_\bullet(F(C(U)(W) \times \Delta^1)) \,.$$

Using first that the free simplicial group functor is monoidal and then that the normalized Moore complex functor is lax monoidal (as described at Dold-Kan correspondence)

what to do about laxness versus pseudoness??

we get a morphism to that from

$$N_\bullet(F(C(U)(W))) \otimes N_\bullet(F(\Delta^1)) \,.$$

Using that the normalized Moore complex is isomorphic to the Moore complex divided by the part generated by degenerate cells, on the right we identify the interval object complex

$$( \cdots \to 0 \to I \stackrel{1 \oplus -1}{\to} I \oplus I) \,.$$

To see that one just needs to observe that the normalized Moore complex of the 1-simplex serves as an interval object in chain complexes.

Relation to abelian sheaf cohomology

For $A$ a complex of sheaves, there is a canonical morphism

$$\check{H}(X,A) \to H(X,A)$$

from the Čech cohomology to the full (hypercompleted) cohomology, which is abelian sheaf cohomology in the case that $A$ is in the image of the Dold-Kan map from chain complexes. Using the description of abelian sheaf cohomology in terms of morphisms out of hypercovers described at the beginning of this entry, this morphism is the obvious one coming from the inclusion of Čech covers into all hypercovers.

Theorem

If $X$ is a paracompact space the canonical morphism

$$\check{H}(X,A) \to H(X,A)$$

from Čech cohomology to abelian sheaf cohomology is an isomorphism from every $A \in Sh(X, Ch_+)$.

Proof

Recalled as theorem 1.3.13 in

• Brylinski, Loop spaces, characteristic classes and geometric quantization

When $X$ is not paracompact, we still have the following condition under which Čech cohomology computes abelian sheaf cohomology-.

For $A$ a complex of sheaves, there is a canonical morphism

$$H_0(C(U,A)_\bullet) \to H(X,A)$$

from the cohomology of the Čech complex with respect to a cover $U$ with coefficients in $A$ to the abelian sheaf cohomology of $X$ with values in $A$. Using the description of abelian sheaf cohomology in terms of morphisms out of hypercovers described at the beginning of this entry, this morphism is the obvious one coming from the inclusion of Čech covers into all hypercovers.

Theorem

Let $A$ be a complex of sheaves on $X$ concentrated in a single degree $p \geq 0$, and let $\{U_i \to X\}$ be a cover such that the $A$-cohomology of all intersections vanishes,

$$\forall U_{i_0 , \cdots, i_k} H(U_{i_0, \cdots, i_k},A) = 0$$

Then the canonical morphism

$$H_0(C(U,A)_\bullet) \to H(X,A)$$

is an isomorphism.

Proof

One considers the spectral sequence associated with the Čech double complex. Details are on page 28 of

• Brylinski, Loop spaces, characteristic classes and geometric quantization

Examples

We now list examples for abelian Čech cohomology expressed in terms of the Čech complex described above.

Line bundles

Let $G = U(1)$ be the circle group, an abelian group. The nerve $N(\mathbf{B}G)$ of its delooping $\mathbf{B}G$ is the bar construction of $G$. This is equivalently the nerve of the chain complex

$$U(1)[1] := ( \cdots \to 0 \to U(1) \to 0 ) \,.$$

Equivalently we get the corresponding simplicial sheaves and complexes of sheaves, eg

$$U(1)[1] : W \mapsto ( \cdots \to 0 \to Hom(W,U(1)) \to 0 ) \,.$$

So given a cover $\{U_i \to X\}$ a Čech cocycle is a collection

$$c = ( \{g_{i j} \in U(1)[1](U_{i j})\}|_{i,j})$$

such that the Čech differential evaluated on it

$$\begin{aligned} (d_{C(U,U(1)[1])} c)_{i j k} &= g_{j k}|_{U_{i j k}} - g_{i k}|_{U_{i j k}} + g_{i j}|_{U_{i j k}} ) \end{aligned}$$

vanishes:

$$g_{i j}|_{U_{i j k}} + g_{j k}|_{U_{i j k}} = g_{i k}|_{U_{i j k}}$$

for all $i, j, k$.

Here now we write plus signs for the operation in the abelian group $U(1)$ which above we have written by juxtaposition or using a dot. So up to notation for the group operation this is

$$g_{i j} g_{j k} = g_{i k}$$

on $U_{i j k}$ as before in the nonabelian case of Čech cocycles for $U(1)$-principal bundles.

Here and from now on we shall notationally suppress the restriction maps $(-)|_{U_{i_0, i_1, \cdots, i_n}}$ as they are unambiguously obviuous in every case.

Line bundle gerbes

Similarly by shifting $U(1)$ ever higher in chain degree, one finds Čech cocycles for bundle gerbes, bundle 2-gerbes, etc.

A Čech cocycle for

$$U(1)[2] := ( \cdots \to 0 \to U(1) \to 0 \to 0 ) \,.$$

is

$$c = ( \{g_{i j k} \in U(1)[2](U_{i j k})\}|_{i,j, k})$$

such that on $U_{i,j,k,l}$ we have

$$g_{i j k} - g_{i j l} + g_{i k l} - g_{j k l} = 0$$

which in the nonabelian context would be written as

$$g_{i j k} g_{i k l} = g_{i j l} g_{j k l} \,.$$

Čech-Deligne cohomology

When refining the complexes of sheaves $U(1)[n]$ to the Deligne complex

$$U(1)[n] \hookrightarrow \mathbb{Z}(n+1)_D^\infty$$

and then evaluating Čech cohomology with coefficients in the Deligne complex, we obtain the formulas for Čech-Deligne cohomology.

• For $n = 1$ a cocycle is a collection

  $$( A_i \in \Omega^1(U_i), g_{i j} \in C^\infty(U_{i j}, U(1)) )$$

  such that for all $i, j$

  $$d log g_{i j} + A_i - A_j - = 0$$

  and for all $i,j ,k$

  $$g_{i j} g_{j k} = g_{i k} \,.$$

  Such cocycles classify $U(1)$-principal bundles with connection.

  These $n=1$ Čech-Deligne cocycles appear naturally in the study of the electromagnetic field.

• For $n = 2$ a cocycle is a collection

  $$( B_i \in \Omega^2(U_i), A_{i j} \in \Omega^1(U_{i j}), g_{i j k} \in C^\infty(U_{i j k}, U(1)) )$$

  such that for all $i, j$

  $$d A_{i j} + B_i - B_j = 0$$

  and for all $i,j ,k$

  $$A_{i j} - A_{i k} + A_{j k} + d log g_{i j k} = 0$$

  and for all $i, j, k ,l$

  $$g_{i j k} g_{i k l} = g_{i j l} g_{j k l} \,.$$

  Such cocycles classify $U(1)$-bundle gerbes with connection.

Related entries

• For discussion of the isomorphism between isomorphism classes of topological vector bundles and Čech cohomology on topological spaces with coefficients in continuous functions with values in the general linear group see there.

References

Classical accounts (in the generality of nonabelian cohomology):

• Alexander Grothendieck, A General Theory of Fibre Spaces With Structure Sheaf, University of Kansas, Report No. 4 (1955, 1958) [pdf, pdf]

• Jean Frenkel, Cohomologie à valeurs dans un faisceau non abélien, C. R. Acad. Se., t. 240 (1955) 2368-2370

• Jean Frenkel, Cohomologie non abélienne et espaces fibrés, Bulletin de la Société Mathématique de France, 85 (1957) 135-220 [numdam:BSMF_1957__85__135_0]

• Roger Godement, Topologie algébrique et theorie des faisceaux, Actualités Sci. Ind. 1252, Hermann, Paris (1958) [webpage, pdf]

Textbook account:

• Torsten Wedhorn, Torsors and Non-abelian Čech Cohomology, chapter 7 in: Manifolds, Sheaves, and Cohomology, Springer (2016) [doi:10.1007/978-3-658-10633-1]

A historical survey (of some aspects) is in

• David Edwards, Harold M. Hastings, Cech Theory: Its past, present, and future, Rocky Mountain J. Math. Volume 10, Number 3 (1980), 429-468. (Euclid)

A motivational introduction from within complex analytic geometry is in

• Michael Weiss, The Search for a Global Primitive – Cech Cohomology with Coefficients in a Sheaf (pdf)

A discussion of Čech cohomology in the wider context of cohomology particularly realized in terms of the model structure on simplicial presheaves and with an emphasis on the shades of notions between Čech cover and hypercover is

• Tibor Beke, Higher Čech theory (journal, pdf)

Abelian Čech cohomology is discussed in some detail in section I.3 of

• Jean-Luc Brylinski, Loop spaces, characteristic classes and geometric quantization

A discussion of the model structure on simplicial presheaves with an eye towards the distinction between Čech and hyperlocalization is in

• Daniel Dugger, Sheaves and Homotopy theory (web, pdf)

(But beware that, while providing useful insights, these are unfinished abandoned notes that seem to have some gaps right at this point.)

A long list of reasons why higher Čech cohomology might is after all better behaved than its hypercompletion where a cycle is with respect to an arbitrary hypercover is in section 6.5.4, Descent versus Hyperdescent of

• Jacob Lurie, Higher Topos Theory

The relation between smooth and continuous Čech cohomology is discussed in

• Christoph Müller, Christoph Wockel, Equivalences of smooth and continuous principal bundles with infinite-dimensional structure groups, Advances in Geometry. Volume 9, Issue 4, Pages 605–626 (2009)