group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
Čech cohomology is a way of computing cohomology in (infinity,1)-topos of infinity-stacks that are localized with respect to Čech covers.
Čech cohomology is more an algorithm for computing cohomology (see Čech methods) than a cohomology theory in itself.
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
where the colimit is over all acyclic fibrations $Y \stackrel{\in W \cap F}{\to} X$ 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
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
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.
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
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
that make all diagrams of the form
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
and
and
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
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
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
on $U_{i j}$
a collection of 3-morphisms
We now plug in some concrete coefficient object $n$-types for low $n$ and reproduce some concrete formulas from this.
For $G$ a group let $\mathbf{B} G$ by abuse of notation denote
then the corresponding nerve
then finally the corresponding simplicial sheaf that for each object $U$ assigns the nerve for the group $Hom(U,G)$:
The above then says that
a $\mathbf{B}G$-Čech cocycle is
a collection of $G$-valued functions
a collection of identities between $G$-valued functions
a $\mathbf{B}G$-Čech coboundary is
a collection of $G$-valued functions
a collection of identites between $G$-valued functions
Remembering that the Čech cohomology is the colimit over refinement of covers over cohomology classes defined this way, one sees the standard
Čech cohomology with coefficients in $\mathbf{B}G$ (as above) classies $G$-principal 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
One finds that a Čech-cocycle now is (see also the diagrams at group cohomology for this)
a collection of functions
a collection of functions
a collection of identities
and a collection of identities
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
this is the nonabelian cohomology classifying gerbes.
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
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.
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.
The following structure arises, when one computes Čech cohomology in this context, as shown below.
(Čech complex)
Let $\{U_i \to X\}$ be a collection of open subsets of $X$ and let $A_\bullet$ be a sheaf with coefficients in non-negatively graded chain complexes $Ch_+$.
Then define the Čech chain complex $C(U,A_\bullet)$ of $A_\bullet$ relative to $U$ by
where $U_{i_1, \cdots, i_n} = U_{i_0} \cap \cdots \cap U_{i_n}$ if all indices are pairwise different, and empty otherwise, and with differential given by
where on the right we sum over all components of $a$ obtained by discarding one of the original $(n+1)$ subscripts.
When $A_\bullet$ is not concentrated in a single degree, the above is often called the Čech hypercomplex and its cohomology is called Čech hypercohomology.
As the following derivation will show, this complex encodes a linearization of the simplicial nonabelian cocycle depicted above. 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:
In dimension 2 we have:
Here the relation on the right is the Dold-Kan correspondence relating coboundaries in the complex $A_\bullet$ to simplices .
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 $A_\bullet$:
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
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
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:
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
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
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
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
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.
For $A$ a complex of sheaves, there is a canonical morphism
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.
If $X$ is a paracompact space the canonical morphism
from Čech cohomology to abelian sheaf cohomology is an isomorphism from every $A \in Sh(X, Ch_+)$.
Recalled as theorem 1.3.13 in
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
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.
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,
Then the canonical morphism
is an isomorphism.
One considers the spectral sequence associated with the Čech double complex. Details are on page 28 of
We now list examples for abelian Čech cohomology expressed in terms of the Čech complex described above.
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
Equivalently we get the corresponding simplicial sheaves and complexes of sheaves, eg
So given a cover $\{U_i \to X\}$ a Čech cocycle is a collection
such that the Čech differential evaluated on it
vanishes:
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
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.
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
is
such that on $U_{i,j,k,l}$ we have
which in the nonabelian context would be written as
When refining the complexes of sheaves $U(1)[n]$ to the Deligne complex
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
such that for all $i, j$
and for all $i,j ,k$
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
such that for all $i, j$
and for all $i,j ,k$
and for all $i, j, k ,l$
Such cocycles classify $U(1)$-bundle gerbes with connection.
A classical reference is
A historical survey (of some aspects) is in
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
Abelian Čech cohomology is discussed in some detail in section I.3 of
A discussion of the model structure on simplicial presheaves with an eye towards the distinction between Čech and hyperlocalization is in
(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
The relation between smooth and continuous Cech cohomology is discussed in