group cohomology, nonabelian group cohomology, Lie group cohomology
Hochschild cohomology, cyclic cohomology?
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
From the nPOV, a cohomology set $H(X,A)$ of cohomology classes
with coefficients in an object $A$
in an (∞,1)-topos $\mathbf{H}$
is the connected components/decategorification of
The objects in this cocycle space ∞-groupoid $\mathbf{H}(X,A)$ are called cocycles; the morphisms are called coboundaries.
For suitable choices of $\mathbf{H}$ this applies notably to cocycles in
general abelian sheaf cohomology
general nonabelian cohomology
and in particular to
It also encompasses variations such as cocycles in
For a detailed discussion of this and how it relates to various familiar realizations of cocycles, see cohomology and the links provided there.
The objects in the hom ∞-groupoid $\mathbf{H}(X,A)$ are often expressed in terms of various 1-categorical models for $\mathbf{H}$, such as a homotopical category $C$ equipped with
the extra structure of a calculus of fractions,
or with the structure of a category of fibrant objects
or even with the full structure of a model category.
In all of these cases, cocycles $c$ on $X$ with coefficients in $A$ may be modeled by spans of the form
in the ordinary category $C$, where the morphism on the left is taken from a special class of morphisms (for instance from the class of acyclic fibrations in the case that $C$ is a category of fibrant objects). In each case the relevant hom-set in the homotopy category $H(X,A) = \pi_0 \mathbf{H}(X,A)$ is given by the collection of cocycles module an equivalence relation given by coboundaries. In formulas
For details see the respective discussion at homotopy category, calculus of fractions and category of fibrant objects.
In the existing literature on localizations, the spans $X \stackrel{\in FW}{\leftarrow} \tilde X \to X$ are often not called by a dedicated special term. On the other hand, in the existing literature that explicitly uses the term “cocycle”, often more pedestrian definitions are used and it is not made explicit that morphisms in a homotopy category are being represented.
An notable exception to this is the article
that makes both the abstract concept and the terminology of cocycles explicit and manifest. The author is mainly motivated from the model structure on simplicial presheaves and its variants, which in particular models cocycles and cohomology of abelian sheaf cohomology. But more generally it models nonabelian cohomology. Notably when the underlying space is the point, it models ordinary chain cohomology as well as group cohomology and nonabelian group cohomology.
Notice that this article chooses to work with the full structure of a model category but presents constructions for cocycles entirely analogous to and in fact inspired by those used in a category of fibrant objects or in one equipped with a calculus of fractions. The author emphasizes that he can give a definition where the left leg of the cocycle spans are not required to be acyclic fibrations, but can be any weak equivalences. But all this is just a technical question of how exactly to model a cocycle, not a question of principle of concept. For instance in this context every cocycle defined with respect to a weak equivalence over its domain is cohomologous to one defined with respect to an acyclic fibration over its domain.
In the special case that the category $C$ is Cat equipped with the folk model structure on Cat, cocycles out of acyclic fibrations – which are k-surjective functors for all $k$ in this case – have been considered in
There they are called anafunctors.
One could take this as a suggestion to find a dedicated term for spans as above and call generally such a span an anamorphism. An anamorphism would be effectively the same as a cocycle, but the term morphism in it would amplify the nature of cocycles as morphisms.
So
would correspond to
The archetypical example of a notion of cocycles is that of chain cohomology:
a non-negatively graded chain complex $V_\bullet = (V_0 \stackrel{\partial_V}{\leftarrow} V_1 \stackrel{\partial_V}{\leftarrow} V_2 \stackrel{}{\leftarrow} \cdots)$ is given. An element $v \in V_n$ is a chain. A linear dual $\omega : V_n \to k$ on its elements is a cochain. The cochains arrange into the cochain complex $V^\bullet = (V^0 \stackrel{d_V}{\to} V^1 \to \stackrel{d_V}{\to} V^2 \to \cdots) := (V_0^* \stackrel{(\partial_V)^*}{\to} V_1^* \to \stackrel{(\partial_V)^*}{\to} V_2^* \to \cdots)$
A cochain $\omega \in V^n$ is a cocycle if it is closed with respect to the differential $d_V$ in that
One sees that such cocycles are in bijection to morphisms of chain complexes
where on the right we have the Eilenberg-MacLane object of the ground field $k$, which is the chain complex trivial everywhere except in degree $n$, where it is $k$:
By definition such a morphism is a collection of morphisms $\omega_r : V_r \to (\mathbf{B}^n k)_r$, of which by definition only $\omega_n \in V_n^* = V^n$ can be nontrivial.
For the collection of these maps to be a morphism of chain complexes they have to make all squares in sight commute. The only nontrivial one in this case is the one
Its commutativity means in formulas that
which is the cocycle condition from above.
In most cases the morphism $\omega : V_\bullet \to \mathbf{B}^n k$ defined this way is already a morphism in the relevant (∞,1)-category $\mathbf{H}_{Ch_\bullet}$ of chain complexes: this is modeled for instance by the projective model structure on chain complexes. In this every object is fibrant, and the cofibrant objects are those consisting of projective $k$-modules. If we assume that all our modules are projective (for instance in the archetypical case that our modules are simply vector spaces), then $\omega : V_\bullet \to \mathbf{B}^n k$ is a cocycle in $\mathbf{H}_{Ch_\bullet}$ from the above abstract nonsense point of view. For its cohomology class we may write
One says for $\omega \in \mathbf{H}(X,A)$ a cocycle, that the object classified by the cocycle is its homotopy fiber $P \to X$ regarded as an object in the overcategory over $X$.
This homotopy fiber may be thought of as the internal principal ∞-bundle in $\mathbf{H}$ with classifying map $\omega$.
$H_n = Z_n/B_n$ | (chain-)homology | (cochain-)cohomology | $H^n = Z^n/B^n$ |
---|---|---|---|
$C_n$ | chain | cochain | $C^n$ |
$Z_n \subset C_n$ | cycle | cocycle | $Z^n \subset C^n$ |
$B_n \subset C_n$ | boundary | coboundary | $B^n \subset C^n$ |