Contents

Idea

There are various different-looking definitions of the general notion of cohomology in different contexts, some familiar, some more exotic.

The claim is all these notions of cohomology are special cases of – and in many instances special concrete models for – the following general idea:

Cohomology is something associated to a given (∞,1)-topos $H$. For $X,A$ two objects of $H$, the cohomology of $X$ with coefficients in $A$ is the set of connected components of the ∞-groupoid of morphisms from $X$ to $A$ in $H$:

A non-technical introduction to some concepts in cohomology from this perspective is at

• motivation for sheaves, cohomology and higher stacks.

The following section

• Overview

gives a tour through the zoo of cohomology theories traditionally known, indicating how they all fit into this picture. Then the section

• Definition

gives the very general formal definition and discusses very general properties of and constructions in cohomology theory, such as the terminology of cocycles and coboundaries of objects classified by cohomology, of characteristic classes of these objects, and so on. In particular the section

• Extra structure on cohomology

describes addtition stuff, structure, property that may be present for certain choices of coefficient objects – such as gradings , group and ring-structures –_ and aspects of which are in different parts of the traditional literature often required (differently) on cohomology.

The straightforward definition of cohomology in terms of mapping spaces in an (∞,1)-topos has some slight, but similarly straightforward, variants, notably that of twisted cohomology (which includes other cases such as differential cohomology) and of equivariant cohomology (with its different flavors such as Borel-equivariant and Bredon cohomology). These are discussed in the section

• Variants

before the next main section

• Examples

then starts going through concrete examples in detail. The reader uneasy with the abstract generality of our perspective is advised to skip ahead to this section and find from a long list of examples discussed his or her favorite traditional notion of cohomology and how it fits into the general structure.

Tour through notions of cohomology

The statement of this slogan is well familiar for the special case that $H=$ Top is the (∞,1)-topos of topological spaces. In this context for instance for $A:=K(\mathbb{Z},n)$ an Eilenberg-MacLane space, we have that for $X$ any topological space that

coincides with the “ordinary” integral cohomology of $X$.

This definition in Top alone already goes a long way. By the Brown representability theorem all cohomology theories that are called generalized (Eilenberg-Steenrod) cohomology theories are of this form, for $A$ a topological space that is part of a spectrum. This includes everything that is traditionally just called “a cohomology theory”, such as K-theory, elliptic cohomology, tmf, complex cobordism, etc.

Another big complex of notions of cohomology that on first sight maybe does not seem to fit into this pattern is abelian sheaf cohomology. Usually this is introduced and defined in the language of derived functors. However, derived functors are nothing but a tool, or presentation, for encoding (∞,1)-categorical hom-spaces such as $H(X,A)$ in cases where $H$ is presented by a homotopical category or model category.

Indeed, it turns out that an old result from the 1960s, Verdier’s hypercovering theorem effectively shows that what was introduced as abelian sheaf cohomology is really nothing but an instance of the above general setup. A particularly clear-sighted understanding of this fact was presented in

• Ken Brown, Abstract homotopy theory and generalized sheaf cohomology.

Therein Brown considers essnetially the model structure on simplicial presheaves – which today is known to be one of the standard models for ∞-stack (∞,1)-toposes $H$ – rederives Verdier’s hypercovering theorem and shows that ordinary abelian sheaf cohomology is indeed nothing but ${\pi }_{0}H(X,A)$ in such an (∞,1)-topos, for the special case that the simplicial presheaf $A$ happens to be objectwise in the image of the Dold-Kan correspondence, i.e. for the special case that $A$ is a maximally abelian ∞-stack.

One can then understand various “cohomology theories” as nothing but tools for computing ${\pi }_{0}H(X,A)$ using the known presentations of (∞,1)-categorical hom-spaces: for instance Čech cohomology computes these spaces by finding cofibrant models for the domain $X$, called Čech nerves. Dual to that, most texts on abelian sheaf cohomology find fibrant models for the codomain $A$: called injective resolutions. Both algorithms in the end compute the same intrinsically defined $(\mathrm{\infty }\mathrm{,1})$-categorical hom-space.

In other words, abelian sheaf cohomology is of the exact same nature as the familiar cohomology of topological spaces (and hence of spectra) if only we switch from the archetypical (∞,1)-topos Top to a more general ∞-stack (∞,1)-topos. And abelian sheaf cohomology in turn subsumes many special cases, such as Deligne cohomology (hence also deRham cohomology, …), or etale cohomology, … You name it.

But this also shows that abelian sheaf cohomology itself is just a very special case of cohomology in an $\mathrm{\infty }$-stack $(\mathrm{\infty }\mathrm{,1})$-topos: the stable or maximally abelian case. For coefficient objects $A\in H$ that are not maximally abelian (for instancce not degreewise in the image of the Dold-Kan correspondence for sheaf cohomology) the cohomology of an $\mathrm{\infty }$-stack topos is a nonabelian cohomology.

Often in the literature the term “nonabelian cohomology” is restricted to nonabelian group cohomology, which is indeed one special case. Another familiar special case is cohomology in Top with coefficients in the classifying space $ℬG$ of a (possibly nonabelian) group $G$ (which is of course not part of a spectrum, in general). This degree 1 nonabelian cohomology classifies $G$-principal bundles.

If the group $G$ here is generalized to a (possibly nonabelian) 2-group, the coefficient object $ℬG$ gives degree 2 nonabelian cohomology in Top, which classifies nonabelian gerbes and, more generally, principal 2-bundles. The celebrate treatise by Giraud Cohomologie non abélienne is concerned with this case. In fact, Giraud considered gerbes on stacks and hence was implicitly really computing cohomology in a stack 2-topos with both the domain and the coefficient object allowed to have nontrivial homotopy groups of stacks in degree 2.

Conceptually, with higher topos theory in hand, there is no problem in generalizing nonabelian cohomology and its relation to gerbes and principal bundles further from stacks to ∞-stacks. For instance, while the discussion of spin structure on a space/∞-stack requires a 1-stack coefficient object and classifies principal bundles, and the discussion of string structure requires a 2-stack coefficient object and classifies gerbes and principal 2-bundles, the next case of fivebrane structure requires 6-stack coefficient objects and classifies principal 6-bundles. Generally, we may speak of principal ∞-bundles in any (∞,1)-topos $H$: these are nothing but the homotopy fibers of the corresponding (“nonabelian”) cocycles, which are just morphisms $X\to A$ in $H$.

Various other notions of cohomology are special cases of this. For instance group cohomology is nothing but the cohomology in $H=$ ∞Grpd on objects $X=BG$ that are deloopings of groups. What is called nonabelian group cohomology is nothing but the general case of this where there is no restriction on the coefficient object $A$. Here we can once again replace $\mathrm{\infty }\mathrm{Grpd}$ – which is the $(\mathrm{\infty }\mathrm{,1})$-topos of $\mathrm{\infty }$-stacks on the point – by a more general $\mathrm{\infty }$-stack $(\mathrm{\infty }\mathrm{,1})$-topos. For instance if we take the underlying site to be Diff, the category of smooth manifolds, then the objects of $H={\mathrm{Sh}}_{(\mathrm{\infty }\mathrm{,1})}(\mathrm{Diff})$ are Lie ∞-groupoids. Their cohomology is generalized group cohomology that knows about smooth structure: smooth group cohomology . In this context for instance one can give cohomological interpretations of smooth realizations of the string 2-group or the fivebrane 6-group.

Conversely, given an unconstrained (unstable) (∞,1)-topos $H$ with its general notion of nonabelian cohomology, one can systematically find its stable or abelian content by considering objects that are components of spectrum objects in $H$. These form the stabilization of $H$ to a stable (∞,1)-category.

An example of this is motivic cohomology and motivic homotopy: this is the cohomology given by (∞,1)-categorical hom spaces in the (∞,1)-topos of ∞-stacks on the Nisnevich site: motivic cohomology proper is that where the coefficient objects happen to be components of spectrum objects and A1-homotopy invariant. For instance the Chow groups are precisely the cohomology in this sense with coefficients in the Eilenberg-MacLane objects of this (∞,1)-topos. From this perspective, hom-spaces into more general objects in this $(\mathrm{\infty }\mathrm{,1})$-topos could be called nonabelian motivic cohomology .

A noteworthy example the restriction to homotopy invariant objects in an $(\mathrm{\infty }\mathrm{,1})$-topos, hence to its homotopy localization is the internal cohomology of the $(\mathrm{\infty }\mathrm{,1})$-topos of $\mathrm{\infty }$-stacks on the site Top (topological $\mathrm{\infty }$-stacks): when restricted to homotopy local objects this turns out to be just the ordinary cohomology in Top. This is described in more detail at topological ∞-groupoid.

There are some slight variations on the theme that cohomology is all about connected components of hom-spaces in (∞,1)-toposes: by looking at homotopy fibers of such (∞,1)-categorical hom-spaces instead, one finds twisted cohomology. The most prominent example is twisted K-theory: in degree 0 this is the study of the homotopy fiber of the morphism of $(\mathrm{\infty }\mathrm{,1})$-categorical hom-space $\mathrm{Top}(-,ℬ\mathrm{PU}(n))\to \mathrm{Top}(-,{ℬ}^{2}U(1))$ that sends a projective unitary principal bundle (hence its associated vector bundle) to the lifting gerbe for the lift of its structure group to the full unitary group.

Another example of twisted cohomology is differential cohomology: differential cohomology refinements of abelian generalized (Eilenberg-Steenrod) cohomology theories with coefficient objects a spectrum $E$ is the study of the homotopy fibers of the Chern character map $\mathrm{ch}:H(X,E)\to {\Omega }_{\mathrm{dR}}^{•}(X)\otimes {\pi }_{•}(E)$ from $E$-cohomology to deRham cohomology. This classifies (abelian versions of) connections on the underlying bundles, for instance Simons-Sullivan structured bundles (vector bundles with connection).

By generalizing the notion of Chern character to richer $(\mathrm{\infty }\mathrm{,1})$-toposes, one obtains by the same token a notion of differential nonabelian cohomology encoding connections on general principal ∞-bundles and associated ∞-vector bundles.

Definition

We give now the very general definition of cohomology and describe very general properties of and very general constructions in cohomology theory.

General definition

Given an (∞,1)-topos $H$, for any two objects $X$, $A$ of $H$ we have the (∞,1)-categorical hom-space $H(X,A)$ – an ∞-groupoid. For $H={\mathrm{Ho}}_H$ the homotopy category of $H$, its set of connected components is ${\pi }_{0}H(X,A)={\mathrm{Ho}}_H(X,A)$.

• The objects $(c:X\to A)\in H(X,A)$ are called cocycles on $X$ with coefficients in $A$;

• if $A$ is understood to be equipped with the structure $*\to A$ of a pointed object, then the cocycle $X\to *\to A$ is the trivial cocycle $c_{\mathrm{triv}}$;

• the morphisms $\lambda :c_1\to c_2$ in $H(X,A)$ are the coboundaries. Two cocycles connected by a coboundary are cohomologoues. (More specifically, a cocycle cohomologous to the trivial cocycle is called a coboundary.)

• the equivalence classes $[c]\in {\pi }_{0}H(X,A)$ of cohomologous are the cohomology classes;

• the set of cohomology classes is the $A$-cohomology set

  $$H(X,A):={\mathrm{Ho}}_H(X,A)={\pi }_{0}H(X,A)$$H(X,A) := Ho_{\mathbf{H}}(X,A) = \pi_0 \mathbf{H}(X,A)

  of $X$.

• for $c\in H(X,A)$ a cocycle on $X$ and $k\in H(A,B)$ a cocycle on $A$, the class of the composite cocycle

  $$[k(c)]:=[X\stackrel{c}{\to }A\stackrel{k}{\to }B]\in H(X,B)$$[k(c)] := [X \stackrel{c}{\to} A \stackrel{k}{\to} B] \in H(X,B)

  is the characteristic class of $c$ with respect to $k$.

Remark Notice that there is no notion of cochain in this general setup. What are called cochains are specifically components of certain specific models for $H(X,A)$. More on this in the section on abelian cohomology below.

Objects classified by cohomology

For $g:X\to A$ a cocycle, one says that its homotopy fiber $P\to X$ is the object classified by the cohomology class.

Such an object usually has the interpretation of a principal ∞-bundle. Special cases of this are principal bundles, gerbes, principal 2-bundles, etc. If the domain object $X$ itself is a group object, then $P\to X$ is a group extension. For that reason in abelian cohomology $H(X,A)$ is often denoted $\mathrm{Ext}(X,A)$ and a cocycle is then called an Ext?.

Characteristic classes

For $A\in H$ some coefficient object and $\{c_n:A\to E_n\}$ a collection of cocycles on the coefficient object with values in objects $E_n\in H$ – typically chosen to be Eilenberg-MacLane objects – composition of morphism in $H$ induces a map of cohomology ∞-groupoids

and hence of cohomology classes

that sends each $A$-cocycle $g$ to its characteristic class $c_n(g)$. Typically, for $P\to X$ is the principal ∞-bundle clasified by $g$ one speaks of the characteristic class $c_n(P)$ of this principal $\mathrm{\infty }$-bundle.

Extra structure on cohomology

Extra stuff, structure, property on the coefficient object $A$ will induce corresponding stuff, structure or property on the cohomology sets $H(X,A)$.

Gradings

Integer grading

In the case that the coefficient object $A$ admits $(n\in \mathbb{N})$ deloopings to objects $B^{n}A$ one writes

and speaks of $A$-cohomology in degree $n$.

Similarly, looping defines negative degree cohomology:

Because loop space objects are defined by an $(\mathrm{\infty }\mathrm{,1})$-pullback and the (∞,1)-categorical hom – as any hom-functor – preserves limits in its second argument, this is the same as

This means that all the non-positive degree cohomology identifies with the homotopy groups of the ∞-groupoid $H(X,A)$.

Bigrading

If the underlying topos of $H$ is a lined topos, the line object $I$ canonically has the structure of an interval object and induces a cosimplicial object ${\Delta }_I:\Delta \to H$ of geometric $n$-simplices in addition to the categorical standard cellular simplices. Accordingly there are then two differnt loop objects,

• the categorical 1-sphere (or simplicial loop ) $S^1={\Delta }^1/\partial {\Delta }^1$;

• the geometric 1-sphere $S_I^1={\Delta }_I^1/\partial {\Delta }_I^1$.

(Warning: one has to be careful with different ways how to interpret ${\Delta }_I^1/\partial {\Delta }_I^1$. More later. See the example of motivic cohomology.)

The notion of loop space object and of delooping have geometric analogs in this case and so a second integer grading is induced on cohomology, now coming from the geometric loops. Both gradings may be considered at once, which makes the cohomology theory bigraded:

This bigrading is traditionally considered in motivic cohomology where the line object is that of A1-homotopy theory, but the general construction depends only on the presence and choice of an interval object.

Exotic grading

In some cases one considers geometric spheres $S^V$ that do not necessarily arise from a single interval object. One can still follow the general procedure and define a corresponding graded cohomology

This is notably the standard case in Bredon equivariant cohomology used in equivariant stable homotopy theory, where the $S^V$ are one-point compactifications of representation vector spaces of a group $G$.

Abelian and stable cohomology

Often the coefficient object $A\in H$ for cohomology is taken to be indefinitely deloopable – an $\mathrm{\infty }$-loop space object – or, more generally, a component of a spectrum object in the stabilization $\mathrm{Stab}(H)$ of the (∞,1)-topos $H$ to a stable (∞,1)-category.

In terms of the stabilization adjunction

this means that $A$ is of the form

for some spectrum object $E$, and some integer? $n$ (not necessarily a natural number).

One single such spectrum object this way yields a $\mathbb{Z}$-graded tower of cohomologies

which taken together, denoted $H^{•}(X,E)$ is called a cohomology theory. For the case that $H=$ Top this special case of cohomology is called generalized (Eilenberg-Steenrod) cohomology.

As above in the discussion of gradings, the same discussion goes through analogously in the presence of an interval object that induces a notion of geometric loops. Notably in motivic cohomology coefficient objects are taken to be stable with respect to both categorical and geometric looping and delooping.

Cohomology groups and rings

If $A$ happens to be a group object in $H$ then the cohomology set naturally inherits the structure of a group and then $H(X;A)$ is called the $A$-cohomology group of $X$. If $A$ is at least an $E_2$ object, then $H(X;A)$ is abelian.

This is in particular necessarily the case if $A$ is a component of a spectrum object in abelian cohomology in the sense described above, i.e. of the form ${\Omega }^{\mathrm{\infty }}{\Sigma }^{n}A\prime$.

If the corresponding spectrum object $A\prime$ in addition carries the structure of a ring — in which case it is a ring spectrum or E-∞ ring — then we speak of a multiplicative cohomology theory and the cohomology groups $H^{•}(X,A)$ form a graded ring, the cohomology ring of $X$ with coefficients in $A$.

Variants

Twisted cohomology

Given $A\in H$ an object and $k:A\to B$ cocycle on $A$, classifying its homotopy fiber $C\to A$, the $A$-cohomology with specified $B$-characteristic class $[k]$ is $k$-twised $C$-cohomology.

For the moment see

• twisted cohomology

Differential cohomology

A special type characteristic class is the Chern character. The twisted cohomology with respect to the Chern character is differential cohomology.

• differential abelian cohomology

• schreiber:differential nonabelian cohomology]

Equivariant cohomology

Various related but different variations of cohomology are obtained by domain objects, or coefficient objects or both with action groupoids of actions by some group object, or by more general groupoids (“orbifolds”).

For the moment see

• equivariant cohomology

for more details.

Relative cohomology

For the moment see

• relative cohomology

Homotopy

By abstract duality, cohomology is dual to homotopy (as an operation):

the cohomology of $X$ with coefficients in $A$ is the homotopy of $A$ with co-coefficients in $X$.

Notably, when $H$ is a lined topos there is for each $n\in \mathbb{N}$ a sphere object? $S^n$ in $H$.

For any $A\in H$ the set $H(S^n,A)$ is equivalently

1. the $A$-cohomology of $S^n$.

2. the $n$th homotopy group of $A$.

One could argue that a more suitable term for cohomology is cohomotopy. Unfortunately, of course, this term is traditonally used only for a very special case of what it should mean generally…

Examples

Long list of examples

Classes of special cases of cohomologies with their own entries include

• chain homology and cohomology

• groupoid cohomology

  • group cohomology

  • nonabelian group cohomology

  • cohomology of a category

• generalized (Eilenberg-Steenrod) cohomology

  • integral cohomology

  • K-theory

  • elliptic cohomology

  • tmf

  • complex cobordism

• abelian sheaf cohomology

  • Deligne cohomology

    • de Rham cohomology

  • etale cohomology

• motivic cohomology

• nonabelian cohomology

  • principal bundle

  • gerbe/principal 2-bundle

  • principal ∞-bundle

  • orientation, spin structure, string structure, fivebrane structure

• Hochschild cohomology

• Čech cohomology

• twisted cohomology

• equivariant cohomology

• differential cohomology

  • differential nonabelian cohomology

Chain cohomology

The probably most familiar kind of cohomology is that of a cochain complex dual to a chain complex.

Using the Dold-Kan correspondence chain complexes are understood as components of strict spectrum objects in the archetypical (∞,1)-topos ∞Grpd of ∞-groupoids: namely those ∞-groupoids with the structure of a strict abelian group object: as Kan complexes these are abelian simplicial groups.

This way ordinary chain cohomology is seen to be a special case of general cohomology in $H=$ ∞Grpd. A more detailed discussion of how from this perspective the usual formulas for cochains and cocycles appear is at

• chain homology and cohomology.

Cohomology in $\mathrm{Top}$

The archetypical example for nonabelian cohomology theory is the (∞,1)-topos $H=$ Top, the (∞,1)-category of topological spaces. For $X$ and $A$ two topological spaces, the cohomology classes of $X$ with values in $A$ are the homotopy classes of continuous maps $X\to A$. For $A=K(a,n)$ an Eilenberg-Mac Lane space with $a$ an abelian group this reproduces “ordinary cohomology” of spaces. For $n>1$ this special case happens to be actually abelian. For $A=BG$ a classifying space of a topological group $G$, this reproduces degree 1 nonabelian cohomology $H^1(X,G)$. In general, for $A$ an $n$-type, $H(X,A)$ is topological degree-$n$ nonabelian cohomology.

• The archetypical example for abelian cohomology theory is the stable (∞,1)-topos $H=$ Spec, the stable (∞,1)-category of spectra. This is the case in the literature often addressed as generalized cohomology, since it generalizes the entities specified by the Eilenberg–Steenrod axioms. But really, the general concept of cohomology is more general than this “generalized cohomology”.

  • “ordinary” cohomology is cohomology with coefficients in the Eilenberg-MacLane spectrum

  • K-theory is cohomology with coefficients in the K-theory spectrum

  • elliptic cohomology is somehow subsumes by cohomology with coefficients in tmf.

some left-over material, to be merged…

Ordinary nonabelian cohomology in degree 1 of a ‘nice’ topological space $X$ with values in a discrete (and possibly nonabelian) group $G$ can be defined as the pointed set of homotopy classes of maps of topological spaces from $X$ into the classifying space $BG$. The content of nonabelian cohomology is the generalization of this statement to cohomology in higher degree. The content of general nonabelian differential cohomology is moreover the generalization of nonabelian cohomology to generalized spaces with extra structure, in particular with smooth structure.

Henceforth we will refer to * spaces * meaning perhaps some generalization or restriction, e.g. smooth spaces, and occasionally specify the nature of the generalization. For spaces $X$,$A$, we denote by $\mathscr{H}(X,A)=\mathrm{Maps}(X,A)$ the $(\mathrm{\infty }\mathrm{,0})$-category of maps from $X$ to $A$. To emphasize the relation to cohomology, we name these maps as cocycles and refer to $\mathscr{H}(X,A)=\mathrm{Maps}(X,A)$ as the cohomology of X with coefficients in A: the objects in $\mathrm{Maps}(X,A)$ are the $A$-valued cocycles on $X$, the morphisms are homotopies (or coboundaries) between these and the higher morphisms are homotopies between homotopies, etc. The connected components in $\mathrm{Map}(X,A)$ are the cohomology classes, $H(X,A)={\pi }_0\mathrm{Map}(X,A)$. These are the sets of morphisms in the homotopy category $H$ of $\mathscr{H}$.

For instance for $G$ an ordinary abelian group and $X$ a nice topological space, the choice $A=K(G,n)$ (an Eilenberg-Mac Lane space) yields the ordinary cohomology $H^n(X,G)=H(X,K(G,n))={\pi }_0\mathscr{H}(X,A)$.

If $A$ is pointed in that it is equipped with a morphism ${}_{*}\stackrel{{\mathrm{pt}}_A}{\to }A$, then $\mathscr{H}(X,A)$ is naturally pointed with point $X\to {}_{*}\stackrel{{\mathrm{pt}}_A}{\to }A,$ the trivial $A$-cocycle on $X$. In particular, if $A$ is the delooping, $A=BG$, of a group-like space $G$ in $\mathscr{H}$ (an $\mathrm{\infty }$-group or $A_{\mathrm{\infty }}$-space) and if $g:X\to BG$ is a cocycle, then the homotopy fiber of $g$, i.e. the homotopy pullback $P\to X$ of the point of $A$ in

is the $G$-principal bundle classified by the cocycle $g$.

$(\mathrm{\infty }\mathrm{,1})$-Sheaf-cohomology / $\mathrm{\infty }$-stack-cohomology

A Grothendieck–Rezk–Lurie (∞,1)-topos is an (∞,1)-category of (∞,1)-sheaves. Its objects are often called ∞-stacks or derived stacks.

Abelian sheaf cohomology

• Abelian sheaf cohomology for complexes of sheaves in non-negative degree is cohomology of the sub-(∞,1)-topos of $\mathrm{\infty }$-stacks which take values in ∞-groupoids which, under the Dold-Kan correspondence come from chain complexes.

• Abelian sheaf cohomology for unbounded complexes of sheaves is stable cohomology of the stable (∞,1)-topos of spectrum-valued (∞,1)-sheaves.

Several familiar “cohomology theories” are not so much genuine cohomology theories as rather computational techniques for computing certain cohomology classes in an (∞,1)-category by using 1-categorical tools of homotopy coherent category theory such as model categories, derived categories and the like.

• Čech cohomology is the technique of computing $H(X,A)$ by computing 1-categorical hom-sets $C(\hat{X},A)$ on resolutions of the domain object $X$.

• The technique of computing abelian sheaf cohomology by computing the derived global section functor? is similarly a technique of computing $H(X,A)$ in terms of 1-categorical hom-sets $C(X,\hat{A})$ into resolutions of the coefficient object (namely injective resolution?s).

Hochschild and cyclic cohomology

Hochschild cohomology is the cohomology $H(ℒX,C)$ of free loop space objects $ℒX$ in a derived stack (∞,1)-topos $H$ with coefficients in quasicoherent ∞-stacks of modules $C$. There is a natural action of the circle $S^1$ on the free loop space object $ℒX$ and the corresponding $S^1$-equivariant cohomology is cyclic cohomology.

Motivic cohomology

Motivic cohomology is the cohomology of the (∞,1)-topos of ∞-stacks on the Nisnevich site, usually restricted to coefficient objects that are stable and A1-homotopy invariant.

Tools for computing cohomology

Various notions called “cohomology” in the literature are not so much specific examples of cohomology theories (specific choices of ambient (∞,1)-toposes) as rather specific tools or algorithms for constructing $H(X,A)$.

Čech cohomology

For the moment see

• Čech cohomology

$\mathrm{Ext}$-functor and derived global-sections functor

Using a model category presentation for $H$ one can compute $H(X,A)$ using the derived functor of the hom-functor: called the Ext functor?.

Specifically for the model structure on simplicial sheaves and $X$ representable, one has by Yoneda lemma that $\mathrm{Hom}(X,A)\simeq A(X)$ which is often written as $\simeq \Gamma (A,X)$ and called the global section functor $\Gamma (A,-)$ applied to $X$. Accordingly its derived functor is another way to think of $H(X,A)$.

Monadic cohomology

• Monadic cohomology, like Čech cohomology, is a way to compute cocycles using tools from universal algebra in general and the theory of monads in particular in the presence of extra

History and references

The general perspective on cohomology was essentially established 35 years ago in

• Kenneth Brown, Abstract homotopy theory and generalized sheaf cohomology

and apparently known in one form or other before that.

This article establishes that

• all of abelian sheaf cohomology

  • with all its special cases like, say, Deligne cohomology,

• all generalized (Eilenberg-Steenrod) cohomology

  • with simple things like ordinary cohomology and more intricate things like K-theory and tmf),

• as well as nonabelian cohomology

  • classifying gerbes and principal ∞-bundles

• as well as the more mundane special cases of this like group cohomology and, yes, cohomology of cochain complexes itself

are naturally special cases of one single concept: that of hom-sets

in the homotopy category of ∞-groupoid-valued sheaves.

The only fundamental new addition to this insight that is available now and wasn’t available 35 years ago is that

• These categories $H={\mathrm{Ho}}_{\mathrm{SSh}}$ are precisely the hom-wise decategorification of (∞,1)-category of (∞,1)-sheaves otherwise known as the (∞,1)-topoi of ∞-stacks.

This is propositon 6.5.2.1 in Jacob Lurie’s Higher Topos Theory and builds on the fundamental work by K. Brown, Joyal and Jardine and others on the model structure on simplicial presheaves.

For a motivation of these definitions from the point of view of cohomology as a homotopy hom-set of $\mathrm{\infty }$-stacks see for instance the introductory pages of

• Dan Dugger, Sheaves and homotopy theory (web, pdf) .

The general abstract picture of cohomology as connected components of mapping spaces in (∞,1)-toposes is the topic of section 7.2.2 of

• Jacob Lurie, Higher Topos Theory

Notice that the discussion there is, as often in the literature, given from the perspectiv of a petit topos, i.e. where one thinks of the (∞,1)-topos $𝒳$ as that of ∞-stacks on a given space $X$ (instead of as a gros topos of all generalized spaces, as we do in the above entry). Accordingly then from that perspective one wants to study the cohomology of $X$ itself, which corresponds to the terminal object in the $(\mathrm{\infty }\mathrm{,1})$-topos. Accordingly, the cohomology in that section 7.2.2 is defined for the terminal coefficient object and for an Eilenberg-MacLane object $K(A,n)$:

(definition 7.2.2.14).

Another reference with a discussion of cohomology in the general sense discussed above, using tools of model category theory for simplicial objects, is

• Brian Conrad?, Cohomological descent (pdf)

Discussion