cohomological field theory

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## Idea

The term "cohomological field theory" is mostly used for the "Witten style" [[topological quantum field theories]] which arise from a [[topological twist]] of a [[supersymmetry|supersymmetric]] [[quantum field theory]], notable example being the [[A-model]] and the [[B-model]] [[topological string]].

Mathematically these topological field theories came to be known as _[[TCFT|TCFTs]]_. With the proof of the [[cobordism hypothesis]] these were understood to be the "non-compact" 2-dimensional [[extended TQFTs]] with codomain an [[(infinity,1)-category of chain complexes]], where "non-compact" refers to all [[cobordisms]] being required to have non-empty outgoing [[boundary]].

For more on all this see at _[[TCFT]]_ and at _[[2d TQFT]]_.

## References

* {#Witten91} [[Edward Witten]], _Introduction to cohomological field theory_, InternationalJournal of Modern Physics A, Vol. 6,No 6 (1991) 2775-2792 ([[WittenCQFT.pdf:file]])
* M. Kontsevich, Yu. Manin, _Gromov–Witten classes, quantum cohomology, and enumerative geometry_, Comm. Math. Phys. 164.3 (1994) 525–562
* Rahul Pandharipande, _Cohomological field theory calculations_, Proc. ICM 2018 Rio de Janeiro, Vol. 1 (869–898) [pdf](https://eta.impa.br/dl/PL017.pdf)

[[!redirects cohomological field theories]]
[[!redirects cohomological quantum field theories]]

[[!redirects cohomological quantum field theory]]
[[!redirects CQFT]]
[[!redirects cohomology topological field theory]]

[[!redirects cohomological QFT]]
[[!redirects cohomological QFTs]]


cohomological functor

An [[additive category|additive functor]] $F$ from a [[triangulated category]] $A$ (with translation denoted $X\mapsto X[1]$) to an [[abelian category]] $C$ is a **cohomological functor** if for any distinguished triangle
$$
X\stackrel{f}\to Y\stackrel{g}\to Z\stackrel{h}\to X[1],
$$
in $A$, $F X\stackrel{F f}\to F Y\stackrel{F g}\to F Z$ is an [[exact sequence]] in $C$.


cohomological induction

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## Idea

__Cohomological induction__ is a [[derived functor]] version of [[induction functors]] in [[representation theory]]. As the elements in induced representations correspond to [[sections]] of certain equivariant bundles or sheaves, similarly the cohomological induction functors could be interpreted as higher cohomologies of certain [[equivariant sheaves]].

In specific contexts, like real Lie groups, there are specifical versions like [[Zuckerman induction functors]]; they are defined algebraically.

## Related concepts

* [[Zuckerman induction]]

## Literature

* A. Knapp, David A. Vogan, _Cohomological induction and unitary representations_, [MR1330919](http://www.ams.org/mathscinet-getitem?mr=1330919)
* wikipedia [Zuckerman functor](http://en.wikipedia.org/wiki/Zuckerman_functor)
* Pavle Pandžić, _Zuckerman functors between equivariant derived categories_, Trans. Amer. Math. Soc. __359__ (2007), 2191-2220 [MR2276617](http://www.ams.org/mathscinet-getitem?mr=2276617) [pdf](http://www.ams.org/journals/tran/2007-359-05/S0002-9947-06-04013-X/S0002-9947-06-04013-X.pdf)
* Dragan Miličić, Pavle Pandžić, _Equivariant derived categories, Zuckerman functors and localization_,  from  Geometry and Representation Theory of real and p-adic Lie Groups , J. Tirao, D. Vogan, J.A. Wolf, editors, Progress in Mathematics __158__, Birkhäuser, Boston, 1997, 209-242, [pdf](http://www.math.utah.edu/~milicic/Eprints/cordoba.pdf)
* Toshiyuki Kobayashi, _Branching problems of Zuckerman derived functor modules_, [arxiv/1104.4399](http://arxiv.org/abs/1104.4399)
* [[Gregg Zuckerman|Greg J. Zuckerman]], _Construction of representations via derived functors_, unpublished lecture series, Institute for Advanced Study, 1978.
* Jia-jun Ma, _Derived functor modules, dual pairs and $U(\mathfrak{g})^K$-actions_, [arxiv/1310.6378](http://arxiv.org/abs/1310.6378)


cohomological integration

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#### Integration theory
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#### Cohomology
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# Contents
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## Idea

Under suitable circumstances, forming the [[integral]] of a [[differential form]] may be understood as passing to the [[cohomology]]-[[equivalence class]] of that differential form in a suitable [[chain complex]].

Examples of this appear in the constructions discussed at [[Lie integration]] and at [[BV-BRST formalism]], where cohomological integration is used as a way to formalize the idea of the [[path integral]]. (See at _[The BV-complex and homological (path-)integration](BV-BRST+formalism#HomologicalIntegration)_)

While the idea has been around (as witnessed by the references at _[[Lie integration]]_ and _[[BV-BRST formalism]]_) a comprehensive and dedicated theory, or a published account thereof, currently seems to be missing. But see the [References](#References) below.

## Related concepts

* [[geometric integration theory]]

* [[geometric quantization by push-forward]]

## References
 {#References}

* [Diffiety education program](https://sites.google.com/site/levicivitainstitute/dep), _[Differential Cohomology and the Cohomological Theory of Integration](https://sites.google.com/site/levicivitainstitute/dep/ac/dccti)_

[[!redirects cohomological integration]]


cohomology

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## Idea ##
 {#Idea}

There are various different-looking definitions of the general notion of _cohomology_ in different contexts, some familiar, some more exotic.  Most, if not all, of 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)-category]] $\mathbf{H}$. For $X, A$ two objects of $\mathbf{H}$, the **(degree-0) cohomology of $X$ with coefficients in $A$** is the set of connected components of the [[derived hom-space|hom ∞-groupoid]], hence of [[homotopy classes]] of [[morphisms]] from $X$ to $A$ in $\mathbf{H}$:

$$
  H(X;A) = H^0(X;A) \coloneqq \pi_0 \mathbf{H}(X,A)
  \,.
$$

The (∞,1)-category $\mathbf{H}$ is usually an [[(∞,1)-topos]], where the notion of cohomology is [particularly well-behaved](#ToposTheory). However, it is not uncommon to consider cohomology in other contexts, such as in [[stable (∞,1)-categories]].

More generally, if $A$ is equipped with an $n$-fold [[delooping]] $A_n$, then the **degree-$n$ cohomology** of $X$ with coefficients in $A$ is its degree-0 cohomology with coefficients in $A_n$:

$$ H^n(X;A) \coloneqq H^0(X;A_n). $$

Every object $A$ has a unique $n$-fold delooping when $n$ is a negative [[integer]], namely its $(-n)$-fold [[loop object]] $\Omega^{-n}(A)$.  If $A$ has an $n$-fold delooping for positive $n$, then it must be an $n$-[[k-monoidal infinity-group|monoidal group]] --- and conversely, any $n$-monoidal group has a canonical (but not unique) $n$-fold delooping $\mathbf{B}^n A$.  Finally, $n$ could be more general than an integer; see below.

For suitable choices of $\mathbf{H}$, $A$, and $n$, this general definition encompasses (1) the traditional (e.g. [[singular cohomology|singular]]) cohomology of [[topological spaces]] taught in [[algebraic topology]], (2) [[generalized (Eilenberg-Steenrod) cohomology]], (3) [[non-abelian cohomology]], (4) [[twisted cohomology]], (5) [[group cohomology]], (6) [[sheaf cohomology]], (7) sheaf [[hypercohomology]], and (8) [[equivariant cohomology]].  See below for explanations and discussion.

Furthermore, this general notion of cohomology also accurately captures general classification and extension problems ([NSS](#NSS)), such as (1) [[principal ∞-bundles]], (2) [[group extensions]], (3) [[fiber ∞-bundles]], and (4) [[twisted ∞-bundles]].

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](#Overview)

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

* [Definition](#Definition)

gives the general formal definition and discusses general properties of and constructions in cohomology theory, such as the terminology of _[[cocycles]]_ and _coboundaries_ of _[[principal ∞-bundle|objects classified]]_ by cohomology, of _[[characteristic classes]]_ of these objects, of [[Postnikov tower in an (infinity,1)-category|Postnikov towers]] and [[Whitehead tower in an (infinity,1)-topos|Whitehead towers]], and so on. In particular the section

* [Extra structure on cohomology](#ExtraStruc)

describes additional [[stuff, structure, property]] that may be present for certain choices of coefficient objects -- such as _gradings_ , [[cohomology group]]- and ring-structures -- and aspects of which are in different parts of the traditional literature often _required_ (differently) on cohomology.

The general definition of cohomology in terms of mapping spaces in an [[(∞,1)-category]] also encompasses notions that can be considered _variants_ of "honest" cohomology, 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](#Variants)

before the next main section

* [Examples](#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.

Finally we discuss why [[(∞,1)-toposes]] are a particularly nice environment for cohomology in

* [Cohomology in (∞,1)-topoi](#ToposTheory) .

Essentially nothing about this perspective on cohomology is really new, many aspects of it have been made explicit in the literature here and there. In fact, to some extent everything here is just an afterthought of the old seminal article

* [[Kenneth Brown]], _[[BrownAHT|Abstract homotopy theory and generalized sheaf cohomology]]_ (1973),

in the light of fully fledged [[(∞,1)-topos|(∞,1)-topos theory]], of which it is effectively the seed, by noticing that this article secretly discusses precisely the [[homotopy category of an (infinity,1)-category|homotopy categories]] of [[hypercomplete (∞,1)-topos]]es. At the same time, to some extent everything here is also an afterthought of the theory of cohomology in [[category theory|1-categorical]] [[topos theory]] as reviewed for instance in

* [[Ieke Moerdijk]], _Classifying Spaces and Classifying Topoi_ , section I.4

by noticing that the constructions on [[simplicial objects]] in [[toposes]] used there secretly precisely compute the [[derived hom-space|(∞,1)-categorical hom-objects]] of an [[(∞,1)-topos]] as presented by the [[model structure on simplicial sheaves]] on the underlying site.

This and a list of other related references and historical developments is given at

* [References](#References).

### About the nPOV on cohomology {#ToposTheoryHistoricalAspects}

As we will see in the list of examples below, large numbers of examples of notions of cohomology do happen to have a natural interpretation in terms of connected components of hom-spaces in $(\infty,1)$-categories. There are however some definitions of cohomology in the literature that do not fit this principle. But these tend to be _wrong_ definitions, as illustrated by the following example.

In the literature there is a _naive_ definition of Lie group cohomology and topological group cohomology, which is not interpretable in terms of hom-spaces in any natural $(\infty,1)$-category. But later it was found by Segal and then independently by Brylinski that there is a refinement of this definition, which is better behaved. This refinement, it turns out, does have an interpretation in terms of homs in an $(\infty,1)$-topos. This is described at [[group cohomology]].

…

## Tour through notions of cohomology
 {#Overview}

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

$$
  \pi_0 \mathbf{H}(X,K(\mathbb{Z},n)) = H^n(X,\mathbb{Z})
$$

coincides with the "ordinary" [[integral cohomology]] of $X$, modeled as its [[singular cohomology]].

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 cohomology theory|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 functor]]s. However, derived functors are nothing but a tool, or presentation, for encoding [[(∞,1)-categorical hom-space]]s such as $\mathbf{H}(X,A)$ in cases where $\mathbf{H}$ is [[presentable (∞,1)-category|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

* [[Kenneth Brown]], [[BrownAHT|Abstract homotopy theory and generalized sheaf cohomology]].

Therein Brown considers essentially the [[model structure on simplicial presheaves]] -- which today is known to be one of the standard [[models for ∞-stack (∞,1)-toposes]] $\mathbf{H}$ -- rederives Verdier's hypercovering theorem and shows that ordinary [[abelian sheaf cohomology]] is indeed nothing but $\pi_0 \mathbf{H}(X,A)$ 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_ $\pi_0 \mathbf{H}(X,A)$ using the known presentations of [[(∞,1)-categorical hom-space]]s: for instance [[Čech cohomology]] computes these spaces by finding cofibrant models for the domain $X$, called [[Čech nerve]]s. 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 $(\infty,1)$-categorical hom-space.

In other words, [[abelian sheaf cohomology]] is of the exact same nature as the familiar cohomology of [[topological space]]s (and hence of [[spectrum|spectra]]) if only we switch from the archetypical [[(∞,1)-topos]] [[Top]] to a more general [[(∞,1)-category of (∞,1)-sheaves|∞-stack (∞,1)-topos]]. And abelian sheaf cohomology in turn subsumes many special cases, such as [[Deligne cohomology]], [[deRham cohomology]], [[etale cohomology]], [[crystalline cohomology]], [[syntomic cohomology]], etc. You name it.

But this also shows that abelian sheaf cohomology itself is just a very special case of cohomology in an $\infty$-stack $(\infty,1)$-topos: the [[stable (infinity,1)-category|stable]] or _maximally abelian_ case. For coefficient objects $A \in \mathbf{H}$ that are not maximally abelian (for instance not degreewise in the image of the [[Dold-Kan correspondence]] for sheaf cohomology) the cohomology of an $\infty$-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]] $\mathcal{B}G$ 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 bundle]]s.

If the group $G$ here is generalized to a (possibly nonabelian) [[2-group]], the coefficient object $\mathcal{B}G$ gives degree 2 nonabelian cohomology in [[Top]], which classifies nonabelian [[gerbe]]s and, more generally, [[principal 2-bundle]]s. The celebrate treatise by Giraud _Cohomologie non abélienne_ is concerned with this case. In fact, Giraud considered [[gerbe]]s on [[stack]]s 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 group (of an infinity-stack)|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 [[∞-stack]]s. For instance, while the discussion of [[spin group|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 [[gerbe]]s and [[principal 2-bundle]]s, the next case of [[fivebrane group|fivebrane structure]] requires 6-stack coefficient objects and classifies principal 6-bundles. Generally, we may speak of [[principal ∞-bundle]]s in any [[(∞,1)-topos]] $\mathbf{H}$: these are nothing but the [[homotopy fiber]]s of the corresponding ("nonabelian") [[cocycle]]s, which are just morphisms $X \to A$ in $\mathbf{H}$.

Various other notions of cohomology are special cases of this. For instance [[group cohomology]] is nothing but the cohomology in $\mathbf{H} = $ [[∞Grpd]] on objects $X = \mathbf{B}G$ that are [[delooping]]s of [[group]]s. 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 $\infty Grpd$ -- which is the $(\infty,1)$-topos of $\infty$-stacks on the point -- by a more general $\infty$-stack $(\infty,1)$-topos. For instance if we take the underlying [[site]] to be [[Diff]], the category of smooth [[manifold]]s, then the objects of $\mathbf{H} = Sh_{(\infty,1)}(Diff)$ are [[Lie ∞-groupoid]]s. 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)-category]] $\mathbf{H}$ with its general notion of [[nonabelian cohomology]], one can systematically find its [[stable (∞,1)-category|stable]] or _abelian_ content by considering objects that are components of [[spectrum object]]s in $\mathbf{H}$. These form the [[stabilization]] of $\mathbf{H}$ to a [[stable (∞,1)-category]].

In [[stable homotopy theory]] one further considers the cohomology of spectrum objects themselves, which is an example of the notion of cohomology being used in an [[(∞,1)-category]] which is not an [[(∞,1)-topos]]. Another example is the continuous cohomology of [[pro-spaces]] or more generally of [[pro-objects]] in an (∞,1)-topos, which is important in [[shape theory]].

There are some slight variations on the theme that cohomology is all about connected components of hom-spaces in [[(∞,1)-categories]]: by looking at [[homotopy fiber]]s of such  [[derived hom space|(∞,1)-categorical hom-spaces]] instead, one finds [[twisted cohomology]]. It can also be seen as a special case of the general definition by looking at [[slice (∞,1)-categories]]. The most prominent example is [[twisted K-theory]]: in degree 0 this is the study of the homotopy fiber of the morphism of $(\infty,1)$-categorical hom-space $Top(-,\mathcal{B}PU(n)) \to Top(-,\mathcal{B}^2 U(1))$ that sends a projective [[unitary group|unitary]] [[principal bundle]] (hence its [[associated bundle|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 fiber]]s of the [[Chern character]] map $ch : \mathbf{H}(X,E) \to \Omega^\bullet_{dR}(X)\otimes \pi_\bullet(E)$ from $E$-cohomology to [[deRham cohomology]].
This classifies (abelian versions of) [[connection on a bundle|connections]] on the underlying bundles, for instance [[Simons-Sullivan structured bundle]]s (vector bundles with connection).

By generalizing the notion of [[Chern character]] to richer $(\infty,1)$-toposes, one obtains by the same token a notion of [[schreiber:differential cohomology in an (∞,1)-topos]] encoding connections on general [[principal ∞-bundle]]s and associated [[schreiber:∞-vector bundle]]s.

\linebreak

+-- {: .standout}

**Slogan.**

Thousand and one definitions of notions of cohomology and its variants. From the [[nPOV]], just a single concept: an [[derived hom space|∞-categorical hom-space]] in an [[(∞,1)-topos]].

=--

\linebreak

## Definition {#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)-category]] $\mathbf{H}$, for any two [[objects]] $X$, $A$ of $\mathbf{H}$ we have the [[derived hom space|(∞,1)-categorical hom-space]] $\mathbf{H}(X,A)$ -- an [[∞-groupoid]]. For $H = Ho_{\mathbf{H}}$ the [[homotopy category of an (infinity,1)-category|homotopy category]] of $\mathbf{H}$, its set of connected components is $\pi_0 \mathbf{H}(X,A) = Ho_{\mathbf{H}}(X,A)$.

* The [[objects]] $ (c : X \to A) \in \mathbf{H}(X,A)$ are
  called **[[cocycle]]s** 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** $c_{triv}$;

* the [[morphism]]s $\lambda : c_1 \to c_2$ in $\mathbf{H}(X,A)$ are the
  **coboundaries**. Two cocycles connected by a coboundary
  are **cohomologous**. (More specifically, a cocycle cohomologous to the
  trivial cocycle is called a coboundary.)

* the equivalence classes $[c] \in \pi_0 \mathbf{H}(X,A)$ of
  cohomologous are the **cohomology classes**;

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

  $$
    H(X,A) \coloneqq Ho_{\mathbf{H}}(X,A) = \pi_0 \mathbf{H}(X,A)
  $$

  of $X$.

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

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

  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 [[model category|models]] for $\mathbf{H}(X,A)$. More on this in the section on [abelian cohomology](#abelian) 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*.

In an [[(∞,1)-topos]], 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 [[groupoid object in an (infinity,1)-category|group object]], then $P \to X$ is a [[group extension]]. For that reason in abelian cohomology $\mathbf{H}(X,A)$ is often denoted $Ext(X,A)$ and a [[cocycle]] is then called an [[Ext functor|Ext]].

### Characteristic classes

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

$$
  c_n : \mathbf{H}(X,A) \to \mathbf{H}(X,E_n)
$$

and hence of cohomology classes

$$
  c_n : H(X,A) \to H(X,E_n)
$$

that sends each $A$-cocycle $g$ to its **characteristic class** $c_n(g)$. Typically, for $P \to X$, the [[principal ∞-bundle]] classified by $g$, one speaks of the characteristic class $c_n(P)$ of this principal $\infty$-bundle.

### Extra structure on cohomology {#ExtraStruc}

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

#### Gradings {#Grading}

##### Integer grading

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

$$
  H^n(X,A) \coloneqq \pi_0 \mathbf{H}(X, \mathbf{B}^n A)
$$

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

Similarly, [[loop space object|looping]] defines negative degree cohomology:

$$
  H^{-n}(X,A) \coloneqq \pi_0 \mathbf{H}(X, \Omega^n A)
  \,.
$$

Because [[loop space object]]s are defined by an $(\infty,1)$-[[pullback]] and the [[derived hom space|(∞,1)-categorical hom]] -- as any [[hom-functor]] -- preserves [[limit]]s in its second argument, this is the same as

$$
  \begin{aligned}
    H^{-n}(X,A) &\simeq \pi_0 \Omega^n \mathbf{H}(X, A)
    \\
     & \simeq \pi_n \mathbf{H}(X,A)
    \,.
  \end{aligned}
  \,.
$$

This means that all the non-positive degree cohomology identifies with the [[simplicial homotopy group|homotopy group]]s of the [[∞-groupoid]] $\mathbf{H}(X,A)$.

##### Exotic gradings

In many cases, the (∞,1)-category $\mathbf{H}$ is related to a [[symmetric monoidal (∞,1)-category]] $\mathbf{S}$ via a symmetric monoidal adjunction

$$ \Sigma^\infty: \mathbf{H} \leftrightarrows \mathbf{S}: \Omega^\infty $$

which is usually some form of [[stabilization]] of $\mathbf{H}$. Cohomology in $\mathbf{H}$ with coefficients in objects of the form $\Omega^\infty A$, or more generally cohomology in $\mathbf{S}$, is then naturally graded by the [[Picard group]] $Pic(\mathbf{S})$ of $\mathbf{S}$:

$$ H^\star(X, A)=\pi_0\mathbf{S}(X,\star\otimes A), \quad \star\in Pic(\mathbf{S}). $$

The point of the Picard-grading is that it accounts for all possible suspension isomorphisms.

For example, there is always the (∞,1)-category $\mathbf{S}=Stab(\mathbf{H})$ of [[spectrum objects]] in $\mathbf{H}$. The subgroup $\mathbb{Z}\subset Pic(\mathbf{S})$ consisting of the spheres $S^n \coloneqq \Sigma^n(1)$ gives the integer grading discussed above in the special case when the coefficient object is a spectrum object. This is discussed further [below](#abelian).

Examples where some subgroup of the Picard group larger than $\mathbb{Z}$ is commonly used include:

* $\mathbf{H}$ is the (∞,1)-category of $G$-spaces for a compact Lie group $G$ and $\mathbf{S}$ is [[equivariant stable homotopy theory]]. In this context cohomology theories are usually graded by the real representation ring $RO(G)$ which is a subgroup of $Pic(\mathbf{S})$.

* $\mathbf{H}=\infty Grpd/X$ and $\mathbf{S}=[X,E Mod]$ for some $E_\infty$-ring $E$. The Picard-graded cohomology $H^\star(1,E)$ is the same as the [[twisted cohomology|twisted]] $E$-cohomology of $X$.

* $\mathbf{H}=\infty Grpd$ and $\mathbf{S}$ is the $K(n)$-local stable homotopy category for some prime $p$ and some $n\geq 1$. In this case the Picard group contains, among other things, a copy of the [[p-adic integers]] $\mathbb{Z}_p$.

* $\mathbf{H}$ is the [[motivic homotopy category]] over a base scheme $S$ and $\mathbf{S}$ is the associated stable motivic homotopy category. The Picard group contains a copy of $\mathbb{Z}\times K_0(S)$, and one usually considers bigraded cohomology theories via the subgroup $\mathbb{Z}\times\mathbb{Z}$ (with a re-indexing). This recovers for example the bigrading in [[motivic cohomology]].

#### Abelian and stable cohomology {#abelian}

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

In terms of the [[stabilization]] [[adjunction]]

$$
  \mathbf{H} \stackrel{\stackrel{\Omega^\infty}{\leftarrow}}{\underset{\Sigma^\infty}{\to}}
  Stab(\mathbf{H})
$$

this means that $A$ is of the form

$$
  A = E_n \coloneqq \Omega^\infty \circ \Sigma^n E
$$

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

$$
  H^n(X, E) \coloneqq \pi_0 \mathbf{H}(X, \Omega^\infty \Sigma^n E)
$$

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

#### Cohomology groups and rings {#CohomGroup}

If $A$ happens to be a
[[groupoid object in an (infinity,1)-category|group object]]
in $\mathbf{H}$ 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 $\Omega^\infty \Sigma^n A'$.

If the corresponding [[spectrum object]] $A'$ 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 group]]s
$H^\bullet(X,A)$ form a graded [[ring]], the **cohomology ring** of $X$ with coefficients in $A$.

### Variants {#Variants}

#### Twisted cohomology

What is called _[[twisted cohomology]]_ is just the intrinsic cohomology of [[slice toposes]]. In particular if all possible twisting groups are allowed at once, and once considers twisted [[generalized cohomology theories]] then this is the intrinsic cohomology of [[tangent (∞,1)-toposes]].

#### Differential cohomology

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

* [[differential cohomology|differential abelian cohomology]]

* [[schreiber:differential cohomology in an (∞,1)-topos]]

#### Equivariant cohomology

All flavors of $G$-[[equivariant cohomology]]
are obtained from the cohomology of the [[slice (∞,1)-topos]]
$\mathbf{H}_{\mathbf{B}G}$ (which encapsulates [[∞-actions]])

[[!include homotopy type representation theory -- table]]

under [[base change]] down to the [[base (∞,1)-topos]] $\mathbf{H}$:

[[!include equivariant cohomology -- table]]

#### 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 $\mathbf{H}$ is an [[(∞,1)-topos]] there is for
each $n \in \mathbb{N}$ a [[sphere object]] $S^n$ in $\mathbf{H}$.

For any $A \in \mathbf{H}$ 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 {#Examples}

### Long list of examples
 {#LongListOfExamples}

Classes of special cases of cohomologies with their own entries include

* [[chain homology and cohomology]]

* [[groupoid cohomology]]

  * [[group cohomology]]

  * [[nonabelian group cohomology]]

  * [[Galois cohomology]]

  * [[cohomology of a category]]

* [[generalized (Eilenberg-Steenrod) cohomology]]

  * [[integral cohomology]]

  * [[K-theory]]

  * [[elliptic cohomology]]

  * [[tmf]]

  * [[complex cobordism cohomology theory|complex cobordism]]

* [[abelian sheaf cohomology]]

  * [[Deligne cohomology]]

    * [[de Rham cohomology]]

    * [[Dolbeault cohomology]]

  * [[etale cohomology]]

  * [[crystalline cohomology]]

  * [[syntomic cohomology]]

* [[motivic cohomology]]

* [[nonabelian cohomology]]

  * [[principal bundle]]

  * [[gerbe]]/[[principal 2-bundle]]

  * [[principal ∞-bundle]]

  * [[orientation]], [[spin structure]], [[string structure]], [[fivebrane structure]]

* [[cohomology with constant coefficients]]

* [[cohomology with a local system of coefficients]]

* [[cohomology of operads]]

* [[Hochschild cohomology]] / [[cyclic cohomology]]

* [[Čech cohomology]]

* [[hypercohomology]]

* [[twisted cohomology]]

* [[equivariant cohomology]]

* [[differential cohomology]], [[Chern-Weil theory]]

  * [[schreiber:differential cohomology in an (∞,1)-topos]]

  * [[∞-Chern-Weil theory]]

### Chain cohomology {#ChainCohomology}

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

Using the [[Dold-Kan correspondence]] [[chain complex]]es are understood as particular [[spectra]], i.e. [[spectrum object]]s in the archetypical [[(∞,1)-topos]] [[∞Grpd]] of [[∞-groupoid]]s.  Positively graded chain complexes (the "connective" ones) are just [[∞-groupoid]]s with the structure of a _[[group object|strict]]_ abelian [[groupoid object in an (∞,1)-category|group object]]: as [[Kan complex]]es these are abelian [[simplicial group]]s.

This way, ordinary chain cohomology is seen to be a special case of general cohomology in $\mathbf{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 $Top$

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 $n \gt 1$ this special case happens to be actually abelian.
 For $A = B G$ 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 [[spectrum|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 subsumed 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 $B G$. 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 $\mathcal{H}(X,A) = \mathrm{Maps}(X,A)$  the $(\infty,0)$-category of maps from $X$ to $A$. To emphasize the relation to cohomology,  we name these maps as cocycles and refer to $\mathcal{H}(X,A) = \mathrm{Maps}(X,A)$ 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 $\mathcal{H}$.

>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 $H^n(X,G) = H(X,K(G,n)) = \pi_0\mathcal{H}(X,A)$.

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

$$
  \array{
    P          & \rightarrow            & {}_* \\
    \downarrow &                        & \downarrow \\
    X          & \overset{g}\rightarrow & \mathbf{B}G
  }
$$
>is the $G$-principal bundle classified by the cocycle $g$.

### $(\infty,1)$-Sheaf-cohomology / $\infty$-stack-cohomology

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

#### Abelian sheaf cohomology

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

* [[abelian sheaf cohomology|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).

+--{.query}
Zoran: I am not happy with this assertion. First of all the notion of the derived functor is fundamental and it makes sense even in setups when the injective resolutions do not exist. Abelian sheaf cohomology IS a derived functor of the global sections functor, not a specific technique to computing it. On the other hand, the injective resolutions ARE a specific technique to compute the derived functor. It is also not clear in this entry if it is about sheaves on topological spaces or on sites or some more general setup.

Urs: I have posted a reply [here](http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=391&page=1#Item_5). Let's sort this out, improve the entry and remove this query box here.
=--

#### Nonabelian sheaf cohomology with constant coefficients {#NonabelianSheafCohomology}

For $X$ a [[topological space]] and $A$ an [[∞-groupoid]], the standard way to define the [[nonabelian cohomology]] of $X$ with coefficients in $A$ is to define it as the intrinsic cohomology as seen in [[∞Grpd]] $\simeq$ [[Top]]:

$$
  H(X,A) \coloneqq \pi_0 Top(X, |A|) \simeq \pi_0 \infty Func(Sing X, A)
  \,,
$$

where $|A|$ is the [[geometric realization]] of $A$ and $Sing X$ the [[fundamental ∞-groupoid]] of $X$.

But both $X$ and $A$ here naturally can be regarded, in several ways, as objects of [[(∞,1)-category of (∞,1)-sheaves|(∞,1)-sheaf (∞,1)-topos]]es $\mathbf{H} = Sh_{(\infty,1)}(C)$ over nontrivial [[(∞,1)-site]]s $C$. The intrinsic cohomology of such $\mathbf{H}$ is a [[nonabelian cohomology|nonabelian sheaf cohomology]]. The following discusses two such choices for $\mathbf{H}$ such that the corresponding nonabelian sheaf cohomology coincides with $H(X,A)$ (for [[paracompact space|paracompact]] $X$).

##### Petit $(\infty,1)$-sheaf $(\infty,1)$-topos

For $X$ a [[topological space]] and $Op(X)$ its [[category of open subsets]] equipped with the canonical structure of an [[(∞,1)-site]], let

$$
  \mathbf{H}
  \coloneqq
  Sh_{(\infty,1)}(X)
  \coloneqq
  Sh_{(\infty,1)}(Op(X))
$$

be the [[(∞,1)-category of (∞,1)-sheaves]] on $X$. The space $X$ itself is naturally identified with the [[terminal object]] $X = * \in Sh_{(\infty,1)}(X)$. This is the [[petit topos]] incarnation of $X$.

Write

$$
  (LConst \dashv \Gamma) :  Sh_{(\infty,1)}(X)
  \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}
  \infty Grpd
$$

be the [[global section]]s terminal [[geometric morphism]].

Under the [[constant (∞,1)-sheaf]] functor $LConst$ an [[∞-groupoid]] $A \in \infty Grpd$ is regarded as an object $LConst A \in Sh_{(\infty,1)}(X)$.

There is therefore the _intrinsic_ [[cohomology]] of the $(\infty,1)$-topos $Sh_{(\infty,1)}(X)$ with coefficients in the [[constant ∞-stack|constant (∞,1)-sheaf]] on $A$

$$
  H'(X,A) \coloneqq \pi_0 Sh_{(\infty,1)}(X)(X, LConst A)
  \,.
$$

Notice that since $X$ is in fact the [[terminal object]] of $Sh_{(\infty,1)}(X)$ and that $Sh_{(\infty,1)}(X)(X,-)$ is in fact that [[global section]]s functor, this is equivalently

$$
  \cdots \simeq \pi_0 \Gamma LConst A
  \,.
$$

+-- {: .un_theorem }
###### Theorem

If $X$ is a [[paracompact space]], then these two definitions of [[nonabelian cohomology]] of $X$ with [[constant ∞-stack|constant coefficients]] $A \in \infty Grpd$ agree:

$$
  H(X,A)
  \coloneqq
  \pi_0 \infty Grpd(Sing X,A)  \simeq Sh_{(\infty,1)}(X)(X,LConst A)
  \,.
$$

=--

This is [[Higher Topos Theory|HTT, theorem 7.1.0.1]]. See also [[(∞,1)-category of (∞,1)-sheaves]] for more.

##### In terms of covering spaces {#InTermsOfCoveringSpaces}

There is an equivalence between $(\infty,1)$-sheaves on $X$ and topological spaces over $X$, as described in detail at [[(∞,1)-sheaves and over-spaces]].

Suppose that $X$ is a [[locally compact space|locally compact]] [[CW complex]].  In particular, this implies that it is "hereditarily [[m-cofibrant space|m-cofibrant]]," i.e. every open subset of $X$ has the [[homotopy type]] of a CW complex.  That's what you need in order to conclude that taking sheaves of sections of spaces over $X$ is well-behaved homotopically, since only m-cofibrant spaces are good for mapping out of homotopically.

In

* [[Mike Shulman]], _Parametrized spaces model locally constant homotopy sheaves_ ([artXiv:0706.2874](http://arxiv.org/abs/0706.2874))

it is proved that the "sheaf of sections" functor

$$ Top/X \to [Op(X)^{op},sSet] $$

is the [[right adjoint]] in a [[right Quillen embedding]], i.e. a [[Quillen adjunction]] whose  [[derived functor|derived]] right adjoint is [[full and faithful functor|fully faithful]].  In other words, the homotopy theory of spaces over $X$ embeds in the homotopy theory of $(\infty,1)$-sheaves on $X$.

One can also identify its image as consisting of the [[locally constant (∞,1)-sheaves]].  This is a homotopical version of the identification of [[covering space]]s with [[locally constant sheaves]].

Furthermore, if $f\colon X\to Y$ is a map of such spaces, then the pullback functor $f^*\colon Top/Y \to Top/X$ agrees with the [[inverse image]] functor $f^*$ for $(\infty,1)$-sheaves.  In particular, when $Y$ is a point and $A$ a space, then the constant $(\infty,1)$-sheaf $Const(A)$ is identified with (the sheaf of sections of) the space $X^* A = X\times A$ over $X$.  Therefore, the nonabelian cohomology of $X$ with coefficients in $Const(A)$ is the same as the maps in $Top/X$ from $X$ (the terminal object of $Top/X$) to $X^* A$.  Since the left adjoint of $X^*:Top \to Top/X$ just forgets the structure map to $X$, this is the same as maps in $Top$ from $X$ to $A$.

Thereby we recover Lurie's theorem, in the case when $X$ is a locally compact CW complex.

...

##### Gros $(\infty,1)$-sheaf $(\infty,1)$-topos

Another alternative is to regard the space $X$ as an object in the [[gros topos|gros]] [[(∞,1)-sheaf]] topos $Sh_{(\infty,1)}(CartSp)$ over the [[site]] [[CartSp]], as described at [[∞-Lie groupoid]]. This has the special property that it is a [[locally ∞-connected (∞,1)-topos]], which means that the [[global section|terminal]] [[geometric morphism]] is an [[essential geometric morphism]]

$$
  (\Pi \dashv LConst \dashv \Gamma)
  :
  Sh_{(\infty,1)}(CartSp)
  \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}}
  \infty Grpd
  \,,
$$

with the further [[left adjoint]] $\Pi$ to $LConst$ being the intrinsic [[schreiber:path ∞-groupoid]] functor.  The intrinsic [[nonabelian cohomology]] in there also coincides with nonabelian cohomology in [[Top]]; even the full [[cocycle]] [[∞-groupoid]]s are equivalent:

+-- {: .un_theorem }
###### Theorem

For [[paracompact space|paracompact]] $X$ we have an equivalence of [[cocycle]] [[∞-groupoid]]s

$$
  Sh_{(\infty,1)}(CartSp)(X, LConst A)
  \simeq
  Top(X, |A|)
$$

and hence in particular an isomorphism on cohomology

$$
  H(X,A) \simeq \pi_0   Sh_{(\infty,1)}(CartSp)(X, LConst A)
$$

=--

+-- {: .proof}
###### Proof

The key point is that for paracompact $X$, the [[nerve theorem]] asserts that $\Pi(X)$ is [[weak homotopy equivalence|weak homotopy equivalent]] to $Sing X$, the standard [[fundamental ∞-groupoid]] of $X$. This is discussed in detail in the section <a href="http://ncatlab.org/schreiber/show/path+%E2%88%9E-groupoid#GeomReal">geometric realization</a> at [[schreiber:path ∞-groupoid]].

Using this, the statement follows by the [[adjoint (∞,1)-functor|(∞,1)-adjunction]] $(\Pi \dashv LConst)$, that is discussed in detail at <a href="http://ncatlab.org/schreiber/show/path+%E2%88%9E-groupoid#Unstruc">Unstructured homotopy ∞-groupoid</a>.

=--

#### Motivic cohomology

[[motivic cohomology|Motivic cohomology]] of a scheme $X$ can be described as the cohomology of the [[Zariski site|Zariski]] (∞,1)-topos of $X$ with coefficients in particular spectrum objects called _motivic complexes_.

#### Hochschild and cyclic cohomology

[[Hochschild cohomology]]
is the cohomology $\mathbf{H}(\mathcal{L}X , C)$ of
[[free loop space object]]s $\mathcal{L}X$ in a [[derived stack]]
[[(∞,1)-topos]] $\mathbf{H}$ with coefficients in
[[quasicoherent ∞-stack|quasicoherent ∞-stacks of modules]] $C$.
There is a natural [[action]] of the circle $S^1$ on
the [[free loop space object]] $\mathcal{L}X$ and the corresponding $S^1$-[[equivariant cohomology]] is [[cyclic cohomology]].

#### Algebraic K-theory

[[K-theory]] in its general form of [[algebraic K-theory]] is a way of turning a [[stable (∞,1)-category]] (which may be the [[derived category]] induced by an [[abelian category]] or [[Quillen exact category]]) into a [[spectrum]].

Accordingly, an [[∞-stack]] with values in stable $(\infty,1)$-categories induces a spectrum valued $\infty$-stack after passing to its K-theory. Homming objects $X$ into these spectrum-valued $\infty$-stacks then produces the corresponding **K-cohomology** of $X$.

This, too, goes back all the way to [[BrownAHT]], where in the second part the homotopy categories of spectrum-valued $\infty$-stacks is considered.

...

### Twisted generalized bivariant cohomology

[[!include twisted generalized cohomology in linear homotopy type theory -- table]]

## 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)-topos]]es) as rather
specific _tools_ or _algorithms_ for constructing
$\mathbf{H}(X,A)$.

### Čech cohomology

For the moment see

* [[Čech cohomology]]

### $Ext$-functor and derived global-sections functor

Using a [[model category]] [[presentable (infinity,1)-category|presentation]] for $\mathbf{H}$ one can
compute $\mathbf{H}(X,A)$ using the [[derived functor]] of the
[[hom-functor]]: called the [[Ext functor]].

Specifically for the [[model structure on simplicial sheaves]]
and $X$ [[representable functor|representable]], one has
by [[Yoneda lemma]] that $Hom(X,A) \simeq A(X)$ 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 $\mathbf{H}(X,A)$.

### Monadic cohomology

* [[monadic descent|Monadic cohomology]], like [[Čech cohomology]], is a way to compute [[cocycle]]s using tools from [[universal algebra]] in general and the theory of [[monad]]s in particular in the presence of extra

## Properties
 {#Properties}

### Dependence on the ambient $\infty$-category
 {#DependenceOnTheAmbientCategory}

Above in the definition it says that cohomology is [[derived hom space|hom ∞-groupoids]] in some ambient [[(∞,1)-category]]. This is a very general definition. Often (and certainly historically) one is interested in more restrictive cases where certain properties of these hom $\infty$-groupoids are required. These in turn correspond to extra properties of the ambient [[(∞,1)-category]].

Here we discuss which properties of the ambient $(\infty,1)$-category imply which properties of its internal notion of cohomology.

| property of ambient [[(∞,1)-category]] | $\Rightarrow$ | property of cohomology |
|--|--|--|
| [[(∞,1)-topos]] | | equivalent to [[principal ∞-bundles]] |
| [[stable (∞,1)-category]] |  | $\mathbb{Z}$-graded |
| ... | | ... |

#### In $(\infty,1)$-toposes
 {#ToposTheory}

The notion of cohomology is particularly interesting within an [[(∞,1)-topos]], and several of the definitions above are directly motivated by this setting.

Apart from there being cocycles and coboundaries, in order to speak of _cohomology_ we tend to require these to do something: namely to classify something.

Cocycles on some object $X$ do come with a notion of classification of certain structures over $X$ in a $(\infty, 1)$-topos, as described in detail at [[principal ∞-bundle]]. As discussed in the proof there, for that classification to work, however, one needs

* [[pullback]]s,

* [[colimit]]s,

* [[universal colimits]],

* [[groupoid object in an (infinity,1)-category|effective group objects]]

in the ambient [[(∞,1)-category]].

Pullbacks are needed in order to obtain the [[principal ∞-bundle]] classified by a cocycle (as its [[homotopy fiber]]), universal colimits and effective group objects are needed in order to show that every principal $\infty$-bundle does come from a cocycle this way.

But this list of properties is essentially that of the <a href="http://ncatlab.org/nlab/show/(infinity%2C1)-topos#GiraudAxioms">(∞,1)-Giraud axioms</a> that characterize those $(\infty,1)$-categories that are $(\infty,1)$-toposes.

> ... needs to be expanded...

## Related concepts

The relation between homology, cohomology and homotopy:

[[!include homotopy-homology-cohomology]]

The ingredients of homology and cohomology:

[[!include chains and cochains - table]]

See also

* [[cohomological integration]]

* [[cohomology in homotopy type theory]]

## References
 {#References}

[[!include cohomology -- early references]]

For more on the pre-history of the notion of cohomology see

* MO entry, <a href="http://mathoverflow.net/questions/39597/timeline-of-cohomology-1935-to-1938">Timeline of cohomology (1935 to 1938)</a>

A bunch of survey information on types of cohomoloy theories is kept here:

* [[Andreas Holmstrom]], _[[Cohomology Theory Database]]_

### General

The general abstract perspective on cohomology as highlighted in the text above was essentially established in:

* [[Kenneth Brown]], _[[BrownAHT|Abstract homotopy theory and generalized sheaf cohomology]]_ (1973)

but probably 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 [[Eilenberg-MacLane spectrum|ordinary cohomology]] and more intricate things like [[K-theory spectrum|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

$$
  H(X,A)
  \coloneqq
  Ho_{SSh}(X,A)
$$

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

The only fundamental new addition to this insight that is available now and was not available in 1973 is that

* These categories $H = Ho_{SSh}$ are precisely the [[homotopy category of an (infinity,1)-category|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 $\infty$-stacks see for instance the introductory pages of

* [[Dan Dugger]], _Sheaves and homotopy theory_ ([web](http://www.uoregon.edu/~ddugger/cech.html), [pdf](http://ncatlab.org/nlab/files/cech.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 perspective of a [[petit topos]], i.e. where one thinks of the [[(∞,1)-topos]] $\mathcal{X}$ as that of [[∞-stack]]s 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 $(\infty,1)$-topos. Accordingly, the cohomology in that section 7.2.2 is defined for the terminal coefficient object and for an [[Eilenberg-MacLane object]] $K(A,n)$:

$$
  H^n(\mathcal{X},A)
  \coloneqq
  \pi_0\mathcal{X}({*}, K(A,n))
$$

(definition 7.2.2.14).

A comprehensive account of the full non-abelian case and its classification of $G$-[[principal ∞-bundles]], $G$-[[∞-gerbes]] and the corresponding [[twisted cohomology]] is in

* {#NSS} [[Thomas Nikolaus]], [[Urs Schreiber]], [[Danny Stevenson]], _[[schreiber:Principal ∞-bundles -- theory, presentations and applications]]_ (2012)

with generalization to [[equivariant cohomology]] in

* [[Hisham Sati]], [[Urs Schreiber]], *[[schreiber:Proper Orbifold Cohomology]]* [[arXiv:2008.01101](https://arxiv.org/abs/2008.01101)]

and with discussion of the [[Chern-Dold character]] on cohomology understood in this generality:

* {#FSS20} [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:The Character Map in Twisted Non-Abelian Cohomology|The Character Map in Nonabelian Cohomology --- Twisted, Differential, Generalized]]_, World Scientific (2023) [[arXiv:2009.11909](https://arxiv.org/abs/2009.11909), [doi:10.1142/13422](https://doi.org/10.1142/13422)]

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](http://math.stanford.edu/~conrad/papers/hypercover.pdf))

[[!redirects cohomologies]]

[[!redirects cohomology class]]
[[!redirects cohomology classes]]


cohomology - contents

**[[cohomology]]**

* [[cocycle]], [[coboundary]], [[coefficient]]

* [[homology]]

  * [[chain]],  [[cycle]], [[boundary]]

* [[characteristic class]]

  * [[universal characteristic class]]

  * [[secondary characteristic class]]

  * [[differential characteristic class]]

* [[fiber sequence]]/[[long exact sequence in cohomology]]

* [[fiber ∞-bundle]], [[principal ∞-bundle]], [[associated ∞-bundle]],
  [[twisted ∞-bundle]]

* [[∞-group extension]]

* [[obstruction]]

### Special and general types ###

* [[cochain cohomology]]

* [[ordinary cohomology]], [[singular cohomology]]

* [[group cohomology]], [[nonabelian group cohomology]], [[Lie group cohomology]]

* [[Galois cohomology]]

* [[groupoid cohomology]], [[nonabelian groupoid cohomology]]

* [[generalized (Eilenberg-Steenrod) cohomology]]

  * [[cobordism cohomology theory]]

  * [[integral cohomology]]

  * [[K-theory]]

  * [[elliptic cohomology]], [[tmf]]

  * [[taf]]

* [[abelian sheaf cohomology]]

  * [[Deligne cohomology]]

    * [[de Rham cohomology]]

    * [[Dolbeault cohomology]]

  * [[etale cohomology]]

    * [[group of units]], [[Picard group]], [[Brauer group]]

  * [[crystalline cohomology]]

  * [[syntomic cohomology]]

  * [[motivic cohomology]]

* [[cohomology of operads]]

* [[Hochschild cohomology]], [[cyclic cohomology]]

  * [[string topology]]

* [[nonabelian cohomology]]

  * [[principal ∞-bundle]]

    * [[universal principal ∞-bundle]], [[groupal model for universal principal ∞-bundles]]

    * [[principal bundle]], [[Atiyah Lie groupoid]]

    * [[principal 2-bundle]]/[[gerbe]]

  * [[locally constant ∞-stack|covering ∞-bundle]]/[[local system]]

  * [[(∞,1)-vector bundle]] / [[(∞,n)-vector bundle]]

  * [[quantum anomaly]]

    * [[orientation]], [[Spin structure]], [[Spin^c structure]], [[String structure]], [[Fivebrane structure]]

* [[cohomology with constant coefficients]] / [[cohomology with a local system of coefficients|with a local system of coefficients]]

* [[∞-Lie algebra cohomology]]

  * [[Lie algebra cohomology]], [[nonabelian Lie algebra cohomology]], [[Lie algebra extensions]], [[Gelfand-Fuks cohomology]],

* [[Gerstenhaber-Schack cohomology|bialgebra cohomology]]

### Special notions

* [[?ech cohomology]]

* [[hypercohomology]]

### Variants ###

* [[equivariant cohomology]]

  * [[equivariant homotopy theory]]

  * [[Bredon cohomology]]

* [[twisted cohomology]]

  * [[twisted bundle]]

    * [[twisted K-theory]], [[twisted spin structure]], [[twisted spin^c structure]]

  * [[twisted differential c-structures]]

    * [[twisted differential string structure]], [[twisted differential fivebrane structure]]

* differential cohomology

  * [[differential cohomology|differential generalized (Eilenberg-Steenrod) cohomology]]

    * [[differential cobordism cohomology]]

    * [[Deligne cohomology]]

    * [[differential K-theory]]

    * [[differential elliptic cohomology]]

  * [[schreiber:differential cohomology in a cohesive topos]]

    * [[Chern-Weil theory]]

    * [[∞-Chern-Weil theory]]

* [[relative cohomology]]

### Extra structure

* [[Hodge structure]]

* [[orientation]], [[orientation in generalized cohomology|in generalized cohomology]]

### Operations ###

* [[cohomology operations]]

* [[cup product]]

* [[connecting homomorphism]], [[Bockstein homomorphism]]

* [[fiber integration]], [[transgression]]

* [[cohomology localization]]

### Theorems

* [[universal coefficient theorem]]

* [[Künneth theorem]]

* [[de Rham theorem]], [[Poincare lemma]], [[Stokes theorem]]

* [[Hodge theory]], [[Hodge theorem]]

  [[nonabelian Hodge theory]], [[noncommutative Hodge theory]]

* [[Brown representability theorem]]

* [[abelian sheaf cohomology|hypercovering theorem]]

* [[Eckmann-Hilton duality|Eckmann-Hilton-Fuks duality]]

[Edit this sidebar](/nlab/edit/cohomology+-+contents)

[[!redirects cohomology ? contents]]
[[!redirects cohomology -- contents]]


cohomology -- early references

### Early references on (co)homology

The original references on [[chain homology]]/[[cochain cohomology]] and [[ordinary cohomology]] in the form of [[cellular cohomology]]:

* [[Andrei Kolmogoroff]], _Über die Dualität im Aufbau der kombinatorischen Topologie_, Recueil Mathématique 1(43) (1936), 97–102.  ([mathnet](http://mi.mathnet.ru/msb5361))

A footnote on the first page reads as follows, giving attribution to [Alexander 35a](#Alexander35a), [35b](#Alexander35a):

> Die Resultate dieser Arbeit wurden für den Fall gewöhnlicher Komplexe vom Verfasser im Frühling und im Sommer 1934 erhalten und teilweise an der Internationalen Konferenz für Tensoranalysis (Moskau) im Mai 1934 vorgetragen. Die hier dargestellte allgemeinere Theorie bildete den Gegenstand eines Vortrages, den der Verfasser an der Internationalen Topologischen Konferenz (Moskau, September 1935) hielt; bei letzterer Gelegenheit erfuhr er, dass ein grosser Teil dieser Resultate im Falle von Komplexen indessen von Herrn Alexander erhalten worden ist. Vgl. die inzwischen erschienenen Noten von Herrn [[J. W. Alexander|Alexander]] in den «Proceedings of the National Academy of Sciences U.S.A.», 21, (1935), 509—512. Herr Alexander trug über seine Resultate ebenfalls an der Moskauer Topologischen Konferenz vor. Verallgemeinerungen für abgeschlossene Mengen und die Konstruktion eines Homologieringes für Komplexe und abgeschlossene Mengen, über welche der Verfasser ebenso an der Tensorkonferenz 1934 vorgetragen hat, werden in einer weiteren Publikation dargestellt. Diese weitere Begriffsbildungen sind übrigens ebenfalls von Herrn Alexander gefunden und teilweise in den erwähnten Noten publiziert.

* [[Andrei Kolmogoroff]], _Homologiering des Komplexes und des lokal-bicompakten Raumes_, Recueil Mathématique 1(43) (1936), 701–705.  [mathnet](http://mi.mathnet.ru/msb5475).

* {#Alexander35a} [[J. W. Alexander]], _On the chains of a complex and their duals_, Proc. Nat. Acad. Sei. USA, 21(1935), 509–511 ([doi:10.1073/pnas.21.8.50](https://doi.org/10.1073/pnas.21.8.50))

* {#Alexander35b} [[J. W. Alexander]], _On the ring of a compact metric space_, Proc. Nat. Acad. Sci. USA, 21 (1935), 511–512 ([doi:10.1073/pnas.21.8.511](https://doi.org/10.1073/pnas.21.8.511))

* [[J. W. Alexander]], _On the connectivity ring of an abstract space_, Ann. of Math., 37 (1936), 698–708 ([doi:10.2307/1968484](https://doi.org/10.2307/1968484), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/alexcon.pdf))

The term “cohomology” was introduced by [[Hassler Whitney]] in

* {#Whitney37} [[Hassler Whitney]], _On matrices of integers and combinatorial topology_.  Duke Mathematical Journal 3:1 (1937), 35–45 ([doi:10.1215/S0012-7094-37-00304-1](https://projecteuclid.org/journals/duke-mathematical-journal/volume-3/issue-1/On-matrices-of-integers-and-combinatorial-topology/10.1215/S0012-7094-37-00304-1.short))

See also

* [[Hassler Whitney]], _On products in a complex_, Annals of Math. 39 (1938) 397–432 ([doi:10.2307/1968795](https://doi.org/10.2307/1968795))

The notion of [[singular cohomology]] is due to

* [[Samuel Eilenberg]], _Singular homology theory_, Annals of Mathematics 45:3 (1944) ([doi:10.2307/1969185](https://doi.org/10.2307/1969185))

The notion of [[monadic cohomology]] via [[canonical resolutions]]:

* [[Michael Barr]], [[Jon Beck]], _Homology and Standard Constructions_, in: _[[Seminar on Triples and Categorical Homology Theory]]_, Lecture Notes in Maths. **80**, Springer (1969), Reprints in Theory and Applications of Categories __18__ (2008) 186-248 \[tac:18, [pdf](http://www.tac.mta.ca/tac/reprints/articles/18/tr18.pdf)\]

* [[Michael Barr]], *Cartan-Eilenberg cohomology and triples*, J. Pure Applied Algebra **112** 3 (1996) 219–238 \[doi:10.1016/0022-4049(95)00138-7, [pdf](https://www.math.mcgill.ca/barr/papers/coho.pdf), [[Barr-CECohomology.pdf:file]]\]

* [[Michael Barr]], *Algebraic cohomology: the early days*, in *Galois Theory, Hopf Algebras, and Semiabelian Categories*, Fields Institute Communications **43** (2004) 1–26 \[doi:10.1090/fic/043, [pdf](https://www.math.mcgill.ca/barr/papers/algcohom.pdf), [[Barr-AlgebraicCohomology.pdf:file]]\]

The general abstract perspective on [[cohomology]] (subsuming [[sheaf cohomology]], [[hypercohomology]], [[non-abelian cohomology]] and indications of [[Whitehead-generalized cohomology]]) was essentially established in:

* [[Kenneth Brown]], _[[BrownAHT|Abstract homotopy theory and generalized sheaf cohomology]]_ (1973)


cohomology group

Recall that [[cohomology]] in an [[(∞,1)-topos]] $\mathbf{H}$ on an object $X$ with coefficients in an object $A$ is the [[hom-set]] in the [[homotopy category of an (∞,1)-category]]

$$
  H(X,A) = \pi_0 \mathbf{H}(X,A)
  \,.
$$

This is the _cohomology set_ .

It is a [[pointed set]] if $A$ is a [[pointed object]].

In the case that $A$ moreover carries the structure of a [[group object]], the set $H(X,A)$ inherits naturally itself the structure of a [[group]]. In this case one speaks of the _cohomology group_ of $X$ with coefficients in $A$.

+-- {: .query}
Dually, is this why n-spheres are good for homotopy as they are cogroups? ---David
=--

#Examples#

# generalized abelian cohomology #

In all of what is called [[generalized cohomology]]  -- which is really generalized _abelian_ cohomology, compare [[nonabelian cohomology]] -- the coefficient object is taken to be not just a [[group object]] but a "maximally abelian" group object called a [[stable (infinity,1)-category|stable object]] in general and called a [[spectrum]] in the case that $\mathbf{H}$ = [[Top]].

In that case all the [[delooping hypothesis|deloopings]] $\mathbf{B}^n A$ of $A$ exists and are still stably abelian group objects.

So in that case not only is the cohomology set $H(X,A)$ naturally an abelian group, but there is an infinite sequence of such cohomology groups, one for each delooping $\mathbf{B}^n A$. This yields the traditional notation for graded cohomology groups by setting

$$
  H^n(X,A) := H(X, \mathbf{B}^n A)
  \,.
$$

## "ordinary" (integral/Eilenberg-Mac Lane-) cohomology ##

The standard example are the **"ordinary" cohomology groups** that come from taking $\mathbf{H} = $ [[Top]] or = [[∞Grpd]] (see [[homotopy hypothesis]]) and choosing the coefficient object to be the [[Eilenberg-Mac Lane spectrum]]

$$
  A := \mathbf{B} \mathbb{Z}
  \,.
$$

The for $X \in \mathbf{H}$ any object (a [[topological space]] or an [[∞-groupoid]]) the "ordinary" cohimology of $X$ in degree $n$ is

$$
  H^n(X) := H^n(X,\mathbb{Z})
  :=
  H(X, \mathbf{B}^n \mathbb{Z})
  =
  \pi_0 \mathbf{H}(X, \mathbf{B}\mathbb{Z})
  =: [X, K(n, \mathbb{Z})]
  \,.
$$

Here on the left we have the standard notation for the ordinary cohomology groups, and on the right their expression in terms of homotopy classes of maps into an [[Eilenberg-Mac Lane space]].

## Cohomology groups in nonabelian cohomology ##

The standard **counter-example** to keep in mind for a [[nonabelian cohomology]] set that does _not_ carry a group structure is "nonabelian cohomology in degree 1" that classifies $G$-[[principal bundles]], for $G$ some nonabelian group.

This cohomology set

$$
  H^1(X,G)
  := H(X, \mathbf{B}G)
  =:
  [X, \mathbf{B} G]
  \simeq
  G Bund(X)/_\sim
$$

clearly has no natural group structure on it, unless $G$ is in fact abelian (in which case $\mathbf{B}G$ is indeed a [[group object]], namely a [[2-group]]).

But when we pass from group-principal bundles to groupoid-principal bundles, then there may be cohomology sets with group structure even in nonabelian cohomology.

Let for instance $G_{(2)}$ be a [[2-group]], i.e. a [[groupoid]] with group structure, such as the automrophism 2-group $G_{(2)} := AUT(H) := Aut_{Grpd}(\mathbf{B}H)$ of an  ordinary group $H$, then there is the nonabelian cohomology set

$$
  H^1(X, G_{(2)}) := H(X, G_{(2)})
  \simeq G_{(2)} GrpdBund(X)/_\sim
  \,.
$$

and this does carry a _nonabelian_ (in general) group structure.

This is to be distinguished from the cohomology set

$$
  H^2(X, G_{(2)}) := H(X, \mathbf{B} G_{(2)})
  \simeq G_{(2)} Bund(X)/_\sim
$$

that classifies $G_{(2)}$ [[principal 2-bundles]] as opposed to groupoid principal 1-bundles and which is not in general a group (unless $G_{(2)}$ in turn is sufficiently abelian).

For $G_{(2)} = AUT(H)$ both these cohomology sets play a role in the description of [[gerbes]] (see [[gerbe (as a stack)]] and [[gerbe (in nonabelian cohomology)]]).

[[!redirects cohomology groups]]


+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### AQFT
+-- {: .hide}
[[!include AQFT and operator algebra contents]]
=--
#### Cohomology
+-- {: .hide}
[[!include cohomology - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

In the [[AQFT]] formalization of [[quantum field theory]] a system is characterized by its [[local net of observables]] on [[spacetime]], which is in particular a [[copresheaf]] of [[algebra]]s. Accordingly, one can consider notions of [[cohomology]] with coefficients in such a [[local net]].

One such notion was introduced in ([Roberts 76](#Roberts76)), there called **local cohomology** or **net cohomology**.  It has been shown to encode the [[DHR superselection theory]] of local nets ([Roberts 90](#Roberts90)).

## References

### Original articles

The concept was first poposed around

* [[John Roberts]] _Local cohomology and superselection structure_ Comm. Math. Phys. Volume 51, Number 2 (1976), 107-119 ([EUCLID](http://projecteuclid.org/euclid.cmp/1103900345))
  {#Roberts76}

* [[John Roberts]], _Mathematical Aspects of Local Cohomology_ talk at Colloqiumon Operator Algebras and their Applications to Mathematical Physics, Marseille 20-24 June, (1977)

* P. Leyland, [[John Roberts]], _The cohomology of nets over Minkowski space_ ([EUCLID](http://projecteuclid.org/euclid.cmp/1103904354))

Here the idea was put forward that [[local net]]s of [[observable]]s should  carry a notion of [[cohomology]] -- or rather of [[nonabelian cohomology]] -- with coefficients in some kind of  [[∞-category]]. Motivated by this John Roberts was one of the first to consider [[strict ∞-categories]]. He conjectured that these are characterized by their [[∞-nerve]]s being [[complicial sets]]. This led [[Ross Street]] to develop the notion of [[oriental]]s and eventually to prove this conjecture. An account of this development is on pages 9-10 of

* [[Ross Street]], _An Australian conspectus of higher category theory_ ([pdf](http://www.math.uchicago.edu/~may/IMA/Street.pdf))

More comments on the role of cohomology in [[AQFT]] are in

* [[John Roberts]], _A survey of local cohomology_ Mathematical Problems in Theoretical Physics Lecture Notes in Physics, (1978) Volume 80/1978

* [[John Roberts]], _Net cohomology and its applications to field theory, Quantum fields-algebras, processes_ (Proc. Sympos., Univ. Bielefeld, Bielefeld, 1978), pp. 239-268, Springer, Vienna (1980).

* [[John Roberts]], _The Search for Quantum Differential Geometry_ Mathematical Problems in Theoretical Physics, Lecture Notes in Physics, (1982) Volume 153/1982, 374-379

* [[John Roberts]], _The cohomology and homology of quantum field theory_ at Quantum fields and quantum space time (Cargèse, 1996), 357-368, NATO Adv. Sci. Inst. Ser. B Phys., 364, Plenum, New York, (1997)

The description of [[DHR superselection theory]] in terms of net cohomology was given in

* [[John Roberts]], (1990)
  {#Roberts90}

Its generalization to general [[spacetimes]] (curved and with nontrivial topology) is discussed in

* [[Giuseppe Ruzzi]], _Homotopy of posets, net-cohomology and superselection sectors in globally hyperbolic spacetimes_ ([arXiv:math-ph/0412014](http://arxiv.org/abs/math-ph/0412014))

* [[Romeo Brunetti]], [[Giuseppe Ruzzi]], _Superselection Sectors and General Covariance I_ ([arXiv:gr-qc/0511118](http://arxiv.org/abs/gr-qc/0511118))

* [[Romeo Brunetti]], [[Giuseppe Ruzzi]], _Quantum charges and spacetime topology: The emergence of new superselection sectors_ ([arXiv:0801.3365](http://arxiv.org/abs/0801.3365))

The trivial Sectors of the Massless scalar free field in 1 + 3
dimensions was discussed in

* Buchholz, Doplicher, Longo, Roberts (1992)

* Fabio Ciolli, _Massless scalar free Field in $1+1$ dimensions, II: Net Cohomology and Completeness of Superselection Sectors_ ([arXiv:0811.4673](http://arxiv.org/abs/0811.4673))

### Talks

* [[John Roberts]], _Cohomology in the service of AQFT_ , talk at Göttingen, _[AQFT, the first 50 years](http://www.uni-math.gwdg.de/aqft/)_ (2009) ([pdf](http://www.lqp.uni-goettingen.de/events/aqft50/slides/1-5-Roberts.pdf))

* [[Giuseppe Ruzzi]], _Net cohomology and local charges_ , talk at Vietri (2009) ([pdf](http://www.mat.uniroma2.it/~mp/OA/activities/09GENCO/slides/Ruzzi-vietri09.pdf))

* Fabio Ciolli, _Net Cohomology and the Construction of Physical Models_ , talk at Vietri (2009) ([pdf](http://www.mat.uniroma2.it/~mp/OA/activities/09GENCO/slides/Ciolli-vietri09.pdf))

[[!redirects cohomology in a local net of observables]]
[[!redirects cohomology in local nets of observables]]
[[!redirects cohomology in a net of local observables]]
[[!redirects cohomology in nets of local observables]]
[[!redirects cohomology of local net of observables]]
[[!redirects cohomology of a local net of observables]]
[[!redirects cohomology of local nets of observables]]
[[!redirects cohomology of a net of local observables]]
[[!redirects cohomology of nets of local observables]]


cohomology in homotopy type theory

[[!redirects ordinary cohomology in homotopy type theory]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Homotopy theory
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea ##

The [[ordinary cohomology|ordinary]] [[cohomology]] [[cohomology group|groups]] are [[algebra|algebraic]] [[invariants]] of [[homotopy types]], and hence of [[types]] in [[homotopy type theory]]. Typically they are much easier to compute than [[homotopy groups]]. There are many theorems in classical algebraic topology relating them other invariants such as the universal coefficient /theorem and the Hurewicz theorem.

## Definition

There are many different flavours of [[cohomology]], but it's usually best to start simple and add features according to its use.

Let $K(G,n)$ be the [[Eilenberg-MacLane space]] of an [[abelian group]] $G$ for some $n : \mathbb{N}$. The (reduced) **_[[ordinary cohomology|ordinary]]_ cohomology group** (of degree $n$ with coefficients in $G$) of a [[pointed]] space $X$ is the following [[set]]:

$$ \bar{H}^n(X ; G) \equiv \| X \to^* K(G,n) \|_0 $$

Note that there is an [[H-space]] structure on $K(G,n)$ naturally, so for any $|f|,|g| : H^n(X;G)$ we can construct an element $|\lambda x . \mu(f(x),g(x))| : H^n(X; G)$, hence we have a [[group]].

Note for any type $X$ we can make this the **_unreduced_ cohomology** (and call it $H$ instead of $\bar{H}$) by simply adding a disjoint basepoint to $X$ giving us $X_+ \equiv X + 1$ making it pointed.

Let $E$ be a [[spectrum]], we can define the (reduced) **_generalized_ cohomology** group of degree $n$ of a [[pointed]] space $X$ is defined as:

$$ \bar{H}^n (X; E) \equiv \| X \to E_n \|_0 $$

note that $E_n$ has a natural [[H-space]] structure as by definition we have $E_n \simeq \Omega E_{n+1}$ giving us the same group operation as before. In fact, ordinary cohomology becomes a special case of generalized cohomology just by taking coefficients in the [[Eilenberg-MacLane spectrum]] $HG$ with $(HG)_n \equiv K(G,n)$.

## Properties ##

Generalized reduced cohomology satisfies the **Eilenberg-Steenrod axioms**:

* ( _Suspension_ ) There is a natural isomorphism
$$ \bar{H}^{n+1} (\Sigma X; E) \simeq \bar{H}^{n} (X; E).$$

* ( _Exactness_ ) For any cofiber sequence $X \to Y \to Z,$ the sequence
$$\bar{H}^{n} (X; E) \to \bar{H}^{n} (Y; E) \to \bar{H}^{n} (Z; E)$$
is an exact sequence of abelian groups.

* ( _Additivity_ ) Given an indexing type $I$ satisfying $0$-choice (e.g. a finite set) and a family $X: I \to U,$ the canonical homomorphism
$$\bar{H}^{n} (\bigvee_{i:I} X_i; E) \to \prod_{i:I}\bar{H}^{n} (X_i; E)$$
is an isomorphism.

Ordinary cohomology also satisfies the _dimension axiom_:

* $\bar{H}^{n} (X, G) = 0$ if $n \neq 0.$

## See also ##

* [[synthetic homotopy theory]]

* [[homotopy groups of spheres in homotopy type theory]]

* [[Hopf construction in homotopy type theory]]

* [[spectral sequence]]

* [[cohomology]]

* [[homology]]

## References
 {#References}

The notion of [[Whitehead generalized cohomology]], [[Brown representability theorem|i.e.]] with [[coefficients]] in any [[spectrum type]]:

* [[Evan Cavallo]], Section 3.2 of: *Synthetic Cohomology in Homotopy Type Theory* (2015) [[pdf](https://staff.math.su.se/evan.cavallo/works/thesis15.pdf), [[Cavallo-CohomologyInHoTT.pdf:file]]]

In the further generality of [[twisted cohomology]] with [[coefficients]] in any [[spectrum type]]:

* [[Floris van Doorn]], Def. 5.4.2 in: _On the Formalization of Higher Inductive Types and Synthetic Homotopy Theory_ (2018) [[arXiv:1808.10690](https://arxiv.org/abs/1808.10690), [pdf](http://florisvandoorn.com/papers/dissertation.pdf)]

Implementation of [[ordinary cohomology|ordinary]] [[integral cohomology]] (i.e. with [[coefficients]] in the [[Eilenberg-MacLane spaces]]/[[Eilenberg-MacLane spectra|-spectra]]) in [[cubical type theory|cubical]] [[Agda]]:

* [[Guillaume Brunerie]], [[Axel Ljungström]], [[Anders Mörtberg]], *Synthetic Integral Cohomology in Cubical Agda*, 30th EACSL Annual Conference on Computer Science Logic (CSL 2022) **216** (2022) $[$[doi:10.4230/LIPIcs.CSL.2022.11](https://doi.org/10.4230/LIPIcs.CSL.2022.11)$]$

Implementation of [[ordinary cohomology|ordinary]]$\;$[[cohomology rings]] in [[cubical agda]]:

* [[Thomas Lamiaux]], [[Axel Ljungström]], [[Anders Mörtberg]], _Computing Cohomology Rings in Cubical Agda_, CPP 2023: Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs (2023) [[arxiv:2212.04182] (https://arxiv.org/abs/2212.04182), [doi:10.1145/3573105.3575677](https://doi.org/10.1145/3573105.3575677)]

[[!redirects cohomology in homotopy type theory]]


cohomology jump loci

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* automatic table of contents goes here
{:toc}

## Idea

(...)

## Definition

Let $\mathcal{M}$ be a moduli space parametrizing objects $P$ that come with $\mathbb{C}$ vector spaces $H^{i}(P)$ for $i\in \mathbb{Z}$ (e.g. the objects can be studied through a cohomology theory).

The **cohomology jump loci** of $\mathcal{M}$ are defined as

$$
  \mathcal{V}^{i}_{k} = \{P\in \mathcal{M} | \dim H^{i}(P) \geq k\}
  \,.
$$

And for every $i$ there is a chain of inclusions

$$
  \mathcal{M}=\mathcal{V}^{i}_{0}\subset\mathcal{V}^{i}_{1}\subset\mathcal{V}^{i}_{2}\subset
  \,...
$$

## References

Introductions and surveys include

* Nero Budur and Botong Wang, _Recent Results on Cohomology Jump Loci_ ([arXiv:1507.06714](http://arxiv.org/abs/1507.06714))


cohomology localization

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## Definition

Let $C$ be a [[stable (∞,1)-category]] (or [[(∞,1)-topos]]) and $A \in C$ a stable coefficient object. For $X \in C$ and $n \in \mathbb{Z}$ or ($n \in \mathbb{N}$) write

$$
  H^n(X,A) := \pi_0 C(X, \mathbf{B}^n A)
$$

for the [[cohomology]] of $X$ with coefficients in $A$ in degree $n$.

Say a [[morphism]] in $C$ is _$A$-local_ if it induces [[isomorphism]]s on all these [[cohomology group]]s. Let $W$ be the [[class]] of all such morphisms.

Then the $A$-**cohomology localization** of $C$ is -- if it exists -- the [[localization of an (∞,1)-category]] of $C$ at $W$.

## Examples

* At [[function algebras on ∞-stacks]] is discussed a class  of examples of cohomology localizations of [[(∞,1)-sheaf (∞,1)-toposes]] over [[site]]s of [[algebra over a Lawvere theory|algebras over]] an abelian [[Lawvere theory]] at cohomology with coefficients in the canonical [[line object]] $\mathbb{A}^1$.

## Related concepts

* [[homology localization]]

## References

[[set theory|Set theoretic]] issues in cohomology localization -- and their solution using [[large cardinal]] axioms such as [[Vopenka's principle]], is discussed in

* [[Carles Casacuberta]], Dirk Scevenels, [[Jeff Smith]], _Implications of large-cardinal principles in homotopical localization_  Advances in Mathematics Volume 197, Issue 1, 20 October 2005, Pages 120-139

* Joan Bagaria, [[Carles Casacuberta]], A. R. D. Mathias, [[Jiri Rosicky]], _Definable orthogonality classes in accessible categories are small_.  Journal of the European Mathematical Society (2015). Volume: 017, Issue: 3, page 549-589, ([arXiv:1101.2792](http://arxiv.org/abs/1101.2792))

[[!redirects homotopy localizations]]


cohomology of a category

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## Idea

The _[[cohomology]] of a [[category]]_ $C$ is often defined to be the [[groupoid cohomology]] of the [[∞-groupoid]] that is presented by the [[nerve]] of the category.

Hence for $\mathbf{A}$ some coefficient [[∞-groupoid]]  -- typically, but not necessarily, an [[Eilenberg-MacLane object]] $\mathbf{A} = K(A,n) = \mathbf{B}^n A$ -- regarded as a [[Kan complex]], the cohomology of $C$ in this sense is

$$
  H^n(C,A) := \pi_0 \infty Grpd( F(N(C)), \mathbf{A} )
  \,,
$$

where

* $N(C)$ is the [[nerve]] of $C$;

* $F(N(C))$ is the [[Kan fibrant replacement]] of $N(C)$;

* [[∞Grpd]] is the [[(∞,1)-category]] of [[∞-groupoid]]s.

Using the standard [[model structure on simplicial sets]] this is the same as the [[hom-set]] in the [[homotopy category]] of [[SSet]]

$$
  \cdots = Ho_{SSet}(N(C), \mathbf{A})
  \,.
$$

One can also use the [[Thomason model structure]] on [[Cat]] to say the same: due to the [[Quillen equivalence]] $Cat_{Thomason} \stackrel{Quillen}{\simeq} SSet_{Quillen}$ we have for $\alpha$ any category whose groupoidification is equivalent to $\mathbf{A}$, i.e. any cateory such that $F(N(\alpha)) \simeq \mathbf{A}$ in [[∞Grpd]], we have

$$
  \cdots = Ho_{Cat_{Thomason}}(C,\alpha)
  \,.
$$


cohomology of dynamical systems

This is a about an example of a [[nonabelian cohomology]] for [[dynamical system]]s and used in [[ergodic theory]]. It is usually in the general setting of dynamical systems in generality of/decoded in terms of [[action groupoid]]s which directly lead to the cohomology.

* Д. В. Аносов, _Об аддитивном функциональном гомологическом уравнении, связанном с эргодическим поворотом окружности_, Изв. АН СССР. Сер. матем., 37:6 (1973), 1259–1274 [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=im&paperid=2361&what=fullt&option_lang=rus); Eng. transl.: D. V. Anosov, _On an additive functional homology equation connected with an ergodic rotation of the circle_, Math. USSR-Izv., 7:6 (1973), 1257–1271  [doi](http://dx.doi.org/10.1070/IM1973v007n06ABEH002086)

* А. Н. Лившиц, _Когомологии динамических систем_, Изв. АН СССР. Сер. матем., 1972, том 36,	выпуск 6, 1296–1320 [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=im&paperid=2373&what=fullt&option_lang=rus); engl. transl. A. N. Livshits, _Cohomology of dynamical systems_, Izv. Akad. Nauk SSSR Ser. Mat. 36:6 (1972) 1296–1320 [mathnet.ru](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=im&paperid=2373&option_lang=eng)

* [[Terence Tao]], [Cohomology for dynamical systems](https://terrytao.wordpress.com/2008/12/21/cohomology-for-dynamical-systems), What's new blog
* Yu. I. Lyubich, _Axiomatic theory of divergent series and cohomological equations_, Fundamenta Math. 198 (2008) [doi](https://dx.doi.org/10.4064/fm198-3-5)
[arxiv/0705.1578](https://arxiv.org/abs/0705.1578)
* appendix to: Host, Kra, _Nonconventional ergodic averages and nilmanifolds_, Annals Math. 2005 [pdf](http://annals.math.princeton.edu/issues/2005/Jan2005/HostKra.pdf)
* A. A. Kirillov, Динамические системы, факторы и представления групп, УМН (1967) том 22,	в. 5(137) стр. 67–80 [pdf](http://www.mathnet.ru/links/9e0af673c89887a559d6ef7fb42a6354/rm5790.pdf) (Russian); Engl. transl. _Dynamical systems, factors and representations of groups_, Russian Mathematical Surveys, 1967, 22:5, 63–75

[[!redirects cohomology for dynamical systems]]


cohomology of operads

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## Idea

Operads -- or rather [[(∞,1)-operads]] such as given by [[topological operads]] or [[dendroidal sets]] -- form an [[(∞,1)-category]], and hence there is canonically a notion of [[cohomology]] associated with them, given as the sets of [[homotopy]] classes of [[morphism]]s out of a given operad into a given coefficient operad.

Explicitly this has been considered in ([Rezk](#Rezk)).

## References

* [[Charles Rezk]], _Spaces of algebra structures and cohomology of operads_ PhD thesis (1996) ([dvi](http://www.math.uiuc.edu/~rezk/rezk-thesis.dvi))
 {#Rezk}


cohomology operation

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## Idea

A _cohomology operation_ is a family of [[morphisms]] between [[cohomology groups]], which is [[natural transformation|natural]] with respect to the base space.

Equivalently, if the [[cohomology theory]] has a [[classifying space]] (as it does for all usual notions of [[cohomology]], in particular for all [[Whitehead-generalized cohomology theories]]) then, by the [[Yoneda lemma]], (non-abelian/unstable) cohomology operations are in natural bijection with [[homotopy classes]] of maps between classifying spaces.

$$
  \{
    H^k(-, E)
    \to
    H^l(-, F)
  \}
  \simeq
  Ho(E_k, F_l)
  \,.
$$

(This statement is made fully explicit for instance below def. 12.3.22 in ([Aguilar-Gitler-Prieto](#AguilarGitlerPrieto)).

If here $E_\bullet$ and $F_\bullet$ are the component spaces of [[spectra]] and if such cohomology operations are compatible with the [[suspension isomorphism]], then one speaks of a _stable cohomology operation_.

## Examples

### List of examples

* every [[universal characteristic class]] is a cohomology operation.

* [[Bockstein homomorphism]]

* [[cup product]]

* [[Massey product]]

* [[power operation]]

  * [[Steenrod squares]] are the stable cohomology endo-operations on [[ordinary cohomology]] (mod 2)

  * [[Adams operations]] are the endo-cohomology operations on [[K-theory]]

* [[logarithmic cohomology operation]]

### Cup powers in multiplicative cohomology
 {#CupPowersInMultiplicativeCohomology}

\begin{example}
  \label{CupSquareCohomologyOperation}
  [**Cup-square cohomology operation**]

Let $E$ be a [[multiplicative cohomology theory|multiplicative]] [[Whitehead-generalized cohomology theory]] [[Brown representability theorem|represented]] by a [[ring spectrum]]

$$
  \big(
    E, 1^E, m^E
  \big)
  \;\in\;
  CommutativeMonoids
  \Big(
    Ho\big( Spectra\big),
    \mathbb{S},
    \wedge
  \Big)
  \,.
$$

Then for all $n \in \mathbb{N}$ there is an unstable $E$-cohomology operation

\[
  \label{CupSquareAsUnstableCohomologyOperation}
  \big[
    \Omega^\infty \Sigma^n E
    \overset{
      (-)^{2_\cup}
    }{\longrightarrow}
    \Omega^\infty \Sigma^{2 n} E
  \big]
  \;\;\in\;\;
  \widetilde E^{ 2 n }
  \big(
    \Omega^{\infty - n} E
  \big)
\]

defined -- via the [[Yoneda lemma]] on the [[opposite category|opposite]] of the [[classical homotopy category]] of [[pointed topological spaces]] -- by acting over any $X \,\in\, Ho\big(PointedTopologicalSpaces\big)$ as the [cup square on E-cohomology](cup+product#CupProductInWhiteheadGeneralizedCohomologyTheory) on cohomology in degree $n$:

\[
  \label{CupSquareInMultiplicativeCohomologyNaturalTransformation}
  X
  \;\;\mapsto\;\;
  \Big(
  [X, \Omega^\infty \Sigma^n E]
  \,=\,
  {\widetilde E}{}^n(X)
  \overset{
    \;\;
    \Delta_{{\widetilde E}{}^n(X)}
    \;\;
  }{\longrightarrow}
  {\widetilde E}{}^n(X)
  \times
  {\widetilde E}{}^n(X)
  \overset{ \;\;
    \cup^E_X
  \;\; }{\longrightarrow}
  {\widetilde E}{}^n(X)
  \,=\,
  [X, \Omega^\infty \Sigma^{2n} E]
  \Big)
\]

Here the first [[function]] $\Delta_{{\widetilde E}{}^n(X)}$ is the [[diagonal morphism|diagonal]] on the _set_ underlying the degree=$n$ $E$-[[cohomology group]] of $X$, while the second function $\cup^E_X$ is the operation of forming the [E-cup product](cup+product#CupProductInWhiteheadGeneralizedCohomologyTheory) of [[pairs]] of its elements.

Both of these operations are clearly [[natural transformations]] (for the first this is evident, for the second this comes down to the fact that the [smash-monoidal diagonals of suspension spectra](suspension+spectrum#SmashMonoidalDiagonals) are natural, hence are indeed [[monoidal category with diagonals|monoidal diagonals]]). Therefore the [[full subcategory|fully-faithfulness]] of the [[Yoneda embedding]]

$$
  Ho
  \big(
    PointedTopologicalSpaces
  \big)
  \;\overset{ \;\; A \mapsto [-,A]\;\; }{\hookrightarrow}\;
  Functors
  \Big(
    Ho
    \big(
      PointedTopologicalSpaces
    \big)^{op}
    ,
    Set
  \Big)
$$

implies that this natural transformation (eq:CupSquareInMultiplicativeCohomologyNaturalTransformation) between the [[representable functors]] $[-,\Omega^\infty \Sigma^n E]$ and $[-,\Omega^\infty \Sigma^{2n} E]$ is itself represented by a [[morphism]] (eq:CupSquareAsUnstableCohomologyOperation) between the representing classifying spaces, hence by an unstable cohomology operation.

\end{example}

The analogous construction of course exists for the $k$th cup power
$(-)^{k_\cup}$ for any $k \in \mathbb{N}$. Brief mentioning of this is in [Wirthmüller 12, Example (1) on p. 44 (46 of 67) ](#Wirthmuller12).

## References

### General

Steenrod's original colloquium lectures:

* [[Norman Steenrod]], _Cohomology operations, and obstructions to extending continuous functions_,  Advances in Math. 8, 371–416. (1972)  (doi:10.1016/0001-8708(72)90004-7, [pdf](http://www.maths.ed.ac.uk/%7Eaar/papers/steen6.pdf))

Background and review:

* [[John Michael Boardman]], _Stable Operations in Generalized Cohomology_ [[pdf](https://math.jhu.edu/~wsw/papers2/math/28a-boardman-stable.pdf), [[Boardman-StableOperations.pdf:file]]] in: [[Ioan Mackenzie James]] (ed.) _[[Handbook of Algebraic Topology]]_ Oxford 1995 ([doi:10.1016/B978-0-444-81779-2.X5000-7](https://doi.org/10.1016/B978-0-444-81779-2.X5000-7))

* {#May} [[Peter May]], chapter 22, section 5 of: _A concise course in algebraic topology_, University of Chicago Press 1999 (ISBN:978-0226511832, [pdf](https://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf))

On the structure on the collection of all unstable cohomology operations:

* [[John Michael Boardman]], [[David Copeland Johnson]], [[W. Stephen Wilson]], _Unstable Operations in Generalized Cohomology_ ([pdf](https://hopf.math.purdue.edu/Boardman-Johnson-Wilson/bjw.pdf)), in: [[Ioan Mackenzie James]] (ed.) _[[Handbook of Algebraic Topology]]_ Oxford 1995 ([doi:10.1016/B978-0-444-81779-2.X5000-7](https://doi.org/10.1016/B978-0-444-81779-2.X5000-7))

* [[Andrew Stacey]], [[Sarah Whitehouse]], _The Hunting of the Hopf Ring_,     Homology Homotopy Appl. Volume 11, Number 2 (2009), 75-132. ([arXiv:0711.3722](https://arxiv.org/abs/0711.3722), [euclid:hha/1251832594](https://projecteuclid.org/euclid.hha/1251832594))

* [[Tilman Bauer]], _Formal plethories_, Advances in Mathematics Volume 254, 20 March 2014, Pages 497-569 ([arXiv:1107.5745](https://arxiv.org/abs/1107.5745), [doi:10.1016/j.aim.2013.12.023](https://doi.org/10.1016/j.aim.2013.12.023))

* [[William Mycroft]], _Unstable Cohomology Operations: Computational Aspects of Plethories_, 2017 ([pdf](https://etheses.whiterose.ac.uk/20524/1/wm-thesis-final.pdf), [[MycroftUnstableCohomologyOperations.pdf:file]])

### On ordinary cohomology

Operations on [[ordinary cohomology]]:

* [[Robert Mosher]], [[Martin Tangora]], _Cohomology Operations and Application in Homotopy Theory_, Harper and Row (1968), Dover (2008) ([ISBN10:0486466647](https://store.doverpublications.com/0486466647.html), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/moshtang.pdf))

* [[Andrew Stacey]], [[Sarah Whitehouse]], _Stable and unstable operations in mod $p$ cohomology theories_, Algebr. Geom. Topol. Volume 8, Number 2 (2008), 1059-1091 ([arXiv:math/0605471](https://arxiv.org/abs/math/0605471), [euclid:agt/1513796856](https://projecteuclid.org/euclid.agt/1513796856))

### On topological K-theory

Operations on [[topological K-theory]]:

* {#Wirthmuller12} [[Klaus Wirthmüller]], Sections 11 and 13 of: _Vector bundles and K-theory_, 2012 ([[wirthmueller-vector-bundles-and-k-theory.pdf:file]])

For [[Adams operations]]:

* {#AguilarGitlerPrieto} Marcelo Aguilar, [[Samuel Gitler]], Carlos Prieto, Section 10 of: _Algebraic topology from a homotopical viewpoint_, Springer (2002) ([doi:10.1007/b97586](https://link.springer.com/book/10.1007/b97586))

More:

* [[William Mycroft]], [[Sarah Whitehouse]], _The plethory of operations in complex topological K-theory_ ([arXiv:2001.01608](https://arxiv.org/abs/2001.01608))

### On complex-oriented cohomology theories

For [[Steenrod operations]] on [[complex oriented cohomology|complex-oriented]] [[Whitehead-generalized cohomology|generalized cohomology]] (such as [[MU]] and [[BP]]):

* {#Kochman96} [[Stanley Kochman]], section 2.5 and 3.5 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

### On differential cohomology

Refinement to [[differential cohomology]]:

* [[Daniel Grady]], [[Hisham Sati]], _Primary operations in differential cohomology_, Adv. Math. 335 (2018), 519-562 ([arXiv:1604.05988](https://arxiv.org/abs/1604.05988), [doi:10.1016/j.aim.2018.07.019](https://www.sciencedirect.com/science/article/pii/S0001870818302676?via%3Dihub))

[[!redirects cohomology operations]]


cohomology ring

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## Idea

Given any [[generalized (Eilenberg-Steenrod) cohomology theory]] $E$, then for each [[topological space]] $X$, there is, by definition, the [[graded abelian group]]

$$
  E^\bullet(X) \in Ab^{\mathbb{Z}}
  \,.
$$

This is the _$E$-[[cohomology group]]_ of $X$. Now if $E$ is a [[multiplicative cohomology theory]], then these groups inherit the structure of [[rings]]. As such

$$
  E^\bullet(X) \in Ring^{\mathbb{Z}}
$$

is the **$E$-cohomology ring** of $X$.

Analogously for various suitable generalizations of the nature of $E$ and $X$  (see at _[[generalized cohomology]]_).

## Related concepts

* [[quantum cohomology ring]]

* [[genus]]

* [[orientation in generalized cohomology]]

## References

Implementation of [[ordinary cohomology|ordinary]]$\;$[[cohomology rings]] in [[cubical agda]]:

* [[Thomas Lamiaux]], [[Axel Ljungström]], [[Anders Mörtberg]], _Computing Cohomology Rings in Cubical Agda_, CPP 2023: Proceedings of the 12th ACM SIGPLAN International Conference on Certified Programs and Proofs (2023) [[arxiv:2212.04182] (https://arxiv.org/abs/2212.04182), [doi:10.1145/3573105.3575677](https://doi.org/10.1145/3573105.3575677)]

[[!redirects cohomology rings]]

[[!redirects homology ring]]
[[!redirects homology rings]]


cohomology theory > history

< [[cohomology theory]]


Cohomology Theory Database

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A database of types of [[cohomology]] [[cohomology theory|theories]]:

* [[Andreas Holmstrom]]

  _Cohomology Theory Database_

  ([web](http://notes.andreasholmstrom.org/index.php))

See also _[[letter theory]]_.

* [[absolute cohomology]]

* [[algebraic K-theory]]

* ...