group cohomology, nonabelian group cohomology, Lie group cohomology
Hochschild cohomology, cyclic cohomology?
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
algebraic topology – application of higher algebra and higher category theory to the study of (stable) homotopy theory
(2,1)-quasitopos?
structures in a cohesive (∞,1)-topos
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 hom ∞-groupoid, hence of homotopy classes of morphisms from $X$ to $A$ in $\mathbf{H}$:
The (∞,1)-category $\mathbf{H}$ is usually an (∞,1)-topos, where the notion of cohomology is particularly well-behaved. 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 $A_n$:
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 $n$-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 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), 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
The following section
gives a tour through the zoo of cohomology theories traditionally known, indicating how they all fit into this picture. Then the section
gives the general formal definition and discusses general properties of and constructions in cohomology theory, such as the terminology of cocycles and coboundaries of objects classified by cohomology, of characteristic classes of these objects, of Postnikov towers and Whitehead towers, and so on. In particular the section
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
before the next main section
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
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
in the light of fully fledged (∞,1)-topos theory, of which it is effectively the seed, by noticing that this article secretly discusses precisely the homotopy categories of hypercomplete (∞,1)-toposes. At the same time, to some extent everything here is also an afterthought of the theory of cohomology in 1-categorical topos theory as reviewed for instance in
by noticing that the constructions on simplicial objects in toposes used there secretly precisely compute the (∞,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
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.
…
The statement of the above slogan is well familiar for the special case that $\mathbf{H} =$ Top is the (∞,1)-topos of topological spaces. In this context for instance for $A := K(\mathbb{Z}, n)$ an Eilenberg-MacLane space, we have that for $X$ any topological space that
coincides with the “ordinary” integral cohomology of $X$, 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, etc.
Another big complex of notions of cohomology that on first sight maybe does not seem to fit into this pattern is abelian sheaf cohomology. Usually this is introduced and defined in the language of derived functors. However, derived functors are nothing but a tool, or presentation, for encoding (∞,1)-categorical hom-spaces such as $\mathbf{H}(X,A)$ in cases where $\mathbf{H}$ is presented by a homotopical category or model category.
Indeed, it turns out that an old result from the 1960s, Verdier’s hypercovering theorem effectively shows that what was introduced as abelian sheaf cohomology is really nothing but an instance of the above general setup. A particularly clear-sighted understanding of this fact was presented in
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-spaces: for instance Čech cohomology computes these spaces by finding cofibrant models for the domain $X$, called Čech nerves. Dual to that, most texts on abelian sheaf cohomology find fibrant models for the codomain $A$: called injective resolutions. Both algorithms in the end compute the same intrinsically defined $(\infty,1)$-categorical hom-space.
In other words, abelian sheaf cohomology is of the exact same nature as the familiar cohomology of topological spaces (and hence of spectra) if only we switch from the archetypical (∞,1)-topos Top to a more general ∞-stack (∞,1)-topos. And abelian sheaf cohomology in turn subsumes many special cases, such as Deligne cohomology, 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 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 bundles.
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 gerbes and, more generally, principal 2-bundles. The celebrate treatise by Giraud Cohomologie non abélienne is concerned with this case. In fact, Giraud considered gerbes on stacks and hence was implicitly really computing cohomology in a stack 2-topos with both the domain and the coefficient object allowed to have nontrivial homotopy groups of stacks in degree 2.
Conceptually, with higher topos theory in hand, there is no problem in generalizing nonabelian cohomology and its relation to gerbes and principal bundles further from stacks to ∞-stacks. For instance, while the discussion of spin structure on a space/∞-stack requires a 1-stack coefficient object and classifies principal bundles, and the discussion of string structure requires a 2-stack coefficient object and classifies gerbes and principal 2-bundles, the next case of fivebrane structure requires 6-stack coefficient objects and classifies principal 6-bundles. Generally, we may speak of principal ∞-bundles in any (∞,1)-topos $\mathbf{H}$: these are nothing but the homotopy fibers of the corresponding (“nonabelian”) cocycles, 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 deloopings of groups. What is called nonabelian group cohomology is nothing but the general case of this where there is no restriction on the coefficient object $A$. Here we can once again replace $\infty Grpd$ – 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 manifolds, then the objects of $\mathbf{H} = Sh_{(\infty,1)}(Diff)$ are Lie ∞-groupoids. Their cohomology is generalized group cohomology that knows about smooth structure: smooth group cohomology . In this context for instance one can give cohomological interpretations of smooth realizations of the string 2-group or the fivebrane 6-group.
Conversely, given an unconstrained (unstable) (∞,1)-category $\mathbf{H}$ with its general notion of nonabelian cohomology, one can systematically find its stable or abelian content by considering objects that are components of spectrum objects in $\mathbf{H}$. 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 fibers of such (∞,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 principal bundle (hence its associated vector bundle) to the lifting gerbe for the lift of its structure group to the full unitary group.
Another example of twisted cohomology is differential cohomology: differential cohomology refinements of abelian generalized (Eilenberg-Steenrod) cohomology theories with coefficient objects a spectrum $E$ is the study of the homotopy fibers of the Chern character map $ch : \mathbf{H}(X,E) \to \Omega^\bullet_{dR}(X)\otimes \pi_\bullet(E)$ from $E$-cohomology to deRham cohomology. This classifies (abelian versions of) connections on the underlying bundles, for instance Simons-Sullivan structured bundles (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 differential cohomology in an (∞,1)-topos encoding connections on general principal ∞-bundles and associated ∞-vector bundles.
Slogan.
Thousand and one definitions of notions of cohomology and its variants. From the nPOV, just a single concept: an ∞-categorical hom-space in an (∞,1)-topos.
We give now the very general definition of cohomology and describe very general properties of and very general constructions in cohomology theory.
Given an (∞,1)-category $\mathbf{H}$, for any two objects $X$, $A$ of $\mathbf{H}$ we have the (∞,1)-categorical hom-space $\mathbf{H}(X,A)$ – an ∞-groupoid. For $H = Ho_{\mathbf{H}}$ the 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 cocycles on $X$ with coefficients in $A$;
if $A$ is understood to be equipped with the structure ${*} \to A$ of a pointed object, then the cocycle $X \to {*} \to A$ is the trivial cocycle $c_{triv}$;
the morphisms $\lambda : c_1 \to c_2$ 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
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
is the characteristic class of $c$ with respect to $k$.
Remark Notice that there is no notion of cochain in this general setup. What are called cochains are specifically components of certain specific models for $\mathbf{H}(X,A)$. More on this in the section on abelian cohomology below.
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 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.
For $A \in \mathbf{H}$ some coefficient object and $\{c_n : A \to E_n\}$ a collection of cocycles on the coefficient object with values in objects $E_n \in \mathbf{H}$ – typically chosen to be Eilenberg-MacLane objects – composition of morphism in $\mathbf{H}$ induces a map of cohomology ∞-groupoids
and hence of cohomology classes
that sends each $A$-cocycle $g$ to its characteristic class $c_n(g)$. Typically, for $P \to X$, the principal ∞-bundle classified by $g$, one speaks of the characteristic class $c_n(P)$ of this principal $\infty$-bundle.
Extra stuff, structure, property on the coefficient object $A$ will induce corresponding stuff, structure or property on the cohomology sets $H(X,A)$.
In the case that the coefficient object $A$ admits $(n \in \mathbb{N})$ deloopings to objects $\mathbf{B}^n A$ one writes
and speaks of $A$-cohomology in degree $n$.
Similarly, looping defines negative degree cohomology:
Because loop space objects are defined by an $(\infty,1)$-pullback and the (∞,1)-categorical hom – as any hom-functor – preserves limits in its second argument, this is the same as
This means that all the non-positive degree cohomology identifies with the homotopy groups of the ∞-groupoid $\mathbf{H}(X,A)$.
In many cases, the (∞,1)-category $\mathbf{H}$ is related to a symmetric monoidal (∞,1)-category $\mathbf{S}$ via a symmetric monoidal adjunction
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 $\mathbf{S}$:
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 $S^n:=\Sigma^n(1)$ gives the integer grading discussed above in the special case when the coefficient object is a spectrum object. This is discussed further below.
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 $E$-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.
Often the coefficient object $A \in \mathbf{H}$ for cohomology is taken to be indefinitely 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
this means that $A$ is of the form
for some spectrum object $E$, and some integer $n$ (not necessarily a natural number).
One single such spectrum object this way yields a $\mathbb{Z}$-graded tower of cohomologies
which taken together, denoted $H^\bullet(X,E)$ is called a cohomology theory. For the case that $\mathbf{H} =$ Top this special case of cohomology is called generalized (Eilenberg-Steenrod) cohomology.
If $A$ happens to be a 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 groups $H^\bullet(X,A)$ form a graded ring, the cohomology ring of $X$ with coefficients in $A$.
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.
A special type of characteristic class is the Chern character. The twisted cohomology with respect to the Chern character is differential 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)
representation theory and equivariant cohomology in terms of (∞,1)-topos theory/homotopy type theory (FSS 12 I, exmp. 4.4):
homotopy type theory | representation theory |
---|---|
pointed connected context $\mathbf{B}G$ | ∞-group $G$ |
dependent type on $\mathbf{B}G$ | $G$-∞-action/∞-representation |
dependent sum along $\mathbf{B}G \to \ast$ | coinvariants/homotopy quotient |
context extension along $\mathbf{B}G \to \ast$ | trivial representation |
dependent product along $\mathbf{B}G \to \ast$ | homotopy invariants/∞-group cohomology |
dependent product of internal hom along $\mathbf{B}G \to \ast$ | equivariant cohomology |
dependent sum along $\mathbf{B}G \to \mathbf{B}H$ | induced representation |
context extension along $\mathbf{B}G \to \mathbf{B}H$ | restricted representation |
dependent product along $\mathbf{B}G \to \mathbf{B}H$ | coinduced representation |
spectrum object in context $\mathbf{B}G$ | spectrum with G-action (naive G-spectrum) |
under base change down to the base (∞,1)-topos $\mathbf{H}$:
cohomology in the presence of ∞-group $G$ ∞-action:
Borel equivariant cohomology | $\phantom{AAA}\leftarrow\phantom{AAA}$ | general (Bredon) equivariant cohomology | $\phantom{AAA}\rightarrow\phantom{AAA}$ | non-equivariant cohomology with homotopy fixed point coefficients |
---|---|---|---|---|
$\phantom{AA}\mathbf{H}(X_G, A)\phantom{AA}$ | trivial action on coefficients $A$ | $\phantom{AA}[X,A]^G\phantom{AA}$ | trivial action on domain space $X$ | $\phantom{AA}\mathbf{H}(X, A^G)\phantom{AA}$ |
For the moment see
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
the $A$-cohomology of $S^n$.
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…
Classes of special cases of cohomologies with their own entries include
Hochschild cohomology / cyclic cohomology?
The probably most familiar kind of cohomology is that of a cochain complex dual to a chain complex.
Using the Dold-Kan correspondence chain complexes are understood as particular spectra, i.e. spectrum objects in the archetypical (∞,1)-topos ∞Grpd of ∞-groupoids. Positively graded chain complexes (the “connective” ones) are just ∞-groupoids with the structure of a strict abelian group object: as Kan complexes these are abelian simplicial groups.
This way, ordinary chain cohomology is seen to be a special case of general cohomology in $\mathbf{H} =$ ∞Grpd. A more detailed discussion of how from this perspective the usual formulas for cochains and cocycles appear is at
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 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
is the $G$-principal bundle classified by the cocycle $g$.
A Grothendieck–Rezk–Lurie (∞,1)-topos is an (∞,1)-category of (∞,1)-sheaves. Its objects are often called ∞-stacks or derived stacks.
Abelian sheaf cohomology 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 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 resolutions).
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. Let’s sort this out, improve the entry and remove this query box here.
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:
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)-sheaf (∞,1)-toposes $\mathbf{H} = Sh_{(\infty,1)}(C)$ over nontrivial (∞,1)-sites $C$. The intrinsic cohomology of such $\mathbf{H}$ is a 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 $X$).
For $X$ a topological space and $Op(X)$ its category of open subsets equipped with the canonical structure of an (∞,1)-site, let
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
be the global sections 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 (∞,1)-sheaf on $A$
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 sections functor, this is equivalently
If $X$ is a paracompact space, then these two definitions of nonabelian cohomology of $X$ with constant coefficients $A \in \infty Grpd$ agree:
This is HTT, theorem 7.1.0.1. See also (∞,1)-category of (∞,1)-sheaves for more.
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 CW complex. In particular, this implies that it is “hereditarily 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
it is proved that the “sheaf of sections” functor
is the right adjoint in a right Quillen embedding?, i.e. a Quillen adjunction whose derived right adjoint is 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 spaces 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.
…
Another alternative is to regard the space $X$ as an object in the 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 terminal geometric morphism is an essential geometric morphism
with the further left adjoint $\Pi$ to $LConst$ being the intrinsic path ∞-groupoid functor. The intrinsic nonabelian cohomology in there also coincides with nonabelian cohomology in Top; even the full cocycle ∞-groupoids are equivalent:
For paracompact $X$ we have an equivalence of cocycle ∞-groupoids
and hence in particular an isomorphism on cohomology
The key point is that for paracompact $X$, the nerve theorem asserts that $\Pi(X)$ is weak homotopy equivalent to $Sing X$, the standard fundamental ∞-groupoid of $X$. This is discussed in detail in the section geometric realization at path ∞-groupoid.
Using this, the statement follows by the (∞,1)-adjunction $(\Pi \dashv LConst)$, that is discussed in detail at Unstructured homotopy ∞-groupoid.
Motivic cohomology of a scheme $X$ can be described as the cohomology of the Zariski (∞,1)-topos of $X$ with coefficients in particular spectrum objects called motivic complexes.
is the cohomology $\mathbf{H}(\mathcal{L}X , C)$ of free loop space objects $\mathcal{L}X$ in a derived stack (∞,1)-topos $\mathbf{H}$ with coefficients in quasicoherent ∞-stacks of modules $C$. There is a natural action of the circle $S^1$ on the free loop space object $\mathcal{L}X$ and the corresponding $S^1$-equivariant cohomology is cyclic cohomology?.
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 cohomology theory is conjecturally ∞-categorical semantics of linear homotopy type theory:
Various notions called “cohomology” in the literature are not so much specific examples of cohomology theories (specific choices of ambient (∞,1)-toposes) as rather specific tools or algorithms for constructing $\mathbf{H}(X,A)$.
For the moment see
Using a model 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, 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 $\mathbf{H}(X,A)$.
Above in the definition it says that cohomology is 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 | |
… | … |
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
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 (∞,1)-Giraud axioms that characterize those $(\infty,1)$-categories that are $(\infty,1)$-toposes.
… needs to be expanded…
The relation between homology, cohomology and homotopy:
homotopy | cohomology | homology | |
---|---|---|---|
$[S^n,-]$ | $[-,A]$ | $(-) \otimes A$ | |
category theory | covariant hom | contravariant hom | tensor product |
homological algebra | Ext | Ext | Tor |
enriched category theory | end | end | coend |
homotopy theory | derived hom space $\mathbb{R}Hom(S^n,-)$ | cocycles $\mathbb{R}Hom(-,A)$ | derived tensor product $(-) \otimes^{\mathbb{L}} A$ |
The ingredients of homology and cohomology:
$H_n = Z_n/B_n$ | (chain-)homology | (cochain-)cohomology | $H^n = Z^n/B^n$ |
---|---|---|---|
$C_n$ | chain | cochain | $C^n$ |
$Z_n \subset C_n$ | cycle | cocycle | $Z^n \subset C^n$ |
$B_n \subset C_n$ | boundary | coboundary | $B^n \subset C^n$ |
See also
The original references on chain homology/cochain cohomology and ordinary cohomology in the form of cellular cohomology:
A footnote on the first page reads as follows, giving attribution to Alexander 35a, 35b:
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 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.
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)
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)
J. W. Alexander, On the connectivity ring of an abstract space, Ann. of Math., 37 (1936), 698–708 (doi:10.2307/1968484, pdf)
The term “cohomology” was introduced by Hassler Whitney in
See also
The notion of singular cohomology is due to
The general abstract perspective on cohomology (subsuming sheaf cohomology, hypercohomology, non-abelian cohomology and indications of Whitehead-generalized cohomology) was essentially established in:
but probably known in one form or other before that.
The general abstract perspective on cohomology as highlighted in the text above was essentially established in:
but probably known in one form or other before that.
This article establishes that
all of abelian sheaf cohomology
all generalized (Eilenberg-Steenrod) cohomology
as well as nonabelian cohomology
as well as the more mundane special cases of this like group cohomology and, yes, cohomology of cochain complexes itself
are naturally special cases of one single concept: that of hom-sets
in the homotopy category of ∞-groupoid-valued sheaves.
The only fundamental new addition to this insight that is available now and was not available in 1973 is that
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
The general abstract picture of cohomology as connected components of mapping spaces in (∞,1)-toposes is the topic of section 7.2.2 of
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 ∞-stacks on a given space $X$ (instead of as a gros topos of all generalized spaces, as we do in the above entry). Accordingly then from that perspective one wants to study the cohomology of $X$ itself, which corresponds to the terminal object in the $(\infty,1)$-topos. Accordingly, the cohomology in that section 7.2.2 is defined for the terminal coefficient object and for an Eilenberg-MacLane object $K(A,n)$:
(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
Another reference with a discussion of cohomology in the general sense discussed above, using tools of model category theory for simplicial objects, is
For more on the pre-history of the notion of cohomology see
A bunch of survey information on types of cohomoloy theories is kept here: