cohomology

# Contents

## Idea

In the general context of cohomology, as described there, a cocycle representing a cohomology class on an object $X$ with coefficients in an object $A$ is a morphism $c : X \to A$ in a given ambient (∞,1)-topos $\mathbf{H}$.

The same applies with the object $A$ taken as the domain object: for $B$ yet another object, the $B$-valued cohomology of $A$ is similarly $H(A,B) = \pi_0 \mathbf{H}(A,B)$. For $[k] \in H(A,B)$ any cohomology class in there, we obtain an ∞-functor

$[k(-)] : \mathbf{H}(X,A) \to H(X,B)$

from the $A$-valued cohomology of $X$ to its $B$-valued cohomology, simply from the composition operation

$\mathbf{H}(X,A) \times \mathbf{H}(A,B) \to \mathbf{H}(X,B) \,.$

Quite generally, for $[c] \in H(X,A)$ an $A$-cohomology class, its image $[k(c)] \in H(X,B)$ is the corresponding characteristic class.

Notice that if $A = \mathbf{B}G$ is connected, an $A$-cocycle on $X$ is a $G$-principal ∞-bundle. Hence characteristic classes are equivalently characteristic classes of principal $\infty$-bundles.

From the nPOV, where cocycles are elements in an (∞,1)-categorical hom-space, forming characteristic classes is nothing but the composition of cocycles.

In practice one is interested in this notion for particularly simple objects $B$, notably for $B$ an Eilenberg-MacLane object $\mathbf{B}^n K$ for some component $K$ of a spectrum object. This serves to characterize cohomology with coefficients in a complicated object $A$ by a collection of cohomology classes with simpler coefficients. Historically the name characteristic class came a little different way about, however (see also historical note on characteristic classes).

Then with the usual notation $H^n(X,K) := H(X, \mathbf{B}^n K)$ a given characteristic class in degree $n$ assigns

$[k(-)] : \mathbf{H}(X,A) \to H^n(X,K) \,.$

Moreover, recall from the discussion at cohomology that to every cocycle $c : X \to A$ is associated the object $P \to X$ that it classifies – its homotopy fiber – which may be thought of as an $A$-principal ∞-bundle over $X$ with classifying map $X \to A$. One typically thinks of the characteristic class $[k(c)]$ as characterizing this principal ∞-bundle $P$.

## Examples

### Characteristic classes of principal bundles

This is the archetypical example: let $\mathbf{H} =$ Top $\simeq$ ∞Grpd, the canonical (∞,1)-topos of discrete ∞-groupoids, or more generally let $\mathbf{H} =$ ETop∞Grpd, the cohesive (∞,1)-topos of Euclidean-topological ∞-groupoids.

For $G$ topological group write $B G$ for its classifying space: the (geometric realization of its) delooping.

For $A$ any other abelian topological group, similarly write $B^n A$ for its $n$-fold delooping. If $A$ is a discrete group then this is the Eilenberg-MacLane space $K(A,n)$.

Generally,

$H^n(B G, \mathbb{Z}) = \pi_0 \mathbf{H}(B G, B^n A)$

is the cohomology of $B G$ with coefficients in $A$. Every cocycle $c : B G \to B^n A$ represents a characteristic class $[c]$ on $B G$ with coefficients in $A$.

A $G$-principal bundle $P \to X$ is classified by some map $c : X \to B G$. For any $k \in H^n(\mathcal{B}G,\mathbb{Z})$ a degree $n$ cohomology class of the classifying space, the corresponding composite map $X \stackrel{c}{\to} B G \stackrel{k}{\to} \mathcal{B}^n A$ represents a class $[k(c)] \in H^n(X,\mathbb{Z})$. This is the corresponding characteristic class of the bundle.

Notable families of examples include:

### Chern character

The Chern character is a natural characteristic class with values in real cohomology. See there for more details.

### Of Lagrangian submanifolds

A characteristic class of Lagrangian submanifolds is the Maslov index.

### Classes in the sense of Fuks

In (Fuks (1987), section 7) an axiomatization of characteristic classes is proposed. We review the definition and discuss how it is a special case of the one given above.

#### Fuks’s definition

Fuks considers a base category $\mathcal{T}$ of “spaces” and a category $\mathcal{S}$ of spaces with a structure (for example, space together with a vector bundle on it), this category should be a category over $\mathcal{T}$, i.e. at least equipped with a functor $U : \mathcal{S}\to\mathcal{T}$.

A morphism of categories with structures is a morphism in the overcategory Cat$/\mathcal{T}$, i.e. a morphism $U\to U'$ is a functor $F: dom(U)\to dom(U')$ such that $U' F = U$.

Suppose now the category $\mathcal{T}$ is equipped with a cohomology theory which is, for purposes of this definition, a functor of the form $H : \mathcal{T}^{op} \to A$ where $A$ is some concrete category, typically category of T-algebras for some algebraic theory in Set, e.g. the category of abelian groups. Define $\mathcal{H} = \mathcal{H}_H$ as a category whose objects are pairs $(X,a)$ where $X$ is a space (= object in $\mathcal{T}$) and $a\in H(X)$. This makes sense as $A$ is a concrete category. A morphism $(X,a)\to (Y,b)$ is a morphism $f: X\to Y$ such that $H(f)(b) = a$. We also denote $f^* = H(f)$, hence $f^*(b) = a$.

A characteristic class of structures of type $\mathcal{S}$ with values in $H$ in the sense of (Fuks) is a morphism of structures $h: \mathcal{S}\to\mathcal{H}_H$ over $\mathcal{T}$. In other words, to each structure $S$ of the type $\mathcal{S}$ over a space $X$ in $\mathcal{T}$ it assigns an element $h(S)$ in $H(X)$ such that for a morphism $t: S\to T$ in $\mathcal{S}$ the homomorphism $(U(t))^* : H(Y)\to H(X)$, where $Y = U(T)$, sends $h(S)$ to $h(T)$.

#### Discussion

Notice that $\mathcal{H}_H \to \mathcal{T}$ in the above is nothing but the fibered category that under the Grothendieck construction is an equivalent incarnation of the presheaf $H$. In fact, since $A$ in the above is assume to be just a 1-category of sets with structure, $\mathcal{H}_H$ is just its category of elements of $H$.

Similarly in all applications that arise in practice (for instance for the structure of vector bundles) that was mentioned, the functor $\mathcal{S} \to \mathcal{T}$ is a fibered category, too, corresponding under the inverse of the Grothendieck construction to a prestack $F_{\mathcal{S}}$.

Therefore morphisms of fibered categories over $\mathcal{T}$

$c : \mathcal{S} \to \mathcal{H}_H$

are equivalently morphisms of (pre)stacks

$c : F_{\mathcal{S}} \to H \,.$

In either picture, these are morphism in a 2-topos over the site $\mathcal{T}$.

So, as before, for $X \in \mathcal{T}$ some space, a $\mathcal{S}$-structure on $X$ (for instance a vector bundle) is a moprhism in the topos

$g : X \to F_{\mathcal{S}}$

(in this setup simply by the 2-Yoneda lemma) and the characteristic class $[c(g)]$ of that bundle is the bullback of that universal class $c$, hence the class represented by the composite

$c(g) : X \stackrel{g}{\to} F_{\mathcal{S}} \stackrel{c}{\to} H \,.$

## References

A standard textbook is

A concise introduction is in chapter 23

Further texts include

• Jean-Pierre Schneiders, Introduction to characteristic classes and index theory (book), Lisboa (Lisabon) 2000

• Johan Dupont, Fibre bundles and Chern-Weil theory, Lecture Notes Series 69, Dept. of Math., University of Aarhus, 2003, 115 pp. pdf

• Shigeyuki Morita, Geometry of characteristic classes, Transl. Math. Mon. 199, AMS 2001

• Raoul Bott, L. W. Tu, Differential forms in algebraic topology, GTM 82, Springer 1982.

• D. B. Fuks, Непреривные когомологии топологических групп и характеристические классы , appendix to the Russian translation of K. S. Brown, Cohomology of groups, Moskva, Mir 1987.

Revised on November 10, 2013 12:40:02 by Urs Schreiber (82.113.121.220)