nLab
characteristic class of a structure

Entry characteristic class defines characteristic classes from nPOV, within (,1)(\infty,1)-topos and relates the characteristic classes to (,1)(\infty,1)-principal bundles. Here we present an older axiomatics, where characteristic classes characterize any structure on spaces (example: characteristic classes of foliations, bundles, bundles with structure etc).

Thus one considers a base category 𝒯\mathcal{T} of “spaces” and a category 𝒮\mathcal{S} of spaces with a structure (for example, space together with a vector bundle on it), this category should be a category over 𝒯\mathcal{T}, i.e. equipped with a functor U:𝒮𝒯U : \mathcal{S}\to\mathcal{T}. A morphism of categories with structures is a morphism in the overcategory Cat/𝒯Cat/\mathcal{T}, i.e. a morphism UUU\to U' is a functor F:dom(U)dom(U)F: dom(U)\to dom(U') such that UF=UU' F = U.

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

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

This axiomatics can be found e.g. in the section 7 of

  • 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.
Created on May 1, 2011 10:05:23 by Zoran Škoda (77.237.105.8)