nLab
characteristic class of a structure

Entry characteristic class defines characteristic classes from nPOV, within $(\mathrm{\infty },1)$-topos and relates the characteristic classes to $(\mathrm{\infty },1)$-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 $𝒯$ of “spaces” and a category $𝒮$ of spaces with a structure (for example, space together with a vector bundle on it), this category should be a category over $𝒯$, i.e. equipped with a functor $U:𝒮\to 𝒯$. A morphism of categories with structures is a morphism in the overcategory $\mathrm{Cat}/𝒯$, i.e. a morphism $U\to U\prime$ is a functor $F:\mathrm{dom}(U)\to \mathrm{dom}(U\prime )$ such that $U\prime F=U$.

Suppose now the category $𝒯$ is equipped with a cohomology theory which is, for purposes of this definition, a functor of the form $H:𝒯\to A$ where $A$ is some concrete category, typically category of $T$-algebras for some algebraic theory in $\mathrm{Set}$, e.g. the category of abelian groups. Define $\mathscr{H}={\mathscr{H}}_H$ as a category whose objects are pairs $(X,a)$ where $X$ is a space (= object in $𝒯$) 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 $𝒮$ with values in $H$ is a morphism of structures $h:𝒮\to {\mathscr{H}}_H$ over $𝒯$. In other words, to each structure $S$ of the type $𝒮$ over a space $X$ in $𝒯$ it assigns an element $h(S)$ in $H(X)$ such that for a morphism $t:S\to T$ in $𝒮$ the homomorphism $(U(t){)}^{*}:H(Y)\to H(X)$, where $Y=U(T)$, sends $h(S)$ to $h(T)$.

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.