Entry characteristic class defines characteristic classes from nPOV, within -topos and relates the characteristic classes to -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 of “spaces” and a category of spaces with a structure (for example, space together with a vector bundle on it), this category should be a category over , i.e. equipped with a functor . A morphism of categories with structures is a morphism in the overcategory , i.e. a morphism is a functor such that .
Suppose now the category is equipped with a cohomology theory which is, for purposes of this definition, a functor of the form where is some concrete category, typically category of -algebras for some algebraic theory in , e.g. the category of abelian groups. Define as a category whose objects are pairs where is a space (= object in ) and . This makes sense as is a concrete category. A morphism is a morphism such that . We also denote , hence .
A characteristic class of structures of type with values in is a morphism of structures over . In other words, to each structure of the type over a space in it assigns an element in such that for a morphism in the homomorphism , where , sends to .
This axiomatics can be found e.g. in the section 7 of