The same applies with the object taken as the domain object: for yet another object, the -valued cohomology of is similarly . For any cohomology class in there, we obtain an ∞-functor
from the -valued cohomology of to its -valued cohomology, simply from the composition operation
Quite generally, for an -cohomology class, its image is the corresponding characteristic class.
In practice one is interested in this notion for particularly simple objects , notably for an Eilenberg-MacLane object for some component of a spectrum object. This serves to characterize cohomology with coefficients in a complicated object 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 a given characteristic class in degree assigns
Moreover, recall from the discussion at cohomology that to every cocycle is associated the object that it classifies – its homotopy fiber – which may be thought of as an -principal ∞-bundle over with classifying map . One typically thinks of the characteristic class as characterizing this principal ∞-bundle .
A -principal bundle is classified by some map . For any a degree cohomology class of the classifying space, the corresponding composite map represents a class . This is the corresponding characteristic class of the bundle.
Notable families of examples include:
for the orthogonal group:
for the special orthogonal group:
for the unitary group:
for the circle 2-group:
the Dixmier-Douady class.
The Chern character is a natural characteristic class with values in real cohomology. See there for more details.
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 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. at least equipped with a functor .
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 T-algebras for some algebraic theory in Set, 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 in the sense of (Fuks) 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 .
Notice that in the above is nothing but the fibered category that under the Grothendieck construction is an equivalent incarnation of the presheaf . In fact, since in the above is assume to be just a 1-category of sets with structure, is just its category of elements of .
Similarly in all applications that arise in practice (for instance for the structure of vector bundles) that was mentioned, the functor is a fibered category, too, corresponding under the inverse of the Grothendieck construction to a prestack .
Therefore morphisms of fibered categories over
Textbook accounts include
Stanley Kochmann, section 2.3 of Bordism, Stable Homotopy and Adams Spectral Sequences, AMS 1996
A concise introduction is in chapter 23
Further texts include
Jean-Pierre Schneiders, Introduction to characteristic classes and index theory (book), Lisboa (Lisabon) 2000
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.