group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
symmetric monoidal (∞,1)-category of spectra
Given a ring $R$, then there is a natural morphism of spectra
from the algebraic K-theory spectrum to the topological Hochschild homology spectrum and factoring through the topological cyclic homology spectrum (a cyclotomic spectrum) called the cyclotomic trace which much like a Chern character map for algebraic K-theory.
This is a refinement to spectra of the Dennis trace.
Marcel Bökstedt, W.C. Hsiang, Ib Madsen, The cyclotomic trace and algebraic K-theory of spaces, Invent. Math. 111 (1993), 463-539, MR94g:55011, doi
Andrew Blumberg, David Gepner, Goncalo Tabuada, Uniqueness of the multiplicative cyclotomic trace (arXiv:1103.3923)
David Ayala, Aaron Mazel-Gee, Nick Rozenblyum, The geometry of the cyclotomic trace, (arXiv:1710.06409)
For a very brief, very elementary introduction, see
For more see the references at topological Hochschild homology.