symmetric monoidal (∞,1)-category of spectra
Topological Hochschild (resp. topological cyclic) homology is a refinement of Hochschild homology/cyclic homology from commutative rings/algebras to the higher algebra of ring spectra/E-∞ rings/E-∞ algebras. A good survey is in (May).
In particular, given a ring , 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 called the cyclotomic trace which much like a Chern character map for algebraic K-theory.
Marcel Bökstedt, Topological Hochschild homology, Bielefeld, 1985, 1988
and a further generalization is defined in
Review and exposition includes
Anatoly Preygel, Hochschild homology notes, juvitop seminar, pdf
These theories are the target for the trace map from -theory, so they can be viewed as an approximation to algebraic K-theory of a ring spectrum.
Bjørn Ian Dundas, Relative K-theory and topological cyclic homology, Acta Math. 179 (1997), 223-242, pdf
Thomas Geisser, Lars Hesselhoft, Topological cyclic homology of schemes, in: Algebraic -theory (Seattle, WA, 1997), 41–87, Proc. Sympos. Pure Math. 67, Amer. Math. Soc. 1999, MR2001g:19003; K-theory archive
Thomas Geisser, Motivic Cohomology, K-Theory and Topological Cyclic Homology, Handbook of K-theory II.1, pdf
R. McCarthy, Relative algebraic K-theory and topological cyclic homology, Acta Math. 179 (1997), 197-222.
Ib Madsen, Algebraic K-theory and traces, pdf
Dundas, Goodwillie and McCarthy, The local structure of algebraic K-theory, book in progress, pdf
J. McClure, R. Staffeldt, On the topological Hochschild homology of , I, pdf
Daniel Joseph Vera, Topological Hochschild homology of twisted group algebra, MIT Ph. D. thesis 2006, pdf