Special and general types Special notions Variants Extra structure Operations Theorems algebraic theory / 2-algebraic theory / (∞,1)-algebraic theory monad / (∞,1)-monad operad / (∞,1)-operad monoidal (∞,1)-category symmetric monoidal (∞,1)-category of spectra A-∞ algebra C-∞ algebra E-∞ ring, E-∞ algebra L-∞ algebra model structure on simplicial T-algebras / homotopy T-algebra model structure on operads model structure on algebras over an operad 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 (a cyclotomic spectrum). This is called the cyclotomic trace and is 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 Ib Madsen, The cyclotomic trace in algebraic K-theory, In: Joseph, A., Mignot, F., Murat, F., Prum, B., Rentschler, R. (eds) First European Congress of Mathematics Paris, July 6–10, 1992. Progress in Mathematics 120, Birkhäuser 1994 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.