Topological Hochschild homology of ring functors and exact categories

In analogy with Hochschild-Mitchell homology for linear categories topological Hochschild and cyclic homology (THH and TC) are defined for ring functors on a category script c sign. Fundamental properties of THH and TC are proven and some examples are analyzed. A special case of a ring functor on an exact category C is treated separately, and is compared with algebraic K-theory via a Dennis-Bökstedt trace map. Calling THH and TC applied to these ring functors simply THH(C) and TC(C), we get that the iteration of Waldhausen's S construction yields spectra {THH(S⁽ⁿ⁾C)} and {TC(S⁽ⁿ⁾C)}, and the maps from K-theory become maps of spectra. If C is split exact, the THH and TC spectra are Ω-spectra. The inclusion by degeneracies THH₀(S⁽ⁿ⁾C) ⊆ THH(S⁽ⁿ⁾C) is a stable equivalence, and it is shown how this leads to a weak resolution theorem for THH. If ℘_A is the category of finitely generated projective modules over a unital and associative ring A, we get that THH(A) ^≃→ THH(℘_A) and TC(A) ^≃→ TC(℘_A).