Topological Hochschild homology of ring functors and exact categories

Abstract 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 β. Fundamental properties of THH and TC are proven and some examples are analyzed. A special case of a ring functor on an exact category is treated separately, and is compared with algebraic K-theory via a Dennis-Bokstedt trace map. Calling THH and TC applied to these ring functors simply THH( ) and TC( ), we get that the iteration of Waldhausen's S construction yields spectra {THH(S(n) )} and {TC(S(n) )}, and the maps from K-theory become maps of spectra. If is split exact, the THH and TC spectra are Ω-spectra. The inclusion by degeneracies THH0(S(n) ) ⊆ THH(S(n) ) 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).