A remark on \(K\)-theory and \(S\)-categories
\textit{M. Schlichting} [Invent. Math. 150, 111--116 (2002; Zbl 1037.18007)] has shown that there exist two Waldhausen categories having equivalent (triangulated) homotopy categories and non-weakly equivalent associated (Waldhausen) \(K\)-theory spectra. The authors' starting point is this fact which implies that the \(K\)-theory spectrum \(K(C)\) of a Waldhausen category \(C\) depends on strictly more than its homotopy category. First the authors introduce a notion of a good Waldhausen category which is built up in Section 2. By a good Waldhausen category the authors mean a Waldhausen category being embeddable in the category of fibrant objects of a pointed model category. One finds that given a Waldhausen category there is a good Waldhausen category with the same \(K\)-theory spectrum up to homotopy. Moreover by an \(S\)-category one means a category enriched over the category of simplicial sets. Then the main purpose of this paper is to prove that if \(C\) and \(D\) are good Waldhausen categories, and if their simplicial localizations \(L^HC\) and \(L^HD\) are equivalent in the homotopy category of \(S\)-categories then the \(K\)-theory spectra \(K(C)\) and \(K(D)\) are isomorphic in the homotopy category of \(K\)-spectra (Corollary 6.6). The key to the authors' approach is defining a \(K\)-theory spectrum \(K(T)\) for an \(S\)-category \(T\) satisfying some properties which depends (up to weak equivalences) on the \(S\)-equivalence class of \(T\) (Section 5). By applying this construction to the simplicial localization \(L^HC\) of a good Waldhausen category \(C\) one can obtain a weak equivalence \(K(C)\simeq K(L^HC)\) (Theorem 6.1) from which the above result follows immediately. The basic results needed for the proof of this theorem are provided in Sections 3 and 4. In addition this theorem yields the following interesting consequence (Corollary 7.1): There exist two stable model categories \(M_1\) and \(M_2\) such that \(L^HM_1\) and \(L^HM_2\) cannot be equivalent, but their homotopy categories are equivalent. The final portion of the paper is devoted to considerations of possible future directions and relations with other works.
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