\(K\)-theory of braided tensor ring categories with higher commutativity
The paper is a sequel of the author's papers ``\(E_n\)-monoidal categories and their group completions'' and ``\(E_n\)-ring categories'' [J. Pure Appl. Algebra 95, No. 1, 27-39 (1994; Zbl 0809.18007) and ibid., to appear] and makes extensive use of the results proved there. The present paper addresses the \(K\)-theory of \(E_n\)-ring categories. An \(E_n\)-ring category is a permutative based topological category \((A, \oplus)\) together with a multiplicative structure encoded by an unbased topological cat-operad \({\mathcal G}\) satisfying the obvious distributivity laws. \({\mathcal G}\) is required to be equivalent to the tensor product of braided cat-operads \({\mathcal B} \otimes {\mathcal C}_1^{\oplus (n - 2)}\), where \({\mathcal B}\) is the canonical braided cat-operad defined by the translation categories of the braid groups and \({\mathcal C}_1\) is the little 1-cubes operad considered as trivial topological cat-operad. A Waldhausen category \((A, coA, wA)\) with cofibrations and weak equivalences is an \(E_n\)-ring category (with cofibrations and weak equivalence) if \(A\) is an \(E_n\)-ring category whose permutative structure is induced by the coproduct in \(A\) and if \(coA\) and \(wA\) are ring subcategories. The main result is that the Waldhausen \(K\)-theory \(K(A)\) of such a category is (as spectrum) equivalent to an \(E_n\)-ring spectrum. The result specializes to symmetric bimonoidal and braided bimonoidal categories with cofibrations and weak equivalences after passage to equivalent permutative versions: their \(K\)-theory spectra are \(E_\infty\)- respectively \(E_2\)-ring spectra. Particular examples are the \(K\)-theories of the categories of crossed \(G\)-sets and free crossed \(G\)-sets, \(G\) a finite group, with injections as cofibrations and isomorphisms as weak equivalences. Their \(K\)-theory spectra are \(E_2\)-ring spectra.
\(K\)-theory of \(E_ n\)-ring categories
scientific article; zbMATH DE number 827895
scientific article