Spectra associated to symmetric monoidal bicategories

Abstract

We show how to construct a -bicategory from a symmetric monoidal bicategory and use that to show that the classifying space is an infinite loop space upon group completion. We also show a way to relate this construction to the classic -category construction for a permutative category. As an example, we use this machinery to construct a delooping of the K-theory of a rig category as defined by Baas, Dundas and Rognes [2].

K-theory for 2-categories

“…In this work we prove that the K-theory functor defined in [Oso12] induces an equivalence of homotopy theories between symmetric monoidal bicategories and connective spectra. One motivation for this level of generality is that a number of interesting symmetric monoidal structures are naturally 2-categorical.…”

“…One important general example is that of the bicategory Mod R of finitely generated modules over a bimonoidal category R, defined in [BDR04]. By results of [BDRR11] and [Oso12], the bicategorical K-theory of Mod R is equivalent to the K-theory of the ring spectrum K R. In the case when R is the topological category of finite dimensional complex vector spaces, we get that the K-theory spectrum of connective complex topological K-theory, K(ku), is equivalent to K(2Vect C ), where 2Vect C is the bicategory of 2-vector spaces. Thus, K(ku) is the classifying spectrum for 2-vector bundles (see [BDR04]).…”

Extending homotopy theories across adjunctions

“…The classifying space functor from categories to topological spaces provides a way of constructing spaces with certain algebraic structure. Of particular importance are infinite loop space machines, which construct spectra out of structured categories such as symmetric monoidal categories [Sta71, Qui73, May74, Seg74, Wal85, EM06, May09,Oso12]. The discussion of the functoriality of these constructions is somewhat nuanced due to the range of possible morphisms one might choose.…”