Angélica M. Osorno

Infinite loop spaces, and coherence for symmetric monoidal bicategories

Abstract This paper proves three different coherence theorems for symmetric monoidal bicategories. First, we show that in a free symmetric monoidal bicategory every diagram of 2-cells commutes. Second, we show that this implies that the free symmetric monoidal bicategory on one object is equivalent, as a symmetric monoidal bicategory, to the discrete symmetric monoidal bicategory given by the disjoint union of the symmetric groups. Third, we show that every symmetric monoidal bicategory is equivalent to a strict one. We give two topological applications of these coherence results. First, we show that the classifying space of a symmetric monoidal bicategory can be equipped with an E ∞ structure. Second, we show that the fundamental 2-groupoid of an E n space, n ≥ 4 , has a symmetric monoidal structure. These calculations also show that the fundamental 2-groupoid of an E 3 space has a sylleptic monoidal structure.

Spectra associated to symmetric monoidal bicategories

We show how to construct a Gamma-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 Gamma-category construction for a bipermutative category. As an example, we use this machinery to construct a delooping of the K-theory of a bimonoidal category as defined by Baas-Dundas-Rognes.

Explicit formulas for 2-characters

Abstract Ganter and Kapranov associated a 2-character to 2-representations of a finite group. Elgueta classified 2-representations in the category of 2-vector spaces 2 Vect k in terms of cohomological data. We give an explicit formula for the 2-character in terms of this cohomological data and derive some consequences.

K -theory for 2-categories

Abstract We establish an equivalence of homotopy theories between symmetric monoidal bicategories and connective spectra. For this, we develop the theory of Γ-objects in 2-categories. In the course of the proof we establish strictification results of independent interest for symmetric monoidal bicategories and for diagrams of 2-categories.

Constructing equivariant spectra via categorical Mackey functors

We give a functorial construction of equivariant spectra from a generalized version of Mackey functors in categories. This construction relies on the recent description of the category of equivariant spectra due to Guillou and May. The key element of our construction is a spectrally-enriched functor from a spectrally-enriched version of permutative categories to the category of spectra that is built using an appropriate version of K-theory. As applications of our general construction, we produce a new functorial construction of equivariant Eilenberg--MacLane spectra for Mackey functors and for suspension spectra for finite G-sets.

Stable Postnikov data of Picard 2–categories

Picard 2-categories are symmetric monoidal 2-categories with invertible 0-, 1-, and 2-cells. The classifying space of a Picard 2-category

Extending homotopy theories across adjunctions

\mathcal{D}

MODELING STABLE ONE-TYPES

is an infinite loop space, the zeroth space of the

2-Segal sets and the Waldhausen construction

K

The edgewise subdivision criterion for 2-Segal objects

-theory spectrum