Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userExplicit Formulas for 2-Characters
142
0
0.0
(
0
)
Authors
Angelica Osorno
Ask ChatGPT about the research
No Arabic 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 2Vect_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.
rate research
Read More
We establish an equivalence of homotopy theories between symmetric monoidal bicategories and connective spectra. For this, we develop the theory of $Gamma$-objects in 2-categories. In the course of the proof we establish strictfication results of independent interest for symmetric monoidal bicategories and for diagrams of 2-categories.
We show that the edgewise subdivision of a $2$-Segal object is always a Segal object, and furthermore that this property characterizes $2$-Segal objects.
We prove that the homotopy theory of Picard 2-categories is equivalent to that of stable 2-types.
Picard 2-categories are symmetric monoidal 2-categories with invertible 0-, 1-, and 2-cells. The classifying space of a Picard 2-category $mathcal{D}$ is an infinite loop space, the zeroth space of the $K$-theory spectrum $Kmathcal{D}$. This spectrum has stable homotopy groups concentrated in levels 0, 1, and 2. In this paper, we describe part of the Postnikov data of $Kmathcal{D}$ in terms of categorical structure. We use this to show that there is no strict skeletal Picard 2-category whose $K$-theory realizes the 2-truncation of the sphere spectrum. As part of the proof, we construct a categorical suspension, producing a Picard 2-category $Sigma C$ from a Picard 1-category $C$, and show that it commutes with $K$-theory in that $KSigma C$ is stably equivalent to $Sigma K C$.
It is known by results of Dyckerhoff-Kapranov and of Galvez--Carrillo-Kock-Tonks that the output of the Waldhausen S.-construction has a unital 2-Segal structure. Here, we prove that a certain S.-functor defines an equivalence between the category of augmented stable double categories and the category of unital 2-Segal sets. The inverse equivalence is described explicitly by a path construction. We illustrate the equivalence for the known examples of partial monoids, cobordism categories with genus constraints and graph coalgebras.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
