2-Segal sets and the Waldhausen construction
Algebraic Topology
2017-05-04 v3 Category Theory
K-Theory and Homology
Abstract
It is known by results of Dyckerhoff-Kapranov and of G\'alvez--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.
Cite
@article{arxiv.1609.02853,
title = {2-Segal sets and the Waldhausen construction},
author = {Julia E. Bergner and Angélica M. Osorno and Viktoriya Ozornova and Martina Rovelli and Claudia I. Scheimbauer},
journal= {arXiv preprint arXiv:1609.02853},
year = {2017}
}
Comments
48 pages. Final version. Will appear in Proceedings of WIT