2-Segal sets and the Waldhausen construction

Julia E. Bergner; Angélica M. Osorno; Viktoriya Ozornova; Martina Rovelli; Claudia I. Scheimbauer

doi:10.1016/j.topol.2017.12.009

2-Segal sets and the Waldhausen construction

Julia E. Bergner, Angélica M. Osorno, Viktoriya Ozornova, Martina Rovelli, Claudia I. Scheimbauer

Research output: Contribution to journal › Article › peer-review

13 Scopus citations

Abstract

It is known by results of Dyckerhoff–Kapranov and of Gálvez-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.

Original language	English
Pages (from-to)	445-484
Number of pages	40
Journal	Topology and its Applications
Volume	235
DOIs	https://doi.org/10.1016/j.topol.2017.12.009
State	Published - 15 Feb 2018
Externally published	Yes

Keywords

2-Segal spaces
Cobordism categories
Double categories
Partial monoids
Waldhausen S-construction

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10.1016/j.topol.2017.12.009

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title = "2-Segal sets and the Waldhausen construction",

abstract = "It is known by results of Dyckerhoff–Kapranov and of G{\'a}lvez-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.",

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AU - Bergner, Julia E.

AU - Osorno, Angélica M.

AU - Ozornova, Viktoriya

AU - Rovelli, Martina

AU - Scheimbauer, Claudia I.

PY - 2018/2/15

Y1 - 2018/2/15

N2 - It is known by results of Dyckerhoff–Kapranov and of Gálvez-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.

AB - It is known by results of Dyckerhoff–Kapranov and of Gálvez-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.

KW - 2-Segal spaces

KW - Cobordism categories

KW - Double categories

KW - Partial monoids

KW - Waldhausen S-construction

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SN - 0166-8641

VL - 235

SP - 445

EP - 484

JO - Topology and its Applications

JF - Topology and its Applications

ER -

2-Segal sets and the Waldhausen construction

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Keywords

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