Incidence bicomodules, Möbius inversion and a Rota formula for infinity adjunctions

Carlier, Louis

doi:10.2140/agt.2020.20.169

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Incidence bicomodules, Möbius inversion and a Rota formula for infinity adjunctions

Louis Carlier

Algebraic & Geometric Topology 20 (2020) 169–213

DOI: 10.2140/agt.2020.20.169

Abstract

In the same way decomposition spaces, also known as unital $2$ –Segal spaces, have incidence (co)algebras, and certain relative decomposition spaces have incidence (co)modules, we identify the structures that have incidence bi(co)modules: they are certain augmented double Segal spaces subject to some exactness conditions. We establish a Möbius inversion principle for (co)modules and a Rota formula for certain more involved structures called Möbius bicomodule configurations. The most important instance of the latter notion arises as mapping cylinders of infinity adjunctions, or more generally of adjunctions between Möbius decomposition spaces, in the spirit of Rota’s original formula.

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Keywords

2–Segal spaces, decomposition spaces, bisimplicial infinity-groupoids, bicomodules, infinity-adjunctions, Möbius inversion

Mathematical Subject Classification 2010

Primary: 18D05, 18G30, 55U10

Secondary: 06A07, 06A15, 06A75, 16D20, 16T15

References

Publication

Received: 28 January 2018

Revised: 29 May 2019

Accepted: 24 June 2019

Published: 23 February 2020

Authors

Louis Carlier
Departament de Matemàtiques Universitat Autònoma de Barcelona Bellaterra Spain
http://mat.uab.cat/~louiscarlier/

Volume 20, issue 1 (2020)