Incidence bicomodules, M\"obius inversion, and a Rota formula for infinity adjunctions

TL;DR

This paper extends incidence algebra concepts to double Segal spaces, establishing a M"obius inversion principle and a Rota formula for complex structures called M"obius bicomodule configurations, especially in the context of infinity adjunctions.

Contribution

It identifies incidence bi(co)modules within augmented double Segal spaces and develops a M"obius inversion and Rota formula framework for these structures, including applications to infinity adjunctions.

Findings

Established a M"obius inversion principle for (co)modules.

Derived a Rota formula for M"obius bicomodule configurations.

Applied the theory to mapping cylinders of infinity adjunctions.

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 M\"obius inversion principle for (co)modules, and a Rota formula for certain more involved structures called M\"obius bicomodule configurations. The most important instance of the latter notion arises as mapping cylinders of infinity adjunctions, or more generally of adjunctions between M\"obius decomposition spaces, in the spirit of Rota's original formula.