Incidence bicomodules, Möbius inversion, and a Rota formula for 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.