In the same way decomposition spaces, also known as unital –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.
"Incidence bicomodules, Möbius inversion and a Rota formula for infinity adjunctions." Algebr. Geom. Topol. 20 (1) 169 - 213, 2020. https://doi.org/10.2140/agt.2020.20.169