Decomposition spaces: theory and applications

Decomposition spaces are simplicial $\infty$-groupoids with a certain exactness condition: they send generic (end--point preserving) and free (distance preserving) pushout squares in the simplicial category $\Delta$ to pullbacks. They encode the information needed for an 'objective' generalisation of the notion of incidence (co)algebra of a poset, and turn out to coincide with the unital 2-Segal spaces of Dyckerhoff and Kapranov.
We establish a general Möbius inversion principle, and construct the universal Möbius decomposition space.

We will also discuss motivating examples of decomposition spaces and the algebras that arise, including the Connes-Kreimer algebra of trees, Schmitt's algebra of graphs and (derived) Hall algebras.

Joint work with Joachim Kock and Andy Tonks.