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.
Links:
[1] https://www.mpim-bonn.mpg.de/de/taxonomy/term/39
[2] https://www.mpim-bonn.mpg.de/de/node/3444
[3] https://www.mpim-bonn.mpg.de/de/node/6477