Decomposition spaces, incidence algebras, and Mšbius inversion I'll survey recent work with Imma G‡lvez and Andy Tonks developing a homotopy version of the theory of incidence algebras and Mšbius inversion. The 'combinatorial objects' playing the role of posets and Mšbius categories are decomposition spaces, simplicial infinity-groupoids satisfying an exactness condition weaker than the Segal condition, expressed in terms of generic and free maps in Delta. Just as the Segal condition expresses up-to-homotopy composition, the new condition expresses decomposition. The role of vector spaces is played by slices over infinity-groupoids, eventually with homotopy finiteness conditions imposed. To any decomposition space, there is associated an incidence (co)algebra with coefficients in infinity-groupoids, which satisfies an objective Mšbius inversion principle in the style of Lawvere-Menni, provided a certain completeness condition is satisfied, weaker than the Rezk condition. Generic examples of decomposition spaces beyond Segal spaces are given by the Waldhausen S-construction (yielding Hall algebras) and by Schmitt restriction species, and many examples from classical combinatorics admit uniform descriptions in this framework. (The notion of decomposition space is equivalent to the notion of unital 2-Segal space of Dyckerhoff-Kapranov.)