Algebra/Topology Seminar by Joachim Kock

Talk by Joachim Kock (Universitat Autònoma de Barcelona)

Title: Decomposition spaces, incidence algebras, and Möbius inversion

Abstract: I'll survey recent work with Imma Gálvez and Andy Tonks developing a homotopy version of the theory of incidence algebras and Möbius inversion.  The 'combinatorial objects' playing the role of posets and Möbius 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 Möbius 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.)