Decomposition Spaces, Incidence Algebras and Möbius Inversion

Abstract: We introduce the notion of decomposition space as a general framework for incidence algebras and M\"obius inversion: it is a simplicial infinity-groupoid satisfying an exactness condition weaker than the Segal condition, which expresses decomposition. We work on the objective level of homotopy linear algebra with coefficients in infinity-groupoids, developed along the way. To any (complete) decomposition space there is associated an incidence (co)algebra (with coefficients in infinity-groupoids), shown to satisfy a sign-free version of the M\"obius inversion principle. Examples of decomposition spaces beyond Segal spaces are given by the Waldhausen S-construction and by Schmitt restriction species. Imposing certain homotopy finiteness conditions yields the notion of M\"obius decomposition space, an extension of the notion of M\"obius category of Leroux. We take a functorial viewpoint throughout, emphasising conservative ULF functors, and show that most reduction procedures in the classical theory are examples of this notion, and in particular that many are examples of decalage of decomposition spaces. Our main theorem concerns the Lawvere-Menni Hopf algebra of M\"obius intervals, which contains the universal M\"obius function (but does not come from a M\"obius category): we establish that M\"obius intervals form a decomposition space, which is in some sense universal. NOTE: The notion of decomposition space was arrived at independently by Dyckerhoff and Kapranov (arXiv:1212.3563) who call it unital 2-Segal space. Our theory is quite orthogonal to theirs.