Decomposition spaces, incidence algebras and Möbius inversion

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By Imma Gálvez, Joachim Kock and Andy Tonks

Abstract

We introduce the notion of decomposition space as a general framework for incidence algebras and Möbius inversion. A decomposition space is a simplicial infinity-groupoid 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. We work as much as possible on the objective level of linear algebra with coefficients in infinity-groupoids, and develop the necessary homotopy linear algebra along the way. Independently of finiteness conditions, to any decomposition space there is associated an incidence (co)algebra (with coefficients in infinity-groupoids), and under a completeness condition (weaker than the Rezk condition) this incidence algebra is shown to satisfy a sign-free version of the Möbius inversion principle. Examples of decomposition spaces beyond Segal spaces are given by the Waldhausen S-construction of an abelian (or stable infinity) category. Their incidence algebras are various kinds of Hall algebras. Another class of examples are Schmitt restriction species. Imposing certain homotopy finiteness conditions yields the notion of Möbius decomposition space, covering the notion of Möbius category of Leroux (itself a common generalisation of locally finite posets (Rota et al.) and finite decomposition monoids (Cartier-Foata)), as well as many constructions of Dür, including the Faà di Bruno and Connes-Kreimer bialgebras. We take a functorial viewpoint throughout, emphasising conservative ULF functors, and show that most reduction procedures in the classical theory of incidence coalgebras are examples of this notion, and in particular that many are an example of decalage of decomposition spaces. Our main theorem concerns the Lawvere-Menni Hopf algebra of Möbius intervals, which contains the universal Möbius function (but does not come from a Möbius category): we establish that Möbius intervals (in the infinity-setting) form a decomposition space, and that it has the universal property also with respect to Möbius inversion in general decomposition spaces.
NOTE: The notion of decomposition space was arrived at independently by Dyckerhoff and Kapranov (arXiv:1212.3563) who call them unital 2-Segal spaces. Our theory is quite orthogonal to theirs: the definitions are different in spirit and appearance, and the theories differ in terms of motivation, examples and directions. For the few overlapping results ('decalage of decomposition is Segal' and 'Waldhausen's S is decomposition'), our approach seems generally simpler.

Last updated: 2014-04-19 by Joachim Kock.