[Paper Review] Decomposition spaces, incidence algebras and Möbius inversion

Maria Immaculada Gálvez Carrillo, Joachim Kock+1 more|arXiv (Cornell University)|Apr 11, 2014

Homotopy and Cohomology in Algebraic Topology49 references15 citations

TL;DR

This paper introduces decomposition spaces—simplicial infinity-groupoids with a weakened Segal condition—as a general framework for incidence algebras and Möbius inversion, unifying classical theory with homotopy-theoretic methods. It establishes that Möbius intervals form a decomposition space, yielding a universal Hopf algebra containing the universal Möbius function, extending the notion of Möbius categories beyond classical settings.

ABSTRACT

We introduce the notion of decomposition space as a general framework for incidence algebras and Mobius 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 Mobius 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 Mobius decomposition space, an extension of the notion of Mobius 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 Mobius intervals, which contains the universal Mobius function (but does not come from a Mobius category): we establish that Mobius 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.

Motivation & Objective

• To generalize incidence algebras and Möbius inversion beyond classical posets by introducing decomposition spaces as a homotopy-theoretic framework.
• To develop a theory of incidence (co)algebras with coefficients in infinity-groupoids, enabling sign-free Möbius inversion.
• To extend the concept of Möbius categories via homotopy-finite decomposition spaces, defining Möbius decomposition spaces.
• To show that classical reduction procedures in incidence theory arise naturally as decalage of decomposition spaces.
• To establish the Lawvere-Menni Hopf algebra of Möbius intervals as arising from a universal decomposition space.

Proposed method

• Introduce decomposition spaces as simplicial infinity-groupoids satisfying a weak exactness condition, generalizing Segal spaces.
• Use homotopy linear algebra with coefficients in infinity-groupoids to construct incidence (co)algebras from decomposition spaces.
• Define Möbius decomposition spaces via homotopy finiteness conditions, extending Leroux’s Möbius categories.
• Employ conservative ULF functors to formalize reduction procedures in incidence theory as decalage operations on decomposition spaces.
• Prove that the category of Möbius intervals forms a decomposition space, using its universal properties.
• Establish the universal Möbius function via the Lawvere-Menni Hopf algebra realized as an incidence algebra of this decomposition space.

Experimental results

Research questions

• RQ1 Can incidence algebras and Möbius inversion be generalized beyond posets using higher categorical structures?
• RQ2 How do classical reduction procedures in incidence theory relate to categorical constructions like decalage?
• RQ3 Is there a universal structure underlying the Lawvere-Menni Hopf algebra of Möbius intervals?
• RQ4 Can the notion of Möbius category be extended to homotopy-theoretic settings via decomposition spaces?
• RQ5 What is the role of conservative ULF functors in unifying incidence-theoretic constructions?

Key findings

• The category of Möbius intervals forms a decomposition space, providing a universal framework for Möbius inversion.
• The Lawvere-Menni Hopf algebra arises as the incidence algebra of this universal decomposition space, containing the universal Möbius function.
• Decomposition spaces generalize Segal spaces and support incidence (co)algebras with coefficients in infinity-groupoids.
• Classical reduction procedures in incidence theory are shown to be instances of decalage of decomposition spaces.
• Möbius decomposition spaces are defined via homotopy finiteness conditions, extending Leroux’s Möbius categories to the homotopy setting.
• The theory achieves a sign-free version of the Möbius inversion principle within the framework of homotopy linear algebra.