2024/09/05 by Imma Gálvez-Carrillo, Joachim Kock, Gálvez-Carrillo, Imma +3
Computer Science · Decision Sciences · Mathematics · #05A19 #18N50 #Category Theory (math.CT) #Combinatorics (math.CO) #FOS: Mathematics #Fuzzy and Soft Set Theory #Optimization and Variational Analysis #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2409.03742
openalex publication_date 2024/09/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/30
We establish a Crapo complementation formula for the Möbius function μX in a general decomposition space X in terms of a convex subspace K and its complement: μX ≃ μX∖ K + μX*ζK*μX. We work at the objective level, meaning that the formula is an explicit homotopy equivalence of ∞-groupoids. Almost all arguments are formulated in terms of (homotopy) pullbacks. Under suitable finiteness conditions on X, one can take homotopy cardinality to obtain a formula in the incidence algebra at the level of ℚ-algebras. When X is the nerve of a locally finite poset, this recovers the Björner--Walker formula, which in turn specialises to the original Crapo complementation formula when the poset is a finite lattice. A substantial part of the work is to introduce and develop the notion of convexity for decomposition spaces, which in turn requires some general preparation in decomposition-space theory, notably some results on reduced covers and ikeo and semi-ikeo maps. These results may be of wider interest. Once this is set up, the objective proof of the Crapo formula is quite similar to that of Björner--Walker.