Convex decomposition spaces and Crapo complementation formula

Abstract

We establish a Crapo complementation formula for the M\"obius function $\mu^X$ in a general decomposition space $X$ in terms of a convex subspace $K$ and its complement: $\mu^X \simeq \mu^{X\setminus K} + \mu^X*\zeta^K*\mu^X$. We work at the objective level, meaning that the formula is an explicit homotopy equivalence of $\infty$-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 $\mathbb{Q}$-algebras. When $X$ is the nerve of a locally finite poset, this recovers the Bj\"orner--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 Bj\"orner--Walker.