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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2409.03742 |
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| _version_ | 1866910591373803520 |
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| author | Gálvez-Carrillo, Imma Kock, Joachim Tonks, Andrew |
| author_facet | Gálvez-Carrillo, Imma Kock, Joachim Tonks, Andrew |
| contents | We establish a Crapo complementation formula for the Möbius function $μ^X$ in a general decomposition space $X$ in terms of a convex subspace $K$ and its complement: $μ^X \simeq μ^{X\setminus K} + μ^X*ζ^K*μ^X$. 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 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_03742 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Convex decomposition spaces and Crapo complementation formula Gálvez-Carrillo, Imma Kock, Joachim Tonks, Andrew Category Theory Combinatorics 05A19, 18N50 We establish a Crapo complementation formula for the Möbius function $μ^X$ in a general decomposition space $X$ in terms of a convex subspace $K$ and its complement: $μ^X \simeq μ^{X\setminus K} + μ^X*ζ^K*μ^X$. 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 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. |
| title | Convex decomposition spaces and Crapo complementation formula |
| topic | Category Theory Combinatorics 05A19, 18N50 |
| url | https://arxiv.org/abs/2409.03742 |