Augmented Simplicial Combinatorics through Category Theory: Cones, Suspensions and Joins (original) (raw)
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For a simplicial complex K on m vertices and simplicial complexes K 1 ,. .. , Km a composed simplicial complex K(K 1 ,. .. , Km) is introduced. This construction generalizes an iterated simplicial wedge construction studied by A. Bahri, M. Bendersky, F. R. Cohen and S. Gitler and allows to describe the combinatorics of generalized joins of polytopes P (P 1 ,. .. , Pm) defined by G. Agnarsson in most important cases. The composition defines a structure of an operad on a set of finite simplicial complexes, in which a complex on m vertices is viewed as an m-adic operation. We prove the following: (1) a composed complex K(K 1 ,. .. , Km) is a simplicial sphere iff K is a simplicial sphere and K i are the boundaries of simplices; (2) a class of spherical nerve-complexes is closed under the operation of composition (3) finally, we express multigraded Betti numbers of K(K 1 ,. .. , Km) in terms of multigraded Betti numbers of K, K 1 ,. .. , Km using a composition of generating functions.
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In this paper we prove a class of equivalences of additive functor categories that are relevant to enumerative combinatorics, representation theory, and homotopy theory. Let mathcalX\mathcal{X}mathcalX denote an additive category with finite direct sums and splitting idempotents. The class includes (a) the Dold-Puppe-Kan theorem that simplicial objects in mathcalX\mathcal{X}mathcalX are equivalent to chain complexes in mathcalX\mathcal{X}mathcalX; (b) the observation of Church-Ellenberg-Farb that mathcalX\mathcal{X}mathcalX-valued species are equivalent to mathcalX\mathcal{X}mathcalX-valued functors from the category of finite sets and injective partial functions; (c) a Dold-Kan-type result of Pirashvili concerning Segal's category Gamma\GammaGamma; and so on. We provide a construction which produces further examples.
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Abstract. For a simplicial complexK onm vertices and simplicial complexesK1,...,Km a composed simplicial complex K(K1,...,Km) is introduced. This construction gener-alizes an iterated simplicial wedge construction studied by A. Bahri, M. Bendersky, F. R. Cohen and S. Gitler and allows to describe the combinatorics of generalized joins of polytopes P (P1,..., Pm) defined by G. Agnarsson in most important cases. The com-position defines a structure of an operad on a set of finite simplicial complexes, in which a complex on m vertices is viewed as an m-adic operation. We prove the following: (1) a composed complex K(K1,...,Km) is a simplicial sphere iff K is a simplicial sphere and Ki are the boundaries of simplices; (2) a class of spherical nerve-complexes is closed under the operation of composition (3) finally, we express multigraded Betti numbers of K(K1,...,Km) in terms of multigraded Betti numbers of K,K1,...,Km using a composition of generating functions. 1.
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