Homology, Homotopy and Applications

Volume 18 (2016)

Number 2

Homotopy colimits of diagrams over posets and variations on a theorem of Thomason

Pages: 233 – 245

DOI: https://dx.doi.org/10.4310/HHA.2016.v18.n2.a13

Authors

Ximena Fernández (Departamento de Matemática, IMAS, FCEyN, Universidad de Buenos Aires, Buenos Aires, Argentina)

Elías Gabriel Minian (Departamento de Matemática, IMAS, FCEyN, Universidad de Buenos Aires, Buenos Aires, Argentina)

Abstract

We use a classical result of McCord and reduction methods of finite spaces to prove a generalization of Thomason’s theorem on homotopy colimits over posets. In particular, this allows us to characterize the homotopy colimits of diagrams of simplicial complexes in terms of the Grothendieck construction on the diagrams of their face posets. We also derive analogues of well known results on homotopy colimits in the combinatorial setting, including a cofinality theorem and a generalization of Quillen’s Theorem A for posets.

Keywords

homotopy colimit, finite topological space, poset, Grothendieck construction, Quillen’s Theorem A

2010 Mathematics Subject Classification

06A06, 18A30, 18B35, 55P15, 55U10

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Published 29 November 2016