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

TL;DR

This paper generalizes Thomason's theorem on homotopy colimits over posets, linking diagrams of simplicial complexes to face posets and extending classical results in combinatorial homotopy theory.

Contribution

It introduces a new characterization of homotopy colimits of simplicial complex diagrams via the Grothendieck construction, generalizing existing theorems.

Findings

Characterization of homotopy colimits in terms of face posets

A cofinality theorem for homotopy colimits

A generalized Quillen's Theorem A for posets

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.