Topological realizations of finite posets

Abstract

I introduce a new class of cell complexes, called CL-complexes, that provide alternative topological realizations of finite posets. Each realization of a given poset has the same homotopy type as the order complex of that, so in some sense these spaces do not provide anything really new. Their value is that they enable short transparent proofs of standard results in poset homotopy. These include Quillen's Theorem A for finite posets, the Nerve Theorem for good covers of a topological space, and Sheehy's multicover generalization of the Nerve Theorem. Order complexes retain a special status: the order complex of a finite poset admits CL-complex structures both for the poset itself and for its opposite poset. The definition of CL-complexes is motivated by an elementary generalization of the Carrier Lemma of J. W. Walker, which in its original form constructs a canonical homotopy class of maps from a simplicial complex to a space covered by a suitable family of contractible sets. In the generalized lemma, the role played by simplicial complexes and the role played by spaces covered by contractible sets can both be fulfilled by CL-complexes. This sets up an equivalence of categories that tacitly underlies our approach.