AATRN Watch Party: A New Kind of Cell-Complex for Poset Topology

I will describe a new kind of cell-complex, the “CL-complex”, whose explicit purpose is to represent the homotopy type of a finite poset. Traditionally a poset is represented by its order complex: the simplicial complex whose simplices are the finite totally-ordered subsets of the poset. The order complex completely solves the problem of converting posets to topological spaces, so why would we need anything else? In fact there are good reasons for wishing to have a wide range of topological representatives to choose from. The notion of CL-complex follows naturally from this wish, with order complexes occurring as a special case. As an application, we will revisit Sheehy’s higher-order nerve theorem from the perspective of CL complexes.