Topological Posets and Tropical Phased Matroids

For a discrete poset \({\mathcal {X}}\), McCord proved that the natural map \(|{{\mathcal {X}}}|\rightarrow {{\mathcal {X}}}\), from the order complex to the poset with the Up topology, is a weak homotopy equivalence. Much later, Živaljević defined the notion of order complex for a topological poset. For a large class of topological posets we prove the analog of McCord’s theorem, namely that the natural map from the order complex to the topological poset with the Up topology is a weak homotopy equivalence. A familiar topological example is the Grassmann poset \(\mathcal {G}_n(\mathbb {{\mathbb {R}}})\) of proper non-zero linear subspaces of \({\mathbb {R}}^{n+1}\) partially ordered by inclusion. But our motivation in topological combinatorics is to apply the theorem to posets associated with tropical phased matroids over the tropical phase hyperfield, and in particular to elucidate the tropical version of the MacPhersonian Conjecture. This is explained in Sect. 2.