Topology Seminar

I'll discuss some applications of Dowker duality in data analysis. I will then present three short, new proofs of Dowker duality using various poset fiber lemmas. I will introduce modifications of joins and products of simplicial complexes called relational join and relational product complexes. These relational complexes can be constructed whenever there is a relation between simplicial complexes, which includes the context of Dowker duality and covers of simplicial complexes. In this more general setting, I will show that the homologies of the simplicial complexes and the relational complexes fit together in a long exact sequence. If time permits, I'll discuss similar results for profunctors, which are generalizations of relations to categories.