Fell bundles over étale groupoids provide a unifying framework for various combinatorial constructions of C*-algebras, e.g. coming from graphs or group actions. In recent years, more general Fell bundles over inverse semigroups have also been considered by people such as Exel, Meyer and Kwaśniewski. But how much more general are they really? Could there be some simple conditions characterising those coming from Fell bundles over étale groupoids? In this case, can we recover the original groupoid bundle from the inverse semigroup bundle? Does this even yield a functorial duality in appropriate categories? Similar questions also arise for general Banach bundles and their corresponding "Banach cosheaves", which are essentially just families of Banach spaces sharing the same linear structure on all overlaps. Here we outline some work in progress to address these issues in collaboration with Alcides Buss.

Bio: Tristan Bice is an Australian mathematician working in the Czech Republic with previous research experience in Poland, Brazil, Canada and Japan. Having begun his career in set theory and C*-algebras, his current research is primarily concerned with dualities between various algebraic and topological structures.