Title: Modules indexed by small categories as generalized bundles
Speaker: Sanjeevi Krishnan - The Ohio State University
Abstract: Representations of a fundamental group are interpretable as flat vector bundles. The advantage of this interpretation is that the algebraic topology of the base space straightforwardly gives some constraints on the decomposability of the representation into simpler sub-representations. The goal of this talk is to develop some algebraic topology for a nascent generalization of bundles [Hofstra], where 1. fibers can vary, 2. structure groups generalize to structure semigroups, and 3. parallel transport generalizes to arbitrary modules indexed by small categories, include multidimensional persistence modules. Tameness and multiflags, introduced in the previous talk, all have interpretations in terms of the the structure semigroup. The base space in this setting is interpretable as a directed space. These vector semibundles with a given structure semigroup S are classifying by a directed classifying space BS. Some basic decomposition theorems for semigroups (e.g. McAllister's Theorem) yield decomposition results for vector semibundles into vector bundles, which are then amenable to tools from classical algebraic topology. In turn, the set of all equivalence classes of principle S-bundles over a directed space X, a semigroup itself under some weak commutativity assumptions on S, forms an interesting invariant on a directed space X itself. This all represents joint work with Crichton Ogle.