Title: Moduli Spaces of Arrows
Abstract: Many moduli spaces parametrize isomorphism classes in some category, and we can wonder if the morphisms of said category can be similarly parametrized. The relevant moduli stack encodes families of morphisms, but its points are still objects and not morphisms. The idea of this talk is to consider a moduli stack whose points are the morphisms involved in the original moduli problem. We will develop this idea for the specific case of the moduli problem of vector bundles over a fixed base and observe that this has a generalization: if we can parametrize arrows of vector bundles, we can parametrize diagrams thereof of more general shapes. We will then discuss some implications of this line of thought for the moduli problem of Higgs bundles and non-Abelian Hodge theory. Time permitting, we will discuss abstract moduli theory and homotopy theory in this light. This talk reports on joint work with Steven Rayan.
