Moduli stacks of quiver bundles with applications to Higgs bundles

We provide a general method for constructing moduli stacks whose points are diagrams of vector bundles over a fixed base, indexed by a fixed simplicial set -- that is, quiver bundles of a fixed shape. We discuss some constraints on the base for these moduli stacks to be algebraic and observe that a large class of interesting schemes satisfy these constraints. Using this construction, we provide an alternate construction of moduli stacks of Higgs bundles along with a proof of algebraicity following readily from the algebraicity of moduli stacks of quiver bundles. One feature of our approach is that, for each of the moduli stacks we discuss, there are algebraic moduli stacks parametrizing morphisms of the objects being classified. We discuss some potential applications of this in categorifying non-abelian Hodge theory in a sense we will make precise. We also discuss potential applications of our methods and perspectives to the subjects of quiver modifications, abstract moduli theory, and homotopy theory.