Matthew Burke

Title: The Calculus of Functors using Sheafification
Abstract:
In classical calculus we approximate an appropriately differentiable function using a sequence of simpler functions called the Taylor polynomials. In an analogous way we can approximate a functor whose domain and codomain are appropriately topological by using a sequence of simpler functors. These simpler functors can be described using a universal property and a condition asserting that certain pullbacks are taken to certain homotopy pushouts. In this talk we present an alternative perspective based on a paper of de Brito and Weiss in the case that the domain of the functor is the category of smooth manifolds. First we describe the approximating ‘polynomial’ functors as generalised sheafifications with respect to a sequence of Grothendieck coverages. Then we explore how this approach generalises when we replace the category of smooth manifolds with more general categories.