Title: CoCalculus from Monads
Abstract: This is a report from team Functor Calculus at the 4th Women in Topology conference. In earlier work, K. Hess and B. Johnson invented a way of producing what they call “calculus towers” from comonads. These calculus towers generalize functor calculus, and recover other important examples of towers in topology such as certain towers of localizations. In joint work with R. Brooks, K. Hess, B. Johnson, J. Rasmusen and B. Schreiner we produced a dualization of the Hess-Johnson machinery. The resulting “cocalculus towers” recover dual functor calculus, an important co-tower approximating homotopy functors to spectra. Both types of functor calculus are used to approximate important functors in homotopy theory in a manner similair to the Taylor series approximations of functions, but an important distinction for functors is that it is not always possible to recover the entire series from it’s homogeneous parts. Using dual calculus, R. McCarthy and several of his students (including me) produced a variety of results which indicated when one can recover the series from the homogeneous pieces. In this talk, I will explain the generalization of dual calculus to “cocalculus” and explain what kind of results we hope this may lead to in the future.