Distilling Monads

Functor calculi have become important tools for understanding structures in algebraic and geometric topology. A functor calculus provides a means of approximating functors by towers of “polynomial” functors, analogous to using Taylor polynomials to approximate functions of real numbers. A number of different types of functor calculi have been developed to address a broad range of questions, but each has been created on an ad-hoc basis. This work is part of a larger program to identify general categorical conditions and processes from which one can build new functor calculi.

As a starting point, we are focused on the methods used to construct the calculus towers of the discrete and abelian functor calculi of [1] and [2], and a dual process used to create the dual calculus cotower of [3]. These methods use comonads that act on a category of functors to construct degree $n$ approximations in the case of the abelian and discrete calculi, and monads to create codegree $n$ approximations in the case of the dual calculus. These monads and comonads are constructed via particular tools – homotopy colimits and limits – that are widely used in algebraic topology for their nice homotopy-theoretic properties. However, these homotopy-theoretic properties are not essential to the (co)monad-building process, suggesting that a much broader range of monads and comonads for building new functor calculi towers can be created by appropriately generalizing these techniques.

In this talk, we will present a generalization of homotopy colimits, which we call a distillation system, that is defined as a type of oplax transformation between two ${\mathrm{Cat}}^{\mathrm{op}}$-actegory structures on $\mathrm{CAT}$. We will explain how these distillation systems encode some of the essential properties of homotopy colimits alluded to above, and the role these properties play in a general functorial process for using a distillation system to transform a strict monoidal functor $\theta :I\to ℬ$ and a $ℬ$-actegory $𝒞$ into a monad that acts on $𝒞$. This process recovers the monads used in the dual calculus, and offers new examples to explore. Future work will focus on using this process to produce sequences of monads to create new types of dual functor calculi.