(Co)module Representations of Second-Order Functionals

In this talk, I will demonstrate that the well-known approach of representing continuous second order functionals in terms of well-founded question-answer trees is an instance of a general compositional category-theoretic framework in which representations of second-order functionals are modulated by a monad on containers (the topic of last week's seminar) and a certain right (co)module for it. By varying the monad and the (co)module we naturally capture in the same framework standard tree-represented functionals, as well as many others, such as: functionals with finite support; functionals that query their input on one instance; functionals that either query the input once or indicate that no query is needed; functionals that also interact with their environment; and instance reductions as studied in reverse constructive mathematics and Weihrauch reducibility.

This is joint ongoing work with Andrej Bauer (from the University of Ljubljana).