Problems with fixpoints of polynomials of polynomials

TL;DR

This paper investigates fixpoints of polynomial endofunctors over categories of containers, providing methods to compute various fixpoints and exploring their applications in computable analysis and Weihrauch complexity.

Contribution

It introduces a framework for computing fixpoints and fixpoint expressions in categories of containers, extending the syntax with $ ext{zeta}$-binders and connecting to computability and complexity.

Findings

Computed initial algebras, terminal coalgebras, and fixpoints in the setting.

Connected fixpoint constructions to Weihrauch degrees and computable maps.

Extended the syntax of $ ext{mu}$-bicomplete categories with $ ext{zeta}$-expressions.

Abstract

Motivated by applications in computable analysis, we study fixpoints of certain endofunctors over categories of containers. More specifically, we focus on fibred endofunctors over the fibrewise opposite of the codomain fibration that can be themselves be represented by families of polynomial endofunctors. In this setting, we show how to compute initial algebras, terminal coalgebras and another kind of fixpoint $\zeta$. We then explore a number of examples of derived operators inspired by Weihrauch complexity and the usual construction of the free polynomial monad. We introduce $\zeta$-expressions as the syntax of $\mu$-bicomplete categories, extended with $\zeta$-binders and parallel products, which thus have a natural denotation in containers. By interpreting certain $\zeta$-expressions in a category of type 2 computable maps, we are able to capture a number of meaningful Weihrauch degrees, ranging from closed choice on $\{0, 1\}$ to determinacy of infinite parity games, via an "answerable part" operator.