🤖 AI Summary
This work investigates fixed points of fibred endofunctors induced by families of polynomial functors in the category of containers and their applications in computable analysis. By introducing ζ-expressions as an extended syntax for μ-bicomplete categories and integrating a “answerable part” operator, the study provides a unified characterization of Weihrauch degrees ranging from closed choice to the decidability of infinite parity games. Methodologically, it constructs initial algebras, final coalgebras, and a novel notion of ζ-fixed points, while synthesizing tools such as fibred functors, polynomial functors, ζ-binders, and parallel products. This framework successfully yields semantic interpretations for several key Weihrauch degrees, significantly enhancing expressive power and applicability in computable analysis and complexity classification.
📝 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.