Weihrauch problems as containers

TL;DR

This paper explores the categorical structure of Weihrauch problems, showing they can be viewed as containers and linking various degrees and operators to the theory of polynomial functors.

Contribution

It introduces a new perspective by modeling Weihrauch problems as containers and connects their degrees and operators to polynomial functors.

Findings

Weihrauch problems can be modeled as containers over projective represented spaces.

Weihrauch reductions correspond to container morphisms.

Operators like composition product arise from polynomial functors.

Abstract

We note that Weihrauch problems can be regarded as containers over the category of projective represented spaces and that Weihrauch reductions correspond exactly to container morphisms. We also show that Bauer's extended Weihrauch degrees and the posetal reflection of containers over partition assemblies are equivalent. Using this characterization, we show how a number of operators over Weihrauch degrees, such as the composition product, also arise naturally from the abstract theory of polynomial functors.