A topos for extended Weihrauch degrees
Abstract
Weihrauch reducibility is a notion of reducibility between computational problems that is useful to calibrate the uniform computational strength of a multivalued function. It complements the analysis of mathematical theorems done in reverse mathematics, as multi-valued functions on represented spaces can be considered as realizers of theorems in a natural way. Despite the rich literature and the relevance of the applications of category theory in logic and realizability, actually there are just a few works starting to study the Weihrauch reducibility from a categorical point of view. The main purpose of this work is to provide a full categorical account to the notion of extended Weihrauch reducibility introduced by A. Bauer, which generalizes the original notion of Weihrauch reducibility. In particular, we present a tripos and a topos for extended Weihrauch degrees. We start by defining a new tripos, abstracting the notion of extended Weihrauch degrees, and then we apply the tripos-to-topos construction to obtain the desired topos. Then we show that the Kleene-Vesley topos is a topos of -sheaves for a certain Lawvere-Tierney topology over the topos of extended Weihrauch degrees.
Cite
@article{arxiv.2505.08697,
title = {A topos for extended Weihrauch degrees},
author = {Samuele Maschio and Davide Trotta},
journal= {arXiv preprint arXiv:2505.08697},
year = {2026}
}