Categorifying computable reducibilities

This paper discusses categorical formulations of the Medvedev, Muchnik and Weirauch reducibilities in Computability Theory and connects these reducibilities to different Lawvere categorical doctrines. These specific doctrines were used in previous work to provide categorical models for Dialectica logical properties. We relate Medvedev and Weyrauch doctrines to the Dialectica doctrine, showing that all these doctrines can be conceptualized in terms of (logic) quantifier (existential and universal) completions.