An ordered framework for partial multivalued functors

TL;DR

This paper develops a categorical framework linking functors, partial multivalued functors, and direct images, extending the relationship known in the category of relations to a broader setting.

Contribution

It introduces an ordered framework that generalizes the connection between functors and partial multivalued functors beyond Rel, using category theory concepts.

Findings

Establishes a categorical structure relating functors and partial multivalued functors.

Provides a generalized framework similar to the Kleisli category of the power set functor.

Enhances understanding of the relationship between relations and functors in category theory.

Abstract

The category Rel of sets and relations intimately ties the notions of function, partial multivalued function, and direct image under a function through the description of Rel as the Kleisli category of the covariant power set functor on Set. We present a suitable framework to obtain a similar relationship between the concepts of functor, partial multivalued functor, and the direct image under a functor.