# Between Markov and restriction: Two more monads on categories for relations

## TL;DR

This paper introduces two new abstract categories for relations that extend Markov and restriction categories, based on axiomatic notions of mass and domain, and explores their properties and applications in semiring-weighted relations.

## Contribution

It proposes two more abstract gs-monoidal categories for relations, characterized by mass and domain, and studies their associated monads and properties.

## Key findings

- New categories preserve mass and domain axioms.
- Kleisli categories of these monads maintain key equations.
- Applications to semiring-weighted relations are demonstrated.

## Abstract

The study of categories abstracting the structural properties of relations has been extensively developed over the years, resulting in a rich and diverse body of work. A previous paper offered a survey providing a modern and comprehensive presentation of these ``categories for relations'' as instances of gs-monoidal categories, showing how they arise as Kleisli categories of suitable symmetric monoidal monads. The end result was a taxonomy that organised numerous related concepts in the literature, including in particular Markov and restriction categories. This paper further enriches the taxonomy: it proposes two categories that are once more instances of gs-monoidal categories, yet more abstract than Markov and restriction categories. They are characterised by an axiomatic notion of mass and domain of an arrow, the latter one of the key ingredient of restriction categories, which generalises the domain of partial functions. The paper then introduces mass and domain preserving monads, proving that the associated Kleisli categories in fact preserve the corresponding equations and that these monads arise naturally for the categories of semiring-weighted relations.

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Source: https://tomesphere.com/paper/2508.20054