Monotone weak distributive laws over the lifted powerset monad in categories of algebras

TL;DR

This paper investigates the conditions under which monotone weak distributive laws over powerset monads can be lifted in categories of algebras, revealing partial automaticity and specific existence criteria.

Contribution

It characterizes when monotone weak distributive laws exist in categories of algebras, especially combining probabilities and nondeterminism in compact Hausdorff spaces.

Findings

Partial automatic lifting of laws observed in sets and compact Hausdorff spaces

Characterization of categories where such laws exist

Existence of laws combining probabilities and nondeterminism in specific categories

Abstract

Noticing the similarity between the monotone weak distributive laws combining two layers of nondeterminism in sets and in compact Hausdorff spaces, we study whether the latter law can be obtained automatically as a weak lifting of the former. This holds partially, but does not generalize to other categories of algebras: we then characterize when exactly monotone weak distributive laws over powerset monads in categories of algebras exist, exhibiting a law combining probabilities and non-determinism in compact Hausdorff spaces and showing on the other hand that such laws do not exist in a lot of other cases.