Exponential Kleisli monoids as Eilenberg-Moore algebras

TL;DR

This paper generalizes the classical characterization of exponentiable spaces by identifying exponentiable objects in a monoidal category of Kleisli monoids as those with algebraic structure, linking monads, monoids, and topology.

Contribution

It introduces a new framework connecting lax monoidal powerset-enriched monads with exponentiable objects as algebraic Kleisli monoids, extending classical topology results.

Findings

Exponentiable objects are characterized as algebraic Kleisli monoids.

The framework generalizes classical topology results to monoidal categories.

Provides a new perspective on monads and exponentiation in categorical structures.

Abstract

Lax monoidal powerset-enriched monads yield a monoidal structure on the category of monoids in the Kleisli category of a monad. Exponentiable objects in this category are identified as those Kleisli monoids with algebraic structure. This result generalizes the classical identification of exponentiable topological spaces as those whose lattice of open subsets forms a continuous lattice.