Theories of analytic monads

TL;DR

This paper characterizes the relationships between various categories of monads, theories, and operads on Set, establishing equivalences and solving a problem related to their classification.

Contribution

It provides a categorical characterization of analytic and polynomial monads, linking them to regular-linear and rigid theories, and identifies their Lawvere theories.

Findings

Category of analytic monads is equivalent to regular-linear theories

Category of polynomial monads is equivalent to rigid theories

Lawvere theories are identified via factorization systems

Abstract

We characterize the equational theories and Lawvere theories that correspond to the categories of analytic and polynomial monads on Set, and hence also the categories of the symmetric and rigid operads in Set. We show that the category of analytic monads is equivalent to the category of regular-linear theories. The category of polynomial monads is equivalent to the category of rigid theories, i.e. regular-linear theories satisfying an additional global condition. This solves a problem A. Carboni and P. T. Johnstone. The Lawvere theories corresponding to these monads are identified via some factorization systems.