Theories of analytic monads

Abstract

In this paper we characterise the categories of Lawvere theories and equational theories that correspond to the categories of analytic and polynomial monads onSet, and hence also to the categories of the symmetric and rigid operads inSet. 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, that is, regular-linear theories satisfying an additional global condition. This solves a problem posed by A. Carboni and P. T. Johnstone. The Lawvere theories corresponding to these monads are identifiedviasome factorisation systems. We also show that the categories of analytic monads and finitary endofunctors onSetare monadic over the category of analytic functors. The corresponding monad for analytic monads distributes over the monad for finitary endofunctors and hence the category of (finitary) monads onSetis monadic over the category of analytic functors. This extends a result of M. Barr.