Theories of analytic monads

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 stated in [CJ2]. The Lawvere theories corresponding to these monads are identified via some factorization systems. 2010 Mathematical Subject Classification Primary: 18C05, 18C10, 18C15, 18D50, 03F25, 08B05, 03G30. Secondary: 18A32, 18D10, 18C20, 08B20.