Abstract
We characterize the categories of semi-analytic monads, regular Lawvere theories, and regular operads that are equivalent to the category of regular equational theories. We also show that the category of all finitary monads on Set is monadic over the category of semi-analytic functors.
Highlights
The category of algebras of a equational theory can be equivalently described as a category of models of a Lawvere theory or as a category of algebras of a finitary monad on the category Set or a category of algebras of a operad
The main objective of this paper is to describe the categories of regular Lawvere theories RegLT, semi-analytic monads SanMnd, and regular operads RegOp that correspond to the category of regular equational theories with regular morphisms
We show that the category of regular Lawvere theories is equivalent to the category of regular operads and to the category of semi-analytic monads and to the category of regular equational theories; cf
Summary
The category of algebras of a (finitary) equational theory can be equivalently described as a category of models of a Lawvere theory or as a category of algebras of a finitary monad on the category Set or a category of algebras of a (generalized) operad. The four categories of (finitary) equational theories, Lawvere theories, finitary monads on Set and (generalized) operads are equivalent. These equivalences induce a correspondence between various subcategories. We show that the category of regular Lawvere theories is equivalent to the category of regular operads and to the category of semi-analytic monads and to the category of regular equational theories; cf Section 6. When Sn acts on a set An on the right and on the set Bn on the left, the set A ⊗n B is the usual tensor product of Sn-sets
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