Monads and theories

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Algebraic theories, monads, and arities

Monads are of interest both in semantics and in higher dimensional algebra. It turns out that the idea behind usual notion finitary monads (whose values on all sets can be computed from their values on finite sets) extends to a more general class of monads called monads with arities, so that not only algebraic theories can be computed from a proper set of arities, but also more general structur...

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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...

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On Rational Monads and Free Iterative Theories

For every finitary endofunctor H of Set a rational algebraic theory (or a rational finitary monad) R is defined by means of solving all finitary flat systems of recursive equations over H. This generalizes the result of Elgot and his coauthors, describing a free iterative theory of a polynomial endofunctor H as the theory R of all rational infinite trees. We present a coalgebraic proof that R i...

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Monads with arities and their associated theories

After a review of the concept of ‘‘monad with arities’’ we show that the category of algebras for such a monad has a canonical dense generator. This is used to extend the correspondence between finitary monads on sets and Lawvere’s algebraic theories to a general correspondence between monads and theories for a given category with arities. As an application we determine arities for the free gro...

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ژورنال

عنوان ژورنال: Advances in Mathematics

سال: 2019

ISSN: 0001-8708

DOI: 10.1016/j.aim.2019.05.016