Abstract
Distributive laws give a way of combining two algebraic structures expressed as monads; in this paper we propose a theory of distributive laws for combining algebraic structures expressed as Lawvere theories. We propose four approaches, involving profunctors, monoidal profunctors, an extension of the free finite-product category 2-monad from Cat to Prof, and factorisation systems respectively. We exhibit comparison functors between CAT and each of these new frameworks to show that the distributive laws between the Lawvere theories correspond in a suitable way to distributive laws between their associated finitary monads. The different but equivalent formulations then provide, between them, a framework conducive to generalisation, but also an explicit description of the composite theories arising from distributive laws.
Highlights
Distributive laws give a way of combining two algebraic structures expressed as monads; in this paper we propose a theory of distributive laws for combining algebraic structures expressed as Lawvere theories
A natural question arises—is there a notion of distributive law for Lawvere theories? given the above correspondence with finitary monads on Set, one could say “a distributive law for Lawvere theories is a distributive law between the associated finitary monads on Set.”
As evidence that our definitions do give the correct notion, we prove that all our definitions of distributive law for Lawvere theories correspond suitably to distributive laws between the associated finitary monads, with the composite Lawvere theories corresponding to the composite monads
Summary
We recall the basic definitions and results about Lawvere theories that we will need in the rest of this paper. A morphism of Lawvere theories from L to L′ is a functor making the obvious triangle commute; note that such a functor necessarily strictly preserves finite products. Let L be the Lawvere theory for monoids, and C = Set. Consider a finite-product preserving functor. Recall in Example 1.4 we saw that the morphisms 2 1 in the Lawvere theory for monoids were given by all the elements of T [2], where T is the free monoid monad and [2] is a 2-element set. (Linton [12]) Given a Lawvere theory L we can construct a finitary monad TL on Set by TLX =. Law ≃ Mndf where Mndf denotes the full subcategory of finitary monads on Set. This paper can be seen as providing several equivalent definitions of distributive law for Lawvere theory that extend the above correspondence
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