Layer by layer - Combining Monads

We develop a method to incrementally construct programming languages. Our approach is categorical: each layer of the language is described as a monad. Our method either (i) concretely builds a distributive law between two monads, i.e. layers of the language, which then provides a monad structure to the composition of layers, or (ii) identifies precisely the algebraic obstacles to the existence of a distributive law and gives a best approximant language. The running example will involve three layers: a basic imperative language enriched first by adding non-determinism and then probabilistic choice. The first extension works seamlessly, but the second encounters an obstacle, which results in a best approximant language structurally very similar to the probabilistic network specification language ProbNetKAT

Distributive laws for Lawvere theories

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.Comment: 30 pages, presented at CT2011, lightly edited 2019 for publication in Compositionalit

Constructing applicative functors

Applicative functors define an interface to computation that is more general, and correspondingly weaker, than that of monads. First used in parser libraries, they are now seeing a wide range of applications. This paper sets out to explore the space of non-monadic applicative functors useful in programming. We work with a generalization, lax monoidal functors, and consider several methods of constructing useful functors of this type, just as transformers are used to construct computational monads. For example, coends, familiar to functional programmers as existential types, yield a range of useful applicative functors, including left Kan extensions. Other constructions are final fixed points, a limited sum construction, and a generalization of the semi-direct product of monoids. Implementations in Haskell are included where possible

Exploring the Boundaries of Monad Tensorability on Set

We study a composition operation on monads, equivalently presented as large equational theories. Specifically, we discuss the existence of tensors, which are combinations of theories that impose mutual commutation of the operations from the component theories. As such, they extend the sum of two theories, which is just their unrestrained combination. Tensors of theories arise in several contexts; in particular, in the semantics of programming languages, the monad transformer for global state is given by a tensor. We present two main results: we show that the tensor of two monads need not in general exist by presenting two counterexamples, one of them involving finite powerset (i.e. the theory of join semilattices); this solves a somewhat long-standing open problem, and contrasts with recent results that had ruled out previously expected counterexamples. On the other hand, we show that tensors with bounded powerset monads do exist from countable powerset upwards