2 research outputs found
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
Commutativity
We describe a general framework for notions of commutativity based on
enriched category theory. We extend Eilenberg and Kelly's tensor product for
categories enriched over a symmetric monoidal base to a tensor product for
categories enriched over a normal duoidal category; using this, we re-find
notions such as the commutativity of a finitary algebraic theory or a strong
monad, the commuting tensor product of two theories, and the Boardman-Vogt
tensor product of symmetric operads.Comment: 48 pages; final journal versio