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 the...

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Bibliographic Details
Published in:arXiv.org 2013-09
Main Authors: Bowler, Nathan, Goncharov, Sergey, Paul Blain Levy, Schröder, Lutz
Format: Article
Language:English
Subjects:
Commutation
Mathematical analysis
Programming languages
Semantics
Tensors
Online Access:Get full text
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.1309.2128