Central Submonads and Notions of Computation
Abstract
The notion of "centre" has been introduced for many algebraic structures in mathematics. A notable example is the centre of a monoid which always determines a commutative submonoid. Monads in category theory are important algebraic structures that may be used to model computational effects in programming languages and in this paper we show how the notion of centre may be extended to strong monads acting on symmetric monoidal categories. We show that the centre of a strong monad $\mathcal T$, if it exists, determines a commutative submonad $\mathcal Z$ of $\mathcal T$, such that the Kleisli category of $\mathcal Z$ is isomorphic to the premonoidal centre (in the sense of Power and Robinson) of the Kleisli category of $\mathcal T$. We provide three equivalent conditions which characterise the existence of the centre of $\mathcal T$ and we show that every strong monad on many wellknown naturally occurring categories does admit a centre, thereby showing that this new notion is ubiquitous. We also provide a computational interpretation of our ideas which consists in giving a refinement of Moggi's monadic metalanguage. The added benefit is that this allows us to immediately establish a large class of contextually equivalent programs for computational effects that are described via monads that admit a nontrivial centre by simply considering the richer syntactic structure provided by the refinement.
 Publication:

arXiv eprints
 Pub Date:
 July 2022
 arXiv:
 arXiv:2207.09190
 Bibcode:
 2022arXiv220709190C
 Keywords:

 Computer Science  Logic in Computer Science;
 Computer Science  Programming Languages;
 Mathematics  Category Theory
 EPrint:
 25 pages + 8 pages references and appendix