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Central Submonads and Notions of Computation

Titouan Carette 1, 2 Louis Lemonnier 1, 3, 2 Vladimir Zamdzhiev 4
2 QuaCS - Quantum Computation Structures
Inria Saclay - Ile de France, LMF - Laboratoire Méthodes Formelles
4 MOCQUA - Designing the Future of Computational Models
Inria Nancy - Grand Est, LORIA - FM - Department of Formal Methods
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) can be seen as generalisations of monoids 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 T , if it exists, determines a commutative submonad Z of T , such that the Kleisli category of Z is isomorphic to the premonoidal centre (in the sense of Power and Robinson) of the Kleisli category of T . We provide three equivalent conditions which characterise the existence of the centre of T . While not every strong monad admits a centre, we show that every strong monad on well-known 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 terms for monads that admit a non-trivial centre by simply looking at the richer syntactic structure provided by the refinement.
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Preprints, Working Papers, ...
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https://hal.archives-ouvertes.fr/hal-03662565
Contributor : Louis Lemonnier Connect in order to contact the contributor
Submitted on : Wednesday, May 11, 2022 - 11:17:54 PM
Last modification on : Thursday, May 19, 2022 - 3:15:16 AM

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  • HAL Id : hal-03662565, version 2

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Titouan Carette, Louis Lemonnier, Vladimir Zamdzhiev. Central Submonads and Notions of Computation. 2022. ⟨hal-03662565v2⟩

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