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Conference Papers Year : 2023

Central Submonads and Notions of Computation: Soundness, Completeness and Internal Languages

Titouan Carette
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  • PersonId : 1119464
Louis Lemonnier
Vladimir Zamdzhiev
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  • PersonId : 1089895

Abstract

Monads in category theory are algebraic structures that can be used to model computational effects in programming languages. We show how the notion of "centre", and more generally "centrality", i.e., the property for an effect to commute with all other effects, may be formulated for strong monads acting on symmetric monoidal categories. We identify three equivalent conditions which characterise the existence of the centre of a strong monad (some of which relate it to the premonoidal centre of Power and Robinson) and we show that every strong monad on many well-known naturally occurring categories does admit a centre, thereby showing that this new notion is ubiquitous. More generally, we study central submonads, which are necessarily commutative, just like the centre of a strong monad. We provide a computational interpretation by formulating equational theories of lambda calculi equipped with central submonads, we describe categorical models for these theories and prove soundness, completeness and internal language results for our semantics.
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Dates and versions

hal-03662565 , version 1 (09-05-2022)
hal-03662565 , version 2 (11-05-2022)
hal-03662565 , version 3 (20-07-2022)
hal-03662565 , version 4 (10-10-2023)

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Titouan Carette, Louis Lemonnier, Vladimir Zamdzhiev. Central Submonads and Notions of Computation: Soundness, Completeness and Internal Languages. 2023 38th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS), Jun 2023, Boston (MA), United States. ⟨10.1109/LICS56636.2023.10175687⟩. ⟨hal-03662565v4⟩
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