Strong Dinatural Transformations and Generalised Codensity Monads
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
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2025
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| _version_ | 1866915823966224384 |
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| author | Piróg, Maciej Sieczkowski, Filip |
| author_facet | Piróg, Maciej Sieczkowski, Filip |
| contents | We introduce dicodensity monads: a generalisation of pointwise codensity monads generated by functors to monads generated by mixed-variant bifunctors. Our construction is based on the notion of strong dinaturality (also known as Barr dinaturality), and is inspired by denotational models of certain types in polymorphic lambda calculi - in particular, a form of continuation monads with universally quantified variables, such as the Church encoding of the list monad in System F. Extending some previous results on Cayley-style representations, we provide a set of sufficient conditions to establish an isomorphism between a monad and the dicodensity monad for a given bifunctor. Then, we focus on the class of monads obtained by instantiating our construction with hom-functors and, more generally, bifunctors given by objects of homomorphisms (that is, internalised hom-sets between Eilenberg--Moore algebras). This gives us, for example, novel presentations of monads generated by different kinds of semirings and other theories used to model ordered nondeterministic computations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_06777 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Strong Dinatural Transformations and Generalised Codensity Monads Piróg, Maciej Sieczkowski, Filip Logic in Computer Science Programming Languages D.3.1 We introduce dicodensity monads: a generalisation of pointwise codensity monads generated by functors to monads generated by mixed-variant bifunctors. Our construction is based on the notion of strong dinaturality (also known as Barr dinaturality), and is inspired by denotational models of certain types in polymorphic lambda calculi - in particular, a form of continuation monads with universally quantified variables, such as the Church encoding of the list monad in System F. Extending some previous results on Cayley-style representations, we provide a set of sufficient conditions to establish an isomorphism between a monad and the dicodensity monad for a given bifunctor. Then, we focus on the class of monads obtained by instantiating our construction with hom-functors and, more generally, bifunctors given by objects of homomorphisms (that is, internalised hom-sets between Eilenberg--Moore algebras). This gives us, for example, novel presentations of monads generated by different kinds of semirings and other theories used to model ordered nondeterministic computations. |
| title | Strong Dinatural Transformations and Generalised Codensity Monads |
| topic | Logic in Computer Science Programming Languages D.3.1 |
| url | https://arxiv.org/abs/2510.06777 |