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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeNov 20th 2024
   • PermaLink
   Author: David_Corfield
   Format: MarkdownItexOne might think that this new article would connect to linear HoTT in some way. * Elies Harington, Samuel Mimram, _Polynomials in homotopy type theory as a Kleisli category_ [[arXiv:2411.09950](https://arxiv.org/abs/2411.09950)] > Polynomials in a category have been studied as a generalization of the traditional notion in mathematics. Their construction has recently been extended to higher groupoids, as formalized in homotopy type theory, by Finster, Mimram, Lucas and Seiller, thus resulting in a cartesian closed bicategory. We refine and extend their work in multiple directions. We begin by generalizing the construction of the free symmetric monoid monad on types in order to handle arities in an arbitrary universe. Then, we extend this monad to the (wild) category of spans of types, and thus to a comonad by self-duality. Finally, we show that the resulting Kleisli category is equivalent to the traditional category of polynomials. This thus establishes polynomials as a (homotopical) model of linear logic. In fact, we explain that it is closely related to a bicategorical model of differential linear logic introduced by Melliès. Taking a look at their exponential comonad, it's forming some kind of sum: $!A = \sum_{E: V} (E \to A}$, extended to the spans that correspond to linear types.

   One might think that this new article would connect to linear HoTT in some way.

   • Elies Harington, Samuel Mimram, Polynomials in homotopy type theory as a Kleisli category [arXiv:2411.09950]

   Polynomials in a category have been studied as a generalization of the traditional notion in mathematics. Their construction has recently been extended to higher groupoids, as formalized in homotopy type theory, by Finster, Mimram, Lucas and Seiller, thus resulting in a cartesian closed bicategory. We refine and extend their work in multiple directions. We begin by generalizing the construction of the free symmetric monoid monad on types in order to handle arities in an arbitrary universe. Then, we extend this monad to the (wild) category of spans of types, and thus to a comonad by self-duality. Finally, we show that the resulting Kleisli category is equivalent to the traditional category of polynomials. This thus establishes polynomials as a (homotopical) model of linear logic. In fact, we explain that it is closely related to a bicategorical model of differential linear logic introduced by Melliès.

   Taking a look at their exponential comonad, it’s forming some kind of sum: , extended to the spans that correspond to linear types.