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Jean-Baptiste Vienney (Mar 21 2023 at 01:44):My first paper, co-authored with @JS PL (he/him) , appeared on the ArXiv :smile:
https://arxiv.org/abs/2303.10586
Title: Graded Differential Categories and Graded Differential Linear Logic
Abstract: In Linear Logic (), the exponential modality brings forth a distinction between non-linear proofs and linear proofs, where linear means using an argument exactly once. Differential Linear Logic () is an extension of Linear Logic which includes additional rules for which encode differentiation and the ability of linearizing proofs. On the other hand, Graded Linear Logic () is a variation of Linear Logic in such a way that is now indexed over a semiring . This -grading allows for non-linear proofs of degree , such that the linear proofs are of degree . There has been recent interest in combining these two variations of together and developing Graded Differential Linear Logic (). In this paper we present a sequent calculus for , as well as introduce its categorical semantics, which we call graded differential categories, using both coderelictions and deriving transformations. We prove that symmetric powers always give graded differential categories, and provide other examples of graded differential categories. We also discuss graded versions of (monoidal) coalgebra modalities, additive bialgebra modalities, and the Seely isomorphisms, as well as their implementations in the sequent calculus of .
John Baez (Mar 21 2023 at 01:46):Congratulations!
Jean-Baptiste Vienney (Mar 21 2023 at 01:46):Thanks, I'm super excited!
Jean-Baptiste Vienney (Mar 21 2023 at 01:48):So much that I mistakenly resubmitted it this week-end and had to wait 24 hours more before it appears.
John Baez (Mar 21 2023 at 02:01)::upside_down:
Jean-Baptiste Vienney (Mar 21 2023 at 02:06):I can say also that a beautiful characterization of symmetric powers in symmetric monoidal -linear categories and the associated string diagrams were born from this project but it will be the subject and the only subject of my next paper (I want to talk about that at CT2023 also). We didn't say anything about this in this paper because it's a secret.
John Baez (Mar 21 2023 at 02:19):Good! I try to avoid discussing future work in my papers, not because I'm trying to keep it secret, but because I like to make sure the details are correct before I talk about something in a published paper. (I talk a lot about future work in blog articles, because I don't feel too bad if I say something wrong in a blog article: I can easily correct it!)
Jean-Baptiste Vienney (Mar 21 2023 at 02:36):I'd love to make a post on the nCafé about that if a nCafé member wants to allow me to make a guest post :smile:
John Baez (Mar 21 2023 at 04:42):Sure, sounds good. Just try to keep it simple: if you explain a small fraction of your ideas very nicely, people will be glad to read the rest. The mistake of first-time bloggers is trying to say too much, and thus writing more than anyone wants to read.
Jean-Baptiste Vienney (Mar 21 2023 at 04:53):Ahh okay, these string diagrams are much more simple than graded differential categories. They verify just two (families) of equations that's why I want to talk about them. They are the most catchy thing I found. A reason why I didn't want to talk about them in this paper is because they can be understood independently from this more complicated logic. So yes, I think I can do it short.
Jean-Baptiste Vienney (Mar 21 2023 at 05:07):So I'd be glad to make this post if you explain to me how to proceed.