arXiv subjects

Jean-Baptiste Vienney

Publications and source records attributed to Jean-Baptiste Vienney.

4 recordsLinked to original sources

arXiv2026

Extracting an $\mathbb{N}$-filtered differential modality from a differential modality

A differential modality is a comonad on an additive symmetric monoidal category $(\mathsf{C},\otimes,I)$, whose underlying functor we denote $!\colon\mathsf{C} \rightarrow \mathsf{C}$, together with some additional structure including a differential operator $\partial\colon!A \otimes A \rightarrow !A$. A morphism $f\colon !A \rightarrow B$ is interpreted as a smooth function from $A$ to $B$. The notion of an $\mathbb{N}$-filtered differential modality is a variant in which a notion of degree is present. Instead of a single functor $!\colon \mathsf{C} \rightarrow \mathsf{C}$, we ask for a family of functors $!_{\le n}\colon\mathsf{C} \rightarrow \mathsf{C}$ where $n \in \mathbb{N}$. Now, a morphism $f\colon !_{\le n} A \rightarrow B$ is interpreted as a smooth function from $A$ to $B$, with degree less than $n$ for some notion of degree. We prove that under mild conditions, every differential modality on an additive symmetric monoidal category with underlying functor $!\colon \mathsf{C} \rightarrow \mathsf{C}$ yields an $\mathbb{N}$-filtered differential modality with underlying functors $!_{\le n}\colon\mathsf{C} \rightarrow \mathsf{C}$. A morphism $f\colon !_{\le n}A \rightarrow B$ corresponds to a polynomial map of degree less than $n$ from $A$ to $B$, in the sense that the $(n+1)$-th derivative of $f$ is $0$.

math.CT↗

arXiv2025

A bialgebraic characterization of symmetric powers in $\mathbb{Q}_{\ge 0}$-linear symmetric monoidal categories

In any symmetric monoidal category, the $n$-th (co)equalizer symmetric power of an object $A$ is the (co)equalizer of all the permutations from $A^{\otimes n}$ to itself. If the symmetric monoidal category is $\mathbb{Q}_{\ge 0}$-linear, that is, enriched over $\mathbb{Q}_{\ge 0}$-modules, the notions of $n$-th equalizer symmetric power and $n$-th coequalizer symmetric power are equivalent. In this context, the $n$-th symmetric power of $A$ can be described as the intermediate object $A_n$ in a splitting of the idempotent $\frac{1}{n!}\underset{σ\in S_n}{\sum}σ\colon A^{\otimes n} \rightarrow A^{\otimes n}$. We define a permutation splitting as a countable family of such splittings. The main goal of this paper is to prove two theorems. The first theorem exhibits in any $\mathbb{Q}_{\ge 0}$-linear symmetric monoidal category a bijection between operations making a graded object $(A_n)_{n \ge 0}$ into a permutation splitting and operations making this graded object into a bialgebraic structure that we call a binomial bimonoid. Binomial bimonoids can be defined in any additive symmetric monoidal category. The second theorem shows that, in any $\mathbb{Q}_{\ge 0}$-linear symmetric monoidal category, the biassociativity and bicommutativity axioms may be omitted from the definition of a binomial bimonoid. We then show that being a binomial bimonoid in a $\mathbb{Q}_{\ge 0}$-linear symmetric monoidal category is a property: two binomial bimonoids are isomorphic whenever their underlying graded objects are isomorphic. This result does not extend to arbitrary additive symmetric monoidal categories since both the one-variable polynomial algebra and the one-variable divided power polynomial algebra over a field $k$ of positive characteristic are non-isomorphic binomial $k$-bialgebras with isomorphic underlying $\mathbb{N}$-graded vector spaces.

math.CT↗

arXiv2025

An algebra modality admitting countably many deriving transformations

A differential category is an additive symmetric monoidal category, that is, a symmetric monoidal category enriched over commutative monoids, with an algebra modality, axiomatizing smooth functions, and a deriving transformation on this algebra modality, axiomatizing differentiation. Lemay proved that a comonoidal algebra modality has at most one deriving transformation, thus differentiation is unique in models of differential linear logic. It was then an open problem whether this result extends to arbitrary algebra modalities. We answer this question in the negative. We build a free "commutative rig with a self-map" algebra modality on the category of commutative monoids, where the self-map can be seen as an arbitrary smooth function. We then define a countable family of distinct deriving transformations $({}_{n}\mathsf{d})_{n \in \mathbb{N}}$ on this algebra modality where the parameter $n$ controls the derivative of the self-map. It shows that in a differential category, a single algebra modality may admit multiple, inequivalent notions of differentiation.

math.CT↗

arXiv2024

Graded Differential Categories and Graded Differential Linear Logic

In Linear Logic ($\mathsf{LL}$), 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 ($\mathsf{DiLL}$) 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 ($\mathsf{GLL}$) is a variation of Linear Logic in such a way that $!$ is now indexed over a semiring $R$. This $R$-grading allows for non-linear proofs of degree $r \in R$, such that the linear proofs are of degree $1 \in R$. There has been recent interest in combining these two variations of $\mathsf{LL}$ together and developing Graded Differential Linear Logic ($\mathsf{GDiLL}$). In this paper we present a sequent calculus for $\mathsf{GDiLL}$, 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 $\mathsf{GDiLL}$.

cs.LO↗