Additive Enrichment from Coderelictions
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918381010026496 |
|---|---|
| author | Lemay, Jean-Simon Pacaud |
| author_facet | Lemay, Jean-Simon Pacaud |
| contents | Differential linear categories provide the categorical semantics of the multiplicative and exponential fragments of Differential Linear Logic. Briefly, a differential linear category is a symmetric monoidal category that is enriched over commutative monoids (called additive enrichment) and has a monoidal coalgebra modality that is equipped with a codereliction. The codereliction is what captures the ability of differentiating non-linear proofs via linearization in Differential Linear Logic. The additive enrichment plays an important role since it allows one to express the famous Leibniz rule. However, the axioms of a codereliction can be expressed without any sums or zeros. Therefore, it is natural to ask if one can consider a possible non-additive enriched version of differential linear categories. In this paper, we show that even if a codereliction can technically be defined in a non-additive setting, it nevertheless induces an additive enrichment via bialgebra convolution. Thus, we obtain a novel characterization of a differential linear category as a symmetric monoidal category with a monoidal bialgebra modality equipped with a codereliction. Moreover, we also show that coderelictions are, in fact, unique. We also introduce monoidal Hopf coalgebra modalities and discuss how antipodes relate to enrichment over Abelian groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_14134 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Additive Enrichment from Coderelictions Lemay, Jean-Simon Pacaud Category Theory 18F40, 19D23, 16T10, 16T05, 18E05, 18C15 Differential linear categories provide the categorical semantics of the multiplicative and exponential fragments of Differential Linear Logic. Briefly, a differential linear category is a symmetric monoidal category that is enriched over commutative monoids (called additive enrichment) and has a monoidal coalgebra modality that is equipped with a codereliction. The codereliction is what captures the ability of differentiating non-linear proofs via linearization in Differential Linear Logic. The additive enrichment plays an important role since it allows one to express the famous Leibniz rule. However, the axioms of a codereliction can be expressed without any sums or zeros. Therefore, it is natural to ask if one can consider a possible non-additive enriched version of differential linear categories. In this paper, we show that even if a codereliction can technically be defined in a non-additive setting, it nevertheless induces an additive enrichment via bialgebra convolution. Thus, we obtain a novel characterization of a differential linear category as a symmetric monoidal category with a monoidal bialgebra modality equipped with a codereliction. Moreover, we also show that coderelictions are, in fact, unique. We also introduce monoidal Hopf coalgebra modalities and discuss how antipodes relate to enrichment over Abelian groups. |
| title | Additive Enrichment from Coderelictions |
| topic | Category Theory 18F40, 19D23, 16T10, 16T05, 18E05, 18C15 |
| url | https://arxiv.org/abs/2502.14134 |