eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-11-08
19:1
19:21
10.4230/LIPIcs.CALCO.2021.19
article
Coderelictions for Free Exponential Modalities
Lemay, Jean-Simon Pacaud
1
https://orcid.org/0000-0003-4124-3722
Mathematics and Computer Science Department, Mount Allison University, Sackville, Canada
In a categorical model of the multiplicative and exponential fragments of intuitionistic linear logic (MELL), the exponential modality is interpreted as a comonad ! such that each cofree !-coalgebra !A comes equipped with a natural cocommutative comonoid structure. An important case is when ! is a free exponential modality so that !A is the cofree cocommutative comonoid over A. A categorical model of MELL with a free exponential modality is called a Lafont category. A categorical model of differential linear logic is called a differential category, where the differential structure can equivalently be described by a deriving transformation !A⊗A →{𝖽_A} !A or a codereliction A →{η_A} !A. Blute, Lucyshyn-Wright, and O'Neill showed that every Lafont category with finite biproducts is a differential category. However, from a differential linear logic perspective, Blute, Lucyshyn-Wright, and O'Neill’s approach is not the usual one since the result was stated in the dual setting and the proof is in terms of the deriving transformation 𝖽. In differential linear logic, it is often the codereliction η that is preferred and that plays a more prominent role. In this paper, we provide an alternative proof that every Lafont category (with finite biproducts) is a differential category, where we construct the codereliction η using the couniversal property of the cofree cocommtuative comonoid !A and show that η is unique. To achieve this, we introduce the notion of an infinitesimal augmentation k⊕A →{𝖧_A} !(k ⊕ A), which in particular is a !-coalgebra and a comonoid morphism, and show that infinitesimal augmentations are in bijective correspondence to coderelictions (and deriving transformations). As such, infinitesimal augmentations provide a new equivalent axiomatization for differential categories in terms of more commonly known concepts. For a free exponential modality, its infinitesimal augmentation is easy to construct and allows one to clearly see the differential structure of a Lafont category, regardless of the construction of !A.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol211-calco2021/LIPIcs.CALCO.2021.19/LIPIcs.CALCO.2021.19.pdf
Differential Categories
Coderelictions
Differential Linear Logic
Free Exponential Modalities
Lafont Categories
Infinitesimal Augmentations