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Differential Categories Revisited

Resource type
Authors/contributors
Title
Differential Categories Revisited
Abstract
Differential categories were introduced to provide a minimal categorical doctrine for differential linear logic. Here we revisit the formalism and, in particular, examine the two different approaches to defining differentiation which were introduced. The basic approach used a deriving transformation, while a more refined approach, in the presence of a bialgebra modality, used a codereliction. The latter approach is particularly relevant to linear logic settings, where the coalgebra modality is monoidal and the Seely isomorphisms give rise to a bialgebra modality. Here, we prove that these apparently distinct notions of differentiation, in the presence of a monoidal coalgebra modality, are completely equivalent. Thus, for linear logic settings, there is only one notion of differentiation. This paper also presents a number of separating examples for coalgebra modalities including examples which are and are not monoidal, as well as examples which do and do not support differential structure. Of particular interest is the observation that—somewhat counter-intuitively—differential algebras never induce a differential category although they provide a monoidal coalgebra modality. On the other hand, Rota–Baxter algebras—which are usually associated with integration—provide an example of a differential category which has a non-monoidal coalgebra modality.
Publication
Applied Categorical Structures
Date
2019-07-04
Journal Abbr
Appl Categor Struct
Language
en
DOI
10/ggdm44
ISSN
1572-9095
Accessed
2019-11-28T16:23:18Z
Library Catalog
Springer Link
Extra
ZSCC: NoCitationData[s1]
Citation
Blute, R. F., Cockett, J. R. B., Lemay, J.-S. P., & Seely, R. A. G. (2019). Differential Categories Revisited. Applied Categorical Structures. https://doi.org/10/ggdm44
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