Cartesian differential categories as skew enriched categories

Richard Garner, Jean-Simon Pacaud Lemay*

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

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    We exhibit the cartesian differential categories of Blute, Cockett and Seely as a particular kind of enriched category. The base for the enrichment is the category of commutative monoids—or in a straightforward generalisation, the category of modules over a commutative rig k. However, the tensor product on this category is not the usual one, but rather a warping of it by a certain monoidal comonad Q. Thus the enrichment base is not a monoidal category in the usual sense, but rather a skew monoidal category in the sense of Szlachányi. Our first main result is that cartesian differential categories are the same as categories with finite products enriched over this skew monoidal base. The comonad Q involved is, in fact, an example of a differential modality. Differential modalities are a kind of comonad on a symmetric monoidal k-linear category with the characteristic feature that their co-Kleisli categories are cartesian differential categories. Using our first main result, we are able to prove our second one: that every small cartesian differential category admits a full, structure-preserving embedding into the cartesian differential category induced by a differential modality (in fact, a monoidal differential modality on a monoidal closed category—thus, a model of intuitionistic differential linear logic). This resolves an important open question in this area.
    Original languageEnglish
    Pages (from-to)1099-1150
    Number of pages52
    JournalApplied Categorical Structures
    Issue number6
    Publication statusPublished - Dec 2021

    Bibliographical note

    Copyright the Author(s) 2021. Version archived for private and non-commercial use with the permission of the author/s and according to publisher conditions. For further rights please contact the publisher.


    • Cartesian differential categories
    • Skew monoidal categories
    • Differential categories


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