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Cartesian differential categories as skew enriched categories

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If you have a question about this talk, please contact José Siqueira.

Cartesian differential categories are an abstraction of the category of smooth maps between Euclidean spaces. Their main feature is an operator assigning to each map f:A → B another map Df: A*A—> B called the differential of f, subject to a list of axioms.

In this talk, we explain the slightly surprising fact that cartesian differential categories are actually a kind of enriched category. The enrichment base is the category of k-vector spaces, but the monoidal structure is not the usual one, but rather a skew-monoidal warping of it with respect to a monoidal comonad. The comonad at issue is not ad hoc, but in fact the initial one imbuing k-vector spaces with the structure of a model of intuitionistic differential linear logic.

This is a report on joint work with JS Lemay.

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This talk is part of the Category Theory Seminar series.

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