Cartesian Differential Categories

By R. Blute, J. Cockett, R. Seely. Theory and Applications of Categories, 2009

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This paper revisits the authors' notion of a dierential category from a dierent perspective. A dierential category is an additive symmetric monoidal category with a comonad (a \coalgebra modality") and a dierential combinator. The morphisms of a dierential category should be thought of as the linear maps; the dierentiable or smooth maps would then be morphisms of the coKleisli category. The purpose of the present paper is to directly axiomatize dierentiable maps and thus to move the emphasis from the linear notion to structures resembling the coKleisli category. The result is a setting with a more evident and intuitive relationship to the familiar notion of calculus on smooth maps. Indeed a primary example is the category whose objects are Euclidean spaces and whose morphisms are smooth maps.

Subfields: Artificial Intelligence, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1f1e57270909217fc319efb8e7b458dbeb6b4d5c