# Differential categories

@article{Blute2006DifferentialC,
title={Differential categories},
author={Richard Blute and J. Robin B. Cockett and Robert A. G. Seely},
journal={Mathematical Structures in Computer Science},
year={2006},
volume={16},
pages={1049 - 1083}
}
• Published 1 November 2006
• Mathematics
• Mathematical Structures in Computer Science
Following work of Ehrhard and Regnier, we introduce the notion of a differential category: an additive symmetric monoidal category with a comonad (a ‘coalgebra modality’) and a differential combinator satisfying a number of coherence conditions. In such a category one should imagine the morphisms in the base category as being linear maps and the morphisms in the coKleisli category as being smooth (infinitely differentiable). Although such categories do not necessarily arise from models of…
102 Citations
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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
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It is proved that these apparently distinct notions of differentiation, in the presence of a monoidal coalgebra modality, are completely equivalent and, for linear logic settings, there is only one notion of differentiation.
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It is shown that a calculus category with a monoidal coalgebra modality has its integral transformation given by antiderivatives and, thus, that the integral structure is uniquely determined by the differential structure.
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• J. Lemay
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Math. Struct. Comput. Sci.
• 2020
This paper shows that Blute, Ehrhard, and Tasson's differential category of convenient vector spaces has antiderivatives and shows that generalizations of the relational model (which are biproduct completions of complete semirings) are also differential categories with antidervatives.
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It is proved that every small cartesian differential category admits a full, structure-preserving embedding into the cartesiandifferential category induced by a differential modality (in fact, a monoidal differentialmodality on amonoidal closed category—thus, a model of intuitionistic differential linear logic).

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