# 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} }

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

### Differential Categories Revisited

- MathematicsApplied Categorical Structures
- 2019

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…

### Differential Categories Revisited

- MathematicsAppl. Categorical Struct.
- 2020

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.

### Derivations in Codifferential Categories

- Mathematics
- 2015

Derivations provide a way of transporting ideas from the calculus of manifolds to algebraic settings where there is no sensible notion of limit. In this paper, we consider derivations in certain…

### Integral categories and calculus categories

- MathematicsMathematical Structures in Computer Science
- 2018

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.

### Convenient antiderivatives for differential linear categories

- MathematicsMathematical Structures in Computer Science
- 2020

Abstract Differential categories axiomatize the basics of differentiation and provide categorical models of differential linear logic. A differential category is said to have antiderivatives if a…

### Convenient antiderivatives for differential linear categories

- MathematicsMath. 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.

### Properties and Characterisations of Cofree Cartesian Differential Categories

- Mathematics
- 2022

Cartesian diﬀerential categories come equipped with a diﬀerential operator which formalises the total derivative from multi-variable calculus. There always exists cofree Cartesian diﬀerential…

### Cartesian Differential Categories as Skew Enriched Categories

- MathematicsApplied Categorical Structures
- 2021

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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