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**Published in:** LIPIcs, Volume 84, 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017)

Differential linear logic was introduced as a syntactic proof-theoretic approach to the analysis of differential calculus. Differential categories were subsequently introduce to provide a categorical model theory for differential linear logic. Differential categories used two different approaches for defining differentiation abstractly: a deriving transformation and a coderiliction. While it was thought that these notions could give rise to distinct notions of differentiation, we show here that these notions, in the presence of a monoidal coalgebra modality, are completely equivalent.

J. Robin B. Cockett and Jean-Simon Lemay. There Is Only One Notion of Differentiation. In 2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 84, pp. 13:1-13:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{cockett_et_al:LIPIcs.FSCD.2017.13, author = {Cockett, J. Robin B. and Lemay, Jean-Simon}, title = {{There Is Only One Notion of Differentiation}}, booktitle = {2nd International Conference on Formal Structures for Computation and Deduction (FSCD 2017)}, pages = {13:1--13:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-047-7}, ISSN = {1868-8969}, year = {2017}, volume = {84}, editor = {Miller, Dale}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2017.13}, URN = {urn:nbn:de:0030-drops-77166}, doi = {10.4230/LIPIcs.FSCD.2017.13}, annote = {Keywords: Differential Categories, Linear Logic, Coalgebra Modalities, Bialgebra Modalities} }

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**Published in:** LIPIcs, Volume 152, 28th EACSL Annual Conference on Computer Science Logic (CSL 2020)

Following the pattern from linear logic, the coKleisli category of a differential category is a Cartesian differential category. What then is the coEilenberg-Moore category of a differential category? The answer is a tangent category! A key example arises from the opposite of the category of Abelian groups with the free exponential modality. The coEilenberg-Moore category, in this case, is the opposite of the category of commutative rings. That the latter is a tangent category captures a fundamental aspect of both algebraic geometry and Synthetic Differential Geometry. The general result applies when there are no negatives and thus encompasses examples arising from combinatorics and computer science.

Robin Cockett, Jean-Simon Pacaud Lemay, and Rory B. B. Lucyshyn-Wright. Tangent Categories from the Coalgebras of Differential Categories. In 28th EACSL Annual Conference on Computer Science Logic (CSL 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 152, pp. 17:1-17:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{cockett_et_al:LIPIcs.CSL.2020.17, author = {Cockett, Robin and Lemay, Jean-Simon Pacaud and Lucyshyn-Wright, Rory B. B.}, title = {{Tangent Categories from the Coalgebras of Differential Categories}}, booktitle = {28th EACSL Annual Conference on Computer Science Logic (CSL 2020)}, pages = {17:1--17:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-132-0}, ISSN = {1868-8969}, year = {2020}, volume = {152}, editor = {Fern\'{a}ndez, Maribel and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2020.17}, URN = {urn:nbn:de:0030-drops-116607}, doi = {10.4230/LIPIcs.CSL.2020.17}, annote = {Keywords: Differential categories, Tangent categories, Coalgebra Modalities} }

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**Published in:** LIPIcs, Volume 152, 28th EACSL Annual Conference on Computer Science Logic (CSL 2020)

The reverse derivative is a fundamental operation in machine learning and automatic differentiation [Martín Abadi et al., 2015; Griewank, 2012]. This paper gives a direct axiomatization of a category with a reverse derivative operation, in a similar style to that given by [Blute et al., 2009] for a forward derivative. Intriguingly, a category with a reverse derivative also has a forward derivative, but the converse is not true. In fact, we show explicitly what a forward derivative is missing: a reverse derivative is equivalent to a forward derivative with a dagger structure on its subcategory of linear maps. Furthermore, we show that these linear maps form an additively enriched category with dagger biproducts.

Robin Cockett, Geoffrey Cruttwell, Jonathan Gallagher, Jean-Simon Pacaud Lemay, Benjamin MacAdam, Gordon Plotkin, and Dorette Pronk. Reverse Derivative Categories. In 28th EACSL Annual Conference on Computer Science Logic (CSL 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 152, pp. 18:1-18:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{cockett_et_al:LIPIcs.CSL.2020.18, author = {Cockett, Robin and Cruttwell, Geoffrey and Gallagher, Jonathan and Lemay, Jean-Simon Pacaud and MacAdam, Benjamin and Plotkin, Gordon and Pronk, Dorette}, title = {{Reverse Derivative Categories}}, booktitle = {28th EACSL Annual Conference on Computer Science Logic (CSL 2020)}, pages = {18:1--18:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-132-0}, ISSN = {1868-8969}, year = {2020}, volume = {152}, editor = {Fern\'{a}ndez, Maribel and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2020.18}, URN = {urn:nbn:de:0030-drops-116611}, doi = {10.4230/LIPIcs.CSL.2020.18}, annote = {Keywords: Reverse Derivatives, Cartesian Reverse Differential Categories, Categorical Semantics, Cartesian Differential Categories, Dagger Categories, Automatic Differentiation} }

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**Published in:** LIPIcs, Volume 82, 26th EACSL Annual Conference on Computer Science Logic (CSL 2017)

Differential categories are now an established abstract setting for differentiation. The paper presents the parallel development for integration by axiomatizing
an integral transformation in a symmetric monoidal category with a coalgebra modality. When integration is combined with differentiation, the two fundamental theorems of calculus are expected to hold (in a suitable sense): a differential category with integration which satisfies
these two theorem is called a calculus category.
Modifying an approach to antiderivatives by T. Ehrhard, it is shown how examples of calculus categories arise as differential categories with antiderivatives in this new sense. Having antiderivatives amounts to demanding that a certain natural transformation K, is invertible. We observe that a differential category having antiderivatives, in this sense, is always a calculus category and we provide examples of such categories.

Robin Cockett and Jean-Simon Lemay. Integral Categories and Calculus Categories. In 26th EACSL Annual Conference on Computer Science Logic (CSL 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 82, pp. 20:1-20:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{cockett_et_al:LIPIcs.CSL.2017.20, author = {Cockett, Robin and Lemay, Jean-Simon}, title = {{Integral Categories and Calculus Categories}}, booktitle = {26th EACSL Annual Conference on Computer Science Logic (CSL 2017)}, pages = {20:1--20:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-045-3}, ISSN = {1868-8969}, year = {2017}, volume = {82}, editor = {Goranko, Valentin and Dam, Mads}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2017.20}, URN = {urn:nbn:de:0030-drops-76687}, doi = {10.4230/LIPIcs.CSL.2017.20}, annote = {Keywords: Differential Categories, Integral Categories, Calculus Categories} }

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