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**Published in:** LIPIcs, Volume 299, 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)

In a differential category and in Differential Linear Logic, the exponential conjunction ! admits structural maps, characterizing quantitative operations and symmetric co-structural maps, characterizing differentiation. In this paper, we introduce the notion of a Laplace distributor, which is an extra structural map which distributes the linear negation operation (_)^∗ over ! and transforms the co-structural rules into the structural rules. Laplace distributors are directly inspired by the well-known Laplace transform, which is all-important in numerical analysis. In the star-autonomous setting, a Laplace distributor induces a natural transformation from ! to the exponential disjunction ?, which we then call a Laplace transformation. According to its semantics, we show that Laplace distributors correspond precisely to the notion of a generalized exponential function e^x on the monoidal unit. We also show that many well-known and important examples have a Laplace distributor/transformation, including (weighted) relations, finiteness spaces, Köthe spaces, and convenient vector spaces.

Marie Kerjean and Jean-Simon Pacaud Lemay. Laplace Distributors and Laplace Transformations for Differential Categories. In 9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 299, pp. 9:1-9:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{kerjean_et_al:LIPIcs.FSCD.2024.9, author = {Kerjean, Marie and Lemay, Jean-Simon Pacaud}, title = {{Laplace Distributors and Laplace Transformations for Differential Categories}}, booktitle = {9th International Conference on Formal Structures for Computation and Deduction (FSCD 2024)}, pages = {9:1--9:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-323-2}, ISSN = {1868-8969}, year = {2024}, volume = {299}, editor = {Rehof, Jakob}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2024.9}, URN = {urn:nbn:de:0030-drops-203382}, doi = {10.4230/LIPIcs.FSCD.2024.9}, annote = {Keywords: Differential Categories, Differential Linear Logic, Laplace Distributor, Laplace Transformation, Exponential Function} }

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**Published in:** LIPIcs, Volume 288, 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)

Previous work has shown that reverse differential categories give an abstract setting for gradient-based learning of functions between Euclidean spaces. However, reverse differential categories are not suited to handle gradient-based learning for functions between more general spaces such as smooth manifolds. In this paper, we propose a setting to handle this, which we call reverse tangent categories: tangent categories with an involution operation for their differential bundles.

Geoffrey Cruttwell and Jean-Simon Pacaud Lemay. Reverse Tangent Categories. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 21:1-21:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{cruttwell_et_al:LIPIcs.CSL.2024.21, author = {Cruttwell, Geoffrey and Lemay, Jean-Simon Pacaud}, title = {{Reverse Tangent Categories}}, booktitle = {32nd EACSL Annual Conference on Computer Science Logic (CSL 2024)}, pages = {21:1--21:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-310-2}, ISSN = {1868-8969}, year = {2024}, volume = {288}, editor = {Murano, Aniello and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2024.21}, URN = {urn:nbn:de:0030-drops-196644}, doi = {10.4230/LIPIcs.CSL.2024.21}, annote = {Keywords: Tangent Categories, Reverse Tangent Categories, Reverse Differential Categories, Categorical Machine Learning} }

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**Published in:** LIPIcs, Volume 211, 9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021)

In a categorical model of the multiplicative and exponential fragments of intuitionistic linear logic (MELL), the exponential modality is interpreted as a comonad ! such that each cofree !-coalgebra !A comes equipped with a natural cocommutative comonoid structure. An important case is when ! is a free exponential modality so that !A is the cofree cocommutative comonoid over A. A categorical model of MELL with a free exponential modality is called a Lafont category. A categorical model of differential linear logic is called a differential category, where the differential structure can equivalently be described by a deriving transformation !A⊗A →{𝖽_A} !A or a codereliction A →{η_A} !A. Blute, Lucyshyn-Wright, and O'Neill showed that every Lafont category with finite biproducts is a differential category. However, from a differential linear logic perspective, Blute, Lucyshyn-Wright, and O'Neill’s approach is not the usual one since the result was stated in the dual setting and the proof is in terms of the deriving transformation 𝖽. In differential linear logic, it is often the codereliction η that is preferred and that plays a more prominent role. In this paper, we provide an alternative proof that every Lafont category (with finite biproducts) is a differential category, where we construct the codereliction η using the couniversal property of the cofree cocommtuative comonoid !A and show that η is unique. To achieve this, we introduce the notion of an infinitesimal augmentation k⊕A →{𝖧_A} !(k ⊕ A), which in particular is a !-coalgebra and a comonoid morphism, and show that infinitesimal augmentations are in bijective correspondence to coderelictions (and deriving transformations). As such, infinitesimal augmentations provide a new equivalent axiomatization for differential categories in terms of more commonly known concepts. For a free exponential modality, its infinitesimal augmentation is easy to construct and allows one to clearly see the differential structure of a Lafont category, regardless of the construction of !A.

Jean-Simon Pacaud Lemay. Coderelictions for Free Exponential Modalities. In 9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 211, pp. 19:1-19:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{lemay:LIPIcs.CALCO.2021.19, author = {Lemay, Jean-Simon Pacaud}, title = {{Coderelictions for Free Exponential Modalities}}, booktitle = {9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021)}, pages = {19:1--19:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-212-9}, ISSN = {1868-8969}, year = {2021}, volume = {211}, editor = {Gadducci, Fabio and Silva, Alexandra}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CALCO.2021.19}, URN = {urn:nbn:de:0030-drops-153742}, doi = {10.4230/LIPIcs.CALCO.2021.19}, annote = {Keywords: Differential Categories, Coderelictions, Differential Linear Logic, Free Exponential Modalities, Lafont Categories, Infinitesimal Augmentations} }

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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 108, 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)

A categorical model of the multiplicative and exponential fragments of intuitionistic linear logic (IMELL), known as a linear category, is a symmetric monoidal closed category with a monoidal coalgebra modality (also known as a linear exponential comonad). Inspired by R. Blute and P. Scott's work on categories of modules of Hopf algebras as models of linear logic, we study Eilenberg-Moore categories of monads as models of IMELL. We define an IMELL lifting monad on a linear category as a Hopf monad - in the Bruguieres, Lack, and Virelizier sense - with a mixed distributive law over the monoidal coalgebra modality. As our main result, we show that the linear category structure lifts to Eilenberg-Moore categories of IMELL lifting monads. We explain how monoids in the Eilenberg-Moore category of the monoidal coalgebra modality can induce IMELL lifting monads and provide sources for such monoids. Along the way, we also define mixed distributive laws of bimonads over coalgebra modalities and lifting differential category structure to Eilenberg-Moore categories of exponential lifting monads.

Jean-Simon Pacaud Lemay. Lifting Coalgebra Modalities and IMELL Model Structure to Eilenberg-Moore Categories. In 3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 108, pp. 21:1-21:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{lemay:LIPIcs.FSCD.2018.21, author = {Lemay, Jean-Simon Pacaud}, title = {{Lifting Coalgebra Modalities and IMELL Model Structure to Eilenberg-Moore Categories}}, booktitle = {3rd International Conference on Formal Structures for Computation and Deduction (FSCD 2018)}, pages = {21:1--21:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-077-4}, ISSN = {1868-8969}, year = {2018}, volume = {108}, editor = {Kirchner, H\'{e}l\`{e}ne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSCD.2018.21}, URN = {urn:nbn:de:0030-drops-91911}, doi = {10.4230/LIPIcs.FSCD.2018.21}, annote = {Keywords: Mixed Distributive Laws, Coalgebra Modalities, Linear Categories, Bimonads, Differential Categories} }

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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 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} }