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Tangent Categories from the Coalgebras of Differential Categories Cockett, Robin Lemay, Jean-Simon Pacaud Lucyshyn-Wright, Rory B. B. Differential categories Tangent categories Coalgebra Modalities 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Robin Cockett and Jean-Simon Pacaud Lemay and Rory B. B. Lucyshyn-Wright 2020 Is Part Of LIPIcs, Volume 152, 28th EACSL Annual Conference on Computer Science Logic (CSL 2020) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.CSL.2020.17 urn:nbn:de:0030-drops-116607 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2020.17 eng https://creativecommons.org/licenses/by/3.0/legalcode