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Integral Categories and Calculus Categories Cockett, Robin Lemay, Jean-Simon Differential Categories Integral Categories Calculus Categories 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Robin Cockett and Jean-Simon Lemay 2017 Is Part Of LIPIcs, Volume 82, 26th EACSL Annual Conference on Computer Science Logic (CSL 2017) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.CSL.2017.20 urn:nbn:de:0030-drops-76687 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2017.20 eng https://creativecommons.org/licenses/by/3.0/legalcode