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Integral Categories and Calculus Categories

Authors Robin Cockett, Jean-Simon Lemay



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Robin Cockett
Jean-Simon Lemay

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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)
https://doi.org/10.4230/LIPIcs.CSL.2017.20

Abstract

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.
Keywords
  • Differential Categories
  • Integral Categories
  • Calculus Categories

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References

  1. Glen Baxter et al. An analytic problem whose solution follows from a simple algebraic identity. Pacific J. Math, 10(3):731-742, 1960. Google Scholar
  2. R. Blute, J. R. B. Cockett, and R. A. G. Seely. Cartesian differential storage categories. Theory and Applications of Categories, 30(18):620-686, 2015. Google Scholar
  3. R. F. Blute, J. Robin B. Cockett, and R. A. G. Seely. Cartesian differential categories. Theory and Applications of Categories, 22(23):622-672, 2009. Google Scholar
  4. Richard Blute, J. R. B. Cockett, Timothy Porter, and R. A. G. Seely. Kähler categories. Cahiers de Topologie et Géométrie Différentielle Catégoriques, 52(4):253-268, 2011. Google Scholar
  5. Richard Blute, Thomas Ehrhard, and Christine Tasson. A convenient differential category. arXiv preprint arXiv:1006.3140, 2010. Google Scholar
  6. Richard Blute, Rory B. B. Lucyshyn-Wright, and Keith O'Neill. Derivations in codifferential categories. arXiv preprint arXiv:1505.00220, 2015. Google Scholar
  7. Richard F. Blute, J. Robin B. Cockett, and Robert A. G. Seely. Differential categories. Mathematical structures in computer science, 16(06):1049-1083, 2006. Google Scholar
  8. Raoul Bott and Loring W. Tu. Differential forms in algebraic topology, volume 82. Springer Science &Business Media, 2013. Google Scholar
  9. J. Robin B. Cockett and Geoff S. H. Cruttwell. Differential Structure, Tangent Structure, and SDG. Applied Categorical Structures, 22(2):331-417, 2014. Google Scholar
  10. J. R. B. Cockett, G. S. H. Cruttwell, and J. D. Gallagher. Differential restriction categories. Theory and Applications of Categories, 25(21):537-613, 2011. Google Scholar
  11. Thomas Ehrhard. An introduction to differential linear logic: proof-nets, models and antiderivatives. Mathematical Structures in Computer Science, pages 1-66, 2017. Google Scholar
  12. Thomas Ehrhard and Laurent Regnier. The differential lambda-calculus. Theoretical Computer Science, 309(1):1-41, 2003. Google Scholar
  13. Thomas Ehrhard and Laurent Regnier. Differential interaction nets. Theoretical Computer Science, 364(2):166-195, 2006. Google Scholar
  14. Marcelo P. Fiore. Differential structure in models of multiplicative biadditive intuitionistic linear logic. In International Conference on Typed Lambda Calculi and Applications, pages 163-177. Springer, 2007. Google Scholar
  15. Li Guo. An introduction to Rota-Baxter algebra, volume 2. International Press Somerville, 2012. Google Scholar
  16. André Joyal and Ross Street. The geometry of tensor calculus, I. Advances in Mathematics, 88(1):55-112, 1991. Google Scholar
  17. Serge Lang. Algebra revised third edition. Graduate Texts in Mathematics, 1(211):ALL-ALL, 2002. Google Scholar
  18. J.-S. P. Lemay. Integral Categories and Calculus Categories. University of Calgary, 2017. Google Scholar
  19. Saunders Mac Lane. Categories for the working mathematician, volume 5. Springer Science &Business Media, 2013. Google Scholar
  20. Gian-Carlo Rota. Baxter algebras and combinatorial identities. I. Bulletin of the American Mathematical Society, 75(2):325-329, 1969. Google Scholar
  21. Peter Selinger. A survey of graphical languages for monoidal categories. In New structures for physics, pages 289-355. Springer, 2010. Google Scholar
  22. Charles A. Weibel. An introduction to homological algebra. Cambridge university press, 1995. Google Scholar
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