Document

# Integral Categories and Calculus Categories

## File

LIPIcs.CSL.2017.20.pdf
• Filesize: 0.54 MB
• 17 pages

## Cite As

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

## Metrics

• Access Statistics
• Total Accesses (updated on a weekly basis)
0

## 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.
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.
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.
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.
5. Richard Blute, Thomas Ehrhard, and Christine Tasson. A convenient differential category. arXiv preprint arXiv:1006.3140, 2010.
6. Richard Blute, Rory B. B. Lucyshyn-Wright, and Keith O'Neill. Derivations in codifferential categories. arXiv preprint arXiv:1505.00220, 2015.
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.
8. Raoul Bott and Loring W. Tu. Differential forms in algebraic topology, volume 82. Springer Science &Business Media, 2013.
9. J. Robin B. Cockett and Geoff S. H. Cruttwell. Differential Structure, Tangent Structure, and SDG. Applied Categorical Structures, 22(2):331-417, 2014.
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.
11. Thomas Ehrhard. An introduction to differential linear logic: proof-nets, models and antiderivatives. Mathematical Structures in Computer Science, pages 1-66, 2017.
12. Thomas Ehrhard and Laurent Regnier. The differential lambda-calculus. Theoretical Computer Science, 309(1):1-41, 2003.
13. Thomas Ehrhard and Laurent Regnier. Differential interaction nets. Theoretical Computer Science, 364(2):166-195, 2006.
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.
15. Li Guo. An introduction to Rota-Baxter algebra, volume 2. International Press Somerville, 2012.
16. André Joyal and Ross Street. The geometry of tensor calculus, I. Advances in Mathematics, 88(1):55-112, 1991.
17. Serge Lang. Algebra revised third edition. Graduate Texts in Mathematics, 1(211):ALL-ALL, 2002.
18. J.-S. P. Lemay. Integral Categories and Calculus Categories. University of Calgary, 2017.
19. Saunders Mac Lane. Categories for the working mathematician, volume 5. Springer Science &Business Media, 2013.
20. Gian-Carlo Rota. Baxter algebras and combinatorial identities. I. Bulletin of the American Mathematical Society, 75(2):325-329, 1969.
21. Peter Selinger. A survey of graphical languages for monoidal categories. In New structures for physics, pages 289-355. Springer, 2010.
22. Charles A. Weibel. An introduction to homological algebra. Cambridge university press, 1995.
X

Feedback for Dagstuhl Publishing