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Integral categories and calculus categories

Published online by Cambridge University Press:  05 February 2018

J. R. B. COCKETT  and
J.-S. LEMAY
J. R. B. COCKETT
Affiliation:
Department of Computer Science, University of Calgary, Calgary, Alberta, Canada Email: robin@ucalgary.ca
J.-S. LEMAY
Affiliation:
Department of Mathematics and Statistics, University of Calgary, Calgary, Alberta, Canada Email: jeansimon.lemay@ucalgary.ca
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Abstract

Differential categories are now an established abstract setting for differentiation. However, not much attention has been given to the process which is inverse to differentiation: integration. This paper presents the parallel development for integration by axiomatizing an integral transformation, sA: !A → !A ⊗ A, 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 theorems is called a calculus category.

Modifying an approach to antiderivatives by T. Ehrhard, we define having antiderivatives as the demand that a certain natural transformation, K: !A → !A, is invertible. We observe that a differential category having antiderivatives, in this sense, is always a calculus category.

When the coalgebra modality is monoidal, it is natural to demand an extra coherence between integration and the coalgebra modality. In the presence of this extra coherence, we show that a calculus category with a monoidal coalgebra modality has its integral transformation given by antiderivatives and, thus, that the integral structure is uniquely determined by the differential structure.

The paper finishes by providing a suite of separating examples. Examples of differential categories, integral categories and calculus categories based on both monoidal and (mere) coalgebra modalities are presented. In addition, differential categories which are not integral categories are discussed and vice versa.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 29 , Issue 2 , February 2019 , pp. 243 - 308
Copyright
Copyright © Cambridge University Press 2018 

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References

Baxter, G. (1960). An analytic problem whose solution follows from a simple algebraic identity. Pacific Journal of Mathematics 10 (3) 731–742.Google Scholar
Bierman, G. (1995). What is a categorical model of intuitionistic linear logic? Typed Lambda Calculi and Applications, Volume 902 of Lecture Notes in Computer Science, Springer Verlag, 78–93.Google Scholar
Blute, R., Cockett, J., Porter, T. and Seely, R. (2011). Kähler categories. Cahiers de Topologie et Géométrie Différentielle Catégoriques 52 (4) 253–268.Google Scholar
Blute, R., Cockett, J. and Seely, R. (2015a). Cartesian differential storage categories. Theory and Applications of Categories 30 (18) 620–686.Google Scholar
Blute, R., Cockett, J.R.B. and Seely, R. (2009). Cartesian differential categories. Theory and Applications of Categories 22 (23) 622–672.Google Scholar
Blute, R., Ehrhard, T. and Tasson, C. (2010). A convenient differential category. Cahiers de Topologie Géométrie Différentielle Catéroqiue, 53 (2012) 211–233.Google Scholar
Blute, R., Lucyshyn-Wright, R.B. and O'Neill, K. (2015b). Derivations in codifferential categories. Cahiers de Topologie Géométrie Différentielle Catéroqiue, 57 (2016) 243–280.Google Scholar
Blute, R.F., Cockett, J.R.B. and Seely, R.A. (2006). Differential categories. Mathematical Sructures in Computer Science 16 (06) 1049–1083.Google Scholar
Bott, R. and Tu, L.W. (2013). Differential Forms in Algebraic Topology, vol. 82, Springer Science & Business Media.Google Scholar
Cockett, J., Cruttwell, G. and Gallagher, J. (2011). Differential restriction categories. Theory and Applications of Categories 25 (21) 537–613.Google Scholar
Cockett, J.R.B. and Cruttwell, G.S. (2014). Differential structure, tangent structure, and sdg. Applied Categorical Structures 22 (2) 331–417.Google Scholar
Cockett, J.R.B. and Lemay, J.S. (2017). There is only one notion of differentiation. In: LIPIcs-Leibniz International Proceedings in Informatics, vol. 84, Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik.Google Scholar
Ehrhard, T. (2017). An introduction to differential linear logic: Proof-nets, models and antiderivatives. Mathematical Structures in Computer Science 1–66.Google Scholar
Ehrhard, T. and Regnier, L. (2003). The differential lambda-calculus. Theoretical Computer Science 309 (1) 1–41.Google Scholar
Ehrhard, T. and Regnier, L. (2006). Differential interaction nets. Theoretical Computer Science 364 (2) 166–195.Google Scholar
Fiore, M.P. (2007). Differential structure in models of multiplicative biadditive intuitionistic linear logic. In: International Conference on Typed Lambda Calculi and Applications, Springer, 163–177.Google Scholar
Golan, J.S. (2013). Semirings and their Applications, Springer Science & Business Media.Google Scholar
Guo, L. (2012). An Introduction to Rota–Baxter Algebra, vol. 2, International Press Somerville.Google Scholar
Guo, L. and Keigher, W. (2008). On differential Rota–Baxter algebras. Journal of Pure and Applied Algebra 212 (3) 522–540.Google Scholar
Joyal, A. and Street, R. (1991). The geometry of tensor calculus, I. Advances in Mathematics 88 (1) 55–112.Google Scholar
Laird, J., Manzonetto, G. and McCusker, G. (2013). Constructing differential categories and deconstructing categories of games. Information and Computation 222 247–264.Google Scholar
Lang, S. (2002). Algebra, revised 3rd ed. Graduate Texts in Mathematics, vol. 211, Springer-Verlag.Google Scholar
Mac Lane, S. (2013). Categories for the Working Mathematician, vol. 5, Springer Science & Business Media, Springer-Verlag.Google Scholar
Rota, G.C. (1969). Baxter algebras and combinatorial identities. I. Bulletin of the American Mathematical Society 75 (2) 325–329.Google Scholar
Selinger, P. (2010). A survey of graphical languages for monoidal categories. In: New Structures for Physics, Coecke, Bob (Ed.), Springer, 289–355.Google Scholar
Weibel, C.A. (1995). An Introduction to Homological Algebra, Cambridge University Press.Google Scholar
Zhang, S., Guo, L. and Keigher, W. (2016). Monads and distributive laws for Rota–Baxter and differential algebras. Advances in Applied Mathematics 72, 139–165.Google Scholar