Reverse Tangent Categories

Authors Geoffrey Cruttwell , Jean-Simon Pacaud Lemay

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Geoffrey Cruttwell
  • Mount Allison University, Sackville, Canada
Jean-Simon Pacaud Lemay
  • Macquarie University, Sydney, Australia


The authors would like to thank Bryce Clark for pointing out [Philip J. Higgins and Kirril C. H. Mackenzie, 1993], which ties in nicely with the story of this paper, as well as Geoff Vooys for useful discussions, answering questions, and helping find references regarding Example 31.

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Geoffrey Cruttwell and Jean-Simon Pacaud Lemay. Reverse Tangent Categories. In 32nd EACSL Annual Conference on Computer Science Logic (CSL 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 288, pp. 21:1-21:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


Previous work has shown that reverse differential categories give an abstract setting for gradient-based learning of functions between Euclidean spaces. However, reverse differential categories are not suited to handle gradient-based learning for functions between more general spaces such as smooth manifolds. In this paper, we propose a setting to handle this, which we call reverse tangent categories: tangent categories with an involution operation for their differential bundles.

Subject Classification

ACM Subject Classification
  • Theory of computation → Categorical semantics
  • Tangent Categories
  • Reverse Tangent Categories
  • Reverse Differential Categories
  • Categorical Machine Learning


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