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Tangent Categories from the Coalgebras of Differential Categories

Authors Robin Cockett, Jean-Simon Pacaud Lemay, Rory B. B. Lucyshyn-Wright

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Author Details

Robin Cockett
  • University of Calgary, Department of Computer Science, Canada
Jean-Simon Pacaud Lemay
  • University of Oxford, Department of Computer Science, UK
Rory B. B. Lucyshyn-Wright
  • Brandon University, Department of of Mathematics and Computer Science, Canada

Funding

  • Cockett, Robin: Partially supported by NSERC (Canada).
  • Lemay, Jean-Simon Pacaud: Thanks to Kellogg College, the Clarendon Fund, and the Oxford Google-DeepMind Graduate Scholarship for financial support.
  • Lucyshyn-Wright, Rory B. B.: Supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).

Acknowledgements

The authors would like thank to thank Steve Lack for pointing us to an adjoint lifting theorem of Butler found in Barr and Wells' book [Barr and Wells, 2005], as well as the anonymous referee for pointing us to Johnstone’s adjoint lifting theorem [Johnstone, 1975].

Cite As Get BibTex

Robin Cockett, Jean-Simon Pacaud Lemay, and Rory B. B. Lucyshyn-Wright. Tangent Categories from the Coalgebras of Differential Categories. In 28th EACSL Annual Conference on Computer Science Logic (CSL 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 152, pp. 17:1-17:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.CSL.2020.17

Abstract

Following the pattern from linear logic, the coKleisli category of a differential category is a Cartesian differential category. What then is the coEilenberg-Moore category of a differential category? The answer is a tangent category! A key example arises from the opposite of the category of Abelian groups with the free exponential modality. The coEilenberg-Moore category, in this case, is the opposite of the category of commutative rings. That the latter is a tangent category captures a fundamental aspect of both algebraic geometry and Synthetic Differential Geometry. The general result applies when there are no negatives and thus encompasses examples arising from combinatorics and computer science.

Subject Classification

ACM Subject Classification
  • Theory of computation → Linear logic
  • Theory of computation → Categorical semantics
Keywords
  • Differential categories
  • Tangent categories
  • Coalgebra Modalities

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