Tangent Categories from the Coalgebras of Differential Categories
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.
Differential categories
Tangent categories
Coalgebra Modalities
Theory of computation~Linear logic
Theory of computation~Categorical semantics
17:1-17:17
Regular Paper
An extended version of the paper with the proof of Proposition 21 is available at https://arxiv.org/abs/1910.05617.
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].
Robin
Cockett
Robin Cockett
University of Calgary, Department of Computer Science, Canada
https://pages.cpsc.ucalgary.ca/~robin/
Partially supported by NSERC (Canada).
Jean-Simon Pacaud
Lemay
Jean-Simon Pacaud Lemay
University of Oxford, Department of Computer Science, UK
https://www.cs.ox.ac.uk/people/jean-simon.lemay/
Thanks to Kellogg College, the Clarendon Fund, and the Oxford Google-DeepMind Graduate Scholarship for financial support.
Rory B. B.
Lucyshyn-Wright
Rory B. B. Lucyshyn-Wright
Brandon University, Department of of Mathematics and Computer Science, Canada
http://lucyshyn-wright.ca/
Supported by the Natural Sciences and Engineering Research Council of Canada (NSERC).
10.4230/LIPIcs.CSL.2020.17
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Robin Cockett, Jean-Simon Pacaud Lemay, and Rory B. B. Lucyshyn-Wright
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