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Cartesian differential categories revisited

Published online by Cambridge University Press:  13 April 2015

G. S. H. CRUTTWELL
G. S. H. CRUTTWELL*
Affiliation:
Department of Mathematics and Computer Science, Mount Allison University, Sackville, NB
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Abstract

We revisit the definition of Cartesian differential categories, showing that a slightly more general version is useful for a number of reasons. As one application, we show that these general differential categories are comonadic over categories with finite products, so that every category with finite products has an associated cofree differential category. We also work out the corresponding results when the categories involved have restriction structure, and show that these categories are closed under splitting restriction idempotents.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 27 , Issue 1 , January 2017 , pp. 70 - 91
Copyright
Copyright © Cambridge University Press 2015 

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References

Blute, R., Cockett, J. and Seely, R. (2006). Differential categories. Mathematics Structures in Computer Science 16 1049–1083.Google Scholar
Blute, R., Cockett, J. and Seely, R. (2009). Cartesian differential categories. Theory and Applications of Categories 22 622–672.Google Scholar
Blute, R., Ehrhard, T. and Tasson, C. (2012). A convenient differential category. Cahiers de Topologie et Geométrie Différential Catégoriques 53 211–232.Google Scholar
Cockett, J. and Lack, S. (2007). Restriction categories III: Colimits, partial limits, and extensivity. Mathematical Structures in Computer Science 17 775–817.Google Scholar
Cockett, R. and Cruttwell, G. (2014). Differential structure, tangent structure, and SDG. Applied Categorical Structures 22 (2) 331–417.CrossRefGoogle Scholar
Cockett, R., Cruttwell, G. and Gallagher, J. (2011). Differential restriction categories. Theory and Applications of Categories 25 537–613.Google Scholar
Cockett, R. and Lack, S. (2002). Restriction categories I: Categories of partial maps. Theoretical Computer Science 270 (2) 223–259.Google Scholar
Cockett, J. and Seely, R. (2011). The Faá di bruno construction. Theory and Applictions of Categories 25 393–425.Google Scholar
Ehrhard, T. and Regnier, L. (2003). The differential lambda-calculus. Theoretical Computer Science 309 (1) 1–41.CrossRefGoogle Scholar
Kock, A. (2006). Synthetic Differential Geometry, 2nd ed. Cambridge University Press. Also (Available at http://home.imf.au.dk/kock/sdg99.pdf.)CrossRefGoogle 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
Manzonetto, G. (2012). What is a categorical model of the differential and the resource λ-calculi? Mathematical Structures in Computer Science 22 (3) 451–520.CrossRefGoogle Scholar
Robinson, E. and Roslini, G. (1988). Categories of partial maps. Information and Computation 79 94–13.Google Scholar
Rosický, J. (1984). Abstract tangent functors. Diagrammes 12, Exp. (3).Google Scholar