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. This is an extended version of a conference paper for CSL2020.

[1]  J. Robin B. Cockett,et al.  Integral categories and calculus categories , 2017, Mathematical Structures in Computer Science.

[2]  S. Lane.  Categories for the Working Mathematician , 1971 .

[3]  Robert Wisbauer,et al.  Algebras Versus Coalgebras , 2008, Appl. Categorical Struct..

[4]  J. Golan.  Semirings and their applications , 1999 .

[5]  James Clift,et al.  Cofree coalgebras and differential linear logic , 2017, Mathematical Structures in Computer Science.

[6]  Li Guo,et al.  An introduction to Rota-Baxter algebra , 2012 .

[7]  R. Blute,et al.  Derivations in Codifferential Categories , 2015, 1505.00220.

[8]  R. B. Lucyshyn-Wright,et al.  Integral and differential structure on the free $C^{\infty}$-ring modality , 2019, 1902.04555.

[9]  Gavin M. Bierman.  What is a Categorical Model of Intuitionistic Linear Logic? , 1995, TLCA.

[10]  Classifying tangent structures using Weil algebras , 2017 .

[11]  F. E. J. Linton,et al.  Coequalizers in categories of algebras , 1969 .

[12]  Paul-André Melliès.  CATEGORICAL SEMANTICS OF LINEAR LOGIC , 2009 .

[13]  Abstract tangent functors , 1984 .

[14]  R. Blute,et al.  CARTESIAN DIFFERENTIAL STORAGE CATEGORIES , 2014, 1405.6973.

[15]  R. A. G. Seely,et al.  Linear Logic, -Autonomous Categories and Cofree Coalgebras , 1989 .

[16]  J. Robin B. Cockett,et al.  Differential Structure, Tangent Structure, and SDG , 2014, Appl. Categorical Struct..

[17]  Antonio Bucciarelli,et al.  Categorical Models for Simply Typed Resource Calculi , 2010, MFPS.

[18]  Jean-Yves Girard,et al.  Linear Logic , 1987, Theor. Comput. Sci..

[19]  J.R.B. Cockett,et al.  Differential Categories Revisited , 2020, Appl. Categorical Struct..

[20]  Marcelo P. Fiore,et al.  Differential Structure in Models of Multiplicative Biadditive Intuitionistic Linear Logic , 2007, TLCA.

[21]  John M. Lee.  Manifolds and Differential Geometry , 2009 .

[22]  Thomas Ehrhard,et al.  An introduction to differential linear logic: proof-nets, models and antiderivatives , 2016, Mathematical Structures in Computer Science.

[23]  Patrick Lincoln,et al.  Linear logic , 1992, SIGA.

[24]  Bart Jacobs,et al.  Structural Induction and Coinduction in a Fibrational Setting , 1998, Inf. Comput..

[25]  M. Barr,et al.  Toposes, Triples and Theories , 1984 .

[26]  H. Porst.  ON CORINGS AND COMODULES , 2003 .

[27]  J. Robin B. Cockett,et al.  Differential categories , 2006, Mathematical Structures in Computer Science.

[29]  A. Kock.  Synthetic Differential Geometry , 1981 .

[30]  Richard Blute,et al.  CARTESIAN DIFFERENTIAL CATEGORIES , 2009 .

[31]  I. Moerdijk,et al.  Models for smooth infinitesimal analysis , 1990 .

[32]  Thomas Ehrhard,et al.  A convenient differential category , 2010, ArXiv.

[33]  A. Agore.  Limits of Coalgebras, Bialgebras and Hopf Algebras , 2010, 1003.0318.

[34]  R. B. Lucyshyn-Wright,et al.  Ju l 2 01 9 Integral and differential structure on the free C ∞-ring modality , 2021 .

[35]  Martin Hyland,et al.  Glueing and orthogonality for models of linear logic , 2003, Theor. Comput. Sci..

[36]  P. Michor,et al.  C∞-algebras from the functional analytic view point , 1987 .

[37]  Giulio Manzonetto,et al.  What is a categorical model of the differential and the resource λ-calculi? , 2010, Mathematical Structures in Computer Science.

[38]  P. T. Johnstone,et al.  Adjoint Lifting Theorems for Categories of Algebras , 1975 .

[39]  Richard Garner.  An embedding theorem for tangent categories , 2017, 1704.08386.

[40]  J.R.B. Cockett,et al.  DIFFERENTIAL RESTRICTION CATEGORIES , 2012, 1208.4068.

[41]  Thomas Ehrhard,et al.  On Köthe sequence spaces and linear logic , 2002, Mathematical Structures in Computer Science.