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Publications:

  1. Reverse tangent categories
    G.S.H. Cruttwell and JS Lemay. To be published in CSL 2024.
  2. Differential bundles in Commutative Algebra and Algebraic Geometry
    G.S.H. Cruttwell and JS Lemay. Theory and Applications of Categories, Volume 39, Number 36, 2023.
  3. Monoidal reverse differential categories
    G.S.H. Cruttwell and JS Lemay. Mathematical Structures in Computer Science, Volume 32, Issue 10, 2022.
  4. Double fibrations
    G.S.H. Cruttwell, M. Lambert, D. Pronk, and M. Szyld. Theory and Applications of Categories, Volume 38, Number 35, 2022.
  5. Categorical Foundations of Gradient-Based Learning
    G.S.H. Cruttwell, B. Gavranovic, N. Ghani, P. Wilson, and F. Zanasi. To be published in the proceedings of ESOP 2022. .
  6. Latent fibrations: fibrations for categories of partial maps
    R. Cockett, G.S.H. Cruttwell, J. Gallagher, and D. Pronk. Theory and Applications of Categories, Volume 36, Number 16, 2021.
  7. Differential equations in a tangent category I
    R. Cockett, G.S.H. Cruttwell, and J.S.P. Lemay. Applied Categorical structures, Issue 5, 2021. .
  8. Integral and differential structure on the free C-infinity-ring modality
    G.S.H. Cruttwell, J.S.P. Lemay, and R.B.B. Lucyshyn-Wright. Cahiers de Topologie et Géométrie Différentielle Catégoriques, Vol. LVII (2), 2021 .
  9. Categorical semantics of a simple differential programming language
    G.S.H. Cruttwell, Jonathan Gallagher, and Dorette Pronk. To appear at ACT 2020 .
  10. Reverse derivative categories
    Robin Cockett, G.S.H. Cruttwell, Jonathan Gallagher, J.S. Lemay, Benjamin MacAdam, Gordon Plotkin, and Dorette Pronk. To appear at CSL 2020 .
  11. Affine geometric spaces in tangent categories
    Rick Blute, G.S.H. Cruttwell and Rory Lucyshyn-Wright. Theory and Applications of Categories, Vol. 34 (15), pg. 405-437, 2019 . (For reference, here is an older version which has more material on monads and comonads in the affine category; this material may be published later in a different form.)
  12. A simplicial foundation for differential and sector forms in tangent categories
    G.S.H. Cruttwell and Rory Lucyshyn-Wright. Journal of Homotopy and Related Structures, 2018 .
  13. Differential bundles and fibrations for tangent categories
    Robin Cockett and G.S.H. Cruttwell. Cahiers de Topologie et Géométrie Différentielle Catégoriques, Vol. LIX (1) pg. 10-92, 2018 .
  14. Connections in tangent categories
    Robin Cockett and G.S.H. Cruttwell. Theory and Applications of Categories, Vol. 32 (26), pg. 835-888, 2017 .
  15. The Jacobi identity for tangent categories.
    J.R.B. Cockett and G.S.H. Cruttwell. Cahiers de Topologie et Géométrie Différentielle Catégoriques, Vol. LVI (4) pg. 301--316, 2015 .
  16. Cartesian differential categories revisited
    G.S.H. Cruttwell. Mathematical Structures in Computer Science, Vol. 27 (1), pg. 70-91 (first published online in 2015) .
  17. Differential structure, tangent structure, and SDG
    J.R.B. Cockett and G.S.H. Cruttwell. Applied Categorical Structures Vol. 22 (2), pg. 331-417, 2014 . (This has been extensively updated to take into account an earlier paper of Rosicky's. For reference, here is the older version.)
  18. Forms and exterior differentiation in Cartesian differential categories
    G.S.H. Cruttwell. Theory and Applications of Categories, Vol. 28, pg. 981-1001, 2013 .
  19. Differential Restriction Categories
    J.R.B. Cockett, G.S.H. Cruttwell, and Jonathan Gallagher. Theory and Applications of Categories, Vol. 25, pp 537-613, 2011 .
  20. A Unified Framework for Generalized Multicategories
    G.S.H. Cruttwell and Michael A. Shulman. Theory and Applications of Categories, Vol. 24, pg. 580-655, 2010 .
  21. CCD lattices in Presheaf Categories
    G.S.H. Cruttwell, F. Marmolejo, and R.J. Wood. Theory and Applications of Categories, Vol. 18, pg. 157-171, 2007 .
  22. Dimensions of Fractals in the Large
    M.R. Allen, G.S.H. Cruttwell, K. Hare, J.O. Ronning. Chaos, Solitons, and Fractals, Volume 31 (1), pg. 5-13, 2007 .

Submissions and other works:

  1. Combinatorial Game Categories
    J.R.B. Cockett, G.S.H. Cruttwell, and Kevin Saff. Under revision.
  2. Normed Spaces and the Change of Base for Enriched Categories
    G.S.H. Cruttwell, PhD Thesis, Dalhousie University, 2008.
  3. A Study of CCD Lattices in a Functor Category
    G.S.H. Cruttwell, MSc Thesis, Dalhousie University, 2005.

Talks:

  1. Algebraic geometry: a different path up the mountain
    Extended notes for a tutorial given at FMCS 29, June 2022.
  2. Algebraic geometry and tangent categories
    A series of two talks given at the Atlantic category theory seminar, March 2022.
  3. Towards cotangent categories
    A talk given at the Mathematical Congress of the Americas (special session on Categories and Topology), July 2021.
  4. Categorical differential structures and their role in abstract machine learning
    A talk given at the Topos Institute, July 2021.
  5. Introduction to tangent categories
    A tutorial give at the BIRS workshop on Tangent categories and their applications, June 2021.
  6. Latent fibrations
    A series of two talks given at the Brandon University Category theory seminar, April 2021.
  7. The dual fibration
    A series of two talks given at the Calgary Peripatetic seminar, June 2020.
  8. Curvature and torsion without negatives
    A talk given at the summer 2019 CMS meeting, June 2019.
  9. Affine objects in a tangent category
    A talk given at Category Theory 2018, July 2018.
  10. Differential equations in tangent categories
    A talk given at Category Theory 2017, July 2017.
  11. General connections in tangent categories
    A talk given at FMCS 2017, June 2017.
  12. A simplicial framework for de Rham cohomology in a tangent category
    A talk given at Category theory 2016, August 2016.
  13. The Jacobi identity for tangent categories
    A talk given at Category theory 2015, June 2015.
  14. Bundles, pullbacks and fibrations for tangent categories
    A talk given at Foundational methods in computer science 23, June 2015.
  15. Counting finite categories
    An invited talk given at the Fall 2014 AMS Eastern Sectional meeting (special session on New Directions in Category Theory), October 2014.
  16. Structures in tangent categories
    A talk given at Category theory 2014, June 2014.
  17. Introducing tangent categories and An alternative view of the tangent category axioms
    Talks given at FMCS 2014, June 2014.
  18. Connections in tangent categories
    A talk given at the Union college math conference, October 2013.
  19. Tangent categories, vector bundles, and connections
    A talk given in the @cat seminar at Dalhousie University, September 2013.
  20. Differential categories and differential algebra
    An invited talk given at the Annual North American meeting of the ASL, May 2013.
  21. An introduction to categories
    A talk given to the Math and Computer Science Society of Mount Allison University, March 2013.
  22. Differential structure, tangent structure, and SDG
    A talk given at Foundational Methods in Computer Science 2012, June 2012.
  23. Reconsidering Cartesian differential categories
    A talk given at Logic and Interactions 2012 (Quantative approaches session), February 2012.
  24. A tale of two tangent bundles
    A talk given at Category theory Octobefest 2011, October 2011.
  25. Combining differential categories and manifolds
    An invited talk, given at Foundational Methods in Computer Science 2011, June 2011.
  26. Embeddings for atlas categories
    A talk given in the Calgary Peripatetic Seminar in Logic and Category Theory, March 2011.
  27. Differential and tangent structure for restriction categories
    A talk given at Octoberfest 2010, October 2010.
  28. Combinatorial Game Categories
    A talk given at Category Theory 2010, June 2010.
  29. Differential Restriction Categories II
    A talk given at FMCS, June 2010.
  30. A Unified Framework for Generalized Multicategories
    An invited talk given at CMS 2010 Summer Meeting (Tensor categories session) , June 2010.
  31. T-Monoids and 2-Dimensional Category Theory
    Notes for a series of talks given in the Calgary Peripatetic Seminar in Logic and Category Theory on enriched categories, internal categories, 2-categories, and double categories, leading to the main construction in "A Unified Framework for Generalized Multicategories" (updated to December 9, 2009).
  32. Combinatorial Game Categories
    A talk given at Novemberfest 2009, November 2009.
  33. A Unified Framework for Generalized Multicategories
    An invited talk given at CT2009, June 2009.
  34. Change of Base for Enriched Categories
    A talk given at CT2008, June 2008.
  35. Change of Base and the Amazing Technicolour Double Categories
    A talk given in the Atlantic Category Theory Seminar, June 2008.
  36. Change of Base for Enriched Categories
    A talk given in the Atlantic Category Theory Seminar, November 2007.
  37. Analysis and Enriched Category Theory
    Notes for my CT2007 talk, June 2007.
  38. A Look At Some Classical Mathematical Structures
    A talk given in the Dalhousie Honours Mathematics Seminar, February 2007.
  39. Summing Cantor Sets: When Does 0+0=2?
    A talk given in the Dalhousie Honours Mathematics Seminar, November 2006.
  40. Normed and Ordered Algebraic Structures
    A talk given in the Atlantic Category Theory Seminar, October 2006.
  41. A Category of Games
    A talk given in the Dalhousie Game Theory Seminar, May 2006.
  42. A Generalization of Normed Linear Spaces
    A talk given in the Atlantic Category Theory Seminar, April 2006.
  43. CCD Lattices in a Functor Category
    A talk given in the Atlantic Category Theory Seminar, September 2005.
  44. A Category Theory View of Products
    A talk given in the Dalhousie Mathematics and Statistics Graduate Student Seminar, July 2005.
  45. Surreal Numbers
    A talk given in the University of Waterloo Short Attention Span Seminars, November 2003.

Do not worry about your problems with mathematics, I assure you mine are far greater. - Albert Einstein