Geoffrey Cruttwell

Assistant professor, Math & Computer Science

Geoffrey Cruttwell

Contact Information

E-mail
gcruttwell@mta.ca
Phone
(506) 364-2533
Office
Dunn 211
Office hours
Other websites

I am an assistant professor in the Department of Mathematics and Computer Science at Mount Allison University, an adjunct Professor at the Dalhousie university mathematics department, and a member of the Atlantic category theory group. I am interested in a wide variety of subjects related to category theory, such as:
      Differential categories and tangent structures
      Synthetic differential geometry
      Constructions of the real numbers
      Generalized multicategories
      Double categories

Teaching

 

Mount Allison University

 

       Math 1151 : Applied Calculus (Fall 2015)

       Math 3111 : Real Analysis (Fall 2015)

       Math 1111: Calculus I (Winter 2015)

       Math 3121: Real Analysis II (Winter 2015)

       Math 3111: Real Analysis (Fall 2014)

       Math 1151: Applied Calculus (Fall 2014)

       Math 4111: Topology (Winter 2014)
       Math 2221: Linear Algebra (Winter 2014)
       Math 3111: Real Analysis (Fall 2013)
       Math 1111: Calculus I (Fall 2013)
       Math 4211: Modern Algebra II (Winter 2013)
       Math 1111: Calculus I (Winter 2013)
       Math 3111: Real Analysis (Fall 2012)

 
University of Calgary
       CPSC 413: Design and Analysis of Algorithms (Fall 2010)
       CPSC 513: Computability (Winter 2010)

 
Saint Mary's University
       MGSC 1205: Introduction to Quantitative Methods I (Winter 2009)

 
Dalhousie University
       Math 2002: Vector Calculus (July 28th - August 20th, 2008)
       Math 1000: Calculus (July 4th - August 19th, 2007)


Research interests

Recent research news:

 

        October 2014: Submitted paper: Differential bundles in tangent categories.

 

        October 2014: Talk at the "New directions in category theory" special session of the AMS 2014 Eastern Sectional meeting: Counting Finite Categories.

 

        June 2014: Talk at Category theory 2014: on Structures in tangent categories.

 

        June 2014: Two talks at FMCS 2014: "Introducing tangent categories" and "An alternative view of the tangent category axioms".

 

       October 2013: Submitted paper Tangent categories are locally Cartesian differential categories (with Robin Cockett).

 
       October 2013: Talk at Union college math conference on Connections in tangent categories.

 
       October 2013: New publication: Forms and exterior differentiation in Cartesian differential categories in Theory and Applications of Categories.

 
       September 2013: Two-part talk in the @cat seminar on Tangent categories, vector bundles, and connections.

Publications

Publications:

 

  1. 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. 
  2. Cartesian differential categories revisited
    G.S.H. Cruttwell. Mathematical Structures in Computer Science, 22 pages, 2015. 
  3. 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).
  4. Forms and exterior differentiation in Cartesian differential categories
    G.S.H. Cruttwell. Theory and Applications of Categories, Vol. 28, pg. 981-1001, 2013. 
  5. 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. 
  6. 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. 
  7. 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. 
  8. 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. Differential bundles in tangent categories.
    Robin Cockett and G.S.H. Cruttwell. Submitted. PDF
  2. Combinatorial Game Categories
    J.R.B. Cockett, G.S.H. Cruttwell, and Kevin Saff. Under revision. PDF
  3. Normed Spaces and the Change of Base for Enriched Categories
    G.S.H. Cruttwell, PhD Thesis, Dalhousie University, 2008.  PDF
  4. A Study of CCD Lattices in a Functor Category
    G.S.H. Cruttwell, MSc Thesis, Dalhousie University, 2005. PDF

 

Talks:

  1. The Jacobi identity for tangent categories
    A talk given at Category theory 2015, June 2015. 
  2. Bundles, pullbacks and fibrations for tangent categories
    A talk given at Foundational methods in computer science 23, June 2015.
  3. 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. 
  4. Structures in tangent categories
    A talk given at Category theory 2014, June 2014. 
  5. Introducing tangent categories and An alternative view of the tangent category axioms
    Talks given at FMCS 2014, June 2014.
  6. Connections in tangent categories
    A talk given at the Union college math conference, October 2013. 
  7. Tangent categories, vector bundles, and connections
    A talk given in the @cat seminar at Dalhousie University, September 2013.
  8. Differential categories and differential algebra
    An invited talk given at the Annual North American meeting of the ASL, May 2013. 
  9. An introduction to categories
    A talk given to the Math and Computer Science Society of Mount Allison University, March 2013. 
  10. Differential structure, tangent structure, and SDG
    A talk given at Foundational Methods in Computer Science 2012, June 2012. 
  11. Reconsidering Cartesian differential categories
    A talk given at Logic and Interactions 2012 (Quantative approaches session), February 2012. 
  12. A tale of two tangent bundles
    A talk given at Category theory Octobefest 2011, October 2011. 
  13. Combining differential categories and manifolds
    An invited talk, given at Foundational Methods in Computer Science 2011, June 2011.
  14. Embeddings for atlas categories
    A talk given in the Calgary Peripatetic Seminar in Logic and Category Theory, March 2011. 
  15. Differential and tangent structure for restriction categories
    A talk given at Octoberfest 2010, October 2010. 
  16. Combinatorial Game Categories
    A talk given at Category Theory 2010, June 2010. 
  17. Differential Restriction Categories II
    A talk given at FMCS, June 2010
  18. A Unified Framework for Generalized Multicategories
    An invited talk given at CMS 2010 Summer Meeting (Tensor categories session) , June 2010. 
  19. 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). 
  20. Combinatorial Game Categories
    A talk given at Novemberfest 2009, November 2009. 
  21. A Unified Framework for Generalized Multicategories
    An invited talk given at CT2009, June 2009.
  22. Change of Base for Enriched Categories
    A talk given at CT2008, June 2008.
  23. Change of Base and the Amazing Technicolour Double Categories
    A talk given in the Atlantic Category Theory Seminar, June 2008.
  24. Change of Base for Enriched Categories
    A talk given in the Atlantic Category Theory Seminar, November 2007.
  25. Analysis and Enriched Category Theory
    Notes for my CT2007 talk, June 2007. 
  26. A Look At Some Classical Mathematical Structures
    A talk given in the Dalhousie Honours Mathematics Seminar, February 2007.
  27. Summing Cantor Sets: When Does 0+0=2?
    A talk given in the Dalhousie Honours Mathematics Seminar, November 2006.
  28. Normed and Ordered Algebraic Structures
    A talk given in the Atlantic Category Theory Seminar, October 2006.
  29. A Category of Games
    A talk given in the Dalhousie Game Theory Seminar, May 2006.
  30. A Generalization of Normed Linear Spaces
    A talk given in the Atlantic Category Theory Seminar, April 2006.
  31. CCD Lattices in a Functor Category
    A talk given in the Atlantic Category Theory Seminar, September 2005.
  32. A Category Theory View of Products
    A talk given in the Dalhousie Mathematics and Statistics Graduate Student Seminar, July 2005.
  33. Surreal Numbers
    A talk given in the University of Waterloo Short Attention Span Seminars, November 2003.