Category theory in general and topos theory in particular provide a basis
for a foundation of mathematics that is not only closer to mathematical
practice than the one based on (classical) set theory, but that is also
unifying, constructive, and provides new insights. Toposes bounded over a
base S (the "set theory") may be thought of as generalized topological
spaces. My work, of a foundational nature, centers on investigations in
areas lof mathematics, such as functional analysis, algebraic topology,
differential topology, where topological spaces are involved. The main
topics of my proposed research for the next five years are: to deal with
several problems left open in the research monograph (written in collaboration with
J. Funk) "Singular Coverings of Toposes", LNM 1890, Springer, 2006; to clarify
the nature of higher stacks for the intrinsic topology of a topos (in
collaboration with C. Hermida); to pursue investigations into the fundamental groupoid of a
general topos and Galolis theory, and to continue the study of smooth mappings using
Penon infinitesimals (initiated in joint work with F. Gago-Couso).
- RECENT CONFERENCE PRESENTATIONS
- Workshop on Categorical Structures for Descent and Galois Theory,
Hopf Algebras, and Semi-abelian Categories. Fields Institute, Toronto,
September 23-28, 2002.
Invited talk: "Galois groupoids and covering morphisms in topos theory"
- Workshop on Ramifications of Category Theory. Florence, Italy,
November 18-22, 2003.
Invited talk: "Totally disconnected reflections in the homotopy theory of toposes"
- Union College Mathematics Conference. Schenectady, N. Y., November
Contributed talk: "Groupoids, locally paths simply connected toposes, and a van Kampen theorem"
- Categories in Algebra, Geometry and Mathematical Physics. Macquarie
University, Sydney, Australia, 11-15 July 2005.
Invited talk: "An intrinsic characterization of branched coverings"
(joint with J. Funk)
- International Conference: "Charles Ehresmann: 100 ans", Universite
de Picardie, Amiens, October 7-9, 2005.
Invited talk: "On two non-discrete generalizations lof \pi_0"
- Canadian Math Society Summer Meeting 2006. Special Session on
Category Theory. University of Calgary, AB, Canada, 3-5 June 2006.
Invited talk: "Michael toposes are comprehensive" (joint with J.Funk)
- Category Theory Octoberfest. University of Ottawa. October 21-22,
Invited talk: "Quasi locally connected toposes" (joint with J. Funk)
- Canadian Math Society Winter Meeting 2007. Special Session on
Mathematical Applications of Category Theory. University of Western
Ontario, London, Ontario, Canada, December 8-10, 2007.
Invited talk: "Fundamental Pushout Toposes"
- International Category Theory Conference CT2007. Carvoeiro, Portugal. June
Contributed talk: "Branched coverings of quasi locally connected toposes "
- International Category Theory Conference CT2008. Calais, France. June
Contributed talk: "Intrinsic n-stack completions " (Joint with C.
- SEMINARS AT THE CRTC MONTREAL
- "Extensive 2-categories and Top". February 5, 2002.
- "A Van Kampen theorem for toposes and applications" February 12, 2002.
- "Locally constant objcets in a Grothendieck topos". February 19, 2002.
- "Stack completions rfevisited" October 29, 2002.
- "Johnstone's theorem and Fox completions" March 2, 2004
- "Michael cov erings are comprehensive" April 4, 2006
- "Locally discrete locales" January 23, 2007