The title of this course paraphrases that of a landmark 1957 paper by R.H. Fox, "Covering spaces with singularities", in which complete spreads were introduced as a suitable setting in which to discuss branched (or ramified) coverings and knot groupoids in topology. The course will center around a topos-theoretic notion of complete spread introduced by Bunge and Funk and on its connections with the Lawvere distributions on toposes. The link between the two is the symmetric topos construction as a KZ-monad M on the 2-category of (Grothendieck) toposes. Of particular interest are its algebras, its discrete fibrations (and opfibrations), and a factorization theorem associated with M, all of which have interesting interpretations. All categorical background material will be briefly reviewed before (or while) it is used for the purposes indicated. This course will be based on notes for a book "Topos Distributions" joint with Jonathon Funk (in preparation).
Part I: Distributions and the Symmetric Topos
Part II: Complete spreads in Topos Theory
Part III: Lattice-theoretic and localic considerations
The course will be based on a number of papers by myself and my collaborators (Aurelio Carboni, Jonathon Funk, Mamuka Jibladze, Thomas Streicher, Susan Niefield, Marcelo Fiore, Steve Lack), published between 1995 and 2002. An ideal companion to the notes are the recently published books by Peter T. Jonhnstone, Sketches of an Elephant: A Topos Theory Compendium, volumes 1 and 2, Oxford University Press, 2002.
Thursdays 2-5 pm, Burnside Hall 920, McGill University. Begins January 9, 2003.