**MATH-724B: TOPICS IN ALGEBRA V (Covering toposes with singularities).**- Instructor: Marta Bunge (McGill University, BURN 925)
*GENERAL DESCRIPTION*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).

- Topos frames and distributions.
- Free constructions, coinverters and the symmetric topos.
- The KZ-monad M and its algebras: Waelbroeck's theorem.
- Related KZ-doctrines: Bag and the classifier C for probability distributions.
- Cosheaves and distributions.
- Complete spreads as discrete fibrations for the KZ-monad M.
- The pure-complete spread factorization theorem.
- Unramified covers, branched covers, the fundamental groupoid, Van Kampen theorems.
- Distributions on locales.
- Distribution algebras.
- The Michael completion of a topos spread.
- Single universes for functions and distributions.

* SYLLABUS *

* Part I: Distributions and the Symmetric Topos*

* Part II: Complete spreads in Topos Theory*

* Part III: Lattice-theoretic and localic considerations*

* REFERENCES *

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.

* TIMETABLE *

Thursdays 2-5 pm, Burnside Hall 920, McGill University. Begins January 9, 2003.