# Topos Theory, spring term 1999

A graduate course (6 course points) in mathematical logic.

Topos theory grew out of the observation that the category of sheaves
over a fixed topological space forms a universe of "continuously
variable sets" which obeys the laws of intuitionistic logic. These
sheaf models, or Grothendieck toposes, turn out to be generalisations
of Kripke and Beth models (which are fundamental for various non-classical
logics) as well as Cohen's forcing models for set theory. The notion of
topos was subsequently extended and given an elementary axiomatisation
by Lawvere and Tierney, and shown to correspond to a certain higher order
intuitionistic logic. Various logics and type theories have been given
categorical characterisations, which are of importance for the
mathematical foundations for programming languages. One of the most
interesting aspects of toposes is that they can provide natural models
of certain theories that lack classical models, viz. synthetic
differential geometry.

This graduate course offers an introduction to topos theory
and categorical logic. In particular the following topics will be covered:
Categorical logic: relation between logics, type theories and categories.
Generalised topologies, including formal topologies. Sheaves. Pretoposes
and toposes. Beth-Kripke-Joyal semantics. Boolean toposes and Cohen
forcing. Barr's theorem and Diaconescu covers. Geometric morphisms.
Classifying toposes. Sheaf models of infinitesimal analysis.

We will assume some familiarity with basic category theory, such
as is obtained in courses in domain theory or algebra.
The course will be given English, in case someone requests this.

### Schedule:

Mondays and Wednesdays, 15.15 - 17.00, in lecture
hall 2:314, Department of Mathematics, Uppsala University, MIC,
Polacksbacken. The course starts on February 3, 1999.

### Course Literature:

S. Mac Lane and I. Moerdijk: Sheaves in Geometry and Logic.
Springer 1992.

### Reference literature:

P.T. Johnstone. Topos Theory. Academic Press 1977.

J. Lambek and P.J. Scott: An introduction to Higher Order Categorical
Logic. Cambridge University Press 1986

S. Mac Lane: Categories for the Working Mathematician. Springer 1971.

A.M. Pitts. Categorical Logic. Chapter in the Handbook of Logic in
Computer Science, vol. VI. Oxford University Press (to appear)

1999-12-18, Erik Palmgren