Carsten Butz (McGill)
Geometric morphisms and logic

Abstract: The classifying topos theorem for geometric theories is certainly
one of the highlights of the relationship between category theory and
logic. Some of my recent research (partly together with Peter T. Johnstone
and Steve Awodey) has centered around the question what can be said for
other logics (like (full) first-order, first-order logic extended by
quantification over functions, etc.).
Though there do not exist classifying toposes (even when one restricts the
class of geometric morphisms) there are good results available. The talk
will survey some of them, including the sketch of topological completeness
theorems for higher order logics. It is based on the following papers and 
preprints:

 Syntax and Semantics of the Logic L^\lambda_{\omega\omega}  
 Notre Dame J. Formal Logic, 38 (1997), 374-384.  

 (With Peter T. Johnstone, Cambridge)
 Classifying toposes for first-order theories. 
 Ann. Pure Appl . Logic, 91 (1998), 33-58.  

 (With Steve Awodey, Pittsburgh)
 Topological Completeness for Higher-Order Logic.  
 To appear in J. Symbolic Logic.

 Steve Awodey
 Topological representation of the lambda-calculus.
 Preprint.