 
Summary: Topological representation of the –calculus
S. Awodey
Carnegie Mellon University
September 1998
Abstract
The –calculus can be represented topologically by assigning certain
spaces to the types and certain continuous maps to the terms. Using a
recent result from category theory, the usual calculus of –conversion
is shown to be deductively complete with respect to such topological
semantics. It is also shown to be functionally complete, in the sense
that there is always a ``minimal'' topological model, in which every
continuous function is –definable. These results subsume earlier ones
using cartesian closed categories, as well as those employing socalled
Henkin and Kripke –models.
Introduction
The –calculus originates with Church [6]; it is intended as a formal calculus
of functional application and specification. In this paper, we are mainly
interested in the version known as simply typed –calculus ; as is now well
known, the untyped version can be treated as a special case of this ([17]).
We present here a topological representation of the –calculus: types are
