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Topological representation of the calculus Carnegie Mellon University
 

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

  

Source: Andrews, Peter B. - Department of Mathematical Sciences, Carnegie Mellon University

 

Collections: Mathematics