## Topological Representation of the &ambda;-Calculus (1998)

### BibTeX

@MISC{Awodey98topologicalrepresentation,

author = {S. Awodey},

title = {Topological Representation of the &ambda;-Calculus},

year = {1998}

}

### OpenURL

### 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 so-called 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 wellknown, the untyped version can be treated as a special case of this ([17]). We present here a topological representation of the -calculus: types are represented by cert...