Abstract

Introduction

The straightforward way to describe the simply typed λ-calculus (denoted here by λ1) categorically is in terms of cartesian closed categories (CCC's), see [10]. On the type theoretic side this caused some discomfort, because one commonly uses only exponent types without assuming cartesian product types — let alone unit (i.e. terminal) types. The typical reply from category theory is that one needs cartesian products in order to define exponents. Below we give a categorical description of exponent types without assuming cartesian product types. We do use cartesian products of contexts; these always exist by concatenation. Thus both sides can be satisfied by carefully distinguishing between cartesian products of types and cartesian products of contexts. We introduce an appropriate categorical structure for doing so.

In [6] one can find a general method to describe typed λ-calculi categorically. One of the main organizing principles used there is formulated below. It deserves the status of a slogan.