Locally cartesian closed categories and type theory

0. Introduction. I t is well known that for much of the mathematics of topos theory, it is in fact sufficient to use a category C whose slice categories C/A are cartesian closed. In such a category, the notion of a 'generalized set', for example an 'Aindexed set', is represented by a morphism B^-A of C, i.e. by an object of C/A. The point about such a category C is that C is a C-indexed category, and more, is a hyperdoctrine, so that it has a full first order logic associated with it. This logic has some peculiar aspects. For instance, the types are the objects of C and the terms are the morphisms of C. For a given type A, the predicates with a free variable of type A are morphisms into A, and 'proofs' are morphisms over A. We see here a certain 'ambiguity' between the notions of type, predicate, and term, of object and proof: a term of type A is a morphism into A, which is a predicate over A; a morphism 1 -> A can be viewed either as an object of type A or as a proof of the proposition A. For a long time now, it has been conjectured that the logic of such categories is given by the type theory of Martin-Lof [5], since one of the features of Martin-Lof's type theory is that it formalizes ' ambiguities' of this sort. However, to the best of my knowledge, no one has ever worked out the details of the relationship, and when the question again arose in the McGill Categorical Logic Seminar in 1981-82, it was felt that making this precise was long overdue. That is the purpose of this paper. We shall describe the system ML, based on Martin-Lof's system, and show how to construct a locally cartesian closed category from an ML theory, and vice versa. Finally, we show these constructions are inverse. A somewhat different approach to the question was taken by John Cartmell[l], who describes a categorical structure suitable for Martin-Lof's type theory. I have taken greater liberties with the type theory, my purpose being to characterize locally cartesian closed categories; the payoff is that these categories are simpler, and perhaps more natural, than Cartmell's contextual categories. For example, since toposes are locally cartesian closed, there are many familiar locally cartesian closed categories: the category of Sets, and more generally Boolean (or Heyting) valued models and Kripke models of set theory, (indeed any other example of a category of sheaves on a site). These results were first presented at the McGill Categorical Logic Seminar: an early draft, based on the seminar notes, appeared as [11] and an abstract was published [12]. 1. The type theory ML. The type theory described here is based on Martin-Lof's, as given in Martin-Lof [5]. We adopt some simplifications of Diller [2]. In the interests of readability, we present the type theory more or less informally, as in the first section of Martin-Lof [5]; a more formal version would follow the second section of