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We present a two-level theory to formalize constructive mathematics as advocated in a previous paper with G. Sambin [MS05]. One level is given by an intensional type theory, called Minimal type theory. This theory extends the set-theoretic version introduced in [MS05] with collections. The other level is given by an extensional set theory that is… (More)

In this paper we analyze an extension of Martin-Löf's intensional set theory by means of a set contructor P such that the elements of P(S) are the subsets of the set S. Since it seems natural to require some kind of extensionality on the equality among subsets, it turns out that such an extension cannot be constructive. In fact we will prove that this… (More)

There are several kinds of linear typed calculus in the literature, some with their associated notion of categorical model. Our aim in this paper is to systematise the relationship between three of these linear typed calculi and their models. We point out that mere sound-ness and completeness of a linear typed calculus with respect to a class of categorical… (More)

We apply some tools developed in categorical logic to give an abstract description of constructions used to formalize constructive mathematics in foundations based on intensional type theory. The key concept we employ is that of a Lawvere hy-perdoctrine for which we describe a notion of quotient completion. That notion includes the exact completion on a… (More)

This paper describes the categorical semantics of a system of mixed intuitionistic and linear type theory (ILT). ILT was proposed by G. Plotkin and also independently by P. Wadler. The logic associated with ILT is obtained as a combination of intuitionistic logic with intuitionistic linear logic, and can be embedded in Barber and Plotkin's Dual… (More)

We define the notion of exact completion with respect to an existential elementary doctrine. We observe that the forgetful functor from the 2-category exact categories to existential elementary doctrines has a left biadjoint that can be obtained as a composite of two others. Finally, we conclude how this notion encompasses both that of the exact completion… (More)