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 typeâ€¦ (More)

We extend the notion of exact completion on a category with weak finite limits to Lawvereâ€™s elementary doctrines. We show how any such doctrine admits an elementary quotient completion, which is theâ€¦ (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 keyâ€¦ (More)

We build a Kleene realizability semantics for the two-level Minimalist Foundation MF, ideated by Maietti and Sambin in 2005 and completed by Maietti in 2009. Thanks to this semantics we prove thatâ€¦ (More)

Suppose in an arithmetic unverse we have two predicates Ï† and Ïˆ for natural numbers, satisfying a base case Ï†(0) â†’ Ïˆ(0) and an induction step that, for generic n, the hypothesis Ï†(n) â†’ Ïˆ(n) allowsâ€¦ (More)

We present a modular correspondence between various categorical structures and their internal languages in terms of extensional dependent type theories Ã la Martin-LÃ¶f. Starting from lex categories,â€¦ (More)

We extend Martin-LÃ¶fâ€™s constructive set theory with effective quotient sets and the rule of uniqueness of propositional equality proofs. We prove that in the presence of at least two universes U0 andâ€¦ (More)

We define the notion of exact completion with respect to an existential elementary doctrine. We observe that the forgetful functor from the 2category of exact categories to existential elementaryâ€¦ (More)