Modular correspondence between dependent type theories and categories including pretopoi and topoi

Abstract

We present a modular correspondence between various categorical structures and their internal languages in terms of extensional dependent type theoriesà la Martin-Löf. Starting from lex categories, through regular ones we provide internal languages of pretopoi and topoi and some variations of them, like for example Heyting pretopoi. With respect to the internal languages already known for some of these categories like topoi, the novelty of these calculi is that formulas corresponding to subobjects can be regained as particular types equipped with proof-terms according to the isomorphism "propositions as mono types", invisible in the previous languages. MSC 2000: 03G30 03B15 18C50

Why topology in the minimalist foundation must be pointfree

“…They range from axiomatic set theories, as Aczel's CZF [1][2][3] or Friedman's IZF [8], to the internal set theory of categorical universes as topoi or pretopoi [29,31,40], to type theories as Martin-Löf's type theory [45] or Coquand's Calculus of Inductive Constructions [17,19]. No existing constructive foundation has yet superseded the others as the standard one, as Zermelo-Fraenkel axiomatic set theory did for classical mathematics.…”

“…Such a conception of subsets and propositions was later specified in [55] as a tool to be added on top of type theory. As noticed in [32,40], working with existential quantifiers with no proof terms means that the axiom of choice no longer holds. This is different from MLTT where existential quantifications are identified with indexed sums, according to the proposition-as-set isomorphism, thus making the axiom of choice derivable (see [42]).…”

Types, Sets, and Categories

“…This prompts the question: what modification needs to be made to the "propositions-as-types" paradigm so as to yield the topos-theoretic interpretation of AC? An illuminating answer to this question has been given by Maietti [2005] through the use of so-called monotypes (or mono-objects), that is, (dependent) types containing at most one entity or having at most one proof. In Set , mono objects are singletons, that is, sets containing at most one element.…”

Quotients over Minimal Type Theory

“…First, we take the extensional version of our Minimal Type Theory, in the same way as the type theory in [Mar84] is the extensional version of intensional Martin-Löf's type theory in [NPS90]. Then we collapse propositions into mono sets in the sense of [Mai05] and, finally, we add effective quotient sets similarly to those in [Mai05].…”

“…Q(mTT) turns out to be a categorical model of qmTT. Categorically speaking, it turns out to be a lextensive list-arithmetic locally cartesian closed category with stable effective quotients of equivalence relations obtained by comprehension from a propositional fibration (for all these categorical properties see, for example, [Jac99,Mai05]). …”