arXiv subjects

Maria Emilia Maietti

Publications and source records attributed to Maria Emilia Maietti.

At least 19 recordsLinked to original sources

arXiv2026

Quasitoposes as elementary quotient completions

The elementary quotient completion of an elementary doctrine in the sense of Lawvere was introduced in previous work by the first and third authors. It generalises the exact completion of a category with finite products and weak equalisers. In this paper we characterise when an elementary quotient completion is a quasi-topos. We obtain as a corollary a complete characterisation of when an elementary quotient completions is an elementary topos. As a byproduct we determine also when the elementary quotient completion of a tripos is equivalent to the doctrine obtained via the tripos-to-topos construction. Our results are reminiscent of other works regarding exact completions and put those under a common scheme: in particular, Carboni and Vitale's characterisation of exact completions in terms of their projective objects, Carboni and Rosolini's characterisation of locally cartesian closed exact completions, also in the revision by Emmenegger, and Menni's characterisation of the exact completions which are elementary toposes.

math.LO↗

arXiv2025

On the Compatibility of Constructive Predicative Mathematics with Weyl's Classical Predicativity

It is well known that most constructive and predicative foundations aiming to develop Bishop's constructive analysis are incompatible with a classical predicative development of analysis as put forward by Weyl in his $\textit{Das Kontinuum}$. Here, we show how this incompatibility arises from the possibility to define sets by quantifying over (the exponentiation of) functional relations. Such a possibility is present in most constructive foundations but it is not allowed in modern reformulations of Weyl's logical system. In particular, we show how in Aczel's Constructive Set Theory, Martin-Löf's type theory and Homotopy Type Theory, the incompatibility with classical predicativity à la Weyl reduces to the fact of being able to interpret Heyting arithmetic in all finite types with the addition of the internal rule of number-theoretic unique choice, identifying functional relations over natural numbers with a primitive notion of function defined as $λ$-terms of type theory. Then, we argue that a possible way out is offered by constructive foundations, such as the Minimalist Foundation, where exponentiation is limited to functions defined as $λ$-terms of (dependent) type theory. The price to pay is to renounce number-theoretic choice principles, including the rule of unique choice, typical of most foundations formalizing Bishop's constructive mathematics. This restriction calls for a point-free constructive development of topology as advocated by P. Martin-Löf and G. Sambin with the introduction of Formal Topology. We then conclude that the Minimalist Foundation promises to be a natural crossroads between Bishop's constructivism and Weyl's classical predicativity provided that a point-free constructive reformulation of analysis is viable.

math.LO↗

arXiv2025

An Algebraic Abstraction of the Localic Sheafification via the Tripos-to-Topos Construction

Localic and realizability toposes are two central classes of toposes in categorical logic, both arising through the Hyland-Johnstone-Pitts tripos-to-topos construction. We investigate their shared geometric features by providing an algebraic abstraction of the notions of localic presheaves, sheafification and their connection to supercompactification of a locale via an instance of the Comparison Lemma. This can be applied to a broad class of toposes obtained to the tripos-to-topos constructions, including all those generated from a tripos based on the classical category of ZFC-sets. These results provide a unified geometric framework for understanding localic and realizability toposes.

math.CT↗

arXiv2024

Fibred sets within a predicative and constructive effective topos

We describe the fibrational structure of sets within the predicative variant $\mathbf{pEff}$ of Hyland's Effective Topos $\mathbf{Eff}$ previously introduced in Feferman's predicative theory of non-iterative fixpoints $\widehat{ID_1}$. Our structural analysis can be carried out in constructive and predicative variants of $\mathbf{Eff}$ within extensions of Aczel's Constructive Zermelo-Fraenkel Set Theory. All this shows that the full subcategory of discrete objects of Hyland's Effective topos $\mathbf{Eff}$ contains already a fibred predicative topos validating the formal Church's thesis, even when both are formalized in a constructive metatheory.

math.LO↗

arXiv2024

Equiconsistency of the Minimalist Foundation with its classical version

The Minimalist Foundation, for short MF, was conceived by the first author with G. Sambin in 2005, and fully formalized in 2009, as a common core among the most relevant constructive and classical foundations for mathematics. To better accomplish its minimality, MF was designed as a two-level type theory, with an intensional level mTT, an extensional one emTT, and an interpretation of the latter into the first. Here, we first show that the two levels of MF are indeed equiconsistent by interpreting mTT into emTT. Then, we show that the classical extension emTT^c is equiconsistent with emTT by suitably extending the Gödel-Gentzen double-negation translation of classical logic in the intuitionistic one. As a consequence, MF turns out to be compatible with classical predicative mathematics à la Weyl, contrary to the most relevant foundations for constructive mathematics. Finally, we show that the chain of equiconsistency results for MF can be straightforwardly extended to its impredicative version to deduce that Coquand-Huet's Calculus of Constructions equipped with basic inductive types is equiconsistent with its extensional and classical versions too.

math.LO↗

arXiv2024

Quotients, pure existential completions and arithmetic universes

We provide a new description of Joyal's arithmetic universes through a characterization of the exact and regular completions of pure existential completions. We show that the regular and exact completions of the pure existential completion of an elementary doctrine $P$ are equivalent to the $\mathsf{reg}/\mathsf{lex}$ and $\mathsf{ex}/\mathsf{lex}$-completions, respectively, of the category of predicates of $P$. This result generalizes a previous one by the first author with F. Pasquali and G. Rosolini about doctrines equipped with Hilbert's $ε$-operators. Thanks to this characterization, each arithmetic universe in the sense of Joyal can be seen as the exact completion of the pure existential completion of the doctrine of predicates of its Skolem theory. In particular, the initial arithmetic universe in the standard category of ZFC-sets turns out to be the completion with exact quotients of the doctrine of recursively enumerable predicates.

math.CT↗

arXiv2024

A minimalist two-level foundation for constructive mathematics

We present a two-level theory to formalize constructive mathematics as advocated in a previous paper with G. Sambin. One level is given by an intensional type theory, called Minimal type theory. This theory extends the set-theoretic version introduced in the mentioned paper with collections. The other level is given by an extensional set theory which is interpreted in the first one by means of a quotient model. This two-level theory has two main features: it is minimal among the most relevant foundations for constructive mathematics; it is constructive thanks to the way the extensional level is linked to the intensional one which fulfills the "proofs-as-programs" paradigm and acts as a programming language.

math.LO↗

arXiv2024

The Compatibility of the Minimalist Foundation with Homotopy Type Theory

The Minimalist Foundation, for short MF, is a two-level foundation for constructive mathematics ideated by Maietti and Sambin in 2005 and then fully formalized by Maietti in 2009. MF serves as a common core among the most relevant foundations for mathematics in the literature by choosing for each of them the appropriate level of MF to be translated in a compatible way, namely by preserving the meaning of logical and set-theoretical constructors. The two-level structure consists of an intensional level, an extensional one, and an interpretation of the latter in the former in order to extract intensional computational contents from mathematical proofs involving extensional constructions used in everyday mathematical practice. In 2013 a completely new foundation for constructive mathematics appeared in the literature, called Homotopy Type Theory, for short HoTT, which is an example of Voevodsky's Univalent Foundations with a computational nature. So far no level of MF has been proved to be compatible with any of the Univalent Foundations in the literature. Here we show that both levels of MF are compatible with HoTT. This result is made possible thanks to the peculiarities of HoTT which combines intensional features of type theory with extensional ones by assuming Voevodsky's Univalence Axiom and higher inductive quotient types. As a relevant consequence, MF inherits entirely new computable models.

math.LO↗

arXiv2023

A topological counterpart of well-founded trees in dependent type theory

Within dependent type theory, we provide a topological counterpart of well-founded trees (for short, W-types) by using a proof-relevant version of the notion of inductively generated suplattices introduced in the context of formal topology under the name of inductively generated basic covers. In more detail, we show, firstly, that in Homotopy Type Theory, W-types and proof relevant inductively generated basic covers are propositionally mutually encodable. Secondly, we prove they are definitionally mutually encodable in the Agda implementation of intensional Martin-Loef's type theory. Finally, we reframe the equivalence in the Minimalist Foundation framework by introducing well-founded predicates as the logical counterpart for predicates of dependent W-types. All the results have been checked in the Agda proof-assistant.

cs.LO↗

arXiv2022

Inductive and Coinductive Topological Generation with Church's thesis and the Axiom of Choice

In this work we consider an extension MFcind of the Minimalist Foundation MF for predicative constructive mathematics with the addition of inductive and coinductive definitions sufficient to generate Sambin's Positive topologies, namely Martin-Löf-Sambin formal topologies equipped with a Positivity relation (used to describe pointfree formal closed subsets). In particular the intensional level of MFcind, called mTTcind, is defined by extending with coinductive definitions another theory mTTind extending the intensional level mTT of MF with the sole addition of inductive definitions. In previous work we have shown that mTTind is consistent with Formal Church's Thesis CT and the Axiom of Choice AC via an interpretation in Aczel's CZF+REA. Our aim is to show the expectation that the addition of coinductive definitions to mTTind does not increase its consistency strength by reducing the consistency of mTTcind+CT+AC to the consistency of CZF+REA through various interpretations. We actually reach our goal in two ways. One way consists in first interpreting mTTcind+CT+AC in the theory extending CZF with the Union Regular Extension Axiom, REA_U, a strengthening of REA, and the Axiom of Relativized Dependent Choice, RDC. The theory CZF+REA_U+RDC is then interpreted in MLS*, a version of Martin-Löf's type theory with Palmgren's superuniverse S. A last step consists in interpreting MLS* back into CZF+REA. The alternative way consists in first interpreting mTTcind+AC+CT directly in a version of Martin-Löf's type theory with Palmgren's superuniverse extended with CT, which is then interpreted back to CZF+REA. A key benefit of the first way is that the theory CZF+REA_U+RDC also supports the intended set-theoretic interpretation of the extensional level of MFcind. Finally, all the theories considered, except mTTcind+AC+CT, are shown to be of the same proof-theoretic strength.

math.LO↗

arXiv2021

Generalized existential completions and their regular and exact completions

This paper aims to apply the tool of generalized existential completions of conjunctive doctrines, concerning a class $Λ$ of morphisms of their base category, to deepen the study of regular and exact completions of existential elementary Lawvere's doctrines. After providing a characterization of generalized existential completions, we observe that both the subobjects doctrine $\mathrm{Sub}_{C}$ and the weak subobjects doctrine $Ψ_{\mathcal{C}}$ of a category $\mathcal{C}$ with finite limits are generalized existential completions of the constant true doctrine, the first along the class of all the monomorphisms of $\mathcal{C}$ while the latter along all the morphisms of $\mathcal{C}$. We then name full existential completion a generalized completion of a conjunctive doctrine along the class of all the morphisms of its base. From this we immediately deduce that both the regular and the exact completion of a finite limit category are regular and exact completions of full existential doctrines since it is known that both the regular completion $(\mathcal{D})_{ reg / lex}$ and the exact completion $(\mathcal{D})_{ex / lex}$ of a finite limit category $\mathcal{D}$ are respectively the regular completion $\mathrm{Reg}(Ψ_{\mathcal{D}})$ and the exact completion $\mathcal{T}_{Ψ_{\mathcal{D}}}$ (as an instance of the tripos-to-topos construction) of the weak subobjects doctrine $Ψ_{\mathcal{D}}$ of $\mathcal{D}$. Here we prove that the condition of being a generic full existential completion is also sufficient to produce a regular/exact completion equivalent to a regular/exact completion of a finite limit category. Then, we show more specialized characterizations from which we derive known results as well as remarkable examples of exact completions of full existential completions, including all realizability toposes and supercoherent localic toposes.

math.CT↗

arXiv2021

A realizability semantics for inductive formal topologies, Church's Thesis and Axiom of Choice

We present a Kleene realizability semantics for the intensional level of the Minimalist Foundation, for short mtt, extended with inductively generated formal topologies, Church's thesis and axiom of choice. This semantics is an extension of the one used to show consistency of the intensional level of the Minimalist Foundation with the axiom of choice and formal Church's thesis in previous work. A main novelty here is that such a semantics is formalized in a constructive theory represented by Aczel's constructive set theory CZF extended with the regular extension axiom.

math.LO↗

arXiv2019

Elementary Quotient Completions, Church's Thesis, and Partioned Assemblies

Hyland's effective topos offers an important realizability model for constructive mathematics in the form of a category whose internal logic validates Church's Thesis. It also contains a boolean full sub-quasitopos of "assemblies" where only a restricted form of Church's Thesis survives. In the present paper we compare the effective topos and the quasitopos of assemblies each as the elementary quotient completions of a Lawvere doctrine based on the partitioned assemblies. In that way we can explain why the two forms of Church's Thesis each category satisfies differ by the way each is inherited from specific properties of the doctrine which determines the elementary quotient completion.

math.LO↗

arXiv2018

A predicative variant of Hyland's Effective Topos

Here, we present a subcategory pEff of Hyland's Effective Topos Eff which can be considered a predicative variant of Eff itself. The construction of pEff is motivated by the desire of providing a "predicative" categorical universe of realizers to model the Minimalist Foundation for constructive mathematics which was ideated by the first author with G. Sambin in 2005 and completed into a two-level formal system by the first author in 2009. pEff is a "predicative" categorical universe because its objects and morphisms can be formalized in Feferman's predicative weak theory of inductive definitions ID1^. Moreover, it is a predicative variant of the Effective Topos for the following reasons. First, pEff is a list-arithmetic locally cartesian closed pretopos of definable objects in dID1 with a fibred category of small objects over pEff and a (non-small) classifier of small subobjects. Second, it happens to coincide with the exact completion on the lex category defined as a predicative rendering in ID1^ of the subcategory of Eff of recursive definitions. As a consequence it validates the Formal Church's thesis and it embeds in Eff by preserving the list-arithmetic locally cartesian closed pretopos structure.

math.LO↗

arXiv2015

An extensional Kleene realizability semantics for the Minimalist Foundation

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 that both levels of MF are consistent with the (Extended) formal Church Thesis CT. MF consists of two levels, an intensional one, called mTT and an extensional one, called emTT, based on versions of Martin-Löf's type theory. Thanks to the link between the two levels, it is enough to build a semantics for the intensional level to get one also for the extensional level. Hence here we just build a realizability semantics for the intensional level mTT. Such a semantics is a modification of the realizability semantics in Beeson 1985 for extensional first order Martin-Löf's type theory with one universe. So it is formalised in Feferman's classical arithmetic theory of inductive definitions. It is called extensional Kleene realizability semantics since it validates extensional equality of type-theoretic functions extFun, as in Beeson 1985. The main modification we perform on Beeson's semantics is to interpret propositions, which are defined primitively in MF, in a proof-irrelevant way. As a consequence, we gain the validity of CT. Recalling that extFun+ CT+ AC are inconsistent over arithmetics with finite types, we conclude that our semantics does not validate the full Axiom of Choice AC. On the contrary, Beeson's semantics does validate AC, being this a theorem of Martin-Löf's theory, but it does not validate CT. The semantics we present here appears to be the best Kleene realizability semantics for the extensional level emTT of MF. Indeed Beeson's semantics is not an option for emTT since the full AC added to it entails the excluded middle.

math.LO↗

arXiv2013

Quotient completion for the foundation of constructive mathematics

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 hyperdoctrine for which we describe a notion of quotient completion. That notion includes the exact completion on a category with weak finite limits as an instance as well as examples from type theory that fall apart from this.

math.LO↗

arXiv2012

Unifying exact completions

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 of a regular category as well as that of the exact completion of a cartesian category with weak pullbacks.

math.CT↗

arXiv2012

Convergence in Formal Topology: a unifying notion

Several variations on the definition of a Formal Topology exist in the literature. They differ on how they express convergence, the formal property corresponding to the fact that open subsets are closed under finite intersections. We introduce a general notion of convergence of which any previous definition is a special case. This leads to a predicative presentation and inductive generation of locales (formal covers), commutative quantales (convergent covers) and suplattices (basic covers) in a uniform way. Thanks to our abstract treatment of convergence, we are able to specify categorically the precise sense according to which our inductively generated structures are free, thus refining Johnstone's coverage theorem. We also obtain a natural and predicative version of a fundamental result by Joyal and Tierney: convergent covers (commutative quantales) correspond to commutative co-semigroups over the category of basic covers (suplattices).

math.LO↗