Searcharxiv⌕ Search

arXiv subjects

Samuele Maschio

Publications and source records attributed to Samuele Maschio.

17 recordsLinked to original sources
arXiv2025

A topos for extended Weihrauch degrees

Weihrauch reducibility is a notion of reducibility between computational problems that is useful to calibrate the uniform computational strength of a multivalued function. It complements the analysis of mathematical theorems done in reverse mathematics, as multi-valued functions on represented spaces can be considered as realizers of theorems in a natural way. Despite the rich literature and the relevance of the applications of category theory in logic and realizability, actually there are just a few works starting to study the Weihrauch reducibility from a categorical point of view. The main purpose of this work is to provide a full categorical account to the notion of extended Weihrauch reducibility introduced by A. Bauer, which generalizes the original notion of Weihrauch reducibility. In particular, we present a tripos and a topos for extended Weihrauch degrees. We start by defining a new tripos, abstracting the notion of extended Weihrauch degrees, and then we apply the tripos-to-topos construction to obtain the desired topos. Then we show that the Kleene-Vesley topos is a topos of $j$-sheaves for a certain Lawvere-Tierney topology over the topos of extended Weihrauch degrees.

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↗
arXiv2023

Implicative models of set theory

In this paper we show that using implicative algebras one can produce models of set theory generalizing Heyting/Boolean-valued models and realizability models of (I)ZF, both in intuitionistic and classical logic. This has as consequence that any topos which is obtained from a Set-based tripos as the result of the tripos-to-topos construction hosts a model of intuitionistic or classical set theory, provided a large enough strongly inaccessible cardinal exists.

cs.LO↗
arXiv2023

On categorical structures arising from implicative algebras: from topology to assemblies

Implicative algebras have been recently introduced by Miquel in order to provide a unifying notion of model, encompassing the most relevant and used ones, such as realizability (both classical and intuitionistic), and forcing. In this work, we initially approach implicative algebras as a generalization of locales, and we extend several topological-like concepts to the realm of implicative algebras, accompanied by various concrete examples. Then, we shift our focus to viewing implicative algebras as a generalization of partial combinatory algebras. We abstract the notion of a category of assemblies, partition assemblies, and modest sets to arbitrary implicative algebras, and thoroughly investigate their categorical properties and interrelationships.

math.CT↗
arXiv2023

Implicative models of intuitionistic set theory

In this paper we will show that using implicative algebras one can produce models of intuitionistic set theory generalizing both realizability and Heyting-valued models. This has as consequence that if one assumes the inaccessible cardinal axiom, then every topos which is obtained from a Set-based tripos as the result of the tripos-to-topos construction hosts a model of intuitionistic set theory.

math.LO↗
arXiv2022

On posetal and complete partial applicative structures

Every partial applicative structure gives rise to an indexed binary relation, that is a contravariant functor from the category of sets to the category of sets endowed with binary relations and maps preserving them. In this paper we characterize those partial applicative structures giving rise to indexed relations satisfying certain elementary properties in terms of algebraic or computational properties. We will then provide a characterization of those partial applicative structures giving rise to indexed preorders and indexed posets, and we will relate the latter ones to some particular classes of unary partial endofunctions. We will analyze the relation between a series of computational and algebraic properties in the posetal case. Finally, we will study the problem of existence of suprema in the case of partial applicative structures giving rise to indexed preorders, by providing some necessary conditions for a partial applicative structure to be complete.

math.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

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

Remarks on abstract structures of propositions and realizers

We present here an abstract notion of structure consisting of propositions and realizers (which we call PR-structures) giving rise to set based contravariant functors taking values in the category of sets endowed with binary relations. We will characterize those PR-structures giving rise to preorderal and posetal doctrines and we will study in particular the case of a PR-structure induced by a partial applicative structure.

math.LO↗
arXiv2019

Natural density and probability, constructively

We give here a constructive account of the frequentist approach to probability, by means of natural density. Using this notion of natural density, we introduce some probabilistic versions of the Limited Principle of Omniscience. Finally we give an attempt general definition of probability structure which is pointfree and takes into account abstractely the process of probability assignment.

math.LO↗
arXiv2018

Factorizing the Top-Loc adjunction through positive topologies

We characterize the category of Sambin's positive topologies as a fibration over the category of locales Loc. The fibration is obtained by applying the Grothendieck construction to a doctrine over Loc. We then construct an adjunction between the category of positive topologies and that of topological spaces Top, and show that the well-known adjunction between Top and Loc factors through the newly constructed adjunction.

math.GN↗
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↗
arXiv2016

Topology as faithful communication through relations

Basic pairs and their morphisms are the most elementary framework in which standard topological notions can be defined. We present here a new interpretation of topological concepts as those which can be communicated faithfully between the two sides of basic pairs. In particular, we prove that the subsets which can be communicated faithfully (in the suitable way) are exactly open subsets and closed subsets. We also prove that a relation (and in particular a function) between two sets of points can be communicated faithfully if and only if it is continuous.

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↗
arXiv2014

Models of Intuitionistic Set Theory in Subtoposes of Nested Realizability Toposes

With every pca $\mathcal{A}$ and subpca $\mathcal{A}_\#$ we associate the nested realizability topos $\mathsf{RT}(\mathcal{A},\mathcal{A}_\#)$ within which we identify a class of small maps $\mathcal{S}$ giving rise to a model of intuitionistic set theory within $\mathsf{RT}(\mathcal{A},\mathcal{A}_\#)$. For every subtopos $\mathcal{E}$ of such a nested realizability topos we construct an induced class $\mathcal{S_E}$ of small maps in $\mathcal{E}$ giving rise to a model of intuitionistic set theory within $\mathcal{E}$. This covers relative realizability toposes, modified relative realizability toposes, the modified realizability topos and van den Berg's recent Herbrand topos.

math.CT↗
arXiv2012

What is the real category of sets?

Category theory provides a powerful tool to organize mathematics. A sample of this descriptive power is given by the categorical analysis of the practice of "classes as shorthands" in ZF set theory. In this case category theory provides a natural way to describe the relation between mathematics and metamathematics: if metamathematics can be described by using categories (in particular syntactic categories), then the mathematical level is represented by internal categories. Through this two-level interpretation we can clarify the relation between classes and sets in ZF and, in particular, we can present two equivalent categorical notions of definable set. Some common sayings about set theory will be interpreted in the light of this representation, emphasizing the distinction between naive and rigorous sentences about sets and classes.

math.LO↗