arXiv subjects

Iosif Petrakis

Publications and source records attributed to Iosif Petrakis.

At least 19 recordsLinked to original sources

arXiv2026

Categories with a Base of Computability

The notion of a base of computability $\mathscr{C}$ in a category $\mathscr{C}$ was introduced as a tool to generate computability models, in the sense of Longley and Normann, from categories. In this paper we introduce the category $\mathsf{CatBaseComp}$ of categories with a base of computability, and we show that $\mathsf{CatBaseComp}$ has all pie limits. We prove that a Grothendieck fibration lifts a base of computability in the base category to a base of computability in the total category of the fibration, and conversely, a pullback-preserving Grothendieck fibration maps a base of computability in the total category to a base of computability in the base category of the fibration. Connecting $\mathsf{CatBaseComp}$ with the semantics of dependent type theory, we show that $\mathsf{CatBaseComp}$ is a type-category, or a (fam, $Σ$)-category with a terminal object. Moreover, we prove that CatBaseComp is a (2-fam, $Σ$)-category, a 2-categorical generalisation of a (fam, $Σ$)-category. Finally, we describe the canonical (2-dep, $Σ$)-structure of CatBaseComp, i.e., the canonical dependent arrows of CatBaseComp that are compatible with its (2-fam, $Σ$)-structure.

math.CT↗

arXiv2026

Constructive Stone representations for separated swap and Boolean algebras

Swap algebras generalise Bishop's complemented powerset as Boolean algebras generalise the powerset. Actually, all Boolean algebras are swap algebras. We prove constructively a Stone representation theorem for separated swap algebras of type (II), where the notion of a separated swap algebra generalises the corresponding notion of a separated Boolean algebra. Moreover, we prove a Stone-Cech theorem for swap algebras of type (II), showing that the restriction to separated swap algebras is not a loss of generality from the point of view of the theory of swap characters. A constructive Stone representation theorem and a Stone-Cech theorem for Boolean algebras follow as special cases. We introduce sets with a Boolean inequality, that is sets with an internal falsum. These sets allow a book-keeping of the use of the Ex falso principle in constructive mathematics. If we restrict to swap algebras with a Boolean inequality, then the proof of the Stone representation theorem for swap algebras of type (II) is within minimal logic.

math.LO↗

arXiv2025

Orthocomplemented subspaces and partial projections on a Hilbert space

We introduce the notion of an orthocomplemented subspace of a Hilbert space H, that is, a pair of orthogonal closed subspaces of H, as a two-dimensional counterpart to the one-dimensional notion of a closed subspace of H. Orthocomplemented subspaces are the Hilbert space-analogue to Bishop's complemented subsets. To complemented subsets correspond their characteristic functions, which are partial, Boolean-valued functions. Similarly, to orthocomplemented subspaces of H correspond partial projections on H. Previous work of Bridges and Svozil on constructive quantum logic is an one-dimensional approach to the subject. The lattice-properties of the orthocomplemented subspaces of a Hilbert space is a two-dimensional approach to constructive quantum logic, that we call complemented quantum logic. Since the negation of an orthocomplemented subspace is formed by swapping its components, complemented quantum logic, although constructive, is closer to classical quantum logic than the constructive quantum logic of Bridges and Svozil. The introduction of orthocomplemented subspaces and their corresponding partial projections allows a new approach to the constructive theory of Hilbert spaces. For example, the partial projection operator of an orthocomplemented subspace and the construction of the quotient Hilbert space bypass the standard restrictive hypothesis of locatedness on a subspace. Located subspaces correspond to total orthocomplemented subspaces.

quant-ph↗

arXiv2025

Coinductive well-foundedness

We introduce a coinductive version of the well-foundedness of N that is used in our proof within minimal logic of the constructive counterpart CLNP to the standard least number principle LNP. According to CLNP, an inhabited complemented subset of N has a least element if and only if it is downset located. The use of complemented subsets of N in the formulation of CLNP, instead of subsets of N, allows a positive approach to the subject that avoids negation. Generalising the coinductive well-foundedness of N, we define $\exists$-well-founded sets and we prove their fundamental properties.

math.LO↗

arXiv2025

Strong negation in the theory of computable functionals TCF

We incorporate strong negation in the theory of computable functionals TCF, a common extension of Plotkin's PCF and Gödel's system $\mathbf{T}$, by defining simultaneously strong negation $A^{\mathbf{N}}$ of a formula $A$ and strong negation $P^{\mathbf{N}}$ of a predicate $P$ in TCF. As a special case of the latter, we get strong negation of an inductive and a coinductive predicate of TCF. We prove appropriate versions of the Ex falso quodlibet and of double negation elimination for strong negation in TCF. We introduce the so-called tight formulas of TCF i.e., formulas implied by the weak negation of their strong negation, and the relative tight formulas. We present various case-studies and examples, which reveal the naturality of our definition of strong negation in TCF and justify the use of TCF as a formal system for a large part of Bishop-style constructive mathematics.

math.LO↗

arXiv2025

Topologies of open complemented subsets

We introduce cs-topologies, or topologies of open complemented subsets, as a new approach to constructive topology that preserves the duality between open and closed subsets of classical topology. Complemented subsets were used successfully by Bishop in his constructive formulation of the Daniell approach to measure and integration. Here we use complemented subsets in topology, in order to describe simultaneously an open set, the first-component of an open complemented subset, together with its given complement as a closed set, the second component of an open complemented subset. We analyse the canonical cs-topology induced by a metric, and we introduce the notion of a modulus of openness for a cs-open subset of a metric space. Pointwise and uniform continuity of functions between metric spaces are formulated with respect to the way these functions inverse open complemented subsets together with their moduli of openness. The addition of moduli of openness in the concept of a complemented open subset, given a base for the cs-topology, makes possible to define the notions of pointwise-like and uniform-like continuity of functions between csb-spaces, that is cs-spaces with a given base.

math.GN↗

arXiv2024

Pre-measure spaces and pre-integration spaces in predicative Bishop-Cheng measure theory

Bishop's measure theory (BMT) is an abstraction of the measure theory of a locally compact metric space $X$, and the use of an informal notion of a set-indexed family of complemented subsets is crucial to its predicative character. The more general Bishop-Cheng measure theory (BCMT) is a constructive version of the classical Daniell approach to measure and integration, and highly impredicative, as many of its fundamental notions, such as the integration space of $p$-integrable functions $L^p$, rely on quantification over proper classes (from the constructive point of view). In this paper we introduce the notions of a pre-measure and pre-integration space, a predicative variation of the Bishop-Cheng notion of a measure space and of an integration space, respectively. Working within Bishop Set Theory (BST), and using the theory of set-indexed families of complemented subsets and set-indexed families of real-valued partial functions within BST, we apply the implicit, predicative spirit of BMT to BCMT. As a first example, we present the pre-measure space of complemented detachable subsets of a set $X$ with the Dirac-measure, concentrated at a single point. Furthermore, we translate in our predicative framework the non-trivial, Bishop-Cheng construction of an integration space from a given measure space, showing that a pre-measure space induces the pre-integration space of simple functions associated to it. Finally, a predicative construction of the canonically integrable functions $L^1$, as the completion of an integration space, is included.

math.LO↗

arXiv2024

The Grothendieck computability model

Translating notions and results from category theory to the theory of computability models of Longley and Normann, we introduce the Grothendieck computability model and the first-projection-simulation. We prove some basic properties of the Grothendieck computability model, and we show that the category of computability models is a type-category, in the sense of Pitts. We introduce the notion of a fibration and opfibration-simulation, and we show that the first-projection-simulation is a split opfibration-simulation.

math.CT↗

arXiv2023

Categories with dependent arrows

We present an abstract, categorical formulation of dependent functions in a fundamental manner and independently from the Sigma-construction. For that, we define first the notion of a category with family-arrows, or a $\f$-category. A $(\f, Σ)$-category is a $\f$-category with Sigma-objects, where a $(\f, Σ)$-category with a terminal object is exactly a type-category of Pitts, or a category with attributes of Cartmell. We introduce categories with dependent arrows, or $\di$-categories, and we show that every $(\f, Σ)$-category is a $\di$-category in a canonical way. The notion of a Sigma-object in a $\di$-Category is affected by the existence of dependent arrows, and we show that every $(\f, Σ)$-category is a $(\di, Σ)$-category in a canonical way.

math.CT↗

arXiv2022

The Role of the Fifth Postulate in the Euclidean Construction of Parallels

We ascribe to the Euclidean Fifth Postulate a genuine constructive role, which makes it absolutely necessary in the parallel construction. For that, we present a reconstruction of the general principles underlying the Euclidean construction of a geometric property. As a consequence, the epistemological role of Euclidean constructions is revealed. We also examine some first implications of our interpretation to the relation between Euclidean and non-Euclidean geometries. The Bolyai construction of limiting parallels is also discussed from the Euclidean point of view, as this is reconstructed here.

math.HO↗

arXiv2022

Sets completely separated by functions in Bishop Set Theory

Within Bishop Set Theory, a reconstruction of Bishop's theory of sets, we study the so-called completely separated sets, that is sets equipped with a positive notion of an inequality, induced by a given set of real-valued functions. We introduce the notion of a global family of completely separated sets over an index-completely separated set, and we describe its Sigma- and Pi-set. The free completely separated set on a given set is also presented. Purely set-theoretic versions of the classical Stone-Čech theorem and the Tychonoff embedding theorem for completely regular spaces are given, replacing topological spaces with function spaces and completely regular spaces with completely separated sets.

math.LO↗

arXiv2022

Constructive Combinatorics of Dickson's Lemma

We study constructively the relations between the finite cases of Dickson's lemma. Although there are many constructive proofs of them, the novel aspect of our proofs is the extraction of a corresponding bound. We provide some new one-step unprovability results i.e., results of the form "a finite case of Dickson's lemma does not prove in one step a stronger case of it". Moreover, we study the infinite cases of Dickson's lemma from the point of view of constructive reverse mathematics. We work within Bishop's informal system of constructive mathematics BISH.

math.CO↗

arXiv2022

Univalent typoids

A typoid is a type equipped with an equivalence relation, such that the terms of equivalence between the terms of the type satisfy certain conditions, with respect to a given equivalence relation between them, that generalise the properties of the equality terms. The resulting weak 2-groupoid structure can be extended to every finite level. The introduced notions of typoid and typoid function generalise the notions of setoid and setoid function. A univalent typoid is a typoid satisfying a general version of the univalence axiom. We prove some fundamental facts on univalent typoids, their product and exponential. As a corollary, we get an interpretation of propositional truncation within the theory of typoids. The couple typoid and univalent typoid is a weak groupoid-analogue to the couple precategory and category in homotopy type theory.

math.CT↗

arXiv2021

Families of Sets in Bishop Set Theory

We develop the theory of set-indexed families of sets and subsets within the informal Bishop Set Theory BST, a reconstruction of Bishop's theory of sets.

math.LO↗

arXiv2021

From the Sigma-type to the Grothendieck construction

We translate properties of the Sigma-type in Martin-Löf Type Theory (MLTT) to properties of the Grothendieck construction in category theory. Namely, equivalences in MLTT that involve the Sigma-type motivate isomorphisms between corresponding categories that involve the Grothendieck construction. The type-theoretic axiom of choice and the "associativity" of the Sigma-type are the main examples of this phenomenon that are treated here.

math.CT↗

arXiv2021

Chu representations of categories related to constructive mathematics

If C is a closed symmetric monoidal category, the Chu category Chu(C, g) over C and an object g of it was defined by Chu, as a *-autonomous category generated from C. Bishop introduced the category of complemented subsets of a set, in order to overcome the problems generated by the use of negation in constructive measure theory. Shulman mentions that Bishop's complemented subsets correspond roughly to the Chu construction. In this paper we explain this correspondence by showing that there is a Chu representation (a full embedding) of the category of complemented subsets of a set X into Chu(Set, X x X). A Chu representation of the category of Bishop spaces into Chu(Set, R) is shown, as the constructive analogue to the standard Chu representation of the category of topological spaces into Chu(Set, 2). In order to represent the category of predicates (with objects pairs (X, A), where A is a subset of X, and the category of complemented predicates (with objects pairs (X, A), where A is a complemented subset of X, we generalise the Chu construction by defining the Chu category over a cartesian closed category C and an endofunctor on C. Finally, we introduce the antiparallel Grothendieck construction over a product category and a contra-variant Set-valued functor on it of which the Chu construction is a special case, in case C is a locally small, cartesian closed category.

math.CT↗

arXiv2021

Computability models over categories

Generalising slightly the notions of a strict computability model and of a simulation between them, which were elaborated by Longley and Normann, we define canonical computability models over categories and appropriate Set-valued functors on them. We study the canonical total computability model over a category, and the partial one over a category with pullbacks. Our notions and results are generalised to categories with a base of computability, connecting Rosolini's theory of dominions with the theory of computability models.

math.CT↗

arXiv2021

Direct spectra of Bishop spaces and their limits

We apply fundamental notions of Bishop set theory (BST), an informal theory that complements Bishop's theory of sets, to the theory of Bishop spaces, a function-theoretic approach to constructive topology. Within BST we develop the notions of a direct family of sets, of a direct spectrum of Bishop spaces, of the direct limit of a direct spectrum of Bishop spaces, and of the inverse limit of a contravariant direct spectrum of Bishop spaces. Within the extension of Bishop's informal system of constructive mathematics BISH with inductive definitions with rules of countably many premises, we prove the fundamental theorems on the direct and inverse limits of spectra of Bishop spaces and the duality principle between them.

math.LO↗