arXiv subjects

Igor Arrieta

Publications and source records attributed to Igor Arrieta.

8 recordsLinked to original sources

arXiv2026

A topos for \'etale-finite Heyting algebras

A longstanding open problem is whether every Heyting algebra is the lattice of truth values (i.e., of subterminal objects) of some elementary topos. A positive answer is known for complete Heyting algebras (i.e., locales) via sheaves, and for Boolean algebras via a construction due to Peter Freyd. We extend Freyd's construction to all \'etale-finite Heyting algebras, in the sense of Evgeny Kuznetsov. These are the Heyting algebras satisfying a generalisation of the law of excluded middle relative to some finite Heyting subalgebra. For every \'etale-finite Heyting algebra $H$, we use Esakia duality to construct an elementary topos whose lattice of truth values is isomorphic to $H$, thereby extending the class of Heyting algebras for which a positive answer to the Heyting-to-topos problem is known. The toposes we construct are categories of certain compact \'etale spaces. As a consequence, they are finitely propositional: every object has a finite cover by subterminal objects. We show that a Heyting algebra occurs as the lattice of truth values of some finitely propositional topos if and only if it is \'etale-finite. This exhibits an obstruction to extending the use of compact \'etale spaces beyond the \'etale-finite case.

math.LO↗

arXiv2026

The lattice of smooth sublocales as a Bruns-Lakser completion

We characterise the frame morphisms $f:L\to M$ that lift to frame maps $\overline{f}:\mathsf{S}_b(L)\to \mathsf{S}_b(M)$, where $\mathsf{S}_b(L)$ is the collection of joins of complemented sublocales of a frame $L$, or equivalently the Booleanization of the collection $\mathsf{S}(L)$ of all its sublocales. We do so by proving that $\mathsf{S}_b(L)$ is isomorphic to the Bruns--Lakser completion of the meet-semilattice formed by the locally closed sublocales, i.e. the sublocales of the form $\mathfrak{c}(a)\cap \mathfrak{o}(b)$ for $a,b\in L$.

math.GN↗

arXiv2025

Joins of closed sublocales are not always a coframe

Given a locale $L$, the collection $\mathsf{S}_c(L)$ of joins of closed sublocales forms a frame--somewhat unexpectedly, as it is naturally embedded in the coframe of all sublocales of $L$, where by coframe we mean the order-theoretic dual of a frame. This construction has attracted attention in point-free topology: as a maximal essential extension in the category of frames, for its (non-)functorial properties, its relation to canonical extensions and exact filters of frames, etc. A central open question of the theory, posed by Picado, Pultr, and Tozzi in 2019, asked whether $\mathsf{S}_c(L)$ is always a coframe, or whether there exists a locale for which this fails. In this paper, we resolve this question in the negative by constructing a locale $L$ such that $\mathsf{S}_c(L)$ is not a coframe. The main challenge in such questions lies in the difficulty of understanding exact infima in $\mathsf{S}_c(L)$; we circumvent this by analysing a certain separation property satisfied by $\mathsf{S}_c(L)$.

math.GN↗

arXiv2025

The DeMorganization of a locale

In 2009, Caramello proved that each topos has a largest dense subtopos whose internal logic satisfies De Morgan law (also known as the law of the weak excluded middle). This finding implies that every locale has a largest dense extremally disconnected sublocale, referred to as its DeMorganization. In this paper, we take the first steps in exploring the DeMorganization in the localic context, shedding light on its geometric nature by showing that it is always a fitted sublocale and by providing a concrete description. Explicit examples of DeMorganizations for toposes that do not satisfy De Morgan law are rather difficult to find. We present a contribution in that direction, with the main result of the paper showing that for any metrizable locale (without isolated points), its DeMorganization coincides with its Booleanization. This, in particular, implies that any extremally disconnected metric locale (without isolated points) must be Boolean, generalizing a well-known result for topological spaces to the localic setting.

math.GN↗

arXiv2024

The Patch Topology in Univalent Foundations

Stone locales together with continuous maps form a coreflective subcategory of spectral locales and perfect maps. A proof in the internal language of an elementary topos was previously given by the second-named author. This proof can be easily translated to univalent type theory using resizing axioms. In this work, we show how to achieve such a translation without resizing axioms, by working with large and locally small frames with small bases. This requires predicative reformulations of several fundamental concepts of locale theory in predicative HoTT/UF, which we investigate systematically.

cs.LO↗

arXiv2023

Localic separation and the duality between closedness and fittedness

There are a number of localic separation axioms which are roughly analogous to the $T_1$-axiom from classical topology. For instance, besides the well-known subfitness and fitness, there are also Rosicky-Smarda's $T_1$-locales, totally unordered locales and, more categorically, the recently introduced $\mathcal{F}$-separated locales (i.e., those with a fitted diagonal) - a property strictly weaker than fitness. It has recently been shown that the strong Hausdorff property and $\mathcal{F}$-separatedness are in a certain sense dual to each other. In this paper, we provide further instances of this duality - e.g., we introduce a new first-order separation property which is to $\mathcal{F}$-separatedness as the Johnstone-Sun-shu-Hao-Paseka-Smarda conservative Hausdorff axiom is to the strong Hausdorff property, and which can be of independent interest. Using this, we tie up the loose ends of the theory by establishing all the possible implications between these properties and other $T_1$-type axioms occurring in the literature. In particular, we show that the strong Hausdorff property does not imply $\mathcal{F}$-separatedness, a question which remained open and shows a remarkable difference with its counterpart in the category of topological spaces.

math.GN↗

arXiv2020

The coframe of D-sublocales of a locale and the $T_D$ duality

The notion of \emph{D-sublocale} is explored. This is the notion analogue to that of sublocale in the duality of $T_D$spaces. A sublocale $S$ of a frame $L$ is a D-sublocale if and only if the corresponding localic map preserves the property of being a covered prime. It is shown that for a frame $L$ the system of those sublocales which are also D-sublocales form a dense sublocale $\mathsf{S}_D(L)$ of the coframe $\mathsf{S}(L)$ of all its sublocales. It is also shown that the spatialization $\mathsf{sp}_D[\mathsf{S}_D(L)]$ of $\mathsf{S}_D(L)$ consists precisely of those D-sublocales of $L$ which are $T_D$-spatial. Additionally, frames such that we have $\mathsf{S}_D(L)\cong \mathcal{P}(\mathsf{pt}_D(L))$ -- that is, those such that D-sublocales perfectly represent subspaces -- are characterized as those $T_D$-spatial frames such that $\mathsf{S}_D(L)$ is the Booleanization of \mathsf{S}(L).

math.CT↗

arXiv2020

On infinite variants of De Morgan law in locale theory

A locale, being a complete Heyting algebra, satisfies De Morgan law $(a\vee b)^*=a^*\wedge b^*$ for pseudocomplements. The dual De Morgan law $(a\wedge b)^*={a^* \vee b^*}$ (here referred to as the second De Morgan law) is equivalent to, among other conditions, $(a\vee b)^{**} =a^{**}\vee b^{**}$, and characterizes the class of extremally disconnected locales. This paper presents a study of the subclasses of extremally disconnected locales determined by the infinite versions of the second De Morgan law and its equivalents.

math.GN↗