Ultracontact algebras and stack systems
We study the class of structures that, in a way, generalize various approaches to the contact relation on Boolean algebras.
math.LO↗
arXiv subjects
Anna Laura Suarez
Publications and source records attributed to Anna Laura Suarez.
A general framework for the faithful pointfree representation of $T_0$-spaces
We introduce a general framework for studying natural contravariant adjunctions that refine the adjunction between frames and spaces so that the fixpoints are $T_0$-spaces. Our objects of study are \textit{spatializable $\mathbf{Frm}$-concrete categories}, or \textit{SFC-categories}. These consist of a faithful functor $\mathcal O:\mathcal C\to \mathbf{Frm}$ equipped with an object $2_{\mathcal C} \in \mathcal C$, satisfying compatibility conditions that ensure that $(2_{\mathcal C},\mathbb{S})$ forms a dualizing object in the sense of Porst and Tholen, where $\mathbb{S}$ denotes the Sierpiński space. Three important instances of pointfree $T_0$ spaces present in the literature fit into this framework: strictly zero-dimensional biframes, MT-algebras, and Raney extensions. We show SFC-categories are assembled in an ordered category -- a category enriched in preordered sets -- whose morphisms are suitable functors which preserve certain initial liftings. SFC-categories induce natural dual adjunctions, and morphisms between them will respectively induce suitable morphisms between these adjunctions. Motivated by the characterization of sober spaces as maximal objects in the fibers of $Ω:\mathbf{Top}\to \mathbf{Frm}^{\mathsf{op}}$, and of $T_D$-spaces as the minimal ones, due to Banaschewski and Pultr, we study initial and terminal objects of fibers for an arbitrary SFC-category. We prove that the natural adjunction for fiber-initials has exactly the sober spaces as fixpoints, while for fiber-terminals contains at most $T_D$-spaces, recovering their results of in a much more general setting.
math.CT↗
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↗
McKinsey-Tarski algebras and Raney extensions
We introduce the notion of Raney morphism between MT-algebras and show that the resulting category is equivalent to the category of Raney extensions. This is done by generalizing the construction of the Funayama envelope of a frame. The resulting notion of the $T_0$-hull of a Raney extension generalizes that of the $T_D$-hull of a frame.
math.CT↗
Strictly zero-dimensional biframes and Raney extensions
Raney extensions and strictly zero-dimensional biframes both faithfully extend the dual of the category of $T_0$ spaces. We use tools from pointfree topology to look at the connection between the two. Raney extensions may be equivalently described as pairs $(L,\mathcal{F})$ where $L$ is a frame and $\mathcal{F}\subseteq \mathcal{S}_{o}(L)$ a subcolocale containing all open sublocales. Here, $\mathcal{S}_o(L)$ is the collection of all intersections of open sublocales of $L$. Similarly, a strictly zero-dimensional biframe is a pair $(L,\mathcal{D})$ where $\mathcal{D}\subseteq \mathcal{S}(L)$ is a codense subcolocale. We show that there is an adjunction between certain subcolocales of $\mathcal{S}_o(L)$ and codense subcolocales of $\mathcal{S}(L)$. We show that the adjunction maximally restricts to an order-isomorphism between the subcolocales of $\mathcal{S}_o(L)$ where the joins of open sublocales distribute over binary meets, which we call the proper subcolocales, and what we call the essential codense subcolocales. As an application of our main result, we establish a bijection between proper Raney extensions and the strictly zero-dimensional biframes $(L_1,L_2,L)$ such that $L$ is an essential extension of $L_2$ in the category of frames. We show that this correspondence cannot be made functorial in the obvious way, as a frame morphism $f:L\to M$ may lift to a map $f:(L,\mathcal{F})\to (L,\mathcal{G})$ of Raney extensions without lifting to a map between the associated strictly zero-dimensional biframes.
math.GN↗
Raney extensions: a pointfree theory of T_0 spaces based on canonical extension
We introduce a pointfree version of Raney duality. Our objects are \emph{Raney extensions} of frames, pairs $(L,C)$ where $C$ is a coframe and $L\subseteq C$ is a subframe that meet-generates it and whose embedding preserves strongly exact meets. We show that there is a dual adjunction between $\mathbf{Raney}$ and $\mathbf{Top}$, with all $T_0$ spaces as fixpoints, assigning to a space $X$ the pair $(Ω(X),\mathcal{U}(X))$, with $\mathcal{U}(X)$ are the intersections of open sets. We show that for every Raney extension $(L,C)$ there are subcolocale inclusions $\mathcal{S}_c(L)^{op}\subseteq C\subseteq \mathcal{S}_o(L)$ where these are the opposite of the frame of joins of closed sublocales and the coframe of intersections of open sublocales. We thus exhibit a symmetry between these two well-studied structures in pointfree topology. The spectra of these are, respectively, the classical spectrum $\mathsf{pt}(L)$ of the underlying frame and its $T_D$ spectrum $\mathsf{pt}_D(L)$. This confirms the view advanced in \cite{banaschewskitd} that sobriety and the $T_D$ property are mirror images of each other, and suggests that the symmetry above is a pointfree view of it. All Raney extensions satisfy some variation of the properties \emph{density} and \emph{compactness} from the theory of canonical extensions. We characterize sobriety, the $T_1$, and the $T_D$ axioms in terms of density and compactness of $(Ω(X),\mathcal{U}(X))$. We characterize frame morphisms $f:L\to M$ that extend to Raney morphisms $\overline{f}:(L,C)\to (M,D)$. We use this result to exhibit the existence of various free and cofree constructions. We use Raney extensions to give a new perspective on canonical extension generalized to frames as well as $T_D$ duality.
math.CT↗
The Funayama envelope as the $T_D$-hull of a frame
We introduce proximity morphisms between MT-algebras and show that the resulting category is equivalent to the category of frames. This is done by utilizing the Funayama envelope of a frame, which is viewed as the $T_D$-hull. Our results have some spatial ramifications, including a generalization of the $T_D$-duality of Banaschewski and Pultr.
math.CT↗
Raney extensions of frames: topological aspects
We explore a pointfree approach to spaces which extends the category of $T_0$ spaces. Our pointfree objects are Raney extensions, pairs $(L,C)$ where $C$ is a coframe, $L\subseteq C$ is a frame which meet-generates it, and the inclusion $L\subseteq C$ preserves the frame operations as well as the strongly exact meets. We show that the category $\mathbf{Raney}$ extends that of $T_0$ spaces, by showing the existence of an adjunction which extends that between frames and spaces. We map a space $X$ to the pair $(Ω(X),\mathcal{U}(X))$, where $Ω(X)$ are its opens and $\mathcal{U}(X)$ its saturated sets. The spectrum functor $\mathsf{pt}_R$ maps a Raney extension $(L,C)$ to the collection of completely join-prime elements of $C$, suitably topologized. For a frame $L$ the spectra of the largest and the smallest Raney extensions over it are, respectively, the classical spectrum $\mathsf{pt}(L)$ and the $T_D$ spectrum $\mathsf{pt}_D(L)$. We characterize sobriety as well as the $T_D$ and the $T_1$ axioms for spaces in terms of algebraic properties of their Raney duals. We use this to define sobriety for general Raney extensions, as well as the $T_D$ and $T_1$ properties, and show that a sober coreflection always exists, whereas a $T_D$ reflection exists when we restrict morphisms to exact maps. We show that a frame is subfit if and only if it admits a $T_1$ Raney extension, and that a subfit frame is scattered if and only if it admits a unique Raney extension. We show that the dual adjunction between frames and spaces restricts to a dual adjunction between the category of $T_D$ spaces and the category of $\mathbf{Frm}_{\mathcal{E}}$ of frames and exact maps, and that exact sublocales (sublocales whose surjection is exact) form a subcolocale of the coframe of all sublocales.
math.CT↗
Canonical extensions via fitted sublocales
We build on a recent result stating that the frame $\mathsf{SE}(L)$ of strongly exact filters for a frame $L$ is anti-isomorphic to the coframe $\mathsf{S}_o(L)$ of fitted sublocales. The collection $\mathsf{E}(L)$ of exact filters of $L$ is known to be a sublocale of this frame. We consider several other subcollections of $\mathsf{SE}(L)$: the collections $\mathcal{J}(\mathsf{CP}(L))$ and $\mathcal{J}(\mathsf{SO}(L))$ of intersections of completely prime and Scott-open filters, respectively, and the collection $\mathsf{R}(L)$ of regular elements of the frame of filters. We show that all of these are sublocales of $\mathsf{SE}(L)$, and as such they correspond to subcolocales of $\mathsf{S}_o(L)$, which all turn out to have a concise description. By using the theory of polarities of Birkhoff, one can show that all of the structures mentioned above enjoy universal properties which are variations of that of the canonical extension. We also show how some of these subcollections can be described as polarities and give three new equivalent definitions of subfitness in terms of the lattice of filters.
math.CT↗
Pervin spaces and Frith frames: bitopological aspects and completion
A Pervin space is a set equipped with a bounded sublattice of its powerset, while its pointfree version, called Frith frame, consists of a frame equipped with a generating bounded sublattice. It is known that the dual adjunction between topological spaces and frames extends to a dual adjunction between Pervin spaces and Frith frames, and that the latter may be seen as representatives of certain quasi-uniform structures. As such, they have an underlying bitopological structure and inherit a natural notion of completion. In this paper we start by exploring the bitopological nature of Pervin spaces and of Frith frames, proving some categorical equivalences involving zero-dimensional structures. We then provide a conceptual proof of a duality between the categories of $T_0$ complete Pervin spaces and of complete Frith frames. This enables us to interpret several Stone-type dualities as a restriction of the dual adjunction between Pervin spaces and Frith frames along full subcategory embeddings. Finally, we provide analogues of Banaschewski and Pultr's characterizations of sober and $T_D$ topological spaces in the setting of Pervin spaces and of Frith frames, highlighting the parallelism between the two notions.
math.GN↗
A pointfree theory of Pervin spaces
We lay down the foundations for a pointfree theory of Pervin spaces. A Pervin space is a set equipped with a bounded sublattice of its powerset, and it is known that these objects characterize those quasi-uniform spaces that are transitive and totally bounded. The pointfree notion of a Pervin space, which we call Frith frame, consists of a frame equipped with a generating bounded sublattice. In this paper we introduce and study the category of Frith frames and show that the classical dual adjunction between topological spaces and frames extends to a dual adjunction between Pervin spaces and Frith frames. Unlike what happens for Pervin spaces, we do not have an equivalence between the categories of transitive and totally bounded quasi-uniform frames and of Frith frames, but we show that the latter is a full coreflective subcategory of the former. We also explore the notion of completeness of Frith frames inherited from quasi-uniform frames, providing a characterization of those Frith frames that are complete and a description of the completion of an arbitrary Frith frame.
math.GN↗
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↗
The assembly of a pointfree bispace and its two variations
The duality of finitary biframes as pointfree bitopological spaces is explored. In particular, for a finitary biframe $\mathcal{L}$ the ordered collection of all its pointfree bisubspaces (i.e. its biquotients) is studied. It is shown that this collection is bitopological in three meaningful ways. In particular it is shown that, apart from the assembly $\mathsf{A}(\mathcal{L})$ of a finitary biframe $\mathcal{L}$, there are two other structures $\mathsf{A}_{cf}(\mathcal{L})$ and $\mathsf{A}_{\pm}(\mathcal{L})$, which both have the same main component as $\mathsf{A}(\mathcal{L})$. The main component of both $\mathsf{A}_{cf}(\mathcal{L})$ and $\mathsf{A}_{\pm}(\mathcal{L})$ is the ordered collection of all biquotients of $\mathcal{L}$. The structure $\mathsf{A}_{cf}(\mathcal{L})$ being a biframe shows that the collection of all biquotients is generated by the frame of the patch-closed biquotients together with that of the patch-fitted ones. The structure $\mathsf{A}_{\pm}(\mathcal{L})$ being a biframe shows the collection of all biquotients is generated by the frame of the positive biquotients together with that of the negative ones. Notions of fitness and subfitness for finitary biframes are introduced, and it is shown that the analogues of both characterization theorems for these axioms appearing in Picado and Pultr (2011) hold. A spatial, bitopological version of these theorems is proven, in which finitary biframes whose spectrum is pairwise $T_1$ are characterized, among other things in terms of the spectrum of $\mathsf{A}_{cf}(\mathcal{L})$.
math.FA↗
The category of finitary biframes as the category of pointfree bispaces
The theory of finitary biframes as order-theoretical duals of bitopological spaces is explored. The category of finitary biframes is a coreflective subcategory of that of biframes. Some of the advantages of adopting finitary biframes as a pointfree notion of bispaces are studied. In particular, it is shown that for every finitary biframe there is a biframe which plays a role analogue to that of the assembly in the theory of frames: for every finitary biframe $\mathcal{L}$ there is a finitary biframe $\mathsf{A}(\mathcal{L})$ with a universal property analogous to that of the assembly of a frame; and such that its main component is isomorphic to the ordered collection of finitary quotients of $\mathcal{L}$ (i.e. its pointfree bisubspaces). Furthermore, in the finitary biframe duality the bispace associated with $\mathsf{A}(\mathcal{L})$ is a natural bitopological analogue of the Skula space of the bispace associated with $\mathcal{L}$. The finitary biframe duality gives us a notion of bisobriety which is weaker than pairwise Hausdorffness, incomparable with the pairwise $T_1$ axiom, and stronger than the pairwise $T_0$ axiom. The notion of pairwise $T_D$ bispaces is introduced, as a natural point-set generalization of the classical $T_D$ axiom. It is shown that in the finitary biframe duality this axiom plays a role analogous to that of the classical $T_D$ axiom for the frame duality.
math.CT↗
Revisiting the relation between subspaces and sublocales
We revisit results concerning the connection between subspaces of a space and sublocales of its locale of open sets. The approach we present is based on the observation that for every locale $L$ its spatial sublocales $\mathsf{sp}[\mathsf{S}(L)]$ form a coframe which is isomorphic to the coframe $\mathsf{sob}[\mathcal{P}(\mathsf{pt}(L))]$ of sober subspaces of $\mathsf{pt}(L)$. We characterize the frames $L$ such that the spatial sublocales of $\mathsf{S}(L)$ perfectly represent the subspaces of $\mathsf{pt}(L)$. We prove choice-free, weak versions of the results by Niefield and Rosenthal characterizing those frames such that all their sublocales are spatial (i.e., those such that the sober subspaces of $\mathsf{pt}(L)$ perfectly represent the sublocales of $L$). We do so by using a notion of essential prime which does not rely on the existence of enough minimal primes above every element. We will re-prove Simmons' result that spaces such that the sublocales of $Ω(X)$ perfectly represent their subspaces are exactly the scattered spaces. We will characterize scattered spaces in terms of a strong form of essentiality for primes. We apply these characterizations to show that, when $L$ is a spatial frame and a coframe, $\mathsf{pt}(L)$ is scattered if and only if it is $T_D$, and this holds if and only if all the primes of $L$ are completely prime.
math.FA↗