المؤلفون: Suarez, Anna Laura

مصطلحات موضوعية: Mathematics - Category Theory, 06D22, 18F70

الوصف: 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 $(\Omega(X),\mathcal{U}(X))$, where $\Omega(X)$ 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.

الوصول الحر: http://arxiv.org/abs/2405.13437Test

المؤلفون: Suarez, Anna Laura

مصطلحات موضوعية: Mathematics - Category Theory, 06D22, 18F70

الوصف: We draw from Raney duality and generalize the notion of canonical extension for distributive lattices to the context of frames. This is a generalization because for distributive lattices we are in the context of coherent spaces, whereas frames represent arbitrary spaces. We introduce Raney extensions of frames, pairs $(L,C)$ where $C$ is a coframe, $L\subseteq C$ is a frame that meet-generates it, and the inclusion $L\subseteq C$ preserves the frame operations as well as the strongly exact meets. We think of these as being algebraic versions of the embedding $\Omega(X)\subseteq \mathcal{U}(X)$ of a frame of opens into a lattice of saturated sets (upper sets in the specialization order). We show that a frame may have several Raney extensions, and all satisfy a generalization of the properties of density and compactness from the theory of canonical extensions. We show that every frame $L$ has the largest and the smallest Raney extension, and that these are, respectively, the coframe of fitted sublocales $\mathsf{S}_{\mathfrak{o}}(L)$ and the opposite of the frame $\mathsf{S}_{\mathfrak{c}}(L)$ of joins of closed sublocales. We thus show that these structures have universal properties which are dual of one another. For Raney extensions $(L,C)$ and $(M,D)$, we characterize the frame morphisms $f:L\to M$ which can be extended to a morphism of Raney extensions. We apply this result to obtain a characterization of morphisms of frames $f:L\to M$ which can be lifted to frame morphisms $\mathsf{S}_{\mathfrak{c}}(L)\to \mathsf{S}_{\mathfrak{c}}(M)$. We also show that the canonical extension of a locally compact frame is the free Raney extension over it such that it is algebraic (generated by its compact elements). We give a characterization of sobriety and of strict sobriety based on a variation of the compactness property of the extension $(\Omega(X),\mathcal{U}(X))$.

الوصول الحر: http://arxiv.org/abs/2405.02990Test

المؤلفون: Jakl, Tomáš, Suarez, Anna Laura

مصطلحات موضوعية: Mathematics - Category Theory, Mathematics - General Topology

الوصف: 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.

الوصول الحر: http://arxiv.org/abs/2404.18325Test

المؤلفون: Borlido, Célia, Suarez, Anna Laura

مصطلحات موضوعية: Mathematics - General Topology, 06D22, 54E55, 06D50

الوصف: 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.
Comment: 25 pages

الوصول الحر: http://arxiv.org/abs/2303.00443Test

المؤلفون: Borlido, Célia, Suarez, Anna Laura

مصطلحات موضوعية: Mathematics - General Topology, 54E15, 06D22

الوصف: 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.

الوصول الحر: http://arxiv.org/abs/2201.06266Test

المؤلفون: Arrieta, Igor, Suarez, Anna Laura

مصطلحات موضوعية: Mathematics - Category Theory

الوصف: 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).

الوصول الحر: http://arxiv.org/abs/2011.08897Test

المؤلفون: Suarez, Anna Laura

مصطلحات موضوعية: Mathematics - Functional Analysis, Mathematics - Category Theory

الوصف: 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 $\mathsf{A}_{cf}(\mathcal{L})$.
Comment: arXiv admin note: text overlap with arXiv:2010.04622

الوصول الحر: http://arxiv.org/abs/2011.01547Test

المؤلفون: Suarez, Anna Laura

مصطلحات موضوعية: Mathematics - Functional Analysis, Mathematics - Category Theory, 06D22

الوصف: 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 $\Omega (X)$ 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.

الوصول الحر: http://arxiv.org/abs/2010.05284Test

المؤلفون: Suarez, Anna Laura

مصطلحات موضوعية: Mathematics - Category Theory, 06D22

الوصف: 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.

الوصول الحر: http://arxiv.org/abs/2010.04622Test

المؤلفون: Suarez, Anna Laura

الوصف: This thesis is concerned with the study of pointfree bispaces, and in particular with the pointfree notion of inclusion of bisubspaces. We mostly work in the context of d-frames. We study quotients of d-frames as pointfree analogues of the topological notion of bisubspace. We show that for every d-frame L there is a d-frame A(L) such that it plays the role of the assembly of a frame, in the sense that it has the analogue of the universal property of the assembly and that its spectrum is a bitopological version of the Skula space of the bispace dpt(L), the spectrum of L. Furthermore, we show that this bitopological version of the Skula space of dpt(L) is the coarsest topology in which the d-sober bisubspaces of dpt(L) are closed. We also show that there are two free constructions in the category of d-frames Act(L) and A_(L), such that they represent two variations of the bitopological version of the Skula topology. In particular, we show that in dpt(Act) the positive closed sets are exactly those d-sober subspaces of dpt(L) that are spectra of quotients coming from an increase in the con component, and that the negative closed ones are those that come from increases in the tot component. For dpt(A_(L)), we show that the positive closed sets are exactly those bisubspaces of dpt(L) that are spectra of quotients coming from a quotient of L+, and that the negative closed sets come in the same way from quotients of L.

الوصول الحر: https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.861605Test