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Title
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A study of topological properties in approach theory using monoidal topology
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Author
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Abstract
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| Monoidal topology is a research area in mathematics providing a common frame-work for convergence. Two parameters, a Set-monad T and a quantale V, to-gether with an extension of the monad T to V-Rel, provide us with a category of lax algebras, denoted (T, V)-Cat. Suitable choices for T and V result in a lax algebraic description of ordered spaces, metric spaces, topological spaces and ap-proach spaces. Our interest goes out to approach spaces and a first lax algebraic description of approach spaces was given by Clementino and Hofmann [CH03] by defin-ing an extension β of the ultrafilter monad β to numerical relations resulting in (β, P+)-Cat ∼= App. In this work we look for relational representations of App, i.e. lax alge-braic representations only using the quantale 2. We introduce the functional ideal monad I, which is power-enriched, and using the theory of power-enriched monads developed in [HST14] we are able to prove that (I, 2)-Cat ∼= App. We also look at the prime functional ideals and their corresponding monad B. We show that B is a submonad of I satisfying those properties needed in order to conclude (I, 2)-Cat ∼= (B, 2)-Cat. We also turn our attention to NA-App, the full subcategory of App consisting of non-Archimedean approach spaces. We answer the question of determining which parameters T and V should be used in order to capture NA-App as a category of lax algebras. It turns out that the answer lies in switching the quantale P+ in the presentation of App as lax algebras by Clementino and Hofmann to P∨, which results in the isomorphism NA-App ∼= (β, P∨)-Cat. The relational descriptions of App by means of the functional ideal monad and the prime functional ideal monad, and the description of NA-App as a category of lax algebras are the main instruments for an in depth study of new approach invariants. These approach invariants will arise as topological properties in lax algebras depending on the monad T and the quantale V. We study Hausdorff separation, compactness and regularity to name only a few. |
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Language
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English
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Publication
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Brussel
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Vrije Universiteit Brussel/Universiteit Antwerpen
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2019
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ISBN
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978-94-93079-15-1
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Volume/pages
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172 p.
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Full text (open access)
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