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Register a new userInternal Neighbourhood Structures II: Closure and closed morphisms
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Authors
Partha Pratim Ghosh
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Internal preneighbourhood spaces were first conceived inside any finitely complete category with finite coproducts and proper factorisation structure in my earlier paper. In this paper a closure operation is introduced on internal preneighbourhood spaces and investigated along with closed morphisms and its close allies. Analogues of several well known classes of topological spaces for preneighbourhood spaces are investigated. The approach via preneighbourhood systems is shown to be more general than the closure operators and conveniently allows to identify properties of classes of morphisms which are independent of continuity of morphisms with respect to closure operators.
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The notion of an internal preneighbourhood space on a finitely complete category with finite coproducts and a proper $(mathsf{E}, mathsf{M})$ system such that for each object $X$ the set of $mathsf{M}$-subobjects of $X$ is a complete lattice was initiated in cite{2020}. The notion of a closure operator, closed morphism and its near allies investigated in cite{2021-clos}. The present paper provides structural conditions on the triplet $(mathbb{A}, mathsf{E}, mathsf{M})$ (with $mathbb{A}$ lextensive) equivalent to the set of $mathsf{M}$-subobjects of an object closed under finite sums. Equivalent conditions for the set of closed embeddings (closed morphisms) closed under finite sums is also provided. In case when lattices of admissible subobjects (respectively, closed embeddings) are closed under finite sums, the join semilattice of admissible subobjects (respectively, closed embeddings) of a finite sum is shown to be a biproduct of the component join semilattices. Finally, it is shown whenever the set of closed morphisms is closed under finite sums, the set of proper (respectively, separated) morphisms are also closed under finite sums. This leads to equivalent conditions for the full subcategory of compact (respectively, Hausdorff) preneighbourhood spaces to be closed under finite sums.
The main aim of this paper is to provide a description of neighbourhood operators in finitely complete categories with finite coproducts and a proper factorisation system such that the semilattice of admissible subobjects make a distributive complete lattice. The equivalence between neighbourhoods, Kuratowski interior operators and pseudo-frame sets is proved. Furthermore the categories of internal neighbourhoods is shown to be topological. Regular epimorphisms of categories of neighbourhoods are described and conditions ensuring hereditary regular epimorphisms are probed. It is shown the category of internal neighbourhoods of topological spaces is the category of bitopological spaces, while in the category of locales every locale comes equipped with a natural internal topology.
Under a general categorical procedure for the extension of dual equivalences as presented in this papers predecessor, a new algebraically defined category is established that is dually equivalent to the category $bf LKHaus$ of locally compact Hausdorff spaces and continuous maps, with the dual equivalence extending a Stone-type duality for the category of extremally disconnected locally compact Hausdorff spaces and continuous maps. The new category is then shown to be isomorphic to the category $bf CLCA$ of complete local contact algebras and suitable morphisms. Thereby, a new proof is presented for the equivalence ${bf LKHaus}simeq{bf CLCA}^{rm op}$ that was obtained by the first author more than a decade ago. Unlike the morphisms of $bf CLCA$, the morphisms of the new category and their composition law are very natural and easy to handle.
In this paper, we introduce the notion of a topological groupoid extension and relate it to the already existing notion of a gerbe over a topological stack. We further study the properties of a gerbe over a Serre, Hurewicz stack.
In this article a sequential theory in the category of spaces and proper maps is described and developed. As a natural extension a sequential theory for exterior spaces and maps is obtained.
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