Categorical neighborhood operators

Holgate, David; Šlapal, Josef

doi:10.1016/j.topol.2011.06.031

Cited by 23 publications

(17 citation statements)

References 7 publications

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“…The categorical topogenous order that is defined above allows to obtain a nice relationship between closure and interior operators on a category in the sense that many concepts and definitions that have been studied separately for categorical closure and interior operators can be shown to be exactly the same using topogenous orders (see for example the notion of -strict subobject defined and used in third section of this paper). The next result that we recall from ( [9]) exhibits the clear relationship between closure, interior and topogenous order in a category while it is also known from the same paper that a topogenous order on C is basically equivalent to a neighbourhood operator ( [10]). Proposition 2.1.…”

Section: The Resulting Ordered Conglomerate Of All Topogenous Orders On C Is Denoted By T Ord(c M)mentioning

confidence: 66%

Quasi-uniform structures determined by closure operators

Holgate

Iragi

2021

Topology and its Applications

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We demonstrate a one-to-one correspondence between idempotent closure operators and the so-called saturated quasi-uniform structures on a category C. Not only this result allows to obtain a categorical counterpart P of the Császár-Pervin quasiuniformity P, that we characterize as a transitive quasi-uniformity compatible with an idempotent interior operator, but also permits to describe those topogenous orders that are induced by a transitive quasi-uniformity on C. The categorical counterpart P * of P −1 is characterized as a transitive quasi-uniformity compatible with an idempotent closure operator. When applied to other categories outside topology P allows, among other things, to generate a family of idempotent closure operators on Grp, the category of groups and group homomorphisms, determined by the normal closure.

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Section: The Resulting Ordered Conglomerate Of All Topogenous Orders On C Is Denoted By T Ord(c M)mentioning

confidence: 66%

Quasi-uniform structures determined by closure operators

Holgate

Iragi

2021

Topology and its Applications

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“…Regarding algebraic examples, the type of construction in (4.1) has recently received rejuvenated interest in work of Guiterres and Clementino [8]. In [15] a dierent construction of a closure operator from a neighbourhood operator is considered.…”

Section: From Closure To Interiormentioning

confidence: 99%

“…In [15] and [19] neighbourhood operators are introduced as primitive and used to then study compactness in particular. It is this introduction of a general neighbourhood operator that motivated the current note to analyse more systematically how closure, interior and neighbourhood operators interact, and to provide a framework within which a number of existing investigations can be understood.…”

Section: Introductionmentioning

confidence: 99%

Closure, interior and neighbourhood in a category

Holgate¹,

Šlapal²

2018

HJMS

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The natural correspondences in topology between closure, interior and neighbourhood no longer hold in an abstract categorical setting where subobject lattices are not necessarily Boolean algebras. We analyse three canonical correspondences between closure, interior and neighbourhood operators in a category endowed with a subobject structure. While these correspondences coincide in general topology, the analysis highlights subtle dierences which distinguish dierent approaches taken in the literature.

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“…Originally, only the topological structures on categories given by closure operators were considered and investigated. Later on, also other types of topological structures on categories were introduced and studied, e.g., convergence structures in [13][14] and neighborhood structures in [7][8]. Different types of topological structures on categories are studied to provide convenient tools for investigating topological features of (objects of) the categories.…”

Section: Introductionmentioning

confidence: 99%

A categorical approach to convergence

Šlapal

2016

Filomat

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We define the concept of a convergence class on an object of a given category by using certain generalized nets for expressing the convergence. The resulting topological category, whose objects are the pairs consisting of objects of the original category and convergence classes on them, is then investigated. We study the full subcategories of this category which are obtained by imposing on it some natural convergence axioms. In particular, we find sufficient conditions for the subcategories to be cartesian closed. We also investigate the behavior of the closure operator associated with the convergence in a natural way.

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Categorical neighborhood operators

Cited by 23 publications

References 7 publications

Quasi-uniform structures determined by closure operators

Quasi-uniform structures determined by closure operators

Closure, interior and neighbourhood in a category

A categorical approach to convergence

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