On bornologies, locales and toposes of M-sets

Español, Luis; Lambán, Laureano

doi:10.1016/s0022-4049(02)00047-6

Cited by 5 publications

(12 citation statements)

References 7 publications

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“…An ideal B of P(X) is called bornology into X with extent E(B) = {A ∈ B}. B is also said to be a bornology on E(B), and the pair (E(B), B) is a bornological space [4].…”

Section: The Locale Of All Filters On a Setmentioning

confidence: 99%

“…The set Born(X) of all bornologies in X is the free compact regular locale generated by P(X), and it is equivalent to the locale of open subsets of the Stone-Čech compactification βX of the discrete topological space X [10, p. 93]. The locale Born(X) was extensively used in [4], where it was proved that it is a subobject classifier of a Grothendieck topos of N N -sets for N N = Set(N, N).…”

Section: The Locale Of All Filters On a Setmentioning

confidence: 99%

“…We need more notation about the set Ω of all ideals of an arbitrary monoid M. First observe that Ω an with the action given by the ideal [17,4] with the operators:…”

Section: Characterizing Cofinite Subsets Of Nmentioning

confidence: 99%

“…Any general monoid M (we denote its operation by f • g, recalling maps and compositions) has a locale Ω of right ideals, which is the object of true-values for M-sets [17,3,4]. The double negation topology on M is the M-subset of Ω formed by those ideals such that ¬¬I = M. Then, an ideal I belongs to the double negation topology if, and only if, for any f ∈ M there exists g ∈ M such that f • g ∈ I.…”

Section: Introductionmentioning

confidence: 99%

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Cofinite subsets and double negation topologies on locales of filters and ideals

Español,

García-Calcines,

Mínguez

2015

Preprint

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We study the role of the filter cK(X) of cofinite subsets of X in the locale F ilt(X) of all filters on X, by means of the double negation topology of F ilt(X), and an essential locale morphism P(X) op → F ilt(X). Moreover, in the case X = N, we characterise cofinite subsets by means of the double negation topology on the monoid M of the maps N → N with finite fibers, or on the submonoid E ⊆ M of the monotone and injective maps N → N.

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“…An ideal B of P(X) is called bornology into X with extent E(B) = {A ∈ B}. B is also said to be a bornology on E(B), and the pair (E(B), B) is a bornological space [4].…”

Section: The Locale Of All Filters On a Setmentioning

confidence: 99%

Section: The Locale Of All Filters On a Setmentioning

confidence: 99%

“…We need more notation about the set Ω of all ideals of an arbitrary monoid M. First observe that Ω an with the action given by the ideal [17,4] with the operators:…”

Section: Characterizing Cofinite Subsets Of Nmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Cofinite subsets and double negation topologies on locales of filters and ideals

Español,

García-Calcines,

Mínguez

2015

Preprint

Self Cite

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“…Recently, applications of Lawvere-Tierney topologies in broad topics such as measure theory [7] and quantum Physics [14,15] are observed. In the spacial case, considerable work has been presented that is dedicated to the study of (weak) Lawvere-Tierney topology on a presheaf topos on a small category and especially on a monoid, see [6,5]. It is clear that Lawvere-Tierney sheaves in a topos are exactly injective objects (of course, with respect to dense monomorphisms, not to merely monomorphisms) which are separated too.…”

Section: Introductionmentioning

confidence: 99%

Lawvere-Tierney sheaves, factorization systems, sections and $j$-essential monomorphisms in a topos

Khanjanzadeh¹,

Madanshekaf²

2015

Preprint

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Let j be a Lawvere-Tierney topology (a topology, for short) on an arbitrary topos E, B an object of E, and j B = j × 1 B the induced topology on the slice topos E/B. In this manuscript, we analyze some properties of the pullback functor Π B : E → E/B which have deal with topology. Then for a left cancelable class M of all j-dense monomorphisms in a topos E, we achieve some necessary and sufficient conditions for that (M, M ⊥ ) is a factorization system in E, which is related to the factorization systems in slice topoi E/B, where B ranges over the class of objects of E. Among other things, we prove that an arrow f : X → B in E is a j B -sheaf whenever the graph of f , is a section in E/B as well as the object of sections S(f ) of f , is a j-sheaf in E. Furthermore, we introduce a class of monomorphisms in E, which we call them j-essential. Some equivalent forms of those and some of their properties are presented. Also, we prove that any presheaf in a presheaf topos has a maximal essential extension.Finally, some similarities and differences of the obtained result are discussed if we put a (productive) weak topology j, studied by some authors, instead of a topology.

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Weak ideal topology in the topos of right acts over a monoid

Khanjanzadeh

Madanshekaf

2017

Communications in Algebra

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On bornologies, locales and toposes of M-sets

Cited by 5 publications

References 7 publications

Cofinite subsets and double negation topologies on locales of filters and ideals

Cofinite subsets and double negation topologies on locales of filters and ideals

Lawvere-Tierney sheaves, factorization systems, sections and $j$-essential monomorphisms in a topos

Weak ideal topology in the topos of right acts over a monoid

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