11 research outputs found
On hereditary coreflective subcategories of Top
Let A be a topological space which is not finitely generated and CH(A) denote
the coreflective hull of A in Top. We construct a generator of the coreflective
subcategory SCH(A) consisting of all subspaces of spaces from CH(A) which is a
prime space and has the same cardinality as A. We also show that if A and B are
coreflective subcategories of Top such that the hereditary coreflective kernel
of each of them is the subcategory FG of all finitely generated spaces, then
the hereditary coreflective kernel of their join CH(A \cup B) is again FG
Monotone-light factorisation systems and torsion theories
Given a torsion theory (Y,X) in an abelian category C, the reflector I from C
to the torsion-free subcategory X induces a reflective factorisation system (E,
M) on C. It was shown by A. Carboni, G.M. Kelly, G. Janelidze and R. Par\'e
that (E, M) induces a monotone-light factorisation system (E',M*) by
simultaneously stabilising E and localising M, whenever the torsion theory is
hereditary and any object in C is a quotient of an object in X. We extend this
result to arbitrary normal categories, and improve it also in the abelian case,
where the heredity assumption on the torsion theory turns out to be redundant.
Several new examples of torsion theories where this result applies are then
considered in the categories of abelian groups, groups, topological groups,
commutative rings, and crossed modules.Comment: 12 page
Dual closure operators and their applications
Departing from a suitable categorical description of closure operators, this paper dualizes this notion and introduces some basic properties of dual closure operators. Usually these operators act on quotients rather than subobjects, and much attention is being paid here to their key examples in algebra and topology, which include the formation of monotone quotients (Eilenberg-Whyburn) and concordant quotients (Coffins). In fair categorical generality, these constructions are shown to be factors of the fundamental correspondence that relates connectecinesses and disconnectednesses in topology, as well as torsion classes and torsion-free classes in algebra. Depending on a given cogenerator, the paper also establishes a non-trivial correspondence between closure operators and dual closure operators in the category of R-modules. Dual closure operators must be carefully distinguished from interior operators that have been studied by other author
Topogenous structures and related families of morphisms
In a category with a proper -factorization system, we study the notions of strict, co-strict,
initial and final morphisms with respect to a topogenous order. Besides showing
that they allow simultaneous study of four classes of morphisms obtained
separately with respect to closure, interior and neighbourhood operators, the
initial and final morphisms lead us to the study of topogenous structures
induced by pointed and co-pointed endofunctors. We also lift the topogenous
structures along an -fibration. This permits one to obtain the
lifting of interior and neighbourhood operators along an
-fibration and includes the lifting of closure operators found in
the literature. A number of examples presented at the end of the paper
demonstrates our results.Comment: arXiv admin note: text overlap with arXiv:2302.0275
Neighbourhood operators: additivity, idempotency and convergence
We define and discuss the notions of additivity and idempotency for neighbourhood and interior operators. We then propose an order-theoretic description of the notion of convergence that was introduced by D. Holgate and J. Å lapal with the help of these two properties
Interior operators and their applications
Philosophiae Doctor - PhDCategorical closure operators were introduced by Dikranjan and Giuli in [DG87] and then developed by
these authors and Tholen in [DGT89]. These operators have played an important role in the development
of Categorical Topology by introducing topological concepts, such as connectedness, separatedness and
compactness, in an arbitrary category and they provide a uni ed approach to various mathematical
notions. Motivated by the theory of these operators, the categorical notion of interior operators was
introduced by Vorster in [Vor00]. While there is a notational symmetry between categorical closure and
interior operators, a detailed analysis shows that the two operators are not categorically dual to each
other, that is: it is not true in general that whatever one does with respect to closure operators may be
done relative to interior operators. Indeed, the continuity condition of categorical closure operators can
be expressed in terms of images or equivalently, preimages, in the same way as the usual topological
closure describes continuity in terms of images or preimages along continuous maps. However, unlike the
case of categorical closure operators, the continuity condition of categorical interior operators can not
be described in terms of images. Consequently, the general theory of categorical interior operators is not
equivalent to the one of closure operators. Moreover, the categorical dual closure operator introduced in
[DT15] does not lead to interior operators. As a consequence, the study of categorical interior operators
in their own right is interesting
Noncommutative localization in noncommutative geometry
The aim of these notes is to collect and motivate the basic localization
toolbox for the geometric study of ``spaces'', locally described by
noncommutative rings and their categories of one-sided modules.
We present the basics of Ore localization of rings and modules in much
detail. Common practical techniques are studied as well. We also describe a
counterexample for a folklore test principle. Localization in negatively
filtered rings arising in deformation theory is presented. A new notion of the
differential Ore condition is introduced in the study of localization of
differential calculi.
To aid the geometrical viewpoint, localization is studied with emphasis on
descent formalism, flatness, abelian categories of quasicoherent sheaves and
generalizations, and natural pairs of adjoint functors for sheaf and module
categories. The key motivational theorems from the seminal works of Gabriel on
localization, abelian categories and schemes are quoted without proof, as well
as the related statements of Popescu, Watts, Deligne and Rosenberg.
The Cohn universal localization does not have good flatness properties, but
it is determined by the localization map already at the ring level. Cohn
localization is here related to the quasideterminants of Gelfand and Retakh;
and this may help understanding both subjects.Comment: 93 pages; (including index: use makeindex); introductory survey, but
with few smaller new result