PreT 2 Objects in Topological Categories

In previous papers, two notions of pre-Hausdorff (PreT ₂) objects in a topological category were introduced and compared. The main objective of this paper is to show that the full subcategory of PreT ₂ objects is a topological category and all of T ₀, T ₁, and T ₂ objects in this topological category are equivalent. Furthermore, the characterizations of pre-Hausdorff objects in the categories of filter convergence spaces, (constant) local filter convergence spaces, and (constant) stack convergence spaces are given and as a consequence, it is shown that these categories are homotopically trivial.     

Abstract ordered compact convex sets and algebras of the (sub)probabilistic powerdomain monad over ordered compact spaces

The majority of categories used in denotational semantics are topological in nature. One of these is the category of stably compact spaces and continuous maps. Previously, Eilenberg–Moore algebras were studied for the extended probabilistic powerdomain monad over the category of ordered compact spaces X and order-preserving continuous maps in the sense of Nachbin. Appropriate algebras were characterized as compact convex subsets of ordered locally convex topological vector spaces. In so doing, functional analytic tools were involved. The main accomplishments of this paper are as follows: the result mentioned is re-proved and is extended to the subprobabilistic case; topological methods are developed which defy an appeal to functional analysis; a more topological approach might be useful for the stably compact case; algebras of the (sub)probabilistic powerdomain monad inherit barycentric operations that satisfy the same equational laws as those in vector spaces. Also, it is shown that it is convenient first to embed these abstract convex sets in abstract cones, which are simpler to work with. Lastly, we state embedding theorems for abstract ordered locally compact cones and compact convex sets in ordered topological vector spaces.     

Cartesian Closed Topological Categories and Tensor Products

The projective tensor product in a category of topological R-modules (where R is a topological ring) can be defined in Top, the category of topological spaces, by the same universal property used to define the tensor product of R-modules in Set. In this article, we extend this definition to an arbitrary topological category X and study how the Cartesian closedness of X is related to the monoidal closedness of the category of R-module objects in X. Mathematics Subject Classifications (2000) 18D15, 18D35, 18A40.     

Functorial Duality for Ortholattices and De Morgan Lattices

Relational semantics for nonclassical logics lead straightforwardly to topological representation theorems of their algebras. Ortholattices and De Morgan lattices are reducts of the algebras of various nonclassical logics. We define three new classes of topological spaces so that the lattice categories and the corresponding categories of topological spaces turn out to be dually isomorphic. A key feature of all these topological spaces is that they are ordered relational or ordered product topologies.     

Measurable Categories

We develop the theory of categories of measurable fields of Hilbert spaces and bounded fields of operators. We examine classes of functors and natural transformations with good measure theoretic properties, providing in the end a rigorous construction for the bicategory used in [3] and [4] as the basis for a representation theory of (Lie) 2-groups. Two important technical results are established along the way: first it is shown that all invertible additive bounded functors (and thus a fortiori all invertible *-functors) between categories of measurable fields of Hilbert spaces are induced by invertible measurable transformations between the underlying Borel spaces and second we establish the distributivity of Hilbert space tensor product over direct integrals over Lusin spaces with respect to σ-finite measures. The paper concludes with a general definition of measurable bicategories.     

Extremal Prime Filters and Universality of Some Categories

Abstract

Due to the existence of constants, classical topological categories cannot be universal in the sense of containing each concrete category as a full subcategory. In the point-free case, this obstruction vanishes and the question of universality makes sense again. The main problem, namely that as to whether the category of locales and localic morphisms is universal is still open; we prove, however, the universality of the following categories:

- pairs (locale, sublocale) with the localic morphisms preserving the distinguished sublocales,

- frames with frame homomorphisms reflecting the maximal prime ideals,

- Priestley spaces with f-maps preserving the maximal elements.     

Topological Model Categories Generated by Finite Complexes

 Our main result states that for each finite complex L the category TOP of topological spaces possesses a model category structure (in the sense of Quillen) whose weak equivalences are precisely maps which induce isomorphisms of all [L]-homotopy groups. The concept of [L]-homotopy has earlier been introduced by the first author and is based on Dranishnikov’s notion of extension dimension. As a corollary we obtain an algebraic characterization of [L]-homotopy equivalences between [L]-complexes. This result extends two classical theorems of J. H. C. Whitehead. One of them – describing homotopy equivalences between CW-complexes as maps inducing isomorphisms of all homotopy groups – is obtained by letting L = {point}. The other – describing n-homotopy equivalences between at most (n+1)-dimensional CW-complexes as maps inducing isomorphisms of k-dimensional homotopy groups with k ⩽ n – by letting L = S ⁿ⁺¹ , n ⩾ 0.     

Categories of abstract smooth models and their singular envelopes

We define the categories of (abstract) smooth models (Definition 1.2) and, in the additive case, their singular envelopes (Definition 1.5). The first main result is a relative version of the Yoneda representation theorem (Theorem 1.6), and the second one is an existence and uniqueness theorem for the singular envelope (Theorem 1.7). In fact we prove the existence of a canonical process which associates with each additive smooth-model categoryS a singular envelopeS-an ofS, whose objects are calledS-analytic spaces (Definition 5.1). We notice that most of the fundamental categories of geometry are of the formS-an (up to equivalence). As an application we introduce here two such categories: Banach differentiable spaces and Banach mixed spaces.The author is indebted to the Department of Mathematics of the University of Rome (La Sapienza) for hospitality and financial support and to the referees for their critical comments on the first version of this paper.     

An Elementary Approach to Exponentiable Maps

Originally, exponentiable maps in the category Top of topological spaces were described by Niefield in terms of certain fibrewise Scott-open sets. This generalizes the first characterization of exponentiable spaces by Day and Kelly, which was improved thereafter by Hofmann and Lawson who described them as core-compact spaces.Besides various categorical methods, the Sierpinski-space is an essential tool in Niefield's original proof. Therefore, this approach fails to apply to quotient reflective subcategories of Top like Haus, the category of Hausdorff spaces. A recent generalization of the Hofmann–Lawson improvement to exponentiable maps enables now to reprove the characterization in a completely different and very elementary way. This approach works for any nontrivial quotient reflective subcategory of Top or Top/ _T , the category of all spaces over a fixed base space T, as well as for exponentiable monomorphisms with respect to epi-reflective subcategories.An important special case is the category Sep_Top/ _T of separated maps, i.e. distinct points in the same fibre can be separated in the total space by disjoint open neighbourhoods. The exponentiable objects in Sep turn out to be the open and fibrewise locally compact maps. The same holds for Haus/ _T , T a Hausdorff space. In this case, a similar characterization was obtained by Cagliari and Mantovani.     

Topological Model Categories Generated by Finite Complexes

 Our main result states that for each finite complex L the category TOP of topological spaces possesses a model category structure (in the sense of Quillen) whose weak equivalences are precisely maps which induce isomorphisms of all [L]-homotopy groups. The concept of [L]-homotopy has earlier been introduced by the first author and is based on Dranishnikov’s notion of extension dimension. As a corollary we obtain an algebraic characterization of [L]-homotopy equivalences between [L]-complexes. This result extends two classical theorems of J. H. C. Whitehead. One of them – describing homotopy equivalences between CW-complexes as maps inducing isomorphisms of all homotopy groups – is obtained by letting L = {point}. The other – describing n-homotopy equivalences between at most (n+1)-dimensional CW-complexes as maps inducing isomorphisms of k-dimensional homotopy groups with k ⩽ n – by letting L = S ⁿ⁺¹ , n ⩾ 0. The first author was partially supported by NSERC research grant. Received December 12, 2001; in revised form September 7, 2002 Published online February 28, 2003     

取值于序拓扑向量空间的映射与层空间

In the paper [Monotone countable paracompactness and maps to ordered topological vector spaces, Top. Appl., 2014, 169(3): 51–70], Yamazaki initiated the study on maps with values into ordered topological vector spaces. Characterizations of monotonically countably paracompact spaces and some other spaces in terms of maps to ordered topological vector spaces were obtained. In this paper, following Yamazaki's method, we present some characterizations of stratifiable spaces and k-semi-stratifiable spaces in terms of maps with values into ordered topological vector spaces.     

Closed structures on categories of topological spaces

The category of all topological spaces and continuous maps and its full subcategory of all T_o-spaces admit (up to isomorphism) precisely one structure of symmetric monoidal closed category (see [2]). In this paper we shall prove the same result for any epireflective subcategory of the category of topological spaces (particularly e.g. for the categories of Hausdorff spaces, regular spaces, Tychonoff spaces).     

Dagger Categories of Tame Relations

Within the context of an involutive monoidal category the notion of a comparison relation ${\textsf{cp} : \overline{X} \otimes X \rightarrow \Omega}$ is identified. Instances are equality = on sets, inequality ${\leq}$ on posets, orthogonality ${\perp}$ on orthomodular lattices, non-empty intersection on powersets, and inner product ${\langle {-}|{-} \rangle}$ on vector or Hilbert spaces. Associated with a collection of such (symmetric) comparison relations a dagger category is defined with “tame” relations as morphisms. Examples include familiar categories in the foundations of quantum mechanics, such as sets with partial injections, or with locally bifinite relations, or with formal distributions between them, or Hilbert spaces with bounded (continuous) linear maps. Of one particular example of such a dagger category of tame relations, involving sets and bifinite multirelations between them, the categorical structure is investigated in some detail. It turns out to involve symmetric monoidal dagger structure, with biproducts, and dagger kernels. This category may form an appropriate universe for discrete quantum computations, just like Hilbert spaces form a universe for continuous computation.     

Natural examples of Valdivia compact spaces

We collect examples of Valdivia compact spaces, their continuous images and associated classes of Banach spaces which appear naturally in various branches of mathematics. We focus on topological constructions generating Valdivia compact spaces, linearly ordered compact spaces, compact groups, L¹ spaces, Banach lattices and noncommutative L¹ spaces.     

ON CERTAIN INITIAL COMPLETIONS IN FUZZY TOPOLOGY

Abstract

Following a lead given by I.W. Alderton, it is shown that the MacNeille completion and the universal initial completion coincide for the categories of zero-dimensional fuzzy T₀-topological spaces, T₀-fuzzy closure spaces, 2T ₀-fuzzy bitopological spaces, and T ₁-fuzzy topological spaces and that these turn out to be respectively the categories of zero-dimensional fuzzy topological spaces, fuzzy closure spaces, fussy bitopological spaces, and fuzzy R ₀ topological spaces.     

Lexicographic sums and fibre-faithful maps

The notion of lexicographic sum is introduced in general categories. Existence criteria are derived, particularly for locally cartesian closed categories and for categories with suitable coproducts. Lexicographic sums satisfy a generalized associative law. More importantly, every morphism can be factored through the lexicographic sum of its fibres. This factorization and the two types of maps arising from it, fibre-trivial and fibre-faithful, are studied particularly for partially ordered sets and forT ₁-spaces.     

Realizations of Topological Categories

There are functor-preordering-structured categories S(F,P), defined by the Prague School, in which every concrete category over a concretizable basecategory is realizable. Over nice basecategories there are realizations of all topological categories in some topological S(F,L). This gives rise for a new characterization of those concrete categories having a topological hull.     

Compactness in Categories of S-Net Spaces

The $S$ -net spaces studied are convergence structures whose convergences are expressed by using generalized nets, the so called $S$ -nets, which are obtained from the usual nets by replacing the category of directed sets and cofinal maps with an arbitrary construct $S$ . We investigate compactness in categories of $S$ -net spaces defined by introducing continuous maps in a natural way and imposing some usual convergence axioms.     

On 2-dimensional Simple Sets in n-dimensional Cubic Grids

Preserving topological properties of objects during reduction procedures is an important issue in the field of discrete image analysis. Such procedures are generally based on the notion of simple point, the exclusive use of which may result in the appearance of “topological artifacts.” This limitation leads to consider a more general category of objects, the simple sets, which also enable topology-preserving image reduction. A study of two-dimensional simple sets in two-dimensional spaces has been proposed recently. This article is devoted to the study of two-dimensional simple sets in spaces of higher dimension (i.e., n-dimensional spaces, n≥3). In particular, several properties of minimal simple sets (i.e., which do not strictly include any other simple sets) are proposed, leading to a characterisation theorem. It is also proved that the removal of a two-dimensional simple set from an object can be performed by only considering the minimal ones, thus authorising the development of efficient thinning algorithms.