Hereditary Coreflective Subcategories of Categories of Topological Spaces

Let be an epireflective subcategory of the category Top of topological spaces that is not bireflective (e.g., the category of Hausdorff spaces, the category of Tychonoff spaces) and ℬ be a coreflective subcategory of . Extending the corresponding result obtained for coreflective subcategories of Top we prove that ℬ is hereditary if and only if it is closed under the formation of prime factors. As a consequence we obtain that every hereditary coreflective subcategory ℬ of containing a non-discrete space is generated by a class of prime spaces and if is a quotient-reflective subcategory of Top, then the assignment gives a bijection of the collection of all hereditary coreflective subcategories of Top that contain the class FG of all finitely generated spaces onto the collection of all hereditary coreflective subcategories of that contain . Some applications of these results in the categories of Hausdorff spaces, Tychonoff spaces and zero-dimensional Hausdorff spaces are presented.Mathematics Subject Classifications (2000) 18D15, 54B30.     

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.     

Coreflections Versus Regular Epimorphisms in Categories of Topological Spaces

Let be an epireflective subcategory of the category Top of topological spaces which is not contained in the category of indiscrete spaces (e.g. Top, the category of Hausdorff spaces, the category of Tychonoff spaces) and be a coreflective subcategory of . In this paper we prove that the coreflector preserves regular epimorphisms if and only if or is contained in the category of discrete spaces.     

Heredity and Coreflective Subcategories of the Category of Topological Spaces

Every hereditary coreflective subcategory of Top containing the category of finitely-generated spaces is shown to be generated by a class of spaces having a unique accumulation point. It is also shown that the coreflective hull of a union of two hereditary coreflective subcategories of Top need not be hereditary so that a coreflective subcategory of Top need not have a hereditary coreflective kernel.     

On the Reflectivity and Coreflectivity of L-Fuzzifying Topological Spaces in L-Topological Spaces

In this paper it is proved that for all completely distributive lattices L, the category of L-fuzzifying topological spaces can be embedded in the category of L-topological spaces (stratified Chang-Goguen spaces) as a simultaneously bireflective and bicoreflective full subcategory. Received April 2, 1999, Revised January 31, 2000, Accepted February 2, 2000     

Topological Hopf Algebras and Braided Monoidal Categories

The relation between a monoidal category which has an exact faithful monoidal functor to a category of finite rank projective modules over a Dedekind domain, and the category of continuous modules over a topological bialgebra is discussed. If the monoidal category is braided, the bialgebra is topologically quasitriangular. If the monoidal category is rigid monoidal, the bialgebra is a Hopf algebra.     

A(∞)TH上的辫子Monoidal范畴

Let A and H be Hopf algebra,T-smash product A (∞)T H generalizes twisted smash product A*H.This paper shows a necessary and sufficient condition for T-smash product module category A(∞)T H M to be braided monoidal category.     

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(AB) is again FG.     

Tensor Products of Non-Archimedean Weighted Spaces of Continuous Functions

It is shown that the completion of the tensor product of two non-Archimedean weighted spaces of continuous functions is topologically isomorphic to another weighted space. Several applications of this result are given.     

Free Modules over Cartesian Closed Topological Categories

The construction of free R-modules over a Cartesian closed topological category X is detailed (where R is a ring object in X), and it is shown that the insertion of generators is an embedding. This result extends the well-known construction of free groups, and more generally of free algebras over a Cartesian closed topological category. Mathematics Subject Classifications (2000) 18D15, 18D35, 18A40.     

Lattice Tensor Products i. Coordinatization

G. Grätzer and F. Wehrung introduced the lattice tensor product, A B, of the lattices A and B. One of the most important properties is that for a simple and bounded lattice A, the lattice A B is a congruence-preserving extension of B. The lattice A B is defined as the set of certain subsets of A B; there is no easy test when a subset belongs to A B. A special case, M ₃B, was earlier defined by G. Gräatzer and F. Wehrung as M ₃, the it Boolean triple construct, defined as a subset of B ³, with a simple criterion when a triple belongs. A~recent paper of G. Grätzer and E. T. Schmidt illustrates the importance of this Boolean triple arithmetic. In this paper we show that for any finite lattice A, we can ``coordinatize" A B, that is, represent A B as a subset of B ⁿ (where n is the number of join-irreducible elements of A), and provide an effective criteria to recognize the n-tuples of elements of B that occur in this representation. To show the utility of this coordinatization, we reprove a special case of the above result: for a finite simple lattice A, the lattice A B is a congruence-preserving extension of B.     

Mutation Pairs in Abelian Categories

A notion of mutation pairs of subcategories in an abelian category is defined in this article. For an extension closed subcategory 𝒵 and a rigid subcategory 𝒟 ? 𝒵, the subfactor category 𝒵/[𝒟] is also a triangulated category whenever (𝒵, 𝒵) forms a 𝒟-mutation pair. Moreover, if 𝒟 and 𝒵 satisfy certain conditions in modΛ, the category of finitely generated Λ-modules over an artin algebra Λ, the triangulated category 𝒵/[𝒟] has a Serre functor.     

K-Theory of Braided Monoidal Categories

We show that the commutativity constraint of a braided monoidal category gives rise to an algebraic structure on its K-theory known as a Gerstenhaber algebra. If, in addition, the braiding has a compatible balanced structure the Gerstenhaber bracket on the K-theory is generated by a Batalin–Vilkovisky differential. We use these algebraic structures to prove a generalization of the Anderson–Moore–Vafa theorem which says that the order of the twist, in a semi-simple balanced monoidal category with duals and finitely many simple objects, is finite.     

Algebras and Modules in Monoidal Model Categories

In recent years the theory of structured ring spectra (formerlyknown as A- and E-ring spectra) has been simplified by the discoveryof categories of spectra with strictly associative and commutativesmash products. Now a ring spectrum can simply be defined asa monoid with respect to the smash product in one of these newcategories of spectra. In this paper we provide a general methodfor constructing model category structures for categories ofring, algebra, and module spectra. This provides the necessaryinput for obtaining model categories of symmetric ring spectra,functors with smash product, Gamma-rings, and diagram ring spectra.Algebraic examples to which our methods apply include the stablemodule category over the group algebra of a finite group andunbounded chain complexes over a differential graded algebra.1991 Mathematics Subject Classification: primary 55U35; secondary18D10.     

Fusion Operators and Cocycloids in Monoidal Categories

The Yang–Baxter equation has been studied extensively in the context of monoidal categories. The fusion equation, which appears to be the Yang–Baxter equation with a term missing, has been studied mainly in the context of Hilbert spaces. This paper endeavours to place the fusion equation in an appropriate categorical setting. Tricocycloids are defined; they are new mathematical structures closely related to Hopf algebras.     

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.     

一类局部定向完备集及其范畴的性质

本文给出了局部定向完备集的概念及其在此结构下的一种新的双小于关系，从而进一步给出了一种新的连续性概念，接着讨论了局部定向完备集，连续的局部定向完备集等对象的一些性质，最后考察了三种范畴的笛卡儿闭性，并证明了范畴LDCPO是范畴ALG的反射满子范畴．     

General twisting of algebras

We introduce the concept of pseudotwistor (with particular cases called twistor and braided twistor) for an algebra (A,μ,u) in a monoidal category, as a morphism satisfying a list of axioms ensuring that (A,μ○T,u) is also an algebra in the category. This concept provides a unifying framework for various deformed (or twisted) algebras from the literature, such as twisted tensor products of algebras, twisted bialgebras and algebras endowed with Fedosov products. Pseudotwistors appear also in other topics from the literature, e.g. Durdevich's braided quantum groups and ribbon algebras. We also focus on the effect of twistors on the universal first order differential calculus, as well as on lifting twistors to braided twistors on the algebra of universal differential forms.     

Homotopy Categories for Simply Connected Torsion Spaces

For each n > 1 and each multiplicative closed set of integers S, we study closed model category structures on the pointed category of topological spaces, where the classes of weak equivalences are classes of maps inducing isomorphism on homotopy groups with coefficients in determined torsion abelian groups, in degrees higher than or equal to n. We take coefficients either on all the cyclic groups with s ∈ S, or in the abelian group where is the group of fractions of the form with s ∈ S. In the first case, for n > 1 the localized category is equivalent to the ordinary homotopy category of (n − 1)-connected CW-complexes whose homotopy groups are S-torsion. In the second case, for n > 1 we obtain that the localized category is equivalent to the ordinary homotopy category of (n − 1)-connected CW-complexes whose homotopy groups are S-torsion and the nth homotopy group is divisible. These equivalences of categories are given by colocalizations , obtained by cofibrant approximations on the model structures. These colocalization maps have nice universal properties. For instance, the map is final (in the homotopy category) among all the maps of the form Y → X with Y an (n − 1)-connected CW-complex whose homotopy groups are S-torsion and its nth homotopy group is divisible. The spaces , are constructed using the cones of Moore spaces of the form M(T, k), where T is a coefficient group of the corresponding structure of models, and homotopy colimits indexed by a suitable ordinal. If S is generated by a set P of primes and S ^p is generated by a prime p ∈ P one has that for n > 1 the category is equivalent to the product category . If the multiplicative system S is generated by a finite set of primes, then localized category is equivalent to the homotopy category of n-connected Ext-S-complete CW-complexes and a similar result is obtained for .     

On the Geometry of 2-Categories and their Classifying Spaces

In this paper we prove that realizations of geometric nerves are classifying spaces for 2-categories. This result is particularized to strict monoidal categories and it is also used to obtain a generalization of Quillen's Theorem A.