Function spaces in metrically generated theories

The theory of metrically generated constructs provides us with an excellent setting for the study of function spaces. In this paper we develop a function space theory for metrically generated constructs and, by considering different metrically generated constructs, we capture interesting examples. For instance, for uniform spaces we retrieve the uniformity of uniform convergence and its generalization to Σ-convergence and for UG-spaces we obtain a quantified version of these structures. Our theory also allows for many applications, in particular we are able to characterize the complete subspaces of these function spaces and we succeed in producing an appropriate Ascoli theorem.     

Bornologies and metrically generated theories

Bornologies axiomatize an abstract notion of bounded sets and are introduced as collections of subsets satisfying a number of consistency properties. Bornological spaces form a topological construct, the morphisms of which are those functions which preserve bounded sets. A typical example is a bornology generated by a metric, i.e. the collection of all bounded sets for that metric. In a recent paper [E. Colebunders, R. Lowen, Metrically generated theories, Proc. Amer. Math. Soc. 133 (2005) 1547-1556] the authors noted that many examples are known of natural functors describing the transition from categories of metric spaces to the “metrizable” objects in some given topological construct such that, in some natural way, the metrizable objects generate the whole construct. These constructs can be axiomatically described and are called metrically generated. The construct of bornological spaces is not metrically generated, but an important large subconstruct is. We also encounter other important examples of metrically generated constructs, the constructs of Lipschitz spaces, of uniform spaces and of completely regular spaces. In this paper, the unified setting of metrically generated theories is used to study the functorial relationship between these constructs and the one of bornological spaces.     

Topological theories and closed objects

Recent work of several authors shows that many categories of interest to topologists can be represented as categories of lax algebras. In this paper we introduce the concept of a topological theory as a syntactical tool to deal with lax algebras, and show the usefulness of our approach by applying it to the study of function spaces.     

Order-adjoint monads and injective objects

Given a monad T on whose functor factors through the category of ordered sets with left adjoint maps, the category of Kleisli monoids is defined as the category of monoids in the hom-sets of the Kleisli category of T. The Eilenberg-Moore category of T is shown to be strictly monadic over the category of Kleisli monoids. If the Kleisli category of T moreover forms an order-enriched category, then the monad induced by the new situation is Kock-Zöberlein. Injective objects in the category of Kleisli monoids with respect to the class of initial morphisms then characterize the objects of the Eilenberg-Moore category of T, a fact that allows us to recuperate a number of known results, and present some new ones.     

Exponential objects and Cartesian closedness in the constructPrtop

We give an internal characterization of the exponential objects in the constructPrtop and investigate Cartesian closedness for coreflective or topological full subconstructs ofPrtop. If $ is the set {0} {1/n;n 1} endowed with the topology induced by the real line, we show that there is no full coreflective subconstruct ofPrtop containing $ and which is Cartesian closed. With regard to topological full subconstructs ofPrtop we give an example of a Cartesian closed one that is large enough to contain all topological Fréchet spaces and allT ₁ pretopological Fréchet spaces.Aspirant NFWO     

On categorical notions of compact objects

Due to the nature of compactness, there are several interesting ways of defining compact objects in a category. In this paper we introduce and study an internal notion of compact objects relative to a closure operator (following the Borel-Lebesgue definition of compact spaces) and a notion of compact objects with respect to a class of morphisms (following Áhn and Wiegandt [2]). Although these concepts seem very different in essence, we show that, in convenient settings, compactness with respect to a class of morphisms can be viewed as Borel-Lebesgue compactness for a suitable closure operator. Finally, we use the results obtained to study compact objects relative to a class of morphisms in some special settings.Partial financial assistance by Centro de Matemática da Universidade de Coimbra and by a NATO Collaborative Grant (CRG 940847) is gratefully acknowledged.     

Towards Stone duality for topological theories

In the context of categorical topology, more precisely that of T-categories (Hofmann, 2007 [8]), we define the notion of T-colimit as a particular colimit in a V-category. A complete and cocomplete V-category in which limits distribute over T-colimits, is to be thought of as the generalisation of a (co-)frame to this categorical level. We explain some ideas on a T-categorical version of “Stone duality”, and show that Cauchy completeness of a T-category is precisely its sobriety.     

LEGENDRE TRANSFORMATIONS AND CARTAN FORMS IN THE CALCULUS OF VARIATIONS OF MULTIPLE INTEGRALS

ABSTRACT

(PART II): In terms of a given Hamiltonian function the 1-form w = dH + ?_j|dπ^j is defined, where {?_j:j = 1,…, n} denotes an invariant basis of the planes of the distribution D_n. The latter is said to be canonical if w = 0 (which is analogous to the definition of Hamiltonian vector fields in symplectic geometry). This condition is equivalent to two sets of canonical equations that are expressed explicitly in term of the derivatives of H with respect to its positional arguments. The distribution D_n is said to be pseudo-Lagrangian if dπ^j(?_j,V_h) = 0; if D_n, is both canonical and pseudo-Lagrangian it is integrable and such that H = const. on each leaf of the resulting foliation. The Cartan form associated with this construction [9] is defined a II = π² ? ? πⁿ. If π is closed, the distribution D_N is integrable, and the exterior system {π^j} admits the representation ψ^j = dS^j in terms of a set of 0-forms S^j on M. If, in addition, the distribution D_N is canonical, these functions satisfy a single first order Hamilton-Jacobi equation, and conversely. Finally, a complete figure is constructed on the basis of the assumptions that (i) the Cartan form be closed, and (ii) that the distribution D_n, be both canonical and integrable. The last of these requirements implies the existence of N functions ψ^A that depend on x^h and N parameters w^B, whose derivatives are given by ?ψ^A (x^h, w^B)/?x^j = B^A _j (x^h, ψ^B (x^h,w^B)). The complete figure then consists of two complementary foliations: the leaves of the first are described by the functions ψ^A and satisfy the standard Euler-Lagrange equations, while the second, that is, the transversal foliation, is represented by the aforementioned solution of the Hamilton-Jacobi equation. The entire configuration then gives rise in a natural manner to a generalized Hilbert independent integral and consequently also to a generalized Weierstrass excess function.     

Zariski closure, completeness and compactness

The categorical theory of closure operators is used to introduce and study separated, complete and compact objects with respect to the Zariski closure operator naturally defined in any category X(A,Ω) obtained by a given complete category X (endowed with a proper factorization structure for morphisms) and by a given X-algebra (A,Ω) by forming the affine X-objects modelled by (A,Ω). Several basic examples are provided.     

Quotient-reflective and bireflective subcategories of the category of preordered sets

In previous papers, the notions of “closedness” and “strong closedness” in set-based topological categories were introduced. In this paper, we give the characterization of closed and strongly closed subobjects of an object in the category Prord of preordered sets and show that they form appropriate closure operators which enjoy the basic properties like idempotency (weak) hereditariness, and productivity.We investigate the relationships between these closure operators and the well-known ones, the up- and down-closures. As a consequence, we characterize each of T₀, T₁, and T₂ preordered sets and show that each of the full subcategories of each of T₀, T₁, T₂ preordered sets is quotient-reflective in Prord. Furthermore, we give the characterization of each of pre-Hausdorff preordered sets and zero-dimensional preordered sets, and show that there is an isomorphism of the full subcategory of zero-dimensional preordered sets and the full subcategory of pre-Hausdorff preordered sets. Finally, we show that both of these subcategories are bireflective in Prord.     

Some remarks on interior operators and the functional property

In this paper functoriality of the notion of categorical interior operator is discussed and as a related topic, the property of ?-modal interior operator is introduced.     

SOME MODULES OVER A B*-ALGEBRA

Abstract

Some modules, ?_p (B) (1 < p < ∞), over a commutative B*-alge= bra with identity are discussed and these are seen to have properties similar to the Banach spaces ?_p (1 < p < ∞).     

Hausdorff separation in categories

Considering subobjects, points and a closure operator in an abstract category, we introduce a generalization of the Hausdorff separation axiom for topological spaces: the notion ofT ₂-object. We discuss the properties ofT ₂-objects, which depend essentially on the behaviour of points, and finally we relate them to the well-known separated objects.The results of this paper are essentially taken from the author's Ph. D. Thesis written under the supervision of Professors M. Sobral and W. Tholen and partially supported by a scholarship of I.N.I.C.-Instituto Nacional de Investigação Científica.     

On the largest coreflective Cartesian closed subconstruct of Prtop

We show that the subconstruct Fing of Prtop, consisting of all finitely generated pretopological spaces, is the largest Cartesian closed coreflective subconstruct of Prtop. This implies that in any coreflective subconstruct of Prtop, exponential objects are finitely generated. Moreover, in any finitely productive, coreflective subconstruct, exponential objects are precisely those objects of the subconstruct that are finitely generated. We give a counterexample showing that without finite productivity the previous result does not hold.     

Exponentiation for unitary structures

Motivated by the observation that both pretopologies and preapproach limits can be characterized as those convergence relations which have a unit for a suitable composition, we introduce the category Algu(T;V) of reflexive and unitary lax algebras, for a symmetric monoidal closed lattice V and a Set-monad T=(T,e,m). For T=U the ultrafilter monad, we characterize exponentiable morphisms in Algu(U;V). Further, we give a sufficient condition for an object to be exponentiable in the category Alg(U;V) of reflexive and transitive lax algebras. This specializes to known and new results for pretopological, preapproach and approach spaces.     

Some open categorical problems inTop

In this paper, a pendant to a recent survey paper, the authors discuss several open problems in categorical topology. The emphasis is on topology-oriented problems rather than on more general category-oriented ones. In fact, most problems deal with full subconstructs or superconstructs of the constructTop of topological spaces and continuous maps.     

Almost coreflective and projective classes

This paper summarizes the situation around the problem of when classes of projective objects are almost coreflective, both in general categories and in Top or similar categories. In addition to known results, several new contributions and examples are added.For the sixtieth birthday of D. PumplünThe paper was written while the second author was visiting the University of Toledo, Ohio.