An order-theoretic perspective on categorial closure operators

This paper deals with an order-theoretic analysis of certain structures studied in category theory. A categorical closure operator (cco in short) is a structure on a category, which mimics the structure on the category of topological spaces formed by closing subspaces of topological spaces. Such structures play a significant role not only in categorical topology, but also in topos theory and categorical algebra. In the case when the category is a poset, as a particular instance of the notion of a cco, one obtains what we call in this paper a binary closure operator (bco in short). We show in this paper that bco’s allow one to see more easily the connections between standard conditions on general cco’s, and furthermore, we show that these connections for cco’s can be even deduced from the corresponding ones for bco’s, when considering cco’s relative to a well-behaved class of monorphisms as in the literature. The main advantage of the approach to such cco’s via bco’s is that the notion of a bco is self-dual (relative to the usual posetal duality), and by applying this duality to cco’s, independent results on cco’s are brought together. In particular, we can unify basic facts about hereditary closure operators with similar facts about minimal closure operators. Bco’s also reveal some new links between categorical closure operators, the usual unary closure and interior operators, modularity law in order and lattice theory, the theory of factorization systems and torsion theory.     

Every topological category is convenient for Gelfand duality

In this paper we generalize our work on Gelfand dualities in cartesian closed topological categories [42] to categories which are only monoidally closed. Using heavily enriched category theory we show that under very mild conditions on the base category function algebra functor and spectral space functor exist, forming a pair of adjoint functors and establishing a duality between function algebras and spectral spaces. Using recent results in connection with semitopological functors, we show that every (E,M)-topological category is endowed with at least oneconvenient monoidal structure admitting a generalized Gelfand duality. So it turns out that there is no need for a cartesian closed structure on a topological category in order to study generalized Gelfand-Naimark dualities.     

Categorical anatomy of closed graph and open mapping theorems

Recent development of the theory of general topological vector spaces (without local convexity, hence without duality theory), as in [1], has exposed more of the essential (distinctly topological) nature of a good deal of the locally convex theory, resulting in a remarkably high survival rate of theorems, with (it is claimed) greater simplicity and elegance of proofs. This is especially true of the closed graph theorem. Associated concepts, such as barrelledness and the various kinds of completeness, can be described and related in a manner which demands extension to uniform spaces and beyond. In the setting of categories and functors, we believe these ideas acquire an elegance and unity which considerably widens their scope while shedding light on their homeground in functional analysis. The aim of this paper, which takes a much broader perspective than Pelletier [10], is to distill some of this categorical essence, and to point out new problems, both topological and categorical, which present themselves along the way.     

Remarks on Priestley duality for distributive lattices

The notion of a Priestley relation between Priestley spaces is introduced, and it is shown that there is a duality between the category of bounded distributive lattices and 0-preserving join-homomorphisms and the category of Priestley spaces and Priestley relations. When restricted to the category of bounded distributive lattices and 0-1-preserving homomorphisms, this duality yields essentially Priestley duality, and when restricted to the subcategory of Boolean algebras and 0-preserving join-homomorphisms, it coincides with the Halmos-Wright duality. It is also established a duality between 0-1-sublattices of a bounded distributive lattice and certain preorder relations on its Priestley space, which are called lattice preorders. This duality is a natural generalization of the Boolean case, and is strongly related to one considered by M. E. Adams. Connections between both kinds of dualities are studied, obtaining dualities for closure operators and quantifiers. Some results on the existence of homomorphisms lying between meet and join homomorphisms are given in the Appendix.     

向量优化问题的对偶性(英文)

Duality framework on vector optimization problems in a locally convex topological vector space are established by using scalarization with a cone-strongly increasing function.The dualities for the scalar convex composed optimization problems and for general vector optimization problems are studied.A general approach for studying duality in vector optimization problems is presented.     

Coalgebraic representations of distributive lattices with operators

We present a framework for extending Stone's representation theorem for distributive lattices to representation theorems for distributive lattices with operators. We proceed by introducing the definition of algebraic theory of operators over distributive lattices. Each such theory induces a functor on the category of distributive lattices such that its algebras are exactly the distributive lattices with operators in the original theory. We characterize the topological counterpart of these algebras in terms of suitable coalgebras on spectral spaces. We work out some of these coalgebraic representations, including a new representation theorem for distributive lattices with monotone operators.     

Functorial methods in the theory of group representations I

We introduce a candidate for the group algebra of a Hausdorff group which plays the same role as the group algebra of a finite group. It allows to define a natural bijection betweenk-continuous representations of the group in a Hilbert space and continuous representations of the group algebra. Such bijections are known, but to our knowledge only for locally compact groups. We can establish such a bijection for more general groups, namely Hausdorff groups, because we replace integration techniques by functorial methods, i.e., by using a duality functor which lives in certain categories of topological Banach balls (resp., unit balls of Saks spaces).This paper was written while the authors were supported by the Swiss National Science Foundation under grant 21-33644.92.     

Group duality with the topology of precompact convergence

The Pontryagin-van Kampen (P-vK) duality, defined for topological Abelian groups, is given in terms of the compact-open topology. Polar reflexive spaces, introduced by Köthe, are those locally convex spaces satisfying duality when the dual space is equipped with the precompact-open topology. It is known that the additive groups of polar reflexive spaces satisfy P-vK duality. In this note we consider the duality of topological Abelian groups when the topology of the dual is the precompact-open topology. We characterize the precompact reflexive groups, i.e., topological groups satisfying the group duality defined in terms of the precompact-open topology. As a consequence, we obtain a new characterization of polar reflexive spaces. We also present an example of a space which satisfies P-vK duality and is not polar reflexive. Some of our results respond to questions appearing in the literature.     

Theory of spectral sequences. II

In this paper, we continue to discuss the theory of spectral sequences in Abelian categories. In this connection, attention is paid to the manifestation of different dualities in the theory of spectral sequences. The duality in locally convex, topological vector spaces is of special interest. We use results of functional analysis for solving (co)homology problems (theorems on nondegenerate pairing (co)homology) of manifolds. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 5, pp. 117–149, 2005.     

Smooth group representations on bornological vector spaces

We develop the basic theory of smooth representations of locally compact groups on bornological vector spaces. In this setup, we are able to formulate better general theorems than in the topological case. Nonetheless, smooth representations of totally disconnected groups on vector spaces and of Lie groups on Fréchet spaces remain special cases of our theory. We identify smooth representations with essential modules over an appropriate convolution algebra. We examine smoothening functors on representations and modules and show that they agree if they are both defined. We establish the basic properties of induction and compact induction functors using adjoint functor techniques. We describe the center of the category of smooth representations.     

CATEGORICAL HOMOTOPY

We present a homotopy theory of small categories. In a work of this nature there is a need to give a theory which is clear and which shows the methods of work in this field. It is also necessary to prove theorems which place the theory within the general framework of homotopy, i.e. particularly to liaise with the homotopy of topological spaces and with abstract homotopy theories. Firstly we define the important notion of finite functor on which the theory is based. Next we introduce a type of fibred category fitting to the work on homotopy. After having studied the paths and loops of a category, we consider homotopy between functors. Finally, we demonstrate the possibility of obtaining homotopy groups before taking into consideration the relations between categorical and topological homotopy.     

Topological duality and lattice expansions,I: A topological construction of canonical extensions

The two main objectives of this paper are (a) to prove purely topological duality theorems for semilattices and bounded lattices, and (b) to show that the topological duality from (a) provides a construction of canonical extensions of bounded lattices. In previously known dualities for semilattices and bounded lattices, the dual spaces are compact 0-dimensional spaces with additional algebraic structure. For example, semilattices are dual to 0-dimensional compact semilattices. Here we establish dual categories in which the spaces are characterized purely in topological terms, with no additional algebraic structure. Thus the results can be seen as generalizing Stone’s duality for distributive lattices rather than Priestley’s. The paper is the first of two parts. The main objective of the sequel is to establish a characterization of lattice expansions, i.e., lattices with additional operations, in the topological setting built in this paper.     

TOPOLOGICAL CATEGORIES AND CLOSURE OPERATORS

Abstract

It is shown that the category CS of closure spaces is a topological category. For each epireflective subcategory A of a topological category X a functor F _A :X → X is defined and used to extend to the general case of topological categories some results given in [4], [5] and [10] for epireflective subcategories of the category Top of topological spaces.     

Duality between compactness and discreteness beyond pontryagin duality

One of the most striking results of Pontryagin’s duality theory is the duality between compact and discrete locally compact abelian groups. This duality also persists in part for objects associated with noncommutative topological groups. In particular, it is well known that the dual space of a compact topological group is discrete, while the dual space of a discrete group is quasicompact (i.e., it satisfies the finite covering theorem but is not necessarily Hausdorff). The converse of the former assertion is also true, whereas the converse of the latter is not (there are simple examples of nondiscrete locally compact solvable groups of height 2 whose dual spaces are quasicompact and non-Hausdorff (they are T ₁ spaces)). However, in the class of locally compact groups all of whose irreducible unitary representations are finite-dimensional, a group is discrete if and only if its dual space is quasicompact (and is automatically a T ₁ space). The proof is based on the structural theorem for locally compact groups all of whose irreducible unitary representations are finite-dimensional. Certain duality between compactness and discreteness can also be revealed in groups that are not necessarily locally compact but are unitarily, or at least reflexively, representable, provided that (in the simplest case) the irreducible representations of a group form a sufficiently large family and have jointly bounded dimensions. The corresponding analogs of compactness and discreteness cannot always be easily identified, but they are still duals of each other to some extent.     

Representation and duality for Hilbert algebras

In this paper we introduce a special kind of ordered topological spaces, called Hilbert spaces. We prove that the category of Hilbert algebras with semi-homomorphisms is dually equivalent to the category of Hilbert spaces with certain relations. We restrict this result to give a duality for the category of Hilbert algebras with homomorphisms. We apply these results to prove that the lattice of the deductive systems of a Hilbert algebra and the lattice of open subsets of its dual Hilbert space, are isomorphic. We explore how this duality is related to the duality given in [6] for finite Hilbert algebras, and with the topological duality developed in [7] for Tarski algebras.      

Closure operators in topological groups related to von Neumann's kernel

We study a family of idempotent categorical closure operators in the category of topological Abelian groups (and continuous homomorphisms) related to the von Neumann's kernel. The prominent role is played by the idempotent closure operator g also related to questions from Diophantine approximations and ergodic theory.