Finite-product-preserving functors,Kan extensions,and strongly-finitary 2-monads

We study those 2-monads on the 2-categoryCat of categories which, as endofunctors, are the left Kan extensions of their restrictions to the sub-2-category of finite discrete categories, describing their algebras syntactically. Showing that endofunctors of this kind are closed under composition involves a lemma on left Kan extensions along a coproduct-preserving functor in the context of cartesian closed categories, which is closely related to an earlier result of Borceux and Day.The first author gratefully acknowledges the support of the Australian Research Council.     

NETS AND GRAPHS AS PINAL COMPLETIONS AND THE RELATIONS BETWEEN THEM

Abstract

Nets and graphs, both used in Computer Science, are studied from a categorical point of view. It is shown that they may be constructed via final completions of very simple small concrete categories and that their nice properties, namely to form topological categories which are quasitopoi with concrete powers such that products of final maps are final, depend on this fact. Furthermore, the relations between them can be described by means of adjoint functors.     

INSTANCES AND RAMIFICATIONS OF THE SEMI-ADJOINT SITUATION II. THE COMPARISON FUNCTOR

Abstract

If a functor U has a left co-unadjoint then U can be factored through a category of semad algebras. An analogue of the Beck monadicity theory is obtained. If R is a ring without a left unit but satisfying R² = R then the category of unitary left R-modules need not be monadic over Set. The forgetful functor has, however, a left co-unadjoint for which a comparison functor is an equivalence of categories. Another example of a semadic functor is obtained by composing the forgetful functor from Abelian groups to Set with the doubling functor. The semi-adjoint situations in the senses of Medvedev and Davis are examined.     

Solid hulls of concrete categories

This paper deals with the problem of the existence of solid hulls for concrete categories. We present sufficient conditions for a concrete category to have a solid hull. For concrete categories over Set with a small finally dense subcategory, we observe that the existence of solid hulls is equivalent to Weak Vopenka's Principle.Research partially supported by TEMPUS JEP 2692 and by Centro de Matemática da Universidade de Coimbra     

A NOTE ON REGULAR AND EXTREYAL SUBOBJECTS

ABSTRACT

In the general setting of a complete, well-powered category A, we define and study two universal closure operations: regularization and extremalization, by means of regular and extremal subobjects of A. respectively. A general theorem of characterization of epimorphisms in A is given. When A is an epireflective subcategory of TOP, such operations are shown to coincide with A-closure [11] and epiclosure [2]. respectively. In the topological contest, regularization and extremalization are studied in detail and compared with r-closure, defined in [13].     

CONNECTEDNESS AND DISCONNECTEDNESS: A DIFFERENT PERSPECTIVE

Abstract

A factorization of a Galois connection investigated earlier is used to give a definition of a connectedness-disconnectedness Galois connection that is free of the notion of constant morphism. A new notion of N-fixed morphism with respect to a class N of monomorphisms is presented. This is used to characterize the connectedness-disconnectedness Galois connection in the case that N is closed under the formation of pullbacks. Some closedness properties of these Galois connections are investigated.     

CONNECTEDNESS,DISCONNECTEDNESS AND CLOSURE OPERATORS: FURTHER RESULTS

Abstract

Let X be an arbitrary category with an (E, M)-factorization structure for sinks. A notion of constant morphism that depends on a chosen class of monomorphisms was previously used to provide a generalization of the connectedness-disconnectedness Galois connection (also called torsion-torsion free in algebraic contexts). This Galois connection was shown to factor through the class of all closure operators on X with respect to M. Here, properties and implications of this factorization are investigated. In particular, it is shown that this factorization can be further factored. Examples are provided.     

A note on effective descent morphisms of topological spaces and relational algebras

We formulate two open problems related to and, in a sense, suggested by the Reiterman-Tholen characterization of effective descent morphisms of topological spaces.     

Two-sided hyperbolic SVD

In this paper, we propose the two-sided hyperbolic SVD (2HSVD) for square matrices, i.e., A=UΣV^[∗], where U and V^[∗] are J-unitary (J=diag(±1)) and Σ is a real diagonal matrix of “double-hyperbolic” singular values. We show that, with some natural conditions, such decomposition exists without the use of hyperexchange matrices. In other words, U and V^[∗] are really J-unitary with regard to J and not some matrix which is permutationally similar to matrix J. We provide full characterization of 2HSVD and completely relate it to the semidefinite J-polar decomposition.     

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.     

SEMI-TOPOLOGICAL FUNCTORS THAT INDUCE IDEMPOTENT MONADS

Abstract

The aim of this paper is to describe some “topological” subclasses of the class of all semi-topological functors. This is done by considering intersections of certain classes: on the one hand, the classes of all semi-topological functors, all topologically-algebraic functors and all regular functors; and on the other hand, the class of all discrete functors (see [6]). The relations between these and other classes of functors are given along with the necessary examples. Finally, the resulting intersections are described by giving representations of the functors that belong to them.     

Some counting questions for matrices with restricted entries

We obtain upper bounds for the number of arbitrary and symmetric matrices with integer entries in a given box (in an arbitrary location) and a given determinant. We then apply these bounds to estimate the number of matrices in such boxes which have an integer eigenvalues. Finally, we outline some open questions.     

FACTORIZATIONS OF CONES ALONG A FUNCTOR

Abstract

First a general Galois correspondence is established, which generalizes at the same time the correspondence between classes of monomorphisms and injective objects and the correspondence between classes of epimorphisms and monomorphisms in a category. This correspondence arises naturally if one tries to generalize some concepts of “topological” or also of “algebraic” functors. Both kinds of functors admit certain factorizations of cones, and just this fact implies some of their common nice properties: lifting limits, continuity and faithfulness, for instance. These properties can be shown without having a left adjoint. Therefore the theory yields also applications to functors which are neither “topological” nor “algebraic”.     

COMPLETIONS AND CATEGORICAL COMPACTNESS FOR NILPOTENT GROUPS

Abstract

We consider reflection functors in the category of nilpotent groups satisfying certain exactness properties for which the Mal'cev completion functor and the p-cotorsion completion functors are prototypical examples. Each of these functors defines a generalized torsion theory, which in turn defines a closure operator on subgroups. This gives rise to the notion of a categorically compact group with respect to the closure operator which we characterize. This approach provides a unified treatment for the categorically compact groups with respect to the Mal'cev completion and with respect to the p-cotorsion completion, the latter being new. We also consider the p-pro-finite completion, suitably restricted to obtain a reflection functor, and characterize the compact groups so arising.     

Exactness of direct limits for abelian categories with an injective cogenerator

We prove that the exactness of direct limits in an abelian category with products and an injective cogenerator J is equivalent to a condition on J which is well-known to characterize pure-injectivity in module categories, and we describe an application of this result to the tilting theory. We derive our result as a consequence of a more general characterization of when inverse limits in the Eilenberg–Moore category of a monad on the category of sets preserve regular epimorphisms.     

ORDER EXTENSIONS AS ADJOINT FUNCTORS

Abstract

A standard extension (resp. standard completion) is a function Z assigning to each poset P a (closure) system ZP of subsets such that x ? y iff x belongs to every Z ε ZP with y ε Z. A poset P is Z -complete if each Z ε 2P has a join in P. A map f: P → P′ is Z—continuous if f^?1 [Z′] ε ZP for all Z′ ε ZP′, and a Z—morphism if, in addition, for all Z ε ZP there is a least Z′ ε ZP′ with f[Z] ? Z′. The standard extension Z is compositive if every map f: P → P′ with {x ε P: f(x) ? y′} ε ZP for all y′ ε P′ is Z -continuous. We show that any compositive standard extension Z is the object part of a reflector from IP_Z, the category of posets and Z -morphisms, to IR_Z, the category of Z -complete posets and residuated maps. In case of a standard completion Z, every Z -continuous map is a Z -morphism, and IR2 is simply the category of complete lattices and join—preserving maps. Defining in a suitable way so-called Z -embeddings and morphisms between them, we obtain for arbitrary standard extensions Z an adjunction between IP_Z and the category of Z -embeddings. Many related adjunctions, equivalences and dualities are studied and compared with each other. Suitable specializations of the function 2 provide a broad spectrum of old and new applications.     

Prefilter spaces and a precompletion of preuniform convergence spaces related to some well-known completions

In non-symmetric Convenient Topology the notion of pre-Cauchy filter is introduced and the construction of a precompletion of a preuniform convergence space is given from which Wyler's completion of a separated uniform limit space [O. Wyler, Ein Komplettierungsfunktor für uniforme Limesräume, Math. Nachr. 46 (1970) 1-12] as well as Weil's Hausdorff completion of a separated uniform space [A. Weil, Sur les Espaces à Structures Uniformes et sur la Topologie Générale, Hermann, Paris, 1937] can be derived (up to isomorphism). By the way, the construct PFil of prefilter spaces, i.e. of those preuniform convergence space which are ‘generated’ by their pre-Cauchy filters, is a strong topological universe filling in a gap in the theory of preuniform convergence spaces.     

Torsion theories and Galois coverings of topological groups

For any torsion theory in a homological category, one can define a categorical Galois structure and try to describe the corresponding Galois coverings. In this article we provide several characterizations of these coverings for a special class of torsion theories, which we call quasi-hereditary. We describe a new reflective factorization system that is induced by any quasi-hereditary torsion theory. These results are then applied to study various examples of torsion theories in the category of topological groups.