DISCRETE FUNCTORS

Abstract

The concept of a T-discrete object is a generalization of the notion of discrete spaces in concrete categories. In this paper. T-discrete objects are used to define discrete functors. Characterizations of discrete functors are given and their relation to other important functors are studied. A faithful functor T: A → X is discrete iff the full subcategory B of A consisting of all T-discrete objects is (X-iso)-coreflective in A. It follows that the existence of bicoreflective subcategories is equivalent to the existence of suitable discrete functors. Finally, necessary and sufficient conditions are found such that for a given functor T: A → X, the full subcategory B of A consisting of all T-discrete A-objects is monocoreflective in A.     

VARIETAL HULLS OF FUNCTORS

Abstract

The well known characterizations of equational classes of algebras with not necessaryly finitary operations by FELSCHER [6.7] and of categories of A-algebras for algebraic theories A in the sense of LINTON [10], esp., by means of their forgetful functors are the foundations of a concept of varietal functors U:K → L over arbitrary basecategories L. They prove to be monadic functors which satisfy an additional HOM-condition [17]. (In the case L = Set this condition is always fulfilled, see LINTON [11].)

Contrary to monadic functors, varietal functors are closed under composition. Pleasent algebraic properties of the base-category L can be ‘lifted’ along varietal functors, such as e.g. factorization properties, (co-) completeness, classical isomorphism theorems, etc.

By means of the well known EILENBERG-MOORE-algebras there is a universal monadic functor U^T:L ^T → L for any functor U: K → L, having a left adjoint F (T: = UF). But, in general, UT is not varietal. Under some suitable conditions, however it is possible, to construct a canonical varietal functor ?:R → L, the varietal hull of U. This hull has much more interesting (algebraic) properties than the EILENBERG-MOORE construction. Moreover, results of BANASCHEWSKI-HERRLICH [2] are extended.     

EPIREFLECTIVE SUBCATEGORIES AND SEMICLOSURE OPERATORS

ABSTRACT

Let C be a category of topological spaces and continuous functions which is full, hereditary and closed under homeomorphisms and products. If A is a subclass of C, let E(A) be the full subcategory of C whose objects are the subspaces in A. In this paper we characterize the epireflective subcategories of C containing A and contained in E(A) by introducing a “semiclosure” operator which is a generalization for the “idempotent semi-limit” operator introduced by S.S. Hong (see [5]) with respect to Top _o. In case A is extensive in C, so that E(A) = C, all the extensive subcategories of C containing A are thus characterized.     

EPIREFLECTIONS VERSUS BIREFLECTIONS FOR TOPOLOGICAL FUNCTORS

Abstract

In this paper it is proved that if T: A → X is a topological functor satisfying certain conditions, then there is a Galois Connection between the class of bireflective subcategories of A and the class of epireflective subcategories of A that are not bireflective and that are contained in the subcategory of separated objects of A. In general such a correspondence is not bijective.     

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.     

LIGHT FACTORIZATION STRUCTURES

Abstract

In this paper we investigate, for connection subcategories A of a topological category K, the concepts of A-monotone quotients and A-light sources, and characterize (1) those A, which give rise to an (A-monotone quotient, A-light)- factorization structure on K, (2) those factorization structures (C,D) on K, which are light, i.e. of the form (A-monotone quotient, A-light) for suitable A. It turns out that light factorization structures are rather rare in Top, but abundant and well-behaved in categories with hereditary quotients.     

Triple Cohomology and the Galois Cohomology of Profinite Groups

Given a cotriple 𝔾 = (G, ε, δ) on a category X and a functor E:X ^Opp→A into an abelian category A, there exists the cohomology theory of Barr and Beck: Hⁿ(X, E) ε |A| (n ≥ 0, X ε |X|), ([1], p.249). Almost all the important cohomology theories in mathematics have been shown to be special instances of such a general theory (see [1], [2] and [3]). Usually E arises from an abelian group object Y in X in the following manner: it is the contravariant functor from X into the category Ab of abelian groups that associates to each object X in X the abelian group X(X, Y) of maps from X to Y. In such a situation we shall write Hⁿ(X, Y)_𝔾 instead of Hⁿ(X, E)_G. Barr and Beck [2] have shown that the Eilenberg-MacLane cohomology groups H?ⁿ(π, A), n ≥ 2, can be re-captured as follows. One considers the free group cotriple 𝔾′ on the category Gps of groups, which induces in a natural manner a cotriple 𝔾 on the category (Gps, π) of groups over a fixed group π.     

On the Homological Dimension of Algebras of Differential Operators

Let A be a commutative algebra over a field k, and V_A be the k-subalgebra of End_k(A) generated by End_A(A) = A and all k-derivations of A. A study of the homological properties of V_A was initiated by Hochschild, Kostant, and Rosenberg in [5], and continued by Rinehart [8], [9], Roos [11], Björk [1], Rinehart and Rosenberg [10], and others. It was proved in [5] that, if k is perfect and A is a regular affine algebra of dimension r, then the global dimension of V_A is between r and 2r. Moreover, if k has positive characteristic, then gl.dim V_A = 2r [8]. By a recent celebrated theorem of Roos [11], gl.dim V_A = r if k has characteristic zero and A = k[x₁, …, x_r]; in this case V_A is the so-called “Weyl algebra on 2r variables”.     

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.     

ON CATEGORIES OF MONOID-ACTIONS

Abstract

Given a monoidal category B and a category S of monoids in B we study the category MOD_S of all actions of monoids from S on B-objects. This is mainly done by investigation of the underlying functor V: MOD_S → SxB. In particular V creates limits; filtered colimits and arbitrary colimits are detected, provided the monoidal structure behaves nicely with respect to these constructions. Moreover MOD_S contains B as a full coreflective subcategory; S is contained as a full reflective (and coreflective) one provided B has a terminal (zero) object. Monadicity of MOD_S over B is discussed as well.     

THE LARGEST NON-TRIVIAL TORSION CLASS OF MODULES

ABSTRACT

In this paper we investigate the following two classes of left R-modules: N(P) ={A|A has no non-zero direct summand P ε P} and H(p) = {A} if B ? A with B ε N(P), then B = 0}, where P is a class of projective R-modules. We demonstrate that N(p) is, in general, not a torsion class but that H(P) is always a torsionfree class. We also investigate those classes P and rings R for which N(P) is the largest non-trivial torsion class of R-modules.     

CONES AND COMPARISONS IN IND-AFFINE HOMOTOPY THEORY

Abstract

Ind-affine schemes over an algebraically closed field k are introduced. The cone functor is then defined and characterized in the based category (ind-aff)* of ind-affine schemes. Homotopy theories, one induced from the monad related to the cone functor and the other via unirational and then singular simplices, are compared. Some homotopy groups vis-a-vis (ind-aff)* taking as our model of the circle the set of points (x,y) in k² satisfying x²+y² = 1 are determined.     

HILBERT SPACES OF FUNCTIONALS ASSOCIATED WITH NONLINEAR OPERATORS

Abstract

In [7] the subject of reproducing kernel Hilbert spaces (RKHSs) of linear functionals associated with linear operators and, in particular, with second-order generalized stochastic processes (GSPs), is pursued. In this work these ideas are extended to nonlinear operators. As an example the characteristic operator of a GSP is pursued. The so-called nonlinear space of the process associated with the characteristic operator is investigated and the RKHS of functionals isometrically isomorphic to it is constructed. Unlike the linear space, the nonlinear analysis is not limited to second order GSPs.     

V-localizations andV-Kleisli algebras

We prove a pushout theorem for localizations and Kleisli categories over a symmetric monoidal closed categoryV. That is, suppose is aV-localizable subcategory of aV-categoryA and thatT=(T,,) is aV-monad onA. Then under suitable relations betweenT and we show that there is aV-monadT induced onA[-¹] such that the Kleisli category ofT is the pushout of the localization functor :AA[-¹] and the free functor F:AK(T). Consequently,K(T)K(T) [S-¹] for some S K(T). We give several examples of this situation.     

A Characterization of Group Rings

In this paper we prove that every coseparable involutory Hopf algebra over the ring of integers Z which is a free Z-module is the group ring of some group. This result was proved independently for Hopf algebras which are finitely generated Z-modules by H.-J. Schneider [6], using similar techniques. We then give some examples of coseparable Hopf algebras over number rings which are not group algebras, and give an example of a cocommutative coseparable coalgebra over a number ring which cannot be given a multiplicative structure making it into a Hopf algebra. The Hopf algebra structure theory required for this paper is found in [1], [4], and [5]. For completeness we give proofs here of the coalgebra analogues to some “well-known” facts about separable algebras.     

AN AXIOMATIC APPROACH TO CATEGORIES OF TOPOLOGICAL ALGEBRAS

Abstract

Consider a commuting square of functors TV = GU where G is an algebraic functor over sets (in the sense of Herrlich), and T and U are (regular epi, monosource)—topological and fibre small. Such a square is called a Topological Algebraic Situation (TAS) when the following two conditions are satisfied:

1. if h: UA → UB and g: VA → VB are morphisms with Gh = Tg, there exists a morphism f: A → B such that Uf = h and Vf = g;

2. V carries U-initial monosources into T-initial mono-sources.

The functor V has many nice properties which shed light on the blending of the “topology” and “algebra”; e.g., V is a topologically algebraic functor in the sense of Y.H. Hong. An ([Etilde],[Mtilde]) version of O. Wyler's “Taut Lift Theorem” is used to show that the existence of a left adjoint to V is related to Condition (ii). It is also shown that certain topological algebraic reflections arise as Topological Algebraic Situations from algebraic and topological surjective reflections.     

Topological Rings of Quotients and Rings Without Open Left Ideals

Simple locally compact rings without open left ideals were considered in [13] and general locally compact rings without open left ideals were studied extensively in [5] and [6]. We remove the hypothesis of local compactness and consider topological rings A without open left ideals but containing an open subring R. In section 4 we show that under these conditions A is completely determined by R. More precisely A can be identified with the topological ring of quotients C(R) introduced in [8]. As an R-module _RA is topologically isomorphic to I_*(_RR), the topological injective hull of _RR. The last statement was proved in [6] for A locally compact and R compact. Section 5 gives a characterization of those linearly topologized rings R that can be openly embedded into a ring A without open left ideals. In particular we shall show that the open left ideals form an idempotent ideal filter with quotient ring A. In section 6 we consider the class ? of all topological rings that can be openly embedded into a topological ring without open left ideals. If we restrict our attention to linearly topologized rings, then ? is Morita-invariant. In section 2 we construct a topological ring of quotients Q_*(R) and prove that it coincides with the ring C(R) of [8].     

Galois Objects of Finitely Generated Projective Hopf Algebras

Let C be the category of cocommutative coalgebras over a commutative ring R and let H be a group object in C, i.e., let H be a cocommutative Hopf algebra. Assume that H is a finitely generated, projective R-module and that the integrals (of [4]) in H* ≡ Hom_R(H, R) are cocommutative elements. We will show that any Galois H-object (as defined in [3, Def. 1.2, p. 8]) is a finitely generated, projective R-module.