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.     

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.     

ANOTHER CHARACTERIZATION OF BICOREFLECTIVE SUBCATEGORIES

Abstract

In this paper, the relation between the notion of a discrete functor (see [4]) and the notion of a fine functor (see [1]) is examined. As a generalization of the notion of a F-fine object (see [1]), discrete functors T: A → X are used to define K-fine objects, where K is a class of A-objects. It is shown that if T is in addition semi-topological, then (as for F-fine objects in a topological category, see [1]) the class of K-fine objects determines a bicoreflective subcategory of A. Moreover, it is shown that in co-complete, co-(well-powered) categories, the existence of bicoreflective subcategories is equivalent to the existence of functors that are both discrete and semi-topological.     

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.     

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.     

Pure ideals quotient categories and infinite group-graded rings

Abstract

Adapting the idea of twisted tensor products to the category of conic algebras (CA), i.e., finitely generated graded algebras, we define a family of objects hom ^?[?, 𝒜] there, one for each pair 𝒜, ? ∈ CA, with analogous properties to its internal coHom objects hom [?, 𝒜], but representing spaces of transformations whose coordinate rings and the ones of their respective domains do not commute among themselves. They give rise to a CA ^op -based category different from that defined by the function (𝒜, ?) ?  hom [?, 𝒜]. The mentioned non commutativity is controlled by a collection of twisting maps τ_{𝒜, ?}. We show, under certain circumstances, that (bi)algebras end ^?[𝒜] ?  hom ^?[𝒜, 𝒜] are counital 2-cocycle twistings of the corresponding coEnd objects end [𝒜]. This fact generalizes the twist equivalence (at a semigroup level) between, for instance, the quantum groups G L _q (n) and their multiparametric versions.     

Preprojective Cluster Variables of Acyclic Cluster Algebras

It is proved that any cluster-tilted algebra defined in the cluster category 𝒞(H) has the same representation type as the initial hereditary algebra H. For any valued quiver (Γ, Ω), an injection from the subset 𝒫?(Ω) of the cluster category 𝒞(Ω) consisting of indecomposable preprojective objects, preinjective objects, and the first shifts of indecomposable projective modules to the set of cluster variables of the corresponding cluster algebra 𝒜_Ω is given. The images are called “preprojective cluster variables”. It is proved that all preprojective cluster variables other than u_i have denominators u ^dim M in their irreducible fractions of integral polynomials, where M is the corresponding preprojective module or preinjective module. In case the valued quiver (Γ, Ω) is of finite type, the denominator theorem holds with respect to any cluster. Namely, let x = (x₁,…, x_n) be a cluster of the cluster algebra 𝒜_Ω, and V the cluster tilting object in 𝒞(Ω) corresponding to x, whose endomorphism algebra is denoted by Λ. Then the denominator of any cluster variable y other than x_i is x ^dim M, where M is the indecomposable Λ-module corresponding to y. This result is a generalization of the corresponding result of Caldero–Chapoton–Schiffler to the non-simply-laced case.     

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.     

Satelliten; Singuläre Erweiterungen und Derivationen

This paper is divided in two parts. In chapter one we generalize the concept of satellite for abelian categories, substituting Ext by an arbitrary functor E:A ^op×BEns. A very close relation to Kan functor extensions turns out. In chapter two we give a comprehensive formulation of the wellknown cohomology theories for groups and algebras-with the recently used modification in taking the group of derivations as the lowest cohomology group- starting with a categorical definition of the concept of singular extension. There is a zeroth chapter with a construction on functors, needed in both parts, and a third with examples.

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Gekürzte Fassung der Dissertation des Autors zur Erlangung des Doktorgrades der Mathematisch-Naturwissenschaftlichen Fakultät der Universität Hamburg.     

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.     

Topological Representation of Sheaf Cohomology of Sites

For a site & (with enough points), we construct a topological space X(&) and a full embedding ^* of the category of sheaves on & into those on X _(&) (i.e., a morphism of toposes :Sh (X_(&)) Sh(&)). The embedding will be shown to induce a full embedding of derived categories, hence isomorphisms H^*(&,A) = H^*(X_(&), ^*A) for any Abelian sheaf A on &. As a particular case, this will give for any scheme Y a topological space X _(Y) and a functorial isomorphism between the étale cohomology H^*(Y _ét,A) and the ordinary sheaf cohomology H^*(X(_(Y),),^*A), for any sheaf A for the étale topology on Y.     

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.     

Simplicial Cohomology with Coefficients in Symmetric Categorical Groups

In this paper we introduce and study a cohomology theory {H ⁿ (–,A)} for simplicial sets with coefficients in symmetric categorical groups A. We associate to a symmetric categorical group A a sequence of simplicial sets {K(A,n)} _n0, which allows us to give a representation theorem for our cohomology. Moreover, we prove that for any n3, the functor K(–,n) is right adjoint to the functor _n , where _n (X ) is defined as the fundamental groupoid of the n-loop complex ⁿ (X ). Using this adjunction, we give another proof of how symmetric categorical groups model all homotopy types of spaces Y with _i (Y)=0 for all in,n+1 and n3; and also we obtain a classification theorem for those spaces: [–,Y]H ⁿ (–, _n (Y)).     

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”.     

Cohomology of Products and Coproducts of Augmented Algebras

We show that the ordinary cohomology functor $\Lambda \mapsto {\operatorname{Ext}} ^* _\Lambda (k,k)$ from the category of augmented k-algebras to itself exchanges coproducts and products, then that Hochschild cohomology is close to sending coproducts to products if the factors are self-injective. We identify the multiplicative structure of the Hochschild cohomology of a product, modulo a certain ideal, in terms of the cohomology of the factors.