A conglomerate of exponential supercategories of the category of finitely generated topological spaces

Summary For every ordinaln > 1 we define a categoryT _n of topological spaces in ech's sense which is isomorphic to a category ofn-ary monorelational systems. We show that every categoryT _n is an exponential supercategory of the categoryB of finitely generated topological spaces, which means that well-behaved function spacesG ^H can be defined inT _n wheneverG B.     

On Homeomorphisms of Effective Topological Spaces

We study effective presentations and homeomorphisms of effective topological spaces. By constructing a functor from the category of computable models into the category of effective topological spaces, we show in particular that there exist homeomorphic effective topological spaces admitting no hyperarithmetical homeomorphism between them and there exist effective topological spaces whose autohomeomorphism group has the cardinality of the continuum but whose only hyperarithmetical autohomeomorphism is trivial. It is also shown that if the group of autohomeomorphisms of a hyperarithmetical topological space has cardinality less than 2 then this group is hyperarithmetical. We introduce the notion of strong computable homeomorphism and solve the problem of the number of effective presentations of T ₀-spaces with effective bases of clopen sets with respect to strong homeomorphisms.     

Topologische Affine Ebenen Mit Nichtstetigem Parallelismus

In this paper, we present a general method of constructing topological affine planes having non-continuous parallelism. We prove that a topological affine plane E with point set L ^k ×L ^k , and with a special K-algebraic slope has a topological affine subplane with non-continuous parallelism (Satz 4.6). Here, K is a real-closed subfield of a real-closed field L. The crucial tools needed to make our method work are the notion of a slope and the notion of K-algebraicity, a concept which is introduced and intensively studied here. As an application of our general method, we obtain in Section 5 affine Salzmann planes with lines being bent countably infinitely often admitting a subplane with non-continuous parallelism. This provides a negative answer to a question posed by H. Salzmann [13, p. 52].     

Verallgemeinerte topogene Ordnungen

Topogenous orders in the sense of Császár are a common generalization of proximity and topology. ech closures are a generalization of the topological closure operators in the sense of Kuratowski. We show that the topogenous orders as well as the ech closures are special cases of the so called compressed operators. Moreover, the now defined categoryCOM (in germanBAL) of compress spaces and compress faithful maps is a properly fibred topological category in the sense of Herrlich which is weakly cartesian closed, that means the product map of two quotient maps inCOM is a quotient map inCOM. Therefore by results of L. D. Nel it is possible to construct a cartesian closed properly fibred topological category in whichCOM can be nicely embedded. Further it turns out that the compressed operators be in a natural connexion with the uniform convergence structures in the sense of Cook and Fischer and in addition with the limit structures in the sense of Fischer. For principal ideal uniform convergence structures we prove that they are precompact and complete iff the properly constructed compressed operator is compact.     

Generalized homology theories and chain complexes

Summary To each generalized homology theory h_* defined on a category of topological spacesK (definition 1.2)a chain functorC _*:K ch (=category of chain complexes) (cf. definition 2.1) is established, which is related to h _* (definition 2.4,theorem 8.1).In subsequent papers this result is used for the construction of a strong homology theory (i.e. an analogue of the Steenrod-Sitnikov homology theory for general topological spaces) cf. [4].To G.S. ogovili on the occasion of his 75th birthday     

Coinverters and categories of fractions for categories with structure

A category of fractions is a special case of acoinverter in the 2-categoryCat. We observe that, in a cartesian closed 2-category, the product of tworeflexive coinverter diagrams is another such diagram. It follows that an equational structure on a categoryA, if given by operationsA ⁿ A forn N along with natural transformations and equations, passes canonically to the categoryA [^–1] of fractions, provided that is closed under the operations. We exhibit categories with such structures as algebras for a class of 2-monads onCat, to be calledstrongly finitary monads.The first and third authors gratefully acknowledge the support of the Australian Research Council.     

Final Lift Actions Associated with Topological Functors

For a topological functor U:EB, the fiber U ^–1(b), bB, is a cocomplete poset and the left action, induced by final lift, of the endomorphism monoid B(b,b) on U ^–1(b) is cocontinuous. It is shown that every cocontinuous left action of B(b,b) on any cocomplete poset can be realized as the final lift action associated to a canonically defined topological functor over B. If B is a Grothendieck topos and b=, the subobject classifier, then B(,) inherits both a monoidal and a cocomplete poset structure. In the case B= Sets, all cocontinuous left actions of B(,) on itself are explicitly described and each is shown to arise as the final lift action associated to a specific subcategory of a certain fixed category, referred to as the category of LR-spaces. Relationships between these LR-spaces and several other well known topological categories are also considered.     

Homotopy Structures for Algebras over a Monad

Let T be a monad over a category A. Then a homotopy structure for A, defined by a cocylinder P : A A, or path-endofunctor, can be lifted to the category A ^T of Eilenberg–Moore algebras over T, provided that P is consistent with T in a natural sense, i.e. equipped with a natural transformation : T P P T satisfying some obvious axioms. In this way, homotopy can be lifted from well-known, basic situations to various categories of algebras for instance, from topological spaces to topological semigroups, or spaces over a fixed space (fibrewise homotopy), or actions of a fixed topological group (equivariant homotopy); from categories to strict monoidal categories; from chain complexes to associative chain algebras. The interest is given by the possibility of lifting the homotopy operations (as faces, degeneracy, connections, reversion, interchange, vertical composition, etc.) and their axioms from A to A ^T , just by verifying the consistency between these operations and : T P P T. When this holds, the structure we obtain on our category of algebras is sufficiently powerful to ensure the main general properties of homotopy.     

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.     

On Compactness with Respect to Countable Extensions of Ideals and the Generalized Banach Category Theorem

We extend a theorem of Hamlett and Jankovi by proving that if a topological space (X, ) is compact with respect to the countable extension of I, then the local function A ^*(I) of every subset A of X with respect to and I is a compact subspace with respect to the extension in A ^* (I). We also give a generalized version of the Banach category theorem.     

Perverse sheaves,Koszul IC-modules,and the quiver for the category $\mathcal{O}$

For a stratified topological space we introduce the category of IC-modules, which are linear algebra devices with the relations described by the equation d ²=0. We prove that the category of (mixed) IC-modules is equivalent to the category of (mixed) perverse sheaves for flag varieties. As an application, we describe an algorithm calculating the quiver underlying the BGG category for arbitrary simple Lie algebra, thus answering a question which goes back to I. M. Gelfand. Dedicated to George Lusztig on the occasion of his 60-th birthdayMathematics Subject Classification (1991)  14F43, 17B10, 32S60     

Groupes Moyennables, Dimension Topologique Moyenne et sous-décalages

Let G be an infinite countable residually finite amenable group. In this paper we construct a continuous action of G on a compact metrisable space X such that the dynamical system (X, G) cannot be embedded in the G-shift on [0,1] ^G . This result generalizes a construction due to E. Lindenstrauss and B. Weiss (Mean topological dimension, Israel J. Math. 115 (2000), 1–24) for .     

Rigidity of planar tilings

Summary Aperturbation of a tiling of a region inR ⁿ is a set of isometries, one applied to each tile, so that the images of the tiles tile the same region.We show that a locally finite tiling of an open region inR ² with tiles which are closures of their interiors isrigid in the following sense: any sufficiently small perturbation of the tiling must have only earthquake-type discontinuities, that is, the discontinuity set consists of straight lines and arcs of circles, and the perturbation near such a curve shifts points along the direction of that curve.We give an example to show that this type of rigidity does not hold inR ⁿ , forn>2.Using rigidity in the plane we show that any tiling problem with a finite number of tile shapes (which are topological disks) is equivalent to a polygonal tiling problem, i.e. there is a set of polygonal shapes with equivalent tiling combinatorics.Oblatum 19-III-1991     

On Colimits in Categories of Relations

We study (finite) coproducts and colimits of -chains in Rel(C), the 2-category of relations over a given category C. The former exist and are the same as in C provided that C is extensive. The latter do not exist for example in Rel(Set). However, the canonical construction of those colimits in the category of sets can be generalized to Rel(Set). The canonical cocone is shown to satisfy a 2-categorical universal property, namely that of an lax adjoint cooplimit. Sufficient conditions for any base category C to admit the construction are given.A necessary and sufficient condition for the construction to yield colimits of -chains in the category of maps of Rel(C) is also given.     

On the structure of commutative affine group schemes over a non-perfect field

Let k be a non-perfect field of characteristic p>O with a p-basisB and k_s the algebraic separable closure of k. Starting from the ring of Schoeller D ^B [3] and the topological Galois group II of k_s over k, we construct a new ring such that the category of commutative affine k-group schemes is anti-equivalent to the category ofeffaceable left -modules. (The effaceability is defined in the text).     

A Homological Lower Bound for Order Dimension of Lattices

We prove that if a finite lattice L has order dimension at most d, then the homology of the order complex of its proper part L vanishes in dimensions d – 1 and higher. If L can be embedded as a join-sublattice in N ^d , then L actually has the homotopy type of a simplicial complex with d vertices.     

Realizing Operadic Plus-constructions as Nullifications

In this paper we generalize the plus-construction given by M. Livernet for algebras over rational differential graded operads to the framework of cofibrant operads over an arbitrary ring (the category of algebras over such operads admits a closed model category structure). We follow the modern approach of J. Berrick and C. Casacuberta defining topological plus-construction as a nullification with respect to a universal acyclic space. We construct a universalH _*^Q-acyclic algebra and we define A A⁺ as the -nullification of the algebra A. This map induces an isomorphism in Quillen homology and quotients out the maximal perfect ideal of _0(A). As an application, we consider for any associative algebra R the plus-constructions of gl(R) in the categories of homotopy Lie and homotopy Leibniz algebras. This gives rise to two new homology theories for associative algebras, namely homotopy cyclic and homotopy Hochschild homologies. Over the rationals these theories coincide with the classical cyclic and Hochschild homologies.Primary: 19D06, 19D55; Secondary: 18D50, 18G55, 55P60, 55U35Received March 2003     

Projective normality of abelian surfaces of type (1, 2<Emphasis Type="Italic">d</Emphasis>)

We show that an abelian surface embedded in P^N by a very ample line bundle of type (1,2d) is projectively normal if and only if d4. This completes the study of the projective normality of abelian surfaces embedded by complete linear systems.Supported by EAGER.Mathematics Subject Classification (2000): Primary, 14K05; Secondary, 14N05, 14E20     

An Elementary Approach to Exponentiable Maps

Originally, exponentiable maps in the category Top of topological spaces were described by Niefield in terms of certain fibrewise Scott-open sets. This generalizes the first characterization of exponentiable spaces by Day and Kelly, which was improved thereafter by Hofmann and Lawson who described them as core-compact spaces.Besides various categorical methods, the Sierpinski-space is an essential tool in Niefield's original proof. Therefore, this approach fails to apply to quotient reflective subcategories of Top like Haus, the category of Hausdorff spaces. A recent generalization of the Hofmann–Lawson improvement to exponentiable maps enables now to reprove the characterization in a completely different and very elementary way. This approach works for any nontrivial quotient reflective subcategory of Top or Top/ _T , the category of all spaces over a fixed base space T, as well as for exponentiable monomorphisms with respect to epi-reflective subcategories.An important special case is the category Sep_Top/ _T of separated maps, i.e. distinct points in the same fibre can be separated in the total space by disjoint open neighbourhoods. The exponentiable objects in Sep turn out to be the open and fibrewise locally compact maps. The same holds for Haus/ _T , T a Hausdorff space. In this case, a similar characterization was obtained by Cagliari and Mantovani.