Universality of Categories of Coalgebras

Under GCH, a set functor F does not preserve finite unions of non-empty sets if and only if the category Coalg F of all F-coalgebras is universal. Independently of GCH, we show that for any non-accessible functor F preserving intersections, the category Coalg F has a large discrete full subcategory, and we give an example of a category of F-coalgebras that is not universal, yet has a large discrete full subcategory.     

The slice classification of categories of coalgebras for comonads

We prove that the category _G Set of all G-coalgebras in s-equivalent to the category Alg(1) of all unary algebras iff the functional part G of G is not a product of the identity functor I and a constant functor. Received June 11, 1998; accepted in final form January 16, 1999.     

Products of coalgebras

We prove that the category of F-coalgebras is complete, that is products and equalizers exist, provided that the type functor F is bounded or preserves mono sources. This generalizes and simplifies a result of Worrell ([Wor98]). We also describe the relationship between the product and the largest bisimulation between and and find an example of two finite coalgebras whose product is infinite. Received January 11, 2000; accepted in final form October 16, 2000.     

Types and coalgebraic structure

We relate weak limit preservation properties of coalgebraic type functors F to structure theoretic properties of the class of all F-coalgebras. In particular, we give coalgebraic characterizations for the condition that F weakly preserves pullbacks, kernel pairs or preimages. We also describe regular monos and epis. In case that |F(1)| ≠ 1 we show that F preserves preimages iff for every class of F-coalgebras. The case |F(1)| = 1 is left as an open problem.Dedicated to the memory of Ivan RivalReceived August 29, 2003; accepted in final form July 13, 2004.This revised version was published online in August 2005 with a corrected cover date.     

On universal categories of coalgebras

A category ${\mathcal{K}}$ is called universal if for every accessible functor F : Set → Set the category of all F-coalgebras and the category of all F-algebras can be fully embedded into ${\mathcal{K}}$ . We prove that for a functor G preserving intersections, the category Coalg G of all G-coalgebras is universal unless the functor G is linear, that is, of the form GX = X × A + B for some fixed sets A and B. Other types of universality are also investigated.     

Tensor products and preservation of limits, for acts over monoids

In 1971, Stenström published one of the first papers devoted to the problem of when, for a monoid S and a right S -act A _S , the functor A? (from the category of left acts over S into the category of sets) has certain limit preservation properties. Attention at first focused on when this functor preserves pullbacks and equalizers but, since that time, a large number of related articles have appeared, most having to do with when this functor preserves monomorphisms of various kinds. All of these properties are often referred to as flatness properties of acts . Surprisingly, little attention has so far been paid to the obvious questions of when A _S ? preserves all limits, all finite limits, all products, or all finite products. The present article addresses these matters.     

The Katětov property for finite degree seminormal functors

A seminormal functor kF enjoys the Katěetov property (K-property) if for every compact set X the hereditary normality of kF(X) implies the metrizability of X. We prove that every seminormal functor of finite degree n>3 enjoys the K-property. On assuming the continuum hypothesis (CH) we characterize the weight preserving seminormal functors with the K-property. We also prove that the nonmetrizable compact set constructed in [1] on assuming CH is a universal counterexample for the K-property in the class of weight preserving seminormal functors.     

On the enright functor in the highest weight category

LetG be a complex semisimple Lie group,B its Borel subgroup andX a flag variety ofG. We define a functor on the category ofB-equivariantD _X-modules that corresponds, under the global section functor, to the Enright functor on the highest weight category. We use this to lift Enright functor to the mixed version of the highest weight category. As an application we obtain that the socle and the cosocle filtration of a primitive quotient of the enveloping algebra coincide.     

Rational surfaces and regular maps into the 2-dimensional sphere

Let X and Y be affine nonsingular real algebraic varieties. A general problem in Real Algebraic Geometry is to try to decide when a mapping, , can be approximated by regular mappings in the space of mappings, , equipped with the topology. In this paper, we obtain some results concerning this problem when the target space is the 2-dimensional standard sphere and X has a complexification that is a rational (complex) surface. To get the results we study the subgroup of the second cohomology group of X with integer coefficients that consists of the cohomology classes that are pullbacks, via the inclusion mapping , of the cohomology classes in represented by complex algebraic hypersurfaces. Received December 1, 1998; in final form August 2, 1999     

Tensor Products and Preservation of Weighted Limits,for S-Posets

We prove that the functor of tensor multiplication by a right S-poset (S is a pomonoid) preserves all small weighted limits if and only if this S-poset is cyclic and projective. We also show that this functor preserves all finite pie-limits if and only if the S-poset is a filtered colimit of S-posets isomorphic to S _S .     

Non-effective deformations of Grothendieck’s Hilbert functor

We show that the Hilbert functor of rank one families on a non-separated scheme X admits deformations that are not effective. For such ambient schemes we have that the Hilbert functor is not representable by a scheme or an algebraic space.     

On hull classes of <Emphasis Type="Italic">ℓ</Emphasis>-groups and covering classes of spaces

W denotes the category of archimedean ℓ-groups with designated weak unit and ℓ-homomorphisms that preserve the weak unit. Comp denotes the category of compact Hausdorff spaces with continuous maps. The Yosida functor is used to investigate the relationship between hull classes in W and covering classes in Comp. The central idea is that of a hull class whose hull operator preserves boundedness. We demonstrate how the Yosida functor may be used to identify hull classes in W and covering classes in Comp. In addition, we exhibit an array of order preserving bijections between certain families of hull classes and all covering classes, one of which was recently produced by Martínez. Lastly, we apply our results to answer a question of Knox and McGovern about the class of all feebly projectable ℓ-groups.     

Brown representability for space-valued functors

In this paper we prove two theorems which resemble the classical cohomological and homological Brown representability theorems. The main difference is that our results classify contravariant functors from spaces to spaces up to weak equivalence of functors. In more detail, we show that every contravariant functor from spaces to spaces which takes coproducts to products up to homotopy, and takes homotopy pushouts to homotopy pullbacks is naturally weakly equivalent to a representable functor. The second representability theorem states: every contravariant continuous functor from the category of finite simplicial sets to simplicial sets taking homotopy pushouts to homotopy pullbacks is equivalent to the restriction of a representable functor. This theorem may be considered as a contravariant analog of Goodwillie’s classification of linear functors [14].     

Geometric Models for Quasicrystals II. Local Rules Under Isometries

The atomic structures of quasicrystalline materials exhibit long range order under translations. It is believed that such materials have atomic structures which approximately obey local rules restricting the location of nearby atoms. These local constraints are typically invariant under rotations, and it is of interest to establish conditions under which such local rules can nevertheless enforce order under translations in any structure that satisfies them. A set of local rules in is a finite collection of discrete sets {Y _i } containing 0, each of which is contained in the ball of radius ρ around 0 in . A set X satisfies the local rules under isometries if the ρ -neighborhood of each is isometric to an element of . This paper gives sufficient conditions on a set of local rules such that if X satisfies under isometries, then X has a weak long-range order under translations, in the sense that X is a Delone set of finite type. A set X is a Delone set of finite type if it is a Delone set whose interpoint distance set X-X is a discrete closed set. We show for each minimal Delone set of finite type X that there exists a set of local rules such that X satisfies under isometries and all other Y that satisfy under isometries are Delone sets of finite type. A set of perfect local rules (under isometries or under translations, respectively) is a set of local rules such that all structures X that satisfy are in the same local isomorphism class (under isometries or under translations, respectively). If a Delone set of finite type has a set of perfect local rules under translations, then it has a set of perfect local rules under isometries, and conversely. Received February 14, 1997, and in revised form February 14, 1998, February 19, 1998, and March 5, 1998.     

On the Cohomology of Relative Hopf Modules

Let H be a Hopf algebra over a field k, and A an H-comodule algebra. The categories of comodules and relative Hopf modules are then Grothendieck categories with enough injectives. We study the derived functors of the associated Hom functors, and of the coinvariants functor, and discuss spectral sequences that connect them. We also discuss when the coinvariants functor preserves injectives.     

On geometric properties of functors of positive-homogenous and semiadditive functionals

We investigate the functor OH of positive-homogenous functionals and the functor OS of semiadditive functionals. We prove that OH(X) is an absolute retract if and only if X is an open-generated compactum, and OS(X) is an absolute retract if and only if X is an opengenerated compactum of weight ≤ ω₁. We investigate the softness of mappings of multiplication of monads generated by these functors.     

Triangulated functors,homological functors,tilts, and lifts

 Let T be a triangulated category and let X be an object of T. This paper studies the questions: Does there exist a triangulated functor G : D(ℤ)  T with G(ℤ)≌X? Does there exist a triangulated functor H : T  D(ℤ) with h⁰ ⊚ H ⋍ Hom_T (X, −)? To what extent are G and H unique? One spin off is a proof that the homotopy category of spectra is not the stable category of any Frobenius category with set indexed coproducts. Received: 8 March 2002 / Revised version: 18 October 2002 Published online: 14 February 2003 Mathematics Subject Classification (2000): 18E30, 55U35     

Strongly etale extensions and weak henseliziations

Abstract

Since the circulation, in 1974, of the first draft of “The construction D + XD _S [X], J. Algebra 53 (1978), 423–439” a number of variations of this construction have appeared. Some of these are: The generalized D + M construction, the A + (X)B[X] construction, with X a single variable or a set of variables, and the D + I construction (with I not necessarily prime). These constructions have proved their worth not only in providing numerous examples and counter examples in commutative ring theory, but also in providing statements that often turn out to be forerunners of results on general pullbacks. The aim of this paper will be to discuss these constructions and the remarkable uses they have been put to. I will concentrate more on the A + XB[X] construction, its basic properties and examples arising from it.     

Continuously controlledA-theory

To anycontrolled space over the metric spaceB we can associate itsboundedly controlled algebraic K-theory, a functor designed to give information about the space of stable bounded concordances of manifolds homotopy equivalent toX. Generalizing a construction of D. R. Anderson, F. X. Connolly, S. Ferry, and E. K. Pedersen, we define another functor, calledcontinuously controlled A-theory, which depends only on thetopology of the control space, not itsmetric properties. In the special case whereB=R ₊, this functor is (more or less by definition) the same asproper A-theory. We prove that under certain conditions on the controlled space the natural transformation from boundedly controlledA-theory to continuously controlledA-theory is a weak homotopy equivalence, and hence defines a generalized homology theory. Continuously controlledK-theory is used in the approaches of G. Carlsson, E. K. Pedersen, and S. Ferry, S. Weinbergervto theK-theory Novikov conjecture.