广义Comma范畴的Recollement

本文研究广义Comma范畴上Recollement问题.利用Abel范畴上Recollement及其伴随函子,诱导出广义Comma范畴,并利用比较函子构造出广义Comma范畴上的Recollement.这些结果推广了一般Abel范畴上的Recollement,丰富了Comma范畴研究.     

On underlying functors in general and topological algebra

In the present paper we compare the concepts of monadic functors and functors with lifting and preservation properties which are essential in general algebra and in topological algebra. In doing this we show that the Linton conditions are sufficient for tripleability under milder assumptions as is known till yet. Using the concept of regular categories and regular functors [4] we give a complete generalization of theSet-based situation to a large class of base categories which admit certain factorizations.     

Cartesian closed initial completions

We construct cartesian closed extensions of concrete categories with special (topological) properties. As a consequence we find a necessary and sufficient condition for a concrete category to have finitely productive, cartesian closed initial completion. Finally, we exhibit a topological category, not satisfying this condition; this gives a negative answer to the problem of Herrlich and Nel whether each topological category has a cartesian closed topological bull [6]. These results have been announced in [1].     

Interpolation categories for homology theories

For a homological functor from a triangulated category to an abelian category satisfying some technical assumptions, we construct a tower of interpolation categories. These are categories over which the functor factorizes and which capture more and more information according to the injective dimension of the images of the functor. The categories are obtained by using truncated versions of resolution model structures. Examples of functors fitting in our framework are given by every generalized homology theory represented by a ring spectrum satisfying the Adams-Atiyah condition. The constructions are closely related to the modified Adams spectral sequence and give a very conceptual approach to the associated moduli problem and obstruction theory. As an application, we establish an isomorphism between certain E(n)-local Picard groups and some Ext-groups.     

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.     

A Note on the Distributivity of the Huq Commutator Over Finite Joins

We show that the Huq commutator distributes over finite joins, in any semi-abelian algebraically cartesian closed category. As a consequence we show that for semi-abelian varieties of universal algebras (more generally for semi-abelian categories with large directed colimits of subobjects preserved, for each object B, by the functor B×??), the distributivity of the Huq commutator over joins is equivalent to algebraic cartesian closedness.     

Biset Functors as Module Mackey Functors and its Relation to Derivators

In this article, we will show that the category of biset functors can be regarded as a reflective monoidal subcategory of the category of Mackey functors on the 2-category of finite groupoids. This reflective subcategory is equivalent to the category of modules over the Burnside functor. As a consequence of the reflectivity, we can associate a biset functor to any derivator on the 2-category of finite categories.     

Stable Projective Homotopy Theory of Modules,Tails, and Koszul Duality

A contravariant functor is constructed from the stable projective homotopy theory of finitely generated graded modules over a finite-dimensional algebra to the derived category of its Yoneda algebra modulo finite complexes of modules of finite length. If the algebra is Koszul with a noetherian Yoneda algebra, then the constructed functor is a duality between triangulated categories. If the algebra is self-injective, then stable homotopy theory specializes trivially to stable module theory. In particular, for an exterior algebra the constructed duality specializes to (a contravariant analog of) the Bernstein–Gelfand–Gelfand correspondence.     

Universal topological algebra needs closed topological categories

It is shown that a development of universal topological algebra, based in the obvious way on the category of topological spaces, leads in general to a pathological situation. The pathology disappears when the base category is changed to a cartesian closed topological category or to a topological category endowed with a compatible closed symmetric monoidal structure, provided that in the latter case, the algebraic operations are expressed in terms of monoidal powers rather than the usual cartesian powers. With such base categories, universal topological algebra becomes virtually as well-behaved as ordinary (setbased) universal algebra.     

On the categoricity of semigroup-theoretical properties

By asemigroup-theoretical property we mean a property of semigroups which is preserved by isomorphism. Such a property iscategorical if it can be expressed in the language of categories: roughly, without using elements. We show that this is always possible with the proviso that in the case of one-sided properties we cannot refer in categorical terms to a specific side. For example, the property of having aleft identity cannot be described categorically in the category of semigroups, since the functor ()^op which takes a semigroup into its “opposite” semigroup is a category automorphism. We show that ()^op is the only non-trivial automorphism of the category of semigroups (up to natural equivalence of functors). In other words, the “automorphism group” of the category of semigroups has order two.     

Splitting and conjoining objects in monotopological categories

Power-sets are defined for any concrete category (over Set) with finite concrete products, and their structure described for monotopological categories. These sets are used to define the notions of splitting object and of conjoining object. Characterizations of the existence of these objects in monotopological categories are given. It is proved that no proper monotopological category can be concretely cartesian closed. Most well-known monotopological categories with splitting objects are topological or are c-categories, but it is shown that there are many proper monotopological categories which are not c-categories, and yet have splitting objects, and may even be cartesian closed. One of the characterizations of the existence of splitting objects is used to prove that a monotopological category with splitting objects is cartesian closed iff the largest initial completion in which it is epireflective is cartesian closed iff its MacNeille completion is cartesian closed.     

Free Internal Groups

In the first part of this note an elementary proof is given of the fact that algebraic functors, that is, functors induced by morphisms of Lawvere theories, have left adjoints provided that the category K\mathcal{K} in which the models of these theories take their values is locally presentable. The main focus however lies on the special cases of the underlying functor of the category Grp(K)\mathsf{Grp}(\mathcal{K}) of internal groups in K\mathcal{K} and the embedding of Grp(K)\mathsf{Grp}(\mathcal{K}) into Mon(K)\mathsf{Mon}(\mathcal{K}), the category of monoids in K\mathcal{K}: Here a unifying construction of the respective left adjoints is provided which not only works in case K\mathcal{K} is a locally presentable category but also when K\mathcal{K} is, for example, a particular category of topological spaces such as the category of Hausdorff or Tychonoff spaces or a cartesian closed topological category.     

A classification of Taylor towers of functors of spaces and spectra

We describe new structure on the Goodwillie derivatives of a functor, and we show how the full Taylor tower of the functor can be recovered from this structure. This new structure takes the form of a coalgebra over a certain comonad which we construct, and whose precise nature depends on the source and target categories of the functor in question. The Taylor tower can be recovered from standard cosimplicial cobar constructions on the coalgebra formed by the derivatives. We get from this an equivalence between the homotopy category of polynomial functors and that of bounded coalgebras over this comonad.     

On the Underlying Lower Order Bundle Functors

For every bundle functor we introduce the concept of subordinated functor. Then we describe subordinated functors for fiber product preserving functors defined on the category of fibered manifolds with m-dimensional bases and fibered manifold morphisms with local diffeomorphisms as base maps. In this case we also introduce the concept of the underlying functor. We show that there is an affine structure on fiber product preserving functors. Tis work was supported by a grant of the GA CR No. 201/02/0225.     

Chu duality theory and coalgebraic representation of quantum symmetries

Building upon Vaughan Pratt's work on applications of Chu space theory to Stone duality, we develop a general theory of categorical dualities on the basis of Chu space theory and closure conditions, which encompasses a variety of dualities for topological spaces, convex spaces, closure spaces, and measurable spaces (some of which are new duality results on their own). It works as a general method to generate analogues of categorical dualities between frames (locales) and topological spaces beyond topology, e.g., for measurable spaces, convex spaces, and closure spaces. After establishing the Chu duality theory, we apply the state-observable duality between quantum lattices and closure spaces to coalgebraic representations of quantum symmetries, showing that the quantum symmetry groupoid fully embeds into a purely coalgebraic category, i.e., the category of Born coalgebras, which refines, through the quantum duality that follows from Chu duality theory, Samson Abramsky's fibred coalgebraic representations of quantum symmetries (which, in turn, builds upon his Chu representations of symmetries).     

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.     

Universal algebra in a closed category

The development of finitary universal algebra is carried out in a suitable closed category called a π-category. The π-categories are characterized by their completeness and cocompleteness and some product-colimit commutativities. We establish the existence of left adjoints to algebraic functors, completeness and cocompleteness of algebraic categories, a structure-semantics adjunction, a characterization theory for algebraic categories and the existence of the theory generated by a presentation. The conditions on the closed category are sufficiently weak to be satisfied by any (complete and cocomplete) cartesian closed category, semi-additive category, commutatively algebraic category and also the categories of semi-normed spaces, normed spaces and Banach spaces.