Z-Quantale及其范畴性质

本文把集系统的概念应用到Quantale理论中,作为Quantale的一般化,引入了Z-quantale的概念,研究了Z-quantale及其范畴的若干性质.主要结果有:证明了Z-quantale范畴是序半群范畴的反射子范畴,凝聚Z-quantale范畴是Z-quantale范畴的余反射子范畴.讨论了Z-quantale范畴中的投射对象,证明了Z-quantale A是E-投射的当且仅当它是稳定Z-连续的.     

Generalised Verdier duality for presheaves of spectra—I

The stable homotopy category, with spectra as objects, resembles in many aspects the derived category whose objects are complexes of abelian groups. In this paper, we show that many of the familiar techniques of homological algebra available on the category of presheaves of abelian group spectra can be carried over (with suitable modifications) to the category of presheaves of spectra. Using A^∞-ring spectra, we show that the duality theory of Verdier for abelian presheaves may be generalised to a general duality (called generalised Verdier duality) for presheaves of spectra. Other common duality results, for example duality in the sense of Spanier-Whitehead or Poincaré-duality for spectra are interpreted in terms of generalised Verdier duality. One of the goals is to set up a broad framework for defining a presheaf theoretic version of generalised intersection cohomology. The remaining aspects of the theory will be dealt with elsewhere.     

Tilting objects on tubular weighted projective lines: A cluster tilting approach

Using the cluster tilting theory,we investigate the tilting objects in the stable category of vector bundles on a weighted projective line of weight type(2,2,2,2).More precisely,a tilting object consisting of rank-two bundles is constructed via the cluster tilting mutation.Moreover,the cluster tilting approach also provides a new method to classify the endomorphism algebras of the tilting objects in the category of coherent sheaves and the associated bounded derived category.     

事件结构与Domain

研究了Domain理论中的事件结构及其对应的domain结构,证明了事件结构生成的L-事件domain恰好是具有性质I的代数L-domain。特别地,本文通过稳定事件生成的事件domain,证明了以线性映射为态射、以DI-domain为对象的范畴是以稳定映射为态射、以具有性质I的代数L-domain为对象的范畴的反射子范畴。     

Compactly Generated Relative Stable Categories

Let G be a finite group. The stable module category of G has been applied extensively in group representation theory. In particular, it has been used to great effect that it is a triangulated category which is compactly generated by the class of finitely generated modules. Let H be a subgroup of G. It is possible to define a stable module category of G relative to H. This is also a triangulated category, but no non-trivial examples have been known where it was compactly generated. While the finitely generated modules are compact objects, they do not necessarily generate the category. We show that the relative stable category is compactly generated if the group algebra of H has finite representation type. In characteristic p, this is equivalent to the Sylow p-subgroups of H being cyclic.     

On Tubular Tilting Objects in the Stable Category of Vector Bundles

The present paper focuses on the study of the stable category of vector bundles for the weighted projective lines of weight triple. We find some important triangles in this category and use them to construct tilting objects with tubular endomorphism algebras for the case of genus one via cluster tilting theory.     

Krull-Schmidt decompositions for thick subcategories

Following H. Krause [Decomposing thick subcategories of the stable module category, Math. Ann. 313 (1) (1999) 95-108], we prove Krull-Schmidt type decomposition theorems for thick subcategories of various triangulated categories including the derived categories of rings, Noetherian stable homotopy categories, stable module categories over Hopf algebras, and the stable homotopy category of spectra. In all these categories, it is shown that the thick ideals of small objects decompose uniquely into indecomposable thick ideals. We also discuss some consequences of these decomposition results. In particular, it is shown that all these decompositions respect K-theory.     

Singularity categories and singular equivalences for resolving subcategories

Let \(\mathcal {X}\) be a resolving subcategory of an abelian category. In this paper we investigate the singularity category \(\mathsf {D_{sg}}(\underline{\mathcal {X}})=\mathsf {D^b}({\mathsf {mod}}\,\underline{\mathcal {X}})/\mathsf {K^b}({\mathsf {proj}}({\mathsf {mod}}\,\underline{\mathcal {X}}))\) of the stable category \(\underline{\mathcal {X}}\) of \(\mathcal {X}\). We consider when the singularity category is triangle equivalent to the stable category of Gorenstein projective objects, and when the stable categories of two resolving subcategories have triangle equivalent singularity categories. Applying this to the module category of a Gorenstein ring, we prove that the complete intersections over which the stable categories of resolving subcategories have trivial singularity categories are the simple hypersurface singularities of type \((\mathsf {A}_1)\). We also generalize several results of Yoshino on totally reflexive modules.     

Smashing subcategories and the telescope conjecture – an algebraic approach

We prove a modified version of Ravenel’s telescope conjecture. It is shown that every smashing subcategory of the stable homotopy category is generated by a set of maps between finite spectra. This result is based on a new characterization of smashing subcategories, which leads in addition to a classification of these subcategories in terms of the category of finite spectra. The approach presented here is purely algebraic; it is based on an analysis of pure-injective objects in a compactly generated triangulated category, and covers therefore also situations arising in algebraic geometry and representation theory. Oblatum 23-XI-1998 & 19-V-1999 / Published online: 5 August 1999     

On Moduli Spaces for Abelian Categories

We show that if 𝒜 is an abelian category satisfying certain mild conditions, then one can introduce the concept of a moduli space of (semi)stable objects which has the structure of a projective algebraic variety. This idea is applied to several important abelian categories in representation theory, like highest weight categories.     

Exact completion of path categories and algebraic set theory: Part I: Exact completion of path categories

We introduce the notion of a “category with path objects”, as a slight strengthening of Kenneth Brown's classical notion of a “category of fibrant objects”. We develop the basic properties of such a category and its associated homotopy category. Subsequently, we show how the exact completion of this homotopy category can be obtained as the homotopy category associated to a larger category with path objects, obtained by freely adjoining certain homotopy quotients. In a second part of this paper, we will present an application to models of constructive set theory. Although our work is partly motivated by recent developments in homotopy type theory, this paper is written purely in the language of homotopy theory and category theory, and we do not presuppose any familiarity with type theory on the side of the reader.     

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.     

Galois Theory for Braided Tensor Categories and the Modular Closure

Given a braided tensor *-category with conjugate (dual) objects and irreducible unit together with a full symmetric subcategory we define a crossed product . This construction yields a tensor *-category with conjugates and an irreducible unit. (A *-category is a category enriched over Vect with positive *-operation.) A Galois correspondence is established between intermediate categories sitting between and and closed subgroups of the Galois group Gal( / )=Aut ( ) of , the latter being isomorphic to the compact group associated with by the duality theorem of Doplicher and Roberts. Denoting by the full subcategory of degenerate objects, i.e., objects which have trivial monodromy with all objects of , the braiding of extends to a braiding of iff . Under this condition, has no non-trivial degenerate objects iff = . If the original category is rational (i.e., has only finitely many isomorphism classes of irreducible objects) then the same holds for the new one. The category ≡ is called the modular closure of since in the rational case it is modular, i.e., gives rise to a unitary representation of the modular group SL(2,  ). If all simple objects of have dimension one the structure of the category can be clarified quite explicitly in terms of group cohomology.     

Balance in Stable Categories

We study when the stable category ${\mathcal A}/\langle{\mathcal T}\rangleWe study when the stable category A/áT?{\mathcal A}/\langle{\mathcal T}\rangle of an abelian category A{\mathcal A} modulo a full additive subcategory T{\mathcal T} is balanced and, in case T{\mathcal T} is functorially finite in A{\mathcal A}, we study a weak version of balance for A/áT?{\mathcal A}/\langle{\mathcal T}\rangle. Precise necessary and sufficient conditions are given in case T{\mathcal T} is either a Serre class or a class consisting of projective objects. The results in this second case apply very neatly to (generalizations of) hereditary abelian categories.     

Stability conditions and extremal contractions

We study extremal contractions from smooth projective varieties via a moduli theoretic approach. In the two dimensional case, we show that any extremal contraction appears as a moduli space of Bridgeland stable objects in the derived category of coherent sheaves. In the three dimensional case, we show that a a similar result holds with respect to conjectural Bridgeland stability conditions.     

PreT 2 Objects in Topological Categories

In previous papers, two notions of pre-Hausdorff (PreT ₂) objects in a topological category were introduced and compared. The main objective of this paper is to show that the full subcategory of PreT ₂ objects is a topological category and all of T ₀, T ₁, and T ₂ objects in this topological category are equivalent. Furthermore, the characterizations of pre-Hausdorff objects in the categories of filter convergence spaces, (constant) local filter convergence spaces, and (constant) stack convergence spaces are given and as a consequence, it is shown that these categories are homotopically trivial.     

n-预加法范畴中的拟核

In this paper, we have defined the quasi-kernels for a category which possesses initial objects or final objects and have obtained several structure theorems of the quasi kernels in an n-preadditive category.     

Order-adjoint monads and injective objects

Given a monad T on whose functor factors through the category of ordered sets with left adjoint maps, the category of Kleisli monoids is defined as the category of monoids in the hom-sets of the Kleisli category of T. The Eilenberg-Moore category of T is shown to be strictly monadic over the category of Kleisli monoids. If the Kleisli category of T moreover forms an order-enriched category, then the monad induced by the new situation is Kock-Zöberlein. Injective objects in the category of Kleisli monoids with respect to the class of initial morphisms then characterize the objects of the Eilenberg-Moore category of T, a fact that allows us to recuperate a number of known results, and present some new ones.