Grothendieck Institutions

We extend indexed categories, fibred categories, and Grothendieck constructions to institutions. We show that the 2-category of institutions admits Grothendieck constructions (in a general 2-categorical sense) and that any split fibred institution is equivalent to a Grothendieck institution of an indexed institution. We use Grothendieck institutions as the underlying mathematical structure for the semantics of multi-paradigm (heterogenous) algebraic specification. We recuperate the so-called extra theory morphisms as ordinary theory morphisms in a Grothendieck institution. We investigate the basic mathematical properties of Grothendieck institutions, such as theory colimits, liberality (free constructions), exactness (model amalgamation), and inclusion systems by globalisation from the local level of the indexed institution to the level of the Grothendieck institution.     

拟变分包含系统(英文)

本文提出一些新的拟变分包含系统,利用向量映射的锥连续性与Park不动点定理,得到解的存在定理与解集的闭性.作为应用,得到对称强向量拟均衡问题与理想向量鞍点的存在性,推广与改进了近期的一些相关研究工作.     

Relativized Grothendieck topoi

In this paper we define a notion of relativization for higher order logic. We then show that there is a higher order theory of Grothendieck topoi such that all Grothendieck topoi relativizes to all models of set theory with choice.     

One-sided Grothendieck quotients

We introduce one-sided thick subcategories of an arbitrary preadditive category and define a quotient category . When is abelian, this concept specializes to Grothendieck’s quotient for two-sided thick . We determine the left noetherian rings for which the injective modules form a left thick subcategory. We exhibit a class of one-sided thick subcategories in categories of coherent functors which are ubiquitous in representation theory. Received: 14 November 2006 Revised: 12 March 2007     

On Grothendieck Subspaces

The modulus of an order bounded functional on a Riesz space is the sum of a pair of Riesz homomorphisms if and only if the kernel of this functional is a Grothendieck subspace of the ambient Riesz space. An operator version of this fact is given.Original Russian Text Copyright © 2005 Kutateladze S. S.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 620–624, May–June, 2005.     

Systems of Set-Valued Quasivariational Inclusion Problems

We propose four kinds of systems of set-valued quasivariational inclusion problems in product spaces, which include many known systems of equilibrium problems and systems of variational inequalities as well as inclusion problems. Sufficient conditions for the solution existence are established. When applied to special cases, these conditions improve many existing results in the literature. To ensure the generality of our problem setting and results, applications in fixed-point theory and quasioptimization problems are included. This work was partially supported by the Vietnam national program on basic research in natural sciences. A part of the work of the first author was completed during his stay at the Department Mathematics of Namur University and its hospitality is gratefully acknowledged. The authors are indebted to an anonymous referee for valuable remarks.     

Quantitative Grothendieck property

A Banach space X is Grothendieck if the weak and the weak^? convergence of sequences in the dual space X^?$X^{?}$ coincide. The space ?^∞${?}^{\infty }$ is a classical example of a Grothendieck space due to Grothendieck. We introduce a quantitative version of the Grothendieck property, we prove a quantitative version of the above-mentioned Grothendieck?s result and we construct a Grothendieck space which is not quantitatively Grothendieck. We also establish the quantitative Grothendieck property of L^∞(μ)$L^{\infty }(\mu )$ for a σ-finite measure μ.     

Curves in Grothendieck Categories

Noncommutative projective geometry studies noncommutative graded rings by replacing the variety by a suitable Grothendieck category. One way of studying the resulting category is to examine the full subcategories which behave like curves on a commutative variety. Smith and Zhang initiated such a study by considering the subcategory generated by a particular type of module they called a pure curve module in good position. This paper generalizes their construction by allowing more general modules. The resulting category is shown to be categorically equivalent to a quotient of the category of graded modules over a graded ring. In the course of defining the category equivalence, several dimensions, including projective, injective and Krull dimensions, are calculated. In particular, this extension allows examination of the category created from a line module over more general AS-regular rings than those considered by Smith and Zhang. For instance, suppose that C is a generic line module over R _d , Stafford's Sklyanin-like algebra. Let C denote the category C generates. Then C is equivalent to the category of graded k[x, y]/(x ² – y ²) modules under the Z × Z/2Z-grading where deg(x) = (–1, 0) and deg(y) = (–1,1).     

Presentations of Grothendieck Constructions

We will give quiver presentations of the Grothendieck constructions of functors from a small category to the 2-category of 𝕜-categories for a commutative ring 𝕜.     

Localization on certain Grothendieck categories

Using categorical techniques we obtain some results on localization and colocalization theory in Grothendieck categories with a set of small projective generators. In particular, we give a sufficient condition for such category to be semiartinian. For semiartinian Grothendieck categories where every simple object has a projective cover, we obtain that every localizing subcategory is a TTF-class. In addition, some applications to semiperfect categories are obtained.     

On a Theorem of Grothendieck

A smooth projective morphism p : T S to a smooth variety S is considered. In particular, the following result is proved. The total direct image Rp _*(/n) of the constant étale sheaf /n is locally (in Zariski topology) quasiisomorphic to a bounded complex on S that consists of locally constant, constructible étale sheaves of /n-modules. Bibliography: 2 titles.     

Uniquely generated Grothendieck space ideals

Starting from a result of Pietsch on the Grothendieck ideal of strictly nuclear locally convex spaces, we classify all those Grothendieck ideals of nuclear locally convex spaces that are generated by a unique operator ideal on the class of Banach spaces.     

Attached atoms in Grothendieck categories

Let 𝒜 be a Grothendieck category. In this paper we define attached atoms of nonzero objects of 𝒜 as a dual of the associated atoms of nonzero objects introduced by Kanda [2 Kanda, R. (2012). Classifying Serre subcategories via atom spectrum. Adv. Math. 231(3–4):1572–1588. [Google Scholar]]. If R is a (non)commutative ring, then we show that this new notion can be corresponded to the attached prime ideals of nonzero modules introduced by Macdonald [5 Macdonald, I. G. (1973). Secondary representation of modules over a commutative ring. Sympos. Math. 11:23–43. [Google Scholar]] and Annin [1 Annin, S. (2008). Attached primes over noncommutative rings. J. Pure Appl. Algebra 212:510–521.[Crossref], [Web of Science ®] , [Google Scholar]].     

Grothendieck环及其应用

本文以Ｇｒｏｔｈｅｎｄｉｅｃｋ环为工具，利用相关环刻划了环的Ｇｒｏｔｈｅｎｄｉｅｃｋ群，得到了环的连通性、稳定自由性等性质的重要特征．最后给出了它们在群环的应用．     

A measure-theoretic Grothendieck inequality

In this note we develop a notion of integration with respect to a bimeasure μ that allows integration of functions in the projective tensor product , where ν₁ and ν₂ are Grothendieck measures for μ. This integral, which agrees with the standard notion of integration with respect to a bimeasure, allows us to integrate inner products and provides a generalization of the Grothendieck inequality to a measure-theoretic setting.