The Orthogonal Subcategory Problem and the Small Object Argument

A classical result of P. Freyd and M. Kelly states that in “good” categories, the Orthogonal Subcategory Problem has a positive solution for all classes of morphisms whose members are, except possibly for a subset, epimorphisms. We prove that under the same assumptions on the base category and on , the generalization of the Small Object Argument of D. Quillen holds—that is, every object of the category has a cellular -injective weak reflection. In locally presentable categories, we prove a sharper result: a class of morphisms is called quasi-presentable if for some cardinal λ every member of the class is either λ-presentable or an epimorphism. Both the Orthogonal Subcategory Problem and the Small Object Argument are valid for quasi-presentable classes. Surprisingly, in locally ranked categories (used previously to generalize Quillen’s result), this is no longer true: we present a class of morphisms, all but one being epimorphisms, such that the orthogonality subcategory is not reflective and the injectivity subcategory Inj is not weakly reflective. We also prove that in locally presentable categories, the injectivity logic and the Orthogonality Logic are complete for all quasi-presentable classes. Financial support by Centre for Mathematics of University of Coimbra and by School of Technology of Viseu is acknowledged by the third author.     

Injectivity radius bound of Ricci flow with positive Ricci curvature and applications

We study the injectivity radius bound for 3-d Ricci flow with bounded curvature. As applications, we show the long time existence of the Ricci flow with positive Ricci curvature and with curvature decay condition at infinity. We partially settle a question of Chow-Lu-Ni [Hamilton’s Ricci Flow, p. 302].     

Graph-theoretic conditions for injectivity of functions on rectangular domains

This paper presents sufficient graph-theoretic conditions for injectivity of collections of differentiable functions on rectangular subsets of Rn. The results have implications for the possibility of multiple fixed points of maps and flows. Well-known results on systems with signed Jacobians are shown to be easy corollaries of more general results presented here.     

Order-preserving reflectors and injectivity

We investigate a Galois connection in poset enriched categories between subcategories and classes of morphisms, given by means of the concept of right-Kan injectivity, and, specially, we study its relationship with a certain kind of subcategories, the KZ-reflective subcategories. A number of well-known properties concerning orthogonality and full reflectivity can be seen as a particular case of the ones of right-Kan injectivity and KZ-reflectivity. On the other hand, many examples of injectivity in poset enriched categories encountered in the literature are closely related to the above connection. We give several examples and show that some known subcategories of the category of T₀-topological spaces are right-Kan injective hulls of a finite subcategory.     

原子映射空间中的广义Hahn-Banach定理

经典的Hahn-Banach定理告诉读者在有界映射空间(B(.,.), \|\cdot\|)中\mathbb{C具有内射性. 在第二节中主要研究在原子映射空间(\n^{B}(\cdot, \cdot), \nu^{B})中的内射性.作者得到任意有限维Banach空间在原子映射空间(\n^{B}(\cdot, \cdot), \nu^{B})中都是内射的. 这可以看作(\n^{B}(\cdot, \cdot), \nu^{B})中的广义Hahn-Banach定理.
在经典的Banach空间理论中, 众所周知一个Banach空间E在(B(\cdot, \cdot), \|\cdot\|)中具有\{\ell_{1}^{n}\}_{n\in\mathbb{N}有限可表示性当且仅当E同构于某个超积\prod\ell_{1}^{n(\alpha)的子空间. 作为第二节的一个应用,第三节中作者研究了在原子映射空间(\n^{B}(\cdot, \cdot), \nu^{B})中的\{\ell_{1}^{n}\}_{n\in\mathbb{N}有限可表示性. 作者得到 \mathbb{C是唯一在原子映射空间(\n^{B}(\cdot, \cdot), \nu^{B})中具有\{\ell_{1}^{n}\}_{n\in\mathbb{N}有限可表示性的Banach空间. 这与Banach空间理论中的经典结果是迥然不同的.     

Modules Characterized by Injectivity Clases

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On the weak-star extensibility

The paper encompasses a complete study of the weak-star extensibility of subspaces of Banach spaces introduced by B.S. Mordukhovich and B. Wang when studying the restrictive metric regularity in variational analysis. Many useful properties are presented; in particular, a new point of view and formulation of the property in the framework of Banach space theory is established. We also propose the concept of weak-star extensible Banach spaces that has interesting applications, and is fairly broad in that it contains many important classes of Banach spaces. Some further applications of this extensibility to the theory of linear operators between Banach spaces are also explored.     

On Some Problems Studied by R. V. Kadison

Several problems studied by professor R. V. Kadison are shown to be closely related. The problems were originally formulated in the contexts of homomorphisms of C^*-algebras, cohomology of von Neumann algebras and perturbations of C^*-algebras. Recent research by G. Pisier has demonstrated that all of the problems considered are related to the question of whether all C^*-algebras have finite length.