Calculus of Functorial Morphisms

We prove that, if F, G: 𝒞 → 𝒟 are two right exact functors between two Grothendieck categories such that they commute with coproducts and U is a generator of 𝒞, then there is a bijection between Nat(F, G) and the centralizer of Hom_𝒟(F(U), G(U)) considered as an Hom_𝒞(U, U)-Hom_𝒞(U, U)-bimodule. We also prove a dual of this result and give applications to Frobenius functors between Grothendieck categories.     

Applications of Duality to the Pure-injective Envelope

Given an R-T-bimodule _R K _T and R-S-bimodule _R M _S , we study how properties of _R K _T affect the K-double dual M** = Hom _T [Hom _R (M, K), K] considered as a right S-module. If _R K is a cogenerator, then for every R-S-bimodule, the natural morphism Φ _M : M → M** is a pure-monomorphism of right S-modules. If _R K is the minimal (injective) cogenerator and K _T is quasi-injective, then M ^** is a pure-injective right S-module. If _R K is the minimal (injective) cogenerator, and T = End _R K it is shown that K _T is quasi-injective if and only if the K-topology on R is linearly compact. If the _R K-topology on R is of finite type, then the natural morphism Φ _R : R → R** is the pure-injective envelope of R _R as a right module over itself. The author is partially supported by NSF Grant DMS-02-00698.     

Bases of topological epi-reflections

In the first part of this paper a characterization of epi-reflective subcategories of the category TOP of topological spaces in terms of initial topologies is given. This characterization enables us to associate to each epi-reflective subcategory of TOP, regular systems of cogenerators. After having examined some properties of these regular systems, the second part of the paper considers an A-closure operator associated to every epi-reflective subcategory A of TOP which for each regular system of cogenerators of A determines an epi-reflective subcategory of A.     

Convergence and Duality

We describe dualities related to the foundations of probability theory in which sequential convergence and sequential continuity play an important role.     

Ext_F-投射生成子与Ext_F-内射余生成子

设A是Abel范畴,F是Ext_A~1(-,-):A~(op)×A→A的加法子双函子.首先研究了Ext_(F-)投射生成子与Ext_F-内射余生成子的同调性质,其次引入了W_F-Gorenstein模的概念.特别地,证明了如果重复W_F-Gorenstein模的定义程序将不会产生新的模类.最后,统一并推广了许多参考文献中的结论.