The Iwasawa decomposition of a connected semisimple complex Lie group or a connected semisimple split real Lie group is one of the most fundamental observations of classical Lie theory. It implies that the geometry of a connected semisimple complex resp. split real Lie group G is controlled by any maximal compact subgroup K. Examples are Weyl’s unitarian trick in the representation theory of Li...

In this paper, we first define the notion of finitely-cosmall quotient (singly-cosmall quotient) morphisms. Then give a characterization new concept. We show that an epimorphism p:Y→U is if and only for any right R-module Z morphism g:Z→Y such pg finitely-copartial isomorphism (singly-copartial isomorphism) from to Y with codomain U finitely (singly) split epimorphism. also investigate relation...

This paper studies the homomorphism of rings of continuous functions ρ:C(X) → C(Y ), Y a subspace of a Tychonoff space X, induced by restriction. We ask when ρ is an epimorphism in the categorical sense. There are several appropriate categories: we look at CR, all commutative rings, and R/N, all reduced commutative rings. When X is first countable and perfectly normal (e.g., a metric space), ρ ...

Let C be a category with strong monomorphic strong coimages, that is, every morphism ƒ of C factors as ƒ = u ° g so that g is a strong epimorphism and u is a strong monomorphism and this factorization is universal. We define the notion of strong Mittag-Leffler property in pro-C. We show that if ƒ : X → Y is a level morphism in pro-C such that ( ) p Y ! " is a strong epimorphism for all β > α, t...

The Iwasawa decomposition of a connected semisimple complex Lie group or a connected semisimple split real Lie group is one of the most fundamental observations of classical Lie theory. It implies that the geometry of a connected semisimple complex resp. split real Lie group G is controlled by any maximal compact subgroup K. Examples are Weyl’s unitarian trick in the representation theory of Li...

Given a topological space X, K(X) denotes the upper semi-lattice of its (Hausdorff) compactifications. Recent studies have asked when, for αX ∈ K(X), the restriction homomorphism ρ : C(αX) → C(X) is an epimorphism in the category of commutative rings. This article continues this study by examining the sub-semilattice, Kepi(X), of those compactifications where ρ is an epimorphism along with two ...

There is a natural epimorphism from the symmetric algebra to the Rees algebra of an ideal. When this epimorphism is an isomorphism, we say that the ideal is of linear type. Given two determinantal rings over a field, we consider the diagonal ideal, kernel of the multiplication map. We prove in many cases that the diagonal ideal is of linear type and recover the defining ideal of the Rees algebr...

It is proved that in any pointed category with pullbacks, coequalizers and regular epi-mono factorizations, the class of regular epimorphisms is stable under pullback along the so-called balanced effective descent morphisms. Here “balanced” can be omitted if the category is additive. A balanced effective descent morphism is defined as an effective descent morphism p : E → B such that any subobj...