deleanu, frei, and hilton have developed the notion of generalized adams completion in a categorical context. in this paper, we have obtained the postnikov-like approximation of a module, with the help of a suitable set of morphisms.

In this paper we give a construction of free 2-crossed modules. By the use of a `step-by-step' method based on the work of Andr e, we will give a description of crossed algebraic models for the steps in the construction of a free simplicial resolution of an algebra. This involves the introduction of the notion of a free 2-crossed module of algebras.

Previous work (Pradines, 1966, Aof and Brown, 1992) has given a setting for a holon-omy Lie groupoid of a locally Lie groupoid. Here we develop analogous 2-dimensional notions starting from a locally Lie crossed module of groupoids. This involves replacing the Ehresmann notion of a local smooth coadmissible section of a groupoid by a local smooth coadmissible homotopy (or free derivation) for t...

The existence of the split extension classifier of a crossed module in the category of associative algebras is investigated. According to the equivalence of categories XAss ' Cat-Ass we consider this problem in Cat-Ass. This category is not a category of interest, it satisfies its all axioms except one. The action theory developed in the category of interest is adapted to the new type of catego...

In this paper, we introduce and discuss the superstability of generalized module left derivations and generalized module derivations on a Banach module.

We discuss the superstability of generalized module left derivations and generalized module derivations on a Banach module. Let A be a Banach algebra and X a Banach A-module, f : X → X and g : A → A. The mappings Δ1 f,g , Δ2 f,g , Δ3 f,g , and Δ4 f,g are defined and it is proved that if ‖Δ1 f,g x, y, z,w ‖ resp., ‖Δ3 f,g x, y, z,w, α, β ‖ is dominated by φ x, y, z,w , then f is a generalized re...

We obtain some explicit calculations of crossedQ-modules induced from a crossed module over a normal subgroup P of Q. By virtue of theorems of Brown and Higgins, this enables the computation of the homotopy 2-types and second homotopy modules of certain homotopy pushouts of maps of classifying spaces of discrete groups. Introduction A crossed module M = (μ : M → P ) has a classifying space BM (...

We obtain some explicit calculations of crossed Q-modules induced from a crossed module over a normal subgroup P of Q. By virtue of theorems of Brown and Higgins, this enables the computation of the homotopy 2-types and second homotopy modules of certain homotopy pushouts of maps of classifying spaces of discrete groups.

This paper is concerned with the algebraic structure of groupoids and crossed modules of groupoids. We describe the group structure of the automorphism group of a finite connected groupoid C as a quotient of a semidirect product. We pay particular attention to the conjugation automorphisms of C, and use these to define a new notion of groupoid action. We then show that the automorphism group of...