On Adams Completion and Cocompletion

Abstract

The minimal model of a 1-connected differential graded Lie algebra is obtained as the Adams cocompletion of the differential graded Lie algebra with respect to a chosen set of morphisms in the category of 1-connected differential graded Lie algebras (d.g.l.a.’s)over the field of rationals and d.g.l.a.-homomorphisms. The Postnikov-like approximation of a module is obtained as the Adams completions of the space with the help of a suitable set of morphisms in the category of some specific modules and module homomorphisms. The Cartan-Whitehead decomposition of topological G-module is obtained as the Adams cocompletion of the space with respect to suitable sets of morphisms. Postnikov-like approximation is obtained for a topological G-module, in terms of Adams completion with respect to a suitable sets of morphisms, using cohomology theory of topological G-modules.The ring of fractions of the algebra of all bounded linear operators on a separable infinite dimensional Banach space is isomorphic to the Adams completion of the algebra with respect to a carefully chosen set of morphisms in the category of separable infinite dimensional Banach spaces and bounded linear norm preserving operators of norms at most 1. The nth tensor algebra and symmetric algebra are each isomorphic to the Adams completions of the algebras. The exterior algebra and Clifford algebra are each isomorphic to the Adams completions of the algebra with respect to a chosen set of morphisms in the category of modules and module homomorphisms.