A Study on Adams Completion and Cocompletion

Abstract

Many algebraic and geometrical constructions from different field of mathematics such as Algebra, Analysis, Topology, Algebraic Topology, Differential Topology, Differentiable Manifolds and so on can be obtained as Adams completion or cocompletion with respect to chosen sets of morphisms in suitable categories. Cayley’s Theorem, ascending central series and descending central series are well known facts in the area of group theory. It is shown how these concepts are identified with Adams completion. We obtain a Whitehead-like tower of a module by considering a suitable set of morphisms in the corresponding homotopy category (that is, category of right modules and homotopy module homomorphisms) whose different stages are the Adams cocompletion of the module. Indeed, the work is carried out in a general framework by considering a Serre class of abelian groups. The minimal model of a simply connected differential graded algebra is obtained as the Adams cocompletion with respect to the suitably chosen set of morphisms in the category of 1-connected differential graded algebras over Q and differential graded algebra homomorphisms. Also with the help of Kopylov and Timofeev result, the relationship between a graph and Adams cocompletion is established.