MOD-C Postnikov Approximation of a 1-Connected Space

Abstract

Deleanu, Frei and Hilton have developed the notion of generalized Adams completion in a categorical context [4]. They have also shown that if the set of morphisms is saturated then the Adams completion of an object is characterized by a certain couniversai property. We want to prove a stronger version of this result by dropping the saturation assumption on the set of morphisms; we also prove that the canonical map from an object to its Adams completion is an element of the set of morphisms under very moderate assumptions. These two results are fairly general in nature and are applicable to most cases of interest. Further using these two results and introducing “modulo a Serre class C of abelian groups” [9] we have obtained the mod-C Postnikov approximation of a 1-connected based CW-complex, with the help of a suitable set of morphisms.

Primary Decomposition of a 0-Connected Nilpotent Space

“…Proof. The proof follows from the dual of Theorem 1.3 [4]. We take 𝑆 𝑛 1 satisfy all conditions of Theorem 1.2; hence 𝑒 ∈ 𝑆 𝑛 .…”

“…Given aset S of morphisms of 𝒞, the saturation of S, denoted as S is the set of all morphisms u in 𝒞 such that F(u) is an isomorphism in 𝒞 S −1 . Furthermore,S is said to be saturated if S = S [4,9]. Deleanu, Frei and Hilton have shown that if the set of morphisms Sis saturated then the Adams cocompletion of a space is characterized by a certain couniversal property ( [9],dual of Theorem 1.2).…”

“…The following Theorem (dual of Theorem 1.3, [4]) shows that under certain conditions the morphisms e: Y S → Y always belongs to S. (ii) fg ∈ S 1 implies thatg ∈ S 1 .…”