Localization, Algebraic Loops and H-Spaces II

In a previous work [6] it was shown that by imposing certain finiteness conditions on a nilpotent loop certain algebraic results yielded properties about [X, Y] where X is finite CW and Y is an H-Space. In this sequel we further restrict the category of nilpotent loops to a full subcategory called H-loops which still contains all loops of the form [X, Y], We prove that on this category there is a unique and universal P-localization if P ≠ ∅ which corresponds to topological localization. We also show that if the H-loop is a group then the two concepts of localization agree. The first section of this paper is devoted to the definition and basic properties of H-loops. In the second section we develop the localization construction and prove uniqueness. Finally, in the third section we consider the topological and group theoretic situations.