Abstract. We first describe Krull-Schmidt theorems decomposing H spaces and simply-connected co-H spaces into atomic factors in the category of pointed nilpotent p-complete spaces of finite type. We use this to construct a 1-1 correspondence between homotopy types of atomic H spaces and homotopy types of atomic co-H spaces, and construct a split fibration which connects them and illuminates the decomposition. Various properties of these constructions are analyzed. The Krull-Schmidt property first arose in the theory of R-modules, and when valid, it states that each object decomposes in a unique way into a sum of indecomposable objects of the same type. Numerous examples of decomposing the loop space on a co-H space can be found in the literature ([Hi], [M], [CM], [AG], [G1], [G2]). Typically what happens is that the loop space of an atomic co-H space is a product of various factors, and the least connected factor is an H space of special interest, while the other factors, in some sense, represent noise. The first general Krull-Schmidt type theorem in homotopy theory was proved by Wilkerson [W] fo
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