Homomorphisms of homotopy coalgebras

In this paper we consider the theory of higher order homotopy coalgebras as a collection of spaces between co-H-spaces and suspensions, which dualizes Stasheff's theory of Ak-spaces when these are defined through Ak-structures. Moreover we extend two Berstein–Hilton theorems which deal with the primitive homotopy type of a suspension and the class of a suspension map, respectively.