Maps into homotopy coalgebras

Let X be a k-fold homotopy coalgebra of order j with respect to the pair of adjoint functors Sigma(k) and Omega(k). We show that, under some connectivity conditions on the map f : Y -> X, Y inherits a k-fold homotopy coalgebra structure of the same order for which f is a morphism of homotopy coalgebras. In particular, this holds for skeleta of homotopy coalgebras under some mild assumptions. As a consequence, we complete results on [M. Arkowitz, M. Golasitiski, Homotopy coalgebras and k-fold suspensions, Hiroshima Math. J. 27 (1997) 209-220] and [T. Ganea, Cogroups and suspensions, Invent. Math. 9 (1970) 185-197] by detecting k-fold suspensions among skeleta of k-fold homotopy coalgebras. (C) 2006 Elsevier B.V. All rights reserved.