Maps into homotopy coalgebras

Let X be a k-fold homotopy coalgebra of order j with respect to the pair of adjoint functors Σ k and Ω 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. Golasiński, 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.