A-cellular homotopy theories Original Research Article

For the categories of pointed spaces, pointed simplicial sets and simplicial groups and for some fixed cofibrant object A there are closed model category structures in which cofibrant objects are built out of “A-cells”. The A-cellular structure coincides with the usual structure when A=S0 for spaces and simplicial sets, or A=Zconst for simplicial groups. Closed-model category structures are also defined for diagrams in such a way that for diagrams over a contractible category under certain conditions the factorization of a map of diagrams into an A-cofibration followed by an A-trivial fibration commutes with holim up to an equivalence.