Closed model categories for the n-type of spaces and simplicial sets

For each integer n ≥ 0, we give a distinct closed model category structure to the categories of spaces and of simplicial sets. Recall that a non-empty map is said to be a weak equivalence if it induces isomorphisms on the homotopy groups for any choice of base point. Putting the condition on dimensions ≥ n, we have the notion of a weak n-equivalence which is at the base of the nth closed model category structure given here.