Updates on homotopical algebra

Question

I quote Quillen in his notes on homotopical algebra:

I wonder if anyone can give an update on the current state-of-the-art of the situation.

Answer 1

Kan has shown that if $\mathbf{M}$ is any model category, then $s\mathbf{M}$ can be given the structure of a Reedy model category. See Hirschhorn, section 15.3. This is an answer to the "inadequacy" brought up in the last sentence of the quote.

It also seems that pointed simply-connected spaces do form a kind of model category in the sense that the full subcategory of simply-connected spaces in the homotopy category of based spaces is the homotopy category of some model category, namely the one obtained by colocalizing the model category of based spaces using the fact that every object has a simply-connected cover. See the answer from here. I think this is easier to do rationally.