In Strøm's (no relation) paper "The Homotopy Category is a Homotopy Category" he proves that the category of unpointed topological spaces, with Hurewicz fibrations and ordinary cofibrations and homotopy equivalences, is a model category.

Then he has a fairly long section on the pointed case, and his results are not as good: he restricts to well-pointed spaces, and doesn't get all the model category axioms.

But if $\mathcal{M}$ is a model category, then for any object $A\in \mathcal{M}$, the category $A \downarrow \mathcal{M}$ inherits a model category structure in perfectly straightforward way. Since $\mathbf{Top}_* = * \downarrow \mathbf{Top}$, we get a model structure right away.

So: what is going on here? Did he simply miss an easy extension to the pointed case? Is it possible that the natural model structure on $* \downarrow \mathbf{Top}$ is not the one he wants (i.e., the received cofibrations, fibrations or weak equivalences differ somehow from pointed cofibrations, fibrations and pointed homotopy equivalences)?

simplicialclosed model category, then so is $M\downarrow X$. Maybe Str$\emptyset$m just missed it? $\endgroup$ – Charles Rezk Nov 30 '10 at 5:02