In Str\o m'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)?

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)?

In Str\o m'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.

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 $A \downarrow \mathcal{M}$ 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)?

In Str\o m'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)?