About the Cole-Ström model category structure with a locally presentable category

Question

In "Many homotopy categories are homotopy categories" (M. Cole / Topology and its Applications 153 (2006) 1084–1099), Cole generalizes the construction of the model category of Ström to any bicomplete category which is enriched, tensored and cotensored over compactly generated spaces and which satisfies an additional condition which is called the cofibration hypothesis (the mapping path object preserves colimits of towers of strong acyclic cofibrations). Now here are my questions :

1) where is exactly used in the proof the fact that the topological spaces are weak Hausdorff (i.e. for any continuous map $f:K\to X$ with $K$ compact Hausdorff, $f(K)$ is closed in $X$) ?

2) I ask the question since I would like to use this theorem for a locally presentable category which is enriched, tensored and cotensored over $\Delta$-generated spaces (i.e. colimits of simplices). Suppose that the weak Hausdorff condition above can be dropped, we then just have to check the cofibration hypothesis ; but the mapping path object turns out to be accessible so there is nothing to check.

3) If the weak Hausdorff condition is necessary in Cole's theorem, do you think that it can be replaced in the context of $\Delta$-generated spaces by the following separation condition (I don't know its name) : a $\Delta$-generated space $X$ is $\Delta$-Hausdorff if for any continuous map $f:\Delta^n \to X$, $f(\Delta^n)$ is closed in $X$.

Concerning $\Delta$-generated spaces, a short bibliography :

1. Notes on Delta-generated spaces by Dugger : http://math.uoregon.edu/~ddugger/delta.html
2. The proof that they assemble to a locally presentable category : A convenient category for directed homotopy by Fajstrup-Rosicky : http://www.tac.mta.ca/tac/volumes/21/1/21-01abs.html
3. A survey of their properties : Section 2 of Homotopical interpretation of globular complex by multipointed d-space http://www.tac.mta.ca/tac/volumes/22/22/22-22abs.html

Answer 1

Actually, there is an error in the Cole paper you're talking about, as recently discovered by Richard Williamson and very recently corrected by Tobi Barthel and Emily Riehl in "On the Construction of Functorial Factorizations for model categories". The introduction to this paper is very good, and discusses how the error comes in the point-set topology, and the correct fix for this error is to use categorical algebra to do that work instead. I have not read Cole's paper and haven't finished the Barthel-Riehl paper, but it seems probable that if there's some point-set topology in Cole's paper which is causing you confusion then perhaps that's where the error occurs. Even if it's not, the Barthel-Riehl philosophy and machinery might help you get the same result without need for point-set topological arguments.