In view of discussion in another thread (here), I have added (here) the following warning:

Beware that section 4.4 claims a new proof of the Strøm model structure, but relying on a statement in

- Michael Cole,
*Many homotopy categories are homotopy categories*, Topology and its Applications 153 (2006) 1084–1099 (doi:10.1016/j.topol.2005.02.006)

which later was noticed to be false, by Richard Williamson, for details see p. 2 and Rem 5.12 and Sec. 6.1 in:

- Tobias Barthel, Emily Riehl,
*On the construction of functorial factorizations for model categories*, Algebr. Geom. Topol. 13 (2013) 1089-1124 (arXiv:1204.5427, doi:10.2140/agt.2013.13.1089, euclid:agt/1513715550)

It remains unclear, to me anyways, what this implies for the sliced generalization which is the core claim of May & Sigurdsson’s book.(?)

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