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I notice that in W. G. Dwyer and D. M. Kan, Singular functors and realization functors , Nederl. Akad. Wetensch. Indag. Math., 87, (1984), 147 – 153, they look at a generalised version of a an orbit category in a simplicial model category, M. This gives an equivalence between a new (finer) model category structure on M and the projective structure on the simplicial presheaves on the generalised orbit category. Some of their detail is ’lacking’ and I am wondering what the current best approach to this is (as the paper is nearly 30 years old). This does seem very relevant for the recent discussions on global homotopy theory, but is more general it seems. It also plays a role in the Dwyer-Kan-Smith paper on obstructions to rectifying/rigidifying homotopy commutative diagrams.
Do we have anything on that construction of Dwyer and Kan in the Lab? I could not find it in any of the obvious places.
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