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On p. 5 of the above article it says that all dgc-algebras are assumed to be equal to the ground field in degree 0.
But is this really what is meant?
I don’t see it being used. On the contrary: Later on the simplicial sets seem to be meant to be just pointed, not reduced, so that the PL-de Rham functor would not take them to dgc-algebras with that property.
The discussion on p. suggests that what is actually used is that the dgc-algebras are unital.
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I am vaguely wondering:
Might the construction from Scull 2008 of a model category of dg-algebras parameterized over an orbit category generalize to parameterization over any “fundamental category” of a $G$-space (in the sense here)?
I haven’t really thought about it yet, except for noticing that Golasinski 1997a, 1997b amplifies that the existence of injective minimal models in this context depends only on the parameter category being an EI-category – which is still the case for those “fundamental categories”.
I have now forwarded this question (#4) to MathOverflow:q/412833.
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