@BenWieland I thought $BHomeo(\mathbb{R}^n)$ works. I see your point about the stability, i'll fix that.

@skupers Having the stable versions is part of the statement which I think shiuld be true due to Product Structure theorems. Could you perhaps point at the exact statements from Kirby and Siebenmann you have in mind?

Did you have any progress with this problem since then? One of the main problems I have with this is that most of what's in HA (which is the main modern reference for general statements in higher algebra) focuses on algebras/modules. Passing to Coalgebras/Comodules requires one to take 'op's but some of the statements have a presentability assumption on the categories which makes them non applicable to comodules/coalgebras without a reproof of some sort.

@HenriJohnston No worries. I'm happy that you posted this answer if only for this very nice and readable paper. I would accept both answers if I could.

@HenriJohnston If I understand correctly the paper shows that it is possible for $Rep_{\mathbb{Q}}(G) / Perm(G)$ to be non-trivial in general (i.e. the quotient of the ring of virtual rational representations by $Perm(G)$). Though it doesn't assume (as in my question) that $Rep_{\mathbb{Q}}(G) = Rep_{\mathbb{C}}(G)$. I tried to look there whether there's a counter example to this as well but I didn't find it. No doubt its there and I just missed it...

@WilleLiou I believe this is true. You can just take the sum of all translates of a lattice to get a $G$-stable lattice (because it is projective and projectives are free over $\mathbb{Z}$).

Isn't this almost equivalent to the property that the point $[X] \in \mathcal{Hilb}_{Y}$ in the corresponding hilbert scheme isolated?

That's cool! Just to make sure I understood correctly the linked proposition. In particular the Joyal model structure on simplicial sets satisfies the conditions of 5.3.1. as well, is this correct?