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Jul 13, 2020 at 15:22 comment added Simon Henry Yes, I should have linked it as a reference for example (1). But I can't really find an answer in this model. I tend to think that there is an actual homotopy theoretic obstruction to all three properties together, but I can't find it.
Jul 13, 2020 at 15:16 comment added David White I'm assuming you know this paper by Steve Lack? It's the best technology I know for this kind of question: arxiv.org/abs/math/0607646
Jul 13, 2020 at 13:31 history asked Simon Henry CC BY-SA 4.0