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Jul 3, 2019 at 20:54 comment added Simon Henry Nice example ! I think I can prove the converse to some the implication you are mentioning, but your answer really unlocked me. I've written a sketch of this as a separate answer. I assume that when you say "weakly contractible" it is because you have the $\infty$-categorical case in mind, but that in the $1$-categorical case it should be replaced by connected right ?
Jul 3, 2019 at 20:51 vote accept Simon Henry
Jul 3, 2019 at 10:30 history edited Yonatan Harpaz CC BY-SA 4.0
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Jul 3, 2019 at 10:15 history answered Yonatan Harpaz CC BY-SA 4.0