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Aug 20, 2018 at 9:56 comment added Ben And I don't know how to weaken the continuity condition. Maybe it would be enough that (in $K^I$) pull-backs of weak equivalences are weak equivalences.
Aug 20, 2018 at 8:33 comment added Philippe Gaucher I am just discovering and reading your answer. In my case, $I$ is connected. I have no idea concerning the other condition.
Aug 14, 2018 at 12:53 history answered Ben CC BY-SA 4.0