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Nov 8, 2016 at 11:14 answer added Eduardo Pareja Tobes timeline score: 4
Nov 7, 2016 at 23:06 answer added Mike Shulman timeline score: 5
Nov 7, 2016 at 11:48 comment added Simon Henry A few not very deep remarks: 1) the condition that $F^*$ preserve power object is relatively easy to write down explicitely and give an explicit criterion on $f$ equivalent to your question. 2) The answer of John below can be extended to the case where the the foncteur $F$ is (equivalent to) a fibration in groupoid. 3) the criterion obtained in (1) is very strong and I found it very surprising that there is an many example as (2) suggest. In fact I wouldn't be surprised if those were the only examples.
Nov 5, 2016 at 14:01 answer added john timeline score: 4
Nov 4, 2016 at 1:00 history edited David Spivak CC BY-SA 3.0
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Nov 4, 2016 at 0:30 history asked David Spivak CC BY-SA 3.0