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Jul 27, 2014 at 7:17 comment added Charles Rezk Just a remark. Both the six functors and the topological setups associate a pullback functor f^* to a map f. As i understand it, only the topological setup comes with a left adjoint to f^*. In my view, it is very confusing to call this left adjoint f_!, since in the six functors game it means some thing entirely different (and which also seems to have a topological analog). I really wish people wouldn't do this. I prefer something like f_# for the left adjoint.
Jul 23, 2014 at 22:33 history edited Qiaochu Yuan CC BY-SA 3.0
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Jul 23, 2014 at 10:12 comment added Urs Schreiber That's the right point of view! This is discussed a bit more here: arxiv.org/abs/1402.7041
Jul 23, 2014 at 9:33 history answered Qiaochu Yuan CC BY-SA 3.0