A very short and easy question. Why do people write "locally constant constructible sheaf" (e.g. everywhere in SGA) instead of just "locally constant sheaf"? A constructible sheaf is by definition locally constant on a stratification, hence locally constant implies locally constant constructible. And locally constant constructible implies locally constant. So the two notions should be the same. Have I misread the definitions?
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asked Feb 24, 2014 at 7:36
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1$\begingroup$ Apparently "constructible" has a second meaning: mathoverflow.net/questions/69422/… $\endgroup$– Qiaochu YuanCommented Feb 24, 2014 at 7:40
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1$\begingroup$ Indeed, if we impose finiteness in the definition of constructibility, then "locally constant constructible sheaf" is a bit shorter than "locally constant sheaf of finite groups". On the other hand, the Stacks Project seems to be using "finite locally constant", which sounds nicer. $\endgroup$– Piotr AchingerCommented Feb 24, 2014 at 7:45
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$\begingroup$ If I recall it correctly constructible does require locally constant (with respect to a stratification) of finite rank (local system on each stratum)! $\endgroup$– Oliver StraserCommented Mar 26, 2014 at 9:51
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Locally constant means locally constant on the whole space, not on a stratification, so constructible doesn't imply locally constant (or locally constant constructible). Depending on which SGA you are talking about, constructible imposes some finiteness conditions, so locally constant doesn't imply constructible.
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1$\begingroup$ But locally constant constructible does imply locally constant! $\endgroup$ Commented Feb 24, 2014 at 8:10