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Let $X$ be a nice enough topological space, perhaps a complex algebraic variety with its analytic topology. I'm hoping someone could help me prove that the category $\text{Constr}(X)$ of constructible sheaves on $X$ is abelian.

I think one could use that subsheaves and quotients of local systems are local systems. However, conceptually what is confusing me is that given two constructible sheaves $\mathcal{F}, \mathcal{G}$, they might be constructible with respect to completely different Whitney stratifications of X, right?

So given a morphism $\mathcal{F} \to \mathcal{G}$ of constructible sheaves, can you always "refine" the two Whitney stratifications to get one such that the morphism restricts to a morphism of local systems?

Or should I be thinking along another lines?

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    $\begingroup$ There is always a stratification that refines any two given stratifications. Isn’t this almost obvious from the definition? $\endgroup$
    – Will Sawin
    Jan 23, 2020 at 1:29
  • $\begingroup$ @WillSawin I suppose, but is it clear the refinement will be Whitney? Or is it not true that a constructible sheaf must be locally constant over a Whitney stratification? I've been confused whether you need Whitney in the definition of constructible, because I've seen it both ways. $\endgroup$
    – Benighted
    Jan 23, 2020 at 2:04
  • $\begingroup$ I think every stratification of a reasonable space has a refinement that is a Whitney stratification. This would make the two definitions nicely equivalent, and answer this question as well. $\endgroup$
    – Will Sawin
    Jan 23, 2020 at 2:14
  • $\begingroup$ For instance Theorem 2.2 of Verdier's paper seems to be the desired statement eudml.org/doc/142424 $\endgroup$
    – Will Sawin
    Jan 23, 2020 at 2:17
  • $\begingroup$ @WillSawin Great that makes sense. Thanks a lot. If it matters, I would accept it if you wrote this into an answer. $\endgroup$
    – Benighted
    Jan 23, 2020 at 2:27

1 Answer 1

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Verdier has proved (for multiple classes of spaces, including in particular complex varieties) that for any finite set of analytic subsets, there exists a Whitney stratification for which all these analytic subsets are unions of strata.

In particular, given two constructible sheaves, and a stratification on which they are lisse, there exists a Whitney stratification that refines both of them.

This suffices to check that kernels and cokernels of maps of constructible sheaves are constructible, as well as direct sums of constructible sheaves.

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