# Prove category of constructible sheaves is abelian

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?

• There is always a stratification that refines any two given stratifications. Isn’t this almost obvious from the definition? Jan 23, 2020 at 1:29
• @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. Jan 23, 2020 at 2:04
• 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. Jan 23, 2020 at 2:14
• For instance Theorem 2.2 of Verdier's paper seems to be the desired statement eudml.org/doc/142424 Jan 23, 2020 at 2:17
• @WillSawin Great that makes sense. Thanks a lot. If it matters, I would accept it if you wrote this into an answer. Jan 23, 2020 at 2:27