4
$\begingroup$

I'm trying to understand the category-theoretic proof that sheafification preserves stalks, using adjoints, as outlined e.g. in this mathoverflow answer:

https://mathoverflow.net/questions/77581/sheaf-valued-functors-how-much-can-you-prove-about-them-just-using-category-theo

What is bugging me is that I'm having a hard time seeing why the colimit described is actually isomorphic to the stalk (as a skyscraper sheaf based at x using the stalk as the fixed set).

Specifically, in his answer, Ryan Reich claims that for a sheaf $F$ on $X$, the stalk $F_x$ is the colimit over opens $U$ containing $x$ of the sheaves $j_* j^* F$, where $j: U \rightarrow X$ is the inclusion map.

Every time I try to show that the skyscraper sheaf satisfies the universal property I end up in a seeming dead end. I feel like it shouldn't be THAT hard, and that I must be missing something or confusing myself somehow. Does anyone know a reference for this, or be willing to explain why it works? I'd be very grateful.

  • 0
    Ah, ok, this is making sense now. I was always thinking of $j_* j^*F(V)$ as some awkward colimit, but in fact it simplifies like you say to $F(U \cap V)$. From there I was able to convince myself that the colimit of sheaves coincides with the skyscraper sheaf desired. Thanks! If you wish to make your comment an answer, I'll happily accept it.2012-07-15

0 Answers 0