I hope someone can help me with the following:
Say I am given two sheaves $F,G$ on a topological space $X$. How can i proof the Isomorphism of $Hom(F,G) \cong \varprojlim Hom(F\mid_U, G\mid_U)$ And how does this look like.
Thanks everyone
Limit of morphism of sheaves
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sheaf-theory
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2Is the limit taken for every open set $U$ ? In that case, it follows simply from the fact that $X$ is a terminal object in $Open(X)$. – 2017-02-05
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0Thats right, its taken over every open subset. could you please right that i a little more formal way, that would be awesome – 2017-02-06
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1@E.J.K. : this is already very formal. For a directed limit, if you have a terminal object in the indexing set (i.e a $j_0$ such that $ j_0 \leq j, \forall j \in J$) then $\lim_{j \in J} F_j = F_{j_0}$. – 2017-02-06