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I find myself in the following situation:

Let $(\mathcal{F}_i, f_{i,j})$ be a directed system of sheaves, defined on an exhausting family of closed topological subspaces $$X_i \subset X_{i+1} \subset ... \subset X$$.

Suppose that we know how the limit of restrictions $$\underset{j}{\lim} \iota_{i,j}^* \mathcal{F}_j, \quad i \leq j$$ looks on each subset $X_i$, where $\iota_{j,i} : X_j \hookrightarrow X_i$ is the closed inclusion (so the maps defining the limit process look something like the composition $\iota_{i,j}^* f_{i,j}$).

Is it enough to know the whole limit sheaf $$\mathcal{F} := \underset{i}{\lim}\mathcal{F}_i ?$$

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  • $\begingroup$ Is $\mathcal F_i$ a sheaf on $X$ or a sheaf on $X_i$? $\endgroup$
    – Tom Goodwillie
    Commented Aug 27, 2018 at 12:34
  • $\begingroup$ $\mathcal{F}_i$ is a sheaf on $X_i$, that's the problem $\endgroup$
    – BrianT
    Commented Aug 27, 2018 at 12:49
  • $\begingroup$ When you say "exhausting", do you simply mean that the $X$ is the union of the point sets $X_i$, or are you assuming that as a topological space $X$ is the direct limit of $X_i$? $\endgroup$
    – Tom Goodwillie
    Commented Aug 27, 2018 at 12:58
  • $\begingroup$ Don't the inclusions $X_i \subset X_j$, $i \leq j$ imply that the direct limit is the union ? If not, then sorry for my lack of precision. I assume that the space $X$ is the direct limit of $X_i$. $\endgroup$
    – BrianT
    Commented Aug 27, 2018 at 13:02
  • $\begingroup$ That is, the direct limit in the category of topological spaces. So that a set in $X$ is open (or closed) iff for each $i$ its intersection with $X_i$ is open (or closed) in $X_i$. $\endgroup$
    – Tom Goodwillie
    Commented Aug 27, 2018 at 13:57

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