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I recently read an answer on this MO post explaining that one reason people are interested in higher category theory is to make reasonable sense of something like a "sheaf of categories" on a topological space $X$, which leads one to the notion of stacks etc. The answer explained why a certain "sheaf of groupoids" naturally arises when considering the presheaf of isomorphism classes of principal $G$-bundles on a space.

I'm not that far into my study of algebraic geometry/topology but recently I have been working quite hard trying to understand sheaves at various levels of generality as I find them really interesting in themselves. So are there other contexts in which we might require something like a sheaf of categories/groupoids on a space?

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  • $\begingroup$ Categories of sheaves can be assembled to form a "sheaf" of categories. $\endgroup$
    – Zhen Lin
    Dec 27, 2015 at 10:12
  • $\begingroup$ @ZhenLin could you elaborate on what you mean by "assembled"? $\endgroup$
    – Alex Saad
    Dec 27, 2015 at 11:56
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    $\begingroup$ The assignment $X \mapsto \mathbf{Sh} (X)$ extends to form a "sheaf" of categories. $\endgroup$
    – Zhen Lin
    Dec 27, 2015 at 14:27
  • $\begingroup$ @ZhenLin OK, I'm guessing the difference here is that the "restriction diagrams" of inverse image functors only commute up to natural isomorphism or something like that? $\endgroup$
    – Alex Saad
    Dec 27, 2015 at 15:31
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    $\begingroup$ Look up stacks. See e.g. here. $\endgroup$
    – Zhen Lin
    Dec 27, 2015 at 16:54

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