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?

Alex Saad
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  • Categories of sheaves can be assembled to form a "sheaf" of categories. – Zhen Lin Dec 27 '15 at 10:12
  • @ZhenLin could you elaborate on what you mean by "assembled"? – Alex Saad Dec 27 '15 at 11:56
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    The assignment $X \mapsto \mathbf{Sh} (X)$ extends to form a "sheaf" of categories. – Zhen Lin Dec 27 '15 at 14:27
  • @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? – Alex Saad Dec 27 '15 at 15:31
  • That's half of it. The other half is that the sheaf axioms have to be adapted to account for that. – Zhen Lin Dec 27 '15 at 15:41
  • @ZhenLin OK great - thanks for mentioning this example. Do you know of any good references to learn more about what you're talking about? – Alex Saad Dec 27 '15 at 15:44
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    Look up stacks. See e.g. [here](http://math.stackexchange.com/a/254999/5191). – Zhen Lin Dec 27 '15 at 16:54
  • You might be interested to know that in certain areas people prefer to speak about sheaves as set-valued, because a sheaf of groups corresponds to an internal group in the set-valued sheaves. – neptun Oct 07 '17 at 12:41
  • Also sheaves require some kind of topology to make sense. Often a more natural thing to ask for than sheaves of special objects is fibering the category of sheaves in objects. Stacks, which is the higher-categorical formulation of sheaves, use categories fibered in groupoids. I would suggest reading a book on stacks and higher stacks, because it is far to complicated to answer thouroughly here, I think. – neptun Oct 07 '17 at 12:47

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