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I am trying to understand the notion of sheaves and stacks. Intuitively, the former, sheaves are bit easy to understand as a gluing of compatible families of sets assign to opens sets of a topological space. In other words, it is a contravariant functor $\mathcal{F}:\mathbf{Open}(X)^{\operatorname{op}}\to\mathbf{Set}$ such that $$\mathcal{F}(U)=\lim\left(\prod_{i\in I}\mathcal{F}(U_i) \rightrightarrows \prod_{j,k\in I^2}\mathcal{F}(U_j\cap U_k)\right)$$ where $X$ is a topological space, $U$ is an open set of $X$ and $\{U_i\}_{i\in I}$ is any open cover of $U.$ Further, it is easy to imagine a sheaf as a étalé space over $X.$

Then I started reading about stacks using this and nlab as my primary sources. I learned that a stack is a contravariant functor $\mathcal{F}:\mathcal{C}^{\operatorname{op}}\to\mathbf{Grpd}$ satisfying a descent property and, categories fibered in groupoids over $\mathcal{C}$ is an intuitive way to think about stacks, where $\mathcal{C}$ is a site (category equipped with a coverage). Now I have following questions:

  • How can I understand the descent property for stack? To be more specific, how can triple fiber products (intersections) appear in the equalizer fork diagram?

  • What categories fibered in groupoids over $\mathcal{C}$ corresponds to stacks? Is this the correct analogue of étalé space of a stack?

Bumblebee
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    I don't know enough about stacks to make this an answer, but just as an intuition for why we may need triple intersections : for a sheaf, if you have sections over each open subset of the covering, you need the intersections to make sure they agree. In a stack, you don't want to ask "are they equal on the intersection ?", you want to ask "are they isomorphic ?". But again, that is the wrong question to ask : you want a specific isomorphism over each intersection. But then you have to check that those specific isomorphisms are compatible, so you need triple intersections – Maxime Ramzi Mar 20 '20 at 10:36
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    (you don't need more because groupoids have no higher homotopy groups, but if you were looking at, say, stacks in $\infty$-groupoids (or even $2$-groupoids), you'd need more intersections : in fact if your groupoids have possibly arbitrarily high nontrivial homotopy groups ,you need the full Cech nerve of the covering) – Maxime Ramzi Mar 20 '20 at 10:37
  • @Max: Thank you for your helpful comment. I posted this on mathoverflow seeking a detailed answer. – Bumblebee Mar 21 '20 at 20:07

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