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I have seen the grothendieck construction referenced in the literature several times, but never have found a good clean overview of how it works. How can I go from a stack which is a category fibered in groupoids over a site, let's say $(Sch/S)_{et}$, to a groupoid in $(Sch/S)$, and then back? For example, consider the quotient stack associated to the ramified etale cover $$ \text{Spec}(\mathbb{C}[x,y]/(x^5 - y)) \to \text{Spec}(\mathbb{C}[y]) $$ with isotropy group $\mathbb{Z}/5$ at the origin.

Mikhail Katz
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54321user
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  • You just mean a groupoid in sheaves on $Sch/S$, no? As phrased it seems you're looking for this groupoid to be representable. – Kevin Carlson May 06 '17 at 07:59
  • No, category fibered in groupoids – 54321user May 06 '17 at 18:49
  • Yes, you started with a category fibered in groupoid, but I was asking for clarification on the other side of the correspondence. You shouldn't be expecting to get a groupoid object in the category of schemes. – Kevin Carlson May 06 '17 at 23:51
  • Oh, you're right. I should get a groupoid of algebraic spaces, and I should consider those as sheaves on the etale site. – 54321user May 07 '17 at 06:13

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