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Given a topological space $X$, and a cover $\mathcal{U} :=\cup_{\alpha \in I}U_{\alpha}$ of $X$, one can define a groupoid called Čech groupoid $C(\mathcal{U})$ of the cover $\mathcal{U}$ by $\sqcup_{i,j \in I} U_i \cap U_j \rightrightarrows \sqcup_{i \in I} U_i$ whose structure maps are obvious to define.

Now given a site $(C,J)$ and an object $c \in C$, one has a cover $J_c$ of $c$ induced from $J$.

My question:

Is there an analogous notion of Čech Groupoid corresponding to $J_c$? Or the investigation in this direction may not be fruitful?

I will also be very grateful if someone can provide some literature references regarding these.

Thanks in advance.

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    $\begingroup$ What kind of site? (Does it have pullbacks? Disjoint unions?) What do you hope to do with the “groupoid” you get? $\endgroup$
    – Zhen Lin
    Commented Oct 17, 2020 at 2:40
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    $\begingroup$ @ZhenLin No, I am not assuming that pull-backs or disjoint unions to exist . I actually just need the existence of a pretopology on a category $C$ in the sense of definition 2.24 in page 27 of homepage.sns.it/vistoli/descent.pdf. I was thinking about the functors from "such Čech groupoids" (if there is) to $BG$, the delooping of a group. I was thinking whether one can define the notion of locally trivial principal bundles over an object of a site. (I know that Grothendieck topology was introduced for different purpose but I am just curious). Apology in advance if I sound stupid. $\endgroup$
    – Adittya Chaudhuri
    Commented Oct 17, 2020 at 2:55

1 Answer 1

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Take $U=\coprod_{i∈I}Y(U_i)$, where $Y\colon C\to\mathop{\rm Presh}(C,{\rm Set})$ is the Yoneda embedding. We have a canonical morphism $U→Y(X)$.

The Čech groupoid of $J_c$ can now be defined as the groupoid with objects $U$ and morphisms $U⨯_{Y(X)}U$, with source, target, composition, and identity maps defined in the usual manner.

In the case of a site coming from a topological space, this construction recovers the usual Čech groupoid.

In fact, iterating fiber products produces a simplicial presheaf, namely, the Čech nerve of $J_c$, which is used to define Čech descent for simplicial presheaves.

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    $\begingroup$ Thank you Sir for the answer. If we consider functors from such Čech groupoid of $J_c$ to $BG$, the delooping of a group object in $C$.(Assuming group objects exist in $C$ ) then analogous to the classical case I am expecting a notion of principal bundle over the object $c \in C$ . Is this "principal bundle" worth studying for any purpose? $\endgroup$
    – Adittya Chaudhuri
    Commented Oct 17, 2020 at 4:59
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    $\begingroup$ Though I am not sure whether the notion of delooping of a group object in an arbitrary category makes sense or not. In the last comment I just vaguely imagined that such notion may exist in an appropriate sense. (Apology in advance if I am not making much sense) $\endgroup$
    – Adittya Chaudhuri
    Commented Oct 17, 2020 at 5:28
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    $\begingroup$ @AdittyaChaudhuri: Yes, this is how principal bundles over algebraic groups can be defined (for example) for the Zariski site of a scheme. $\endgroup$
    – Dmitri Pavlov
    Commented Oct 17, 2020 at 5:35
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    $\begingroup$ Ohh!! I did not know that. Thank you Sir. $\endgroup$
    – Adittya Chaudhuri
    Commented Oct 17, 2020 at 5:38

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