# Is there a notion of Čech groupoid of a cover of an object in a Grothendieck site?

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.

• What kind of site? (Does it have pullbacks? Disjoint unions?) What do you hope to do with the “groupoid” you get? – Zhen Lin Oct 17 at 2:40
• @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. – Adittya Chaudhuri Oct 17 at 2:55

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 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.
• 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? – Adittya Chaudhuri Oct 17 at 4:59