# 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.

Take $$U=coprod_{i∈I}Y(U_i)$$, where $$Ycolon Ctomathop{rm Presh}(C,{rm Set})$$ 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.

Correct answer by Dmitri Pavlov on January 19, 2021

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