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.**



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


