TITLE: Is there a notion of Čech groupoid of a cover of an object in a Grothendieck site?
QUESTION [5 upvotes]: 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.

REPLY [9 votes]: 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.