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.**