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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 …
Dmitri Pavlov's user avatar