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.

Thanks in advance.

MathOverflow Asked by Adittya Chaudhuri on January 19, 2021

1 Answers

One Answer

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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