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

Question

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.

Dmitri Pavlov · Accepted 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.

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

One Answer

Add your own answers!

Ask a Question