For $\{U_i \to X\}$ a cover of a space $X$, the corresponding Čech groupoid is the internal groupoid
whose set of objects is the disjoint union $\coprod_i U_i$ of the covering patches, and the set of morphisms is the disjoint union of the intersections $U_i \cap U_j$ of these patches.
This is the $2$-coskeleton of the full Čech nerve. See there for more details.
If we speak about generalized points of the $U_i$ (which are often just ordinary points, in applications), then
an object of $C(\{U_i\})$ is a pair $(x,i)$ where $x$ is a point in $U_i$;
there is a unique morphism $(x,i) \to (x,j)$ for all pairs of objects labeled by the same $x$ such that $x \in U_i \cap U_j$;
hence the composition of morphism is of the form
(Cech groupoid)
Let $\mathcal{C}$ be a site, and $X \in \mathcal{C}$ an object of that site. For each covering family $\{ U_i \overset{\iota_i}{\to} X\}$ of $X$ in the given coverage, the Cech groupoid is the presheaf of groupoids
which, regarded as an internal category in the category of presheaves over $\mathcal{C}$, has as presheaf of objects the coproduct
of the presheaves represented (under the Yoneda embedding) by the covering objects $U_i$, and as presheaf of morphisms the coproduct over all fiber products of these:
This means that for any $V \in \mathcal{C}$ the groupoid assigned by $C(\{U_i\})$ has as set of objects pairs consisting of an index $i$ and a morphism $V \overset{\kappa_i}{\to} U_i$ in $\mathcal{C}$, and there is a unique morphism between two such objects
precisely if
We discuss (Prop. below) how the Cech groupoid co-represents matching families as they appear in the definition of sheaves.
For reference, we first recall that definition:
(matching family – descent object)
Let $\mathcal{C}$ be a small category equipped with a coverage, hence a site and consider a presheaf $\mathbf{Y} \in [\mathcal{C}^{op}, Set]$ (Example ) over $\mathcal{C}$.
Given an object $X \in \mathcal{C}$ and a covering $\left\{ U_i \overset{\iota_i}{\to} X \right\}_{i \in I}$ of it we say that a matching family (of probes of $\mathbf{Y}$) is a tuple $(\phi_i \in \mathbf{Y}(U_i))_{i \in I}$ such that for all $i,j \in I$ and pairs of morphisms $U_i \overset{\kappa_i}{\leftarrow} V \overset{\kappa_j}{\to} U_j$ satisfying
we have
We write
for the set of matching families for the given presheaf and covering.
This is also called the descent object of $\mathbf{Y}$ for descent along the covering $\{U_i \overset{\iota_i}{\to}X\}$.
(Cech groupoid co-represents matching families – codescent)
For Grpd regarded as a cosmos for enriched category theory, via its cartesian closed category-structure, and $\mathcal{C}$ a site, let
be a presheaf on $\mathcal{C}$, regarded as a Grpd-enriched presheaf, let $X \in \mathcal{C}$ be any object and $\{U_i \overset{\iota_i}{\to} X\}_i$ a covering family with induced Cech groupoid $C(\{U_i\}_i)$ (Def.).
Then there is an isomorphism
between the hom-groupoid of Grpd-enriched presheaves and the set of matching families (Def. ).
Since therefore the Cech-groupoid co-represents the descent object, it is sometimes called the codescent object along the given covering.
Moreover, under this identification the canonical morphism $C\left( \{U_i\}_i \right) \overset{p_{\{U_i\}_i}}{\longrightarrow} X$ induces the comparison morphism
In conclusion, this means that the presheaf $\mathbf{Y}$ is a sheaf (Def. ) precisely if homming Cech groupoid projections into it produces an isomorphism.
The hom-groupoid is computed as the end
where the “integrand” is the functor category (here: a groupoid) from the Cech groupoid at a given $V$ to the set (regarded as a groupoid) assigned by $\mathbf{Y}$ to $V$.
Since $\mathbf{Y}(V)$ is just a set, that functor groupoid, too, is just a set, regarded as a groupoid. Its elements are the functors $C\left(\{U_i\}_i\right)(V) \longrightarrow \mathbf{Y}(V)$, which are equivalently those functions on sets of objects
which respect the equivalence relation induced by the morphisms in the Cech groupoid at $V$.
Hence the hom-groupoid is a subset of the end of these function sets:
Here we used: first that the internal hom-functor turns colimits in its first argument into limits (see at internal hom-functor preserves limits), then that limits commute with limits, hence that in particular ends commute with products , and finally the enriched Yoneda lemma, which here comes down to just the plain Yoneda lemma . The end result is hence the same Cartesian product set that also the set of matching families is defined to be a subset of, in (4).
This shows that an element in $\int_{V \in \mathcal{C}} \left[ C\left(\{U_i\}_i\right)(V), \, \mathbf{Y}(V) \right]$ is a tuple $(\phi_i \in \mathbf{Y}(U_i))_i$, subject to some condition. This condition is that for each $V \in \mathcal{C}$ the assignment
constitutes a functor of groupoids.
By definition of the Cech groupoid, and since the codomain is a just set regarded as a groupoid, this is the case precisely if
which is exactly the condition (3) that makes $(\phi_i)_i$ a matching family.
For $X$ a smooth manifold and $\{U_i \to X\}$ an atlas by coordinate charts, the Čech groupoid is a Lie groupoid which is equivalent to $X$ as a Lie groupoid: $C(\{U_i\}) \stackrel{\simeq}{\to} X$
For $\mathbf{B}G$ the Lie groupoid with one object coming from a Lie group $G$ morphisms of Lie groupoids of the form
are also called anafunctors from $X$ to $\mathbf{B}G$. They correspond to smooth $G$-principal bundles on $X$.
For instance
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