nLab Čech groupoid

Idea

For $\{U_i \to X\}$ a cover of a space $X$, the corresponding Čech groupoid is the internal groupoid

$$C(\{U_i\}) = (\coprod_{i j} U_i \cap U_j \rightrightarrows \coprod_i U_i)$$

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

  $$\array{ && (x,j) \\ & \nearrow &=& \searrow \\ (x,i) &&\to&& (x,k) } \,.$$

Definition

Definition

(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

$$C(\{U_i\}) \;\in\; [\mathcal{C}^{op}, Grpd] \;\simeq\; Grpd\left( [\mathcal{C}^{op}, Set] \right)$$

which, regarded as an internal category in the category of presheaves over $\mathcal{C}$, has as presheaf of objects the coproduct

$$Obj_{C(\{U_i\})} \;\coloneqq\; \underset{i}{\coprod} y(U_i)$$

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:

$$Mor_{C(\{U_i\})} \;\coloneqq\; \underset{i,j}{\coprod} y(U_i) \times_{y(X)} y(U_j) \,.$$

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

$$\kappa_i \longrightarrow \kappa_j$$

precisely if

(1)$$\iota_i \circ \kappa_i \;=\; \iota_j \circ \kappa_j \phantom{AAAAAAAA} \array{ && V \\ & {}^{\mathllap{\kappa_i}}\swarrow && \searrow^{\mathrlap{\kappa_j}} \\ U_i && && U_j \\ & {}_{\mathllap{\iota_i}}\searrow && \swarrow_{\mathrlap{\iota_j}} \\ && X }$$

Properties

Codescent

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:

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

(2)$$\iota_i \circ \kappa_i \;=\; \iota_j \circ \kappa_j \phantom{AAAAAAAA} \array{ && V \\ & {}^{\mathllap{\kappa_i}}\swarrow && \searrow^{\mathrlap{\kappa_j}} \\ U_i && && U_j \\ & {}_{\mathllap{\iota_i}}\searrow && \swarrow_{\mathrlap{\iota_j}} \\ && X }$$

we have

(3)$$\mathbf{Y}(\kappa_i)(\phi_i) \;=\; \mathbf{Y}(\kappa_j)(\phi_j) \,.$$

We write

(4)$$Match\big( \{U_i\}_{i \in I} \,,\, \mathbf{Y} \big) \subset \underset{i}{\prod} \mathbf{Y}(U_i) \;\in\; Set$$

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 $\{U_i \overset{\iota_i}{\to}X\}$.

Proposition

(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

$$\mathbf{Y} \in [\mathcal{C}^{op}, Set] \hookrightarrow [\mathcal{C}^{op}, Grpd]$$

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

$$[\mathcal{C}^{op},Grpd] \left( C\left(\{U_i\}_i\right), \, \mathbf{Y} \right) \;\simeq\; Match\left( \{U_i\}_i, \, \mathbf{Y} \right)$$

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

$$\array{ [\mathcal{C}^{op}, Grpd]\left( X, \, \mathbf{Y} \right) & \simeq & \mathbf{Y}(X) \\ {}^{ \mathllap{ [\mathcal{C}^{op}, Grpd](p_{\{U_i\}_i}, \mathbf{Y}) } }\downarrow && \downarrow \\ [\mathcal{C}^{op},Grpd] \left( C\left(\{U_i\}_i\right), \, \mathbf{Y} \right) &\simeq& Match\left( \{U_i\}_i, \, \mathbf{Y} \right) } \,.$$

In conclusion, this means that the presheaf $\mathbf{Y}$ is a sheaf (Def. ) precisely if homming Cech groupoid projections into it produces an isomorphism.

Proof

The hom-groupoid is computed as the end

$$[\mathcal{C}^{op},Grpd] \left( C\left(\{U_i\}_i\right), \, \mathbf{Y} \right) \;=\; \int_{V \in \mathcal{C}} \left[ C\left(\{U_i\}_i\right)(V), \, \mathbf{Y}(V) \right] \,,$$

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

$$\underset{i}{\coprod} y(U_i)(V) = Obj_{C\left(\{U_i\}_i\right)(V)} \longrightarrow Obj_{\mathbf{Y}(V)} = \mathbf{Y}(V)$$

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:

$$\begin{aligned} \int_{V \in \mathcal{C}} \left[ C\left(\{U_i\}_i\right)(V), \, \mathbf{Y}(V) \right] & \hookrightarrow \int_{V \in \mathcal{C}} \left[ \underset{i}{\coprod} y(U_i)(V), \, \mathbf{Y}(V) \right] \\ & \simeq \int_{V \in \mathcal{C}} \underset{i}{\prod} \left[ y(U_i)(V), \, \mathbf{Y}(V) \right] \\ & \simeq \underset{i}{\prod} \int_{V \in \mathcal{C}} \left[ y(U_i)(V), \, \mathbf{Y}(V) \right] \\ & \simeq \underset{i}{\prod} \mathbf{Y}(U_i) \end{aligned}$$

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

$$\array{ C\left(\{U_i\}_i\right)(V) & \longrightarrow & \mathbf{Y}(V) \\ (V \overset{\kappa_i}{\to} U_i) &\mapsto& \kappa_i^\ast \phi_i = \mathbf{Y}(\kappa_i)(\phi_i) }$$

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

$$\mathbf{Y}(\kappa_i)(\phi_i) \;=\; \mathbf{Y}(\kappa_j)(\phi_j) \phantom{AAAA} \text{for all}\, i,j \,,$$

which is exactly the condition (3) that makes $(\phi_i)_i$ a matching family.

Examples

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

$$\array{ C(\{U_i\}) &\stackrel{g}{\to}& \mathbf{B}G \\ \downarrow^{\mathrlap{\simeq}} \\ X }$$

are also called anafunctors from $X$ to $\mathbf{B}G$. They correspond to smooth $G$-principal bundles on $X$.

Related concepts

• Cech nerve

References

For instance

• J. F. Jardine, Example 5 in: Homotopy classification of gerbes (arXiv:math/0605200)