Čech groupoid


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

C({U i})=( ijU iU j iU i) C(\{U_i\}) = (\coprod_{i j} U_i \cap U_j \rightrightarrows \coprod_i U_i)

whose set of objects is the disjoint union iU i\coprod_i U_i of the covering patches, and the set of morphisms is the disjoint union of the intersections U iU jU_i \cap U_j of these patches.

This is the 22-coskeleton of the full Čech nerve. See there for more details.

If we speak about generalized points of the U iU_i (which are often just ordinary points, in applications), then

  • an object of C({U i})C(\{U_i\}) is a pair (x,i)(x,i) where xx is a point in U iU_i;

  • there is a unique morphism (x,i)(x,j)(x,i) \to (x,j) for all pairs of objects labeled by the same xx such that xU iU jx \in U_i \cap U_j;

  • hence the composition of morphism is of the form

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



(Cech groupoid)

Let 𝒞\mathcal{C} be a site, and X𝒞X \in \mathcal{C} an object of that site. For each covering family {U iι iX}\{ U_i \overset{\iota_i}{\to} X\} of XX in the given coverage, the Cech groupoid is the presheaf of groupoids

C({U i})[𝒞 op,Grpd]Grpd([𝒞 op,Set]) 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 𝒞\mathcal{C}, has as presheaf of objects the coproduct

Obj C({U i})iy(U i) Obj_{C(\{U_i\})} \;\coloneqq\; \underset{i}{\coprod} y(U_i)

of the presheaves represented (under the Yoneda embedding) by the covering objects U iU_i, and as presheaf of morphisms the coproduct over all fiber products of these:

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

This means that for any V𝒞V \in \mathcal{C} the groupoid assigned by C({U i})C(\{U_i\}) has as set of objects pairs consisting of an index ii and a morphism Vκ iU iV \overset{\kappa_i}{\to} U_i in 𝒞\mathcal{C}, and there is a unique morphism between two such objects

κ iκ j \kappa_i \longrightarrow \kappa_j

precisely if

(1)ι iκ i=ι jκ jAAAAAAAA V κ i κ j U i U j ι i ι j X \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 discuss (Prop. 1 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 familydescent object)

Let 𝒞\mathcal{C} be a small category equipped with a coverage, hence a site and consider a presheaf Y[𝒞 op,Set]\mathbf{Y} \in [\mathcal{C}^{op}, Set] (Example \ref{CategoryOfPresheaves}) over 𝒞\mathcal{C}.

Given an object X𝒞X \in \mathcal{C} and a covering {U iι iX} iI\left\{ U_i \overset{\iota_i}{\to} X \right\}_{i \in I} of it we say that a matching family (of probes of Y\mathbf{Y}) is a tuple (ϕ iY(U i)) iI(\phi_i \in \mathbf{Y}(U_i))_{i \in I} such that for all i,jIi,j \in I and pairs of morphisms U iκ iVκ jU jU_i \overset{\kappa_i}{\leftarrow} V \overset{\kappa_j}{\to} U_j satisfying

(2)ι iκ i=ι jκ jAAAAAAAA V κ i κ j U i U j ι i ι j X \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)Y(κ i)(ϕ i)=Y(κ j)(ϕ j). \mathbf{Y}(\kappa_i)(\phi_i) \;=\; \mathbf{Y}(\kappa_j)(\phi_j) \,.

We write

(4)Match({U i} iI,Y)iY(U i)Set 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 Y\mathbf{Y} for descent along the covering {U iι iX}\{U_i \overset{\iota_i}{\to}X\}.


(Cech groupoid co-represents matching familiescodescent)

For Grpd regarded as a cosmos for enriched category theory, via its cartesian closed category-structire, and 𝒞\mathcal{C} a site, let

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

be a presheaf on 𝒞\mathcal{C}, regarded as a Grpd-enriched presheaf, let X𝒞X \in \mathcal{C} be any object and {U iι iX} i\{U_i \overset{\iota_i}{\to} X\}_i a covering family with induced Cech groupoid C({U i} i)C(\{U_i\}_i) (Def.1).

Then there is an isomorphism

[𝒞 op,Grpd](C({U i} i),Y)Match({U i} i,Y) [\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. 2).

Since therefor 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({U i} i)p {U i} iXC\left( \{U_i\}_i \right) \overset{p_{\{U_i\}_i}}{\longrightarrow} X induces the comparison morphism

[𝒞 op,Grpd](X,Y) Y(X) [𝒞 op,Grpd](p {U i} i,Y) [𝒞 op,Grpd](C({U i} i),Y) Match({U i} i,Y). \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 Y\mathbf{Y} is a sheaf (Def. \ref{Sheaf}) precisely if homming Cech groupoid projections into it produces an isomorphism.


The hom-groupoid is computed as the end

[𝒞 op,Grpd](C({U i} i),Y)= V𝒞[C({U i} i)(V),Y(V)], [\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 VV to the set (regarded as a groupoid) assigned by Y\mathbf{Y} to VV.

Since Y(V)\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({U i} i)(V)Y(V)C\left(\{U_i\}_i\right)(V) \longrightarrow \mathbf{Y}(V), which are equivalently those functions on sets of objects

iy(U i)(V)=Obj C({U i} i)(V)Obj Y(V)=Y(V) \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 VV.

Hence the hom-groupoid is a subset of the end of these function sets:

V𝒞[C({U i} i)(V),Y(V)] V𝒞[iy(U i)(V),Y(V)] V𝒞i[y(U i)(V),Y(V)] i V𝒞[y(U i)(V),Y(V)] iY(U i) \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 V𝒞[C({U i} i)(V),Y(V)] \int_{V \in \mathcal{C}} \left[ C\left(\{U_i\}_i\right)(V), \, \mathbf{Y}(V) \right] is a tuple (ϕ iY(U i)) i(\phi_i \in \mathbf{Y}(U_i))_i, subject to some condition. This condition is that for each V𝒞V \in \mathcal{C} the assignment

C({U i} i)(V) Y(V) (Vκ iU i) κ i *ϕ i=Y(κ i)(ϕ i) \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

Y(κ i)(ϕ i)=Y(κ j)(ϕ j)AAAAfor alli,j, \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 (ϕ i) i(\phi_i)_i a matching family.


For XX a smooth manifold and {U iX}\{U_i \to X\} an atlas by coordinate charts, the Čech groupoid is a Lie groupoid which is equivalent to XX as a Lie groupoid: C({U i})XC(\{U_i\}) \stackrel{\simeq}{\to} X

For BG\mathbf{B}G the Lie groupoid with one object coming from a Lie group GG morphisms of Lie groupoids of the form

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

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


For instance

Last revised on June 13, 2018 at 15:35:34. See the history of this page for a list of all contributions to it.