nLab
Čech groupoid

Context

Category theory

Geometry

Contents

Definition

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) } \,.

Examples

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.

Revised on March 2, 2012 15:56:07 by Andrew Stacey (129.241.15.200)