bisection of a Lie groupoid


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Bisections of Lie groupoids


In components


Let (X 1(d 0,d 1)X 0×X 0)(X_1 \stackrel{(d_0, d_1)}{\to} X_0 \times X_0) be a Lie groupoid.

A bisection of is a smooth function σ:X 0X 1\sigma : X_0 \to X_1 such that

  1. σ\sigma is a section of d 1d_1;

  2. d 0σ:X 0X 0d_0 \circ \sigma : X_0 \to X_0 is a diffeomorphism.

Bisections naturally form a group under pointwise composition in XX, the group of bisections of the Lie groupoid.


Let H=\mathbf{H} = Smooth∞Grpd. Let XHX \in \mathbf{H} be equipped with an atlas, hence with an effective epimorphism UXU \to X out of a 0-truncated object.

We may regard this atlas as an object in the slice (∞,1)-topos XH /X\mathbf{X} \in \mathbf{H}_{/X}


The smooth ∞-group of bisections of X\mathbf{X} is its automorphism ∞-group

BiSect(X,U)Aut /X(X,X). \mathbf{BiSect}(X,U) \coloneqq \mathbf{Aut}_{/X}(\mathbf{X}, \mathbf{X}) \,.

For XX a 1-groupoid as above and U=X 0U = X_0, a bisection is precisely a smooth natural transformation of the form

U U η X. \array{ U &&\stackrel{\simeq}{\to}&& U \\ & \searrow &\swArrow_{\mathrlap{\eta}}& \swarrow \\ && X } \,.

Here the top morphism is a diffeomorphism ϕ:XX\phi : X \to X and since the diagonal morphisms are identities onto the object manifold the component map of η\eta is

x(xη(x)ϕ(x)). x \mapsto (x \stackrel{\eta(x)}{\to} \phi(x)) \,.

This is precisely the bisection in the traditional sense of def. 1.


Relation to Lie-Rinehart algebras

For UXU \to X a Lie groupoid with atlas as above, write 𝔤=Lie(BiSect(X,U))\mathfrak{g} = Lie(\mathbf{BiSect}(X,U)) for the Lie algebra of the group of bisections. Then (C (X),𝔤)(C^\infty(X), \mathfrak{g}) is the Lie-Rinehart algebra corresponding to the Lie algebroid of the Lie groupoid.

Relation to Atiyah groupoids

for the moment see at Atiyah groupoid and higher Atiyah groupoid.

Revised on February 20, 2013 21:30:54 by Urs Schreiber (