nLab smooth groupoid

Context

Cohesive $\infty$-Toposes

cohesive topos

cohesive (∞,1)-topos

cohesive homotopy type theory

Structures in a cohesive $(\infty,1)$-topos

structures in a cohesive (∞,1)-topos

Structures with infinitesimal cohesion

infinitesimal cohesion?

Models

Differential geometry

differential geometry

synthetic differential geometry

∞-Lie theory

Contents

Idea

The notion of smooth groupoid is the first generalization of the notion of smooth space to higher differential geometry. A smooth groupoid is a stack on the site CartSp${}_{smooth}$. This is equivalently an 1-truncated smooth infinity-groupoid.

Definition

A smooth stack or smooth groupoid is a stack on the site SmoothMfd of smooth manifolds or equivalently (and often more conveniently) on its dense subsite CartSp of just Cartesian spaces $\mathbb{R}^n, n \in \mathbb{N}$ and smooth functions between them, equipped with the standard coverage of good open covers.

We write

$\;\;\;$SmoothGrpd $\coloneqq Sh_{(2,1)}(CartSp) \simeq L_{lhe} Func(CartSp^{op}, Grpd)$

for the (2,1)-category of stacks on this site, equivalently the result of taking groupoid-valued presheaves and then universally turning local (as seen by the coverage) equivalences of groupoids into global equivalence in an (infinity,1)-category.

By generalizing here groupoids to general Kan complexes and equivalences of groupoids to homotopy equivalences of Kan complexes, one obtains smooth ∞-stacks or smooth ∞-groupoids, which we write

$\;\;\;$Smooth∞Grpd $\coloneqq Sh_{(\infty,1)}(CartSp) \simeq L_{lhe} Func(CartSp^{op}, KanCplx)$.

We often write $\mathbf{H} \coloneqq$ Smooth∞Grpd for short.

By the (∞,1)-Yoneda lemma there is a sequence of faithful inclusions

$\;\;\;$ SmoothMfd $\hookrightarrow$ SmoothGrpd $\hookrightarrow$ Smooth∞Grpd.

This induces a corresponding inclusion of simplicial objects and hence of groupoid objects

$LieGrpd \hookrightarrow Grpd_\infty(SmoothMfd) \hookrightarrow Grpd_\infty(Smooth\infty Grpd) \,.$

For $\mathcal{G}_\bullet \in Grpd_\infty(\mathbf{H})$ a groupoid object we write

$\mathcal{G}_0 \to \mathcal{G} \coloneqq \underset{\longrightarrow}{\lim}_{n} \mathcal{G}_n$

for its (∞,1)-colimiting cocone, hence $\mathcal{G} \in \mathbf{H}$ (without subscript decoration) denotes the realization of $\mathcal{G}_\bullet$ as a single object in $\mathbf{H}$.

By the Giraud-Rezk-Lurie axioms of the (∞,1)-topos $\mathbf{H}$ this morphism $\mathcal{G}_0 \to \mathcal{G}$ is a 1-epimorphism – an atlas – and its construction establishes is an equivalence of (∞,1)-categories $Grpd_\infty(\mathbf{H}) \simeq \mathbf{H}^{\Delta^1}_{1epi}$, hence morphisms $\mathcal{G}_\bullet \to \mathcal{K}_\bullet$ in $Grpd_\infty(\mathbf{H})$ are equivalently diagrams in $\mathbf{H}$ of the form

$\array{ \mathcal{G}_0 &\to& \mathcal{K}_0 \\ \downarrow &\swArrow& \downarrow \\ \mathcal{G} &\to& \mathcal{K} } \,.$

This is evidently more constrained that just morphisms

$\mathcal{G} \to \mathcal{K}$

by themselves. The latter are the “generalized” or Morita morphisms between the groupoid objects $\mathcal{G}_\bullet$, $\mathcal{K}_\bullet$. These can be modeled as $\mathcal{G}_\bullet$-$\mathcal{K}_\bullet$-bibundles.

Examples

Every Lie groupoid presents a smooth groupoids. Those of this form are also called differentiable stacks.

A 0-truncated smooth groupoid is equivalently a smooth space.

For $G$ a smooth group, its delooping $\mathbf{B}G$ is a smooth groupoid, the moduli stack of smooth $G$-principal bundles.

References

Discussion of smooth stacks as target spaces for sigma-model quantum field theories is in

Discussion of geometric Langlands duality in terms of 2d sigma-models on stacks (moduli stacks of Higgs bundles over a given algebraic curve) is in

Revised on July 1, 2014 04:00:50 by Urs Schreiber (88.128.80.30)