nLab
smooth groupoid

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${}_{\mathrm{smooth}}$. 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 ${ℝ}^n,n\in \mathbb{N}$ and smooth functions between them, equipped with the standard coverage of good open covers.

We write

$$SmoothGrpd $≔{\mathrm{Sh}}_{(2,1)}(\mathrm{CartSp})\simeq L_{\mathrm{lhe}}\mathrm{Func}({\mathrm{CartSp}}^{\mathrm{op}},\mathrm{Grpd})$

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 $≔{\mathrm{Sh}}_{(\mathrm{\infty },1)}(\mathrm{CartSp})\simeq L_{\mathrm{lhe}}\mathrm{Func}({\mathrm{CartSp}}^{\mathrm{op}},\mathrm{KanCplx})$.

We often write $H≔$ 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

$$\mathrm{LieGrpd}\hookrightarrow {\mathrm{Grpd}}_{\mathrm{\infty }}(\mathrm{SmoothMfd})\hookrightarrow {\mathrm{Grpd}}_{\mathrm{\infty }}(\mathrm{Smooth}\mathrm{\infty }\mathrm{Grpd}).$$LieGrpd \hookrightarrow Grpd_\infty(SmoothMfd) \hookrightarrow Grpd_\infty(Smooth\infty Grpd) \,.

For ${𝒢}_{•}\in {\mathrm{Grpd}}_{\mathrm{\infty }}(H)$ a groupoid object we write

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

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

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

$$\begin{array}{ccc}{𝒢}_0& \to & {𝒦}_0\\ \downarrow & ⇙& \downarrow \\ 𝒢& \to & 𝒦\end{array}.$$\array{ \mathcal{G}_0 &\to& \mathcal{K}_0 \\ \downarrow &\swArrow& \downarrow \\ \mathcal{G} &\to& \mathcal{K} } \,.

This is evidently more constrained that just morphisms

$$𝒢\to 𝒦$$\mathcal{G} \to \mathcal{K}$ in $\mathbf{H}

by themselves. The latter are the “generalized” or Morita morphisms between the groupoid objects ${𝒢}_{•}$, ${𝒦}_{•}$. These can be modeled as ${𝒢}_{•}$-${𝒦}_{•}$-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 $BG$ is a smooth groupoid, the moduli stack of smooth $G$-principal bundles.

Related concepts

• moduli stack

• centrally extended groupoid