∞-Lie theory

# Contents

## Idea

The geometric homotopy groups of a Lie groupoid $X$ are those of its geometric realization $\mid X\mid$ when regarded as a simplicial manifold. Equivalently, regarding $X$ as an object in the (∞,1)-topos ∞LieGrpd, its homotopy groups are those of the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos $\Pi \left(X\right)\in$ ∞Grpd.

## Definition

For $X=\left({X}_{1}\stackrel{\to }{\to }{X}_{0}\right)$ a Lie groupoid and $x:*\to X$ a point, let

${X}_{•}=\left(\cdots {X}_{1}{×}_{{X}_{0}}{X}_{1}{×}_{{X}_{0}}{X}_{1}\stackrel{\to }{\stackrel{\to }{\to }}{X}_{1}{×}_{{X}_{0}}{X}_{1}\stackrel{\to }{\to }{X}_{0}\right)$X_\bullet = \left( \cdots X_1 \times_{X_0} X_1 \times_{X_0} X_1 \stackrel{\to}{\stackrel{\to}{\to}}X_1 \times_{X_0} X_1 \stackrel{\to}{\to} X_0 \right)

be its nerve regarded as a simplicial manifold.

###### Remark

When regarding each manifold ${X}_{n}$ as a diffeological space, hence a sheaf on the site CartSp then ${X}_{•}\mathrm{in}\mathrm{PSh}\left(\mathrm{CartSp}{\right)}^{{\Delta }^{\mathrm{op}}}\simeq \left[{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}\right]$ is the simplicial presheaf on CartSp that presents $X$ as an object in the (∞,1)-topos ∞LieGrpd of ∞-Lie groupoids.

###### Definition

Regard ${X}_{•}$ as a simplicial topological space by forgetting the smooth structure. Write $\mid {X}_{•}\mid \in$ Top for its geometric realization as a simplicial topological space.

The geometric homotopy groups of $X$ are defined to be the ordinary homotopy groups of the topological space $\mid {X}_{•}\mid$:

${\pi }_{n}\left(X,x\right):={\pi }_{n}\left(\mid {X}_{•}\mid ,x\right)\phantom{\rule{thinmathspace}{0ex}}.$\pi_n(X,x) := \pi_n(|X_\bullet|,x) \,.

In this form the definition originates in (Segal).

## Properties

Regard $X$ as an ∞-Lie groupoid under the natural embedding $\mathrm{LieGrpd}↪\infty \mathrm{LieGrpd}$. By the discussion at ∞LieGrpd this is a locally ∞-connected (∞,1)-topos, which means that its terminal geometric morphism comes with a further left adjoint $\Pi$

$\left(\Pi ⊣\Delta ⊣\Gamma \right):\infty \mathrm{LieGrpd}\to \infty \mathrm{Grpd}\phantom{\rule{thinmathspace}{0ex}}.$(\Pi \dashv \Delta \dashv \Gamma) : \infty LieGrpd \to \infty Grpd \,.

We say that $\Pi \left(X\right)\in \infty \mathrm{Grpd}\simeq \mathrm{Top}$ is the fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos of $X$.

###### Observation

The geometric homotopy groups of $X$ are those of $\Pi \left(X\right)\in \mathrm{Top}$.

###### Proof

By the discussion at ∞-Lie groupoid we have precisely that $\Pi \left(X\right)$ is presented by the geometric realization of the simplicial topological space underlying the nerve of $X$.

## References

The definition of the homotopy groups of a Lie groupoid as those of its geometric realization appearently goes back to

• Graeme Segal, Classifying spaces and spectral sequences , IHES Publ. Math. 34 (1968) 105–112.

An equivalent definition is in

• A. Haefliger, Groupoïdes d’holonomie et espaces classiants , Astérisque 116 (1984), 70-97

reproduced in section 3 of

• Graeme Segal, Classifying spaces related to foliations , Topology 17 (1978), 367-382.

Revised on January 3, 2011 10:30:41 by Anonymous Coward (81.153.251.158)