nLab
homotopy groups of a Lie groupoid

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 $\Pi (X)\in$ ∞Grpd.

Definition

For $X=(X_1\overrightarrow{\to }{X}_0)$ a Lie groupoid and $x:*\to X$ a point, let

$$X_{•}=\left(\cdots X_1{\times }_{X_0}{X}_1{\times }_{X_0}{X}_1\overrightarrow{\overrightarrow{\to }}{X}_1{\times }_{X_0}{X}_1\overrightarrow{\to }{X}_0\right)$$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 $X_{•}\mathrm{in}\mathrm{PSh}(\mathrm{CartSp}{)}^{{\Delta }^{\mathrm{op}}}\simeq [{\mathrm{CartSp}}^{\mathrm{op}},\mathrm{sSet}]$ 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 $\mid X_{•}\mid$:

$${\pi }_n(X,x):={\pi }_n(\mid X_{•}\mid ,x).$$\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 $\mathrm{LieGrpd}\hookrightarrow \mathrm{\infty }\mathrm{LieGrpd}$. 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 $\Pi$

$$(\Pi ⊣\Delta ⊣\Gamma ):\mathrm{\infty }\mathrm{LieGrpd}\to \mathrm{\infty }\mathrm{Grpd}.$$(\Pi \dashv \Delta \dashv \Gamma) : \infty LieGrpd \to \infty Grpd \,.

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

Observation

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

Proof

By the discussion at ∞-Lie groupoid we have precisely that $\Pi (X)$ 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.