nLab homotopy groups of a Lie groupoid

Idea

The geometric homotopy groups of a Lie groupoid $X$ are those of its geometric realization $|X|$ 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 $X = (X_1 \stackrel{\to}{\to} X_0)$ a Lie groupoid and $x : * \to X$ a point, let

$$X_\bullet = \left( \cdots X_1 \times_{X_0} X_1 \times_{X_0} X_1 \stackrel{\to}{\stackrel{\to}{\to}}X_1 \times_{X_0} X_1 \stackrel{\to}{\to} X_0 \right)$$

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 $X_\bullet in PSh(CartSp)^{\Delta^{op}} \simeq [CartSp^{op},sSet]$ is the simplicial presheaf on CartSp that presents $X$ as an object in the (∞,1)-topos ?LieGrpd? of ∞-Lie groupoids.

Definition

Regard $X_\bullet$ as a simplicial topological space by forgetting the smooth structure. Write $|X_\bullet| \in$ 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 $|X_\bullet|$:

$$\pi_n(X,x) := \pi_n(|X_\bullet|,x) \,.$$

In this form the definition originates in (Segal).

Properties

Regard $X$ as an ∞-Lie groupoid under the natural embedding $LieGrpd \hookrightarrow \infty LieGrpd$. 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 $\Pi$

$$(\Pi \dashv \Delta \dashv \Gamma) : \infty LieGrpd \to \infty Grpd \,.$$

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

Observation

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

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.