homotopy groups of a Lie groupoid

**homotopy theory, (∞,1)-category theory, homotopy type theory**

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…

models: topological, simplicial, localic, …

see also **algebraic topology**

**Introductions**

**Definitions**

**Paths and cylinders**

**Homotopy groups**

**Basic facts**

**Theorems**

**∞-Lie theory** (higher geometry)

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.

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.

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.

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).

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 $X$.

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

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 $X$.

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.