homotopy theory, (∞,1)-category theory, homotopy type theory
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∞-Lie theory (higher geometry)
synthetic differential ∞-groupoid?
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
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 $|X_\bullet|$:
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 $\Pi$
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
An equivalent definition is in
reproduced in section 3 of
Last revised on January 3, 2011 at 10:30:41. See the history of this page for a list of all contributions to it.