In the (∞,1)-toposTop to every object – every topological space – $X$ is associated the set $\pi_0(X)$ of connected components and the homotopy groups $\pi_n(X,x)$ for $x \in X$ and $n \in \mathbb{N}$, $n\gt 0$.

By the general logic of space, we may think of the objects in an arbitrary ∞-stack(∞,1)-topos as generalized spaces of sorts. Accordingly, there is a notion of homotopy groups of an $\infty$-stack .

But care has to be taken. It turns out that there are actually two different notions of homotopy groups in an arbitrary $(\infty,1)$-topos, two notions that accidentally coincide for Top:

there is a notion of categorical homotopy group:

every $(\infty,1)$-topos $\mathbf{H}$ is powered over ∞Grpdusually modeled as SSet, hence for every object $X \in \mathbf{H}$ there is the categorical $n$-sphere object$X^{S^n_c}$, where $S^n_c = \Delta^n/\partial \Delta^n$.

there should be a notion of geometric homotopy group, induced from the monodromy of locally constant ∞-stacks on objects $X \in \mathbf{H}$.

For instance let $\mathbf{H} = Sh_{(\infty,1)}(Diff)$ be the $(\infty,1)$-topos of Lie ∞-groupoids. An ordinary smooth manifold$X$ is represented in $\mathbf{H}$ by a sheaf of sets on Diff. This has no higher nontrivial categorical homotopy groups – $\pi_{n \gt 0}^{cat}(X) = 0$ – reflecting the fact regarded as a smooth ∞-groupoid, $X$ is a categorically discrete groupoid.

But of course the manifold $X$ may have nontrivial homotopy groups in terms of its underlying topological space. For instance if $X = S^1$ is the circle, then the geometric first homotopy group is nontrivial, $\pi_1^{geom}(X) = \mathbb{Z}$.

We discuss below both cases. The case of categorical homotopy groups is fully understood, for the case of geometric homotopy groups at the moment only a few aspects are in the literature, more is in the making. Some authors of this page (U.S.) thank Richard Williamson for pointing this out.