nLab homotopy groups in an (infinity,1)-topos

Idea

In the (∞,1)-topos Top 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 ∞Grpd usually 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.

Categorical homotopy groups

See categorical homotopy groups in an (∞,1)-topos.

Geometric homotopy groups

See geometric homotopy groups in an (∞,1)-topos.