Let $H$ be an $(\infty,1)$-topos (seen as a generalization of the homotopy category of spaces).
You can define the suspension of an object $X$ as the (homotopy) pushout of $*\leftarrow X \to *$, hence you can define inductively the spheres $\mathbb{S}^n$ (the sphere of dimension $-1$ is the initial object of $H$ and the sphere of dimension $n+1$ is the suspension of the sphere of dimension $n$).
You can also define the loop spaces of a pointed object as the (homotopy) pullback of $*\to X \leftarrow *$. It will be itself pointed (because there is an obvious commutative diagram with a $1$ instead of $\Omega{}X$, so there is (I think) an arrow between this $1$ and $\Omega{}X$).
Then, given two integers $n, k$, you can define $\pi_k(\mathbb{S}^n)$ as the set of connected components (global elements up to homotopy) of the $k$-fold loop space of the $n$-sphere (I don’t know if this definition is one of the two described in the nlab)
Is there a natural group structure on $\pi_k(\mathbb{S}^n)$?
Is there something known about these groups in general?
For example,
- Are they completely known for some $H$?
- Is it always true that $\pi_k(\mathbb{S}^n)$ is trivial for
$k<n$and isomorphic to $\mathbb{Z}$ for $k=n$? - Are they isomorphic (or related in some way) to the usual homotopy groups of spheres?
Addition:
What if you assume that $H$ is a cohesive $(\infty,1)$-topos? (see here for the nLab page)
