TITLE: Homotopy groups of spheres in a $(\infty, 1)$-topos
QUESTION [28 upvotes]: 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)

REPLY [9 votes]: If it may be forgiven to resurrect a very old question, it's worth pointing out that these are not the "homotopy groups of spheres" that appear in synthetic homotopy theory / homotopy type theory.  Those are internal group objects in the $\infty$-topos, not external "ordinary" groups.  So in particular it is still true that "$\pi_1(S^1)=\mathbb{Z}$" in the terminal category, because both "$\pi_1(S^1)$" and "$\mathbb{Z}$" denote a (or rather the) object of the terminal category and hence are equal --- it's irrelevant that in that case this internal $\mathbb{Z}$ doesn't have the external $\mathbb{Z}$ as its set of global elements.  Similarly, in $\infty Gpd^A$ for a set $A$, we have "$\pi_1(S^1)=\mathbb{Z}$", but the internal object $\mathbb{Z}$ has the external group $\mathbb{Z}^A$ as its set of global elements.
There is a conjecture that internally all the homotopy groups of spheres are always isomorphic to the classical ones.  This is still open (indeed the meaning of "always" has yet to be formulated precisely), but it seems to be true for all Grothendieck $\infty$-toposes, because inverse image functors preserve spheres, loop spaces, truncations, and the construction of specific finitely presented abelian groups.