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](http://ncatlab.org/nlab/show/homotopy+groups+in+an+%28infinity%2C1%29-topos))


**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?




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**Addition:**


What if you assume that $H$ is a *cohesive* $(\infty,1)$-topos? (see [here](http://ncatlab.org/nlab/show/cohesive+%28infinity%2C1%29-topos) for the nLab page)



Homotopy groups of spheres in a $(\infty, 1)$-topos
* If $H$ is the terminal category (=sheaves on the empty space), then $\pi\_k^HS^n$ (notation for homotopy groups of "spheres" in $H$) is known!
* The slice category $H=\mathrm{Spaces}/B$ is an $(\infty,1)$-topos. The homotopy groups of spheres in this setting amount to the homotopy groups of the space $\mathrm{map}(B,S^n)$ of unbased maps (with basepoint at a constant map $B\to S^n$).
This shows that $\pi\_k^HS^n$ need not be trivial if $k<n$. This also provides non-trivial examples in which $\pi\_k^HS^n$ is isomorphic to the "usual" homotopy groups of spheres (e.g., if $B=BG$ for $G$ a finite group, by Miller's theorem.)
* If $f: H\to H'$ is a geometric morphism, then the pullback functor $f^\*: H'\to H$ induces a homomorphism $\pi\_k^{H'}(S^n)\to \pi\_k^{H}(S^n)$. In particular, if $H$ has a point (a geometric morphism $\mathrm{Spaces}\to H$), then $\pi\_kS^n$ is a summand of $\pi\_k^HS^n$.


**Edit.** As I understand it, if $H$ is cohesive, then $p^\\*: \mathrm{Spaces} \to H$ is supposed to be fully faithful, where $p:H\to\mathrm{Spaces}$ is the unique geometric morphism. Spheres are in the image of $p^\\*$, so it ought to follow that that $\pi\_k^H S^n = \pi\_kS^n$. The only example of cohesive topos I understand is $H=s\mathrm{Spaces}$ (simplicial spaces), and it is certainly true in this case.


