Question

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?

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Addition:

What if you assume that $H$ is a cohesive $(\infty,1)$-topos? (see here for the nLab page)

Answer 1

• 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.

Answer 2

Because Charles has already answered (incredibly nicely!) your other questions, I'll just answer your question about a natural group structure. My answer is really just an elaboration on Sam's comment.

As Sam pointed out, there is in fact an $E_k$ structure on the space of all maps ${ * \to \Omega^k S^n}$; I'll illustrate this in a moment. But then by the usual Eckman-Hilton argument (or drawing pictures), $\pi_0$ of this space will have the structure of a group for $k\geq 1$, and of a commutative group for $k \geq 2$. The fact that we're taking $S^n$ is not so important here, it's true for any object $X$.

In what follows, I'll let $H(A,B)$ denote the space of morphisms from $A$ to $B$ in the $\infty$-category $H$. In particular, when $A=\ast$ we get the space of global elements of $B$.

By the universal property of pullbacks, a map $\ast \to \Omega X$ is the same as a homotopy coherent map from $\ast$ to the diagram $D := \ast \to X \leftarrow \ast$. Without loss of generality we assume that the two maps $\ast \to X$ in $D$ are the same map. We choose this to be the base point in the space of maps $H(\ast,X)$.

Then a map from $\ast$ to the diagram $D$ is precisely a loop in the Hom-space $H(\ast,X)$. (This is the only key observation--it follows easily from the definition of the Hom Kan complex in an $\infty$-category, if you like.) In other words, $$ H(\ast,\Omega X) \cong \Omega H(\ast,X) $$ where $\Omega$ in the right hand side actually means based loop space, in the usual sense of topology. By induction, the space of global elements of $\Omega^k X$ has the structure of a $k$-fold loop space. And we're finished.