[Math] Homotopy groups of spheres in a $(\infty, 1)$-topos

at.algebraic-topologyhigher-category-theoryinfinity-topos-theory

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)

Best Answer

  • 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.

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