Do higher homotopy groups have increasingly more structure?

Question

There is a sequence of forgetful functors $\mathbf{Ab} \to \mathbf{Grp}\to \mathbf{Set}_*$. Further for a pointed topological space $(X,x_0)$ it is $\pi_2(X,x_0) \in \mathbf{Ab}$ and $\pi_1(X,x_0) \in \mathbf{Grp}$ and $\pi_0(X,x_0) \in \mathbf{Set}_*$. Can this sequence be extended on the left side, i.e. are there strictly "better" or smaller categories $\mathbf{C}_n, n\geq 0$ and a sequence of forgetful functors $$\dots \to \mathbf{C}_n \to \dots \to \mathbf{C}_2 = \mathbf{Ab} \to \mathbf{C}_1 = \mathbf{Grp} \to \mathbf{C}_0 = \mathbf{Set}_*$$ such that $\pi_n(X,x_0) \in \mathbf{C}_n$ for all $n \geq 0$? What would be for example the additional (to abelian) special property that all $\pi_3(X,x_0)$ have (like $\pi_1$ are groups and $\pi_2$ have the additional property "abelian")? Can it be described explicitly in a nice way?

Answer 1

As mentioned in the comments, once you quotient by homotopy and look at the mere group $\pi_n X$, these are all abelian groups with no restrictions once $n \geq 2$ so that it doesn't look like these categories are getting more restrictive. But the pattern that you've noticed actually does exist and continue! The caveat is that you have to remember more information!

Recall that one can build a (pointed) mapping space $[X,Y]$ whose points are continuous (pointed) maps $X \to Y$. Then $\pi_0 [X,Y]$ is the set of connected components of this space, which recovers exactly the set of (pointed) maps $X \to Y$ up to homotopy. Then we can think of $\pi_n X = \pi_0 [S^n, X]$ as the homotopy classes of maps from the $n$-sphere into $X$ (as an easy exercise, check this for $\pi_0$, $\pi_1$, and $\pi_2$). You might imagine that when we take connected components of this space we lose a lot of information, and you would be completely correct! It turns out that the category of sets is just not "big" enough (in a homotopical sense) to see much of the interesting structure present in the space $[S^n,X]$. But if we stay in the ($\infty$-)category of spaces, then we can see much more!

In particular, we find that $[S^n, X]$ is an $E_n$-algebra in spaces! This is a homotopical notion of "group" that gets progressively more commutative as $n$ gets large. In particular:

• $E_0$-spaces are just pointed spaces
• $E_1$-spaces are like nonabelian groups up to homotopy
• $E_2$-spaces are like abelian groups that "remember" if you move two elements around each other in 2d
• $E_3$-spaces are like abelian groups that "remember" if you move two elements around each other in 3d
• ...

Since $\mathsf{Set}$ is a 1-category, everything $>1$ looks the same, which explains why truncating $[S^n,X]$ to $\mathsf{Set}$ by taking $\pi_0$ sees that $\pi_0 X$ is a pointed set, $\pi_1 X$ is a group, and then all higher $\pi_{\geq 2} X$ are "just" abelian groups.

If we instead found a way to truncate to $\mathsf{Cat}$, the 2-category of categories, then we could see slightly further! Now an $E_0$-algebra is a pointed category, an $E_1$-algebra is a monoidal category, an $E_2$-algebra is a braided monoidal category and only for $E_{\geq 3}$-algebras do we stabilize to symmetric monoidal categories.

Since spaces form an $\infty$-category, here we can see all the gradations of $E_n$-structure, and there's lots of interesting things to say about this! Though doing so would require some serious homotopy theory.

---

I hope this helps ^_^

Answer 2

I will provide two different interpretations of an answer to this question. The first was already provided by @lulu in the comments and says that there there is no additional property of an abelian group that obstructs it occurring as a homotopy group $\pi_n$ with $n\ge2$ of a pointed space $X$ (even better, for any group $G$ and abelian groups $A_n,\,n\ge2$, there is a pointed space $X$ with $\pi_1(X)\cong G$ and $\pi_n(X)=A_n$ for $n\ge2$).

The second is that this question is the "shadow" of a question with a more positive answer. Namely, there is a sequence of forgetful functors (which, however, are not faithful!) $$ \dotsc\rightarrow\mathrm{Mon}^n(\mathbf{Spc})\rightarrow\dotsc\rightarrow\mathrm{Mon}(\mathbf{Spc})\rightarrow\mathbf{Spc}_{\ast}, $$ where the $n$-th term is the $\infty$-category of $n$-fold monoid objects in the $\infty$-category $\mathbf{Spc}$ of spaces (also known as $\mathbb{E}_n$-algebras). The $n$-fold loop space functor $\mathbf{Spc}_{\ast}\rightarrow\mathbf{Spc}_{\ast}$ has a canonical lift through the $n$-th term of this sequence (the $n$ compatible monoid structures coming from composition of loops in the $n$ directions of $I^n/\partial I^n\cong S^n$), but not in general through the $(n+1)$-th term. The $n$-th homotopy group functor is obtained by applying $\pi_0$ to this functor, which (since it is a cartesian monoidal functor) induces a functor $\mathrm{Mon}^n(\mathbf{Spc})\rightarrow\mathrm{Mon}^n(\mathbf{Set})$. However, $n$-fold monoid objects in $\mathbf{Set}$ are either monoids for $n=1$ or commutative monoids for $n\ge2$ (by the Eckmann-Hilton argument), so you cannot see this interesting & subtle higher structure at the level of homotopy classes!