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 from $\ast \to X$ in this diagram 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.

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.

Hi,

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 from $\ast \to X$ in this diagram 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.) 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.

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 from $\ast \to X$ in this diagram 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.