ScalingStacks

Remark 5.10. In [Fra2], it was proved that ${\sf Bar}^{n}\free_{n}(V)\simeq\uno\oplus\Sigma^{n}V$, the free $0$-disk algebra on the $n$th suspension of $V$. This can now be seen as an application of Proposition 5.9 iterated $n$ times. This result is well-known in the case of $n=1$: the bar construction for the tensor algebra on $V$ is $\uno\oplus\Sigma V$. Our result is also entirely to be expected given the example of $n$-fold loop space, where for a connected pointed space $V$, we can calculate ${\sf Bar}\bigl(\free_{n}(V)\bigr)\simeq\mathsf{B}\Omega^{n}\Sigma^{n}V\simeq\Omega^{n-1}\Sigma^{n}V\simeq\free_{n-1}(\Sigma V)$. As such, this result could have been proved longed ago, as it fits naturally into works such as [Ma] and [Coh]. We note lastly that the limiting statement as $n$ increases gives the well-known equivalence ${\sf Bar}\bigl({\sf Sym}(V)\bigr)\simeq{\sf Sym}(\Sigma V)$.