ScalingStacks

Remark 2.3.2. We note two things about this definition:

1. 1.

   The inclusion of the full subcategory $\mathcal{S}_{/\mathbf{Fin}^{\simeq}}^{Kan}\subseteq\mathcal{S}_{/\mathbf{Fin}^{\simeq}}$ spanned by Kan fibrations is an equivalence of $\infty$-categories and the straightening functor of [Lur09] induces an equivalence of $\infty$-categories $\mathcal{S}_{/\mathbf{Fin}^{\simeq}}^{Kan}\simeq\operatorname{Fun}\left(\mathbf{Fin}^{\simeq},\mathcal{S}\right)$. Since $\mathbf{Fin}^{\simeq}$ is equivalent to the disjoint union of classifying spaces of the symmetric groups $\Sigma_{n}$, we get

   $$\mathbf{SSeq}\simeq\operatorname{Fun}\left(\coprod\limits_{n\geq 0}B\Sigma_{n},\mathcal{S}\right)\simeq\prod_{n\geq 0}\operatorname{Fun}\left(B\Sigma_{n},\mathcal{S}\right).$$

   More explicitly, given a symmetric sequence $p\colon S\to\mathbf{Fin}^{\simeq}$, taking pullback along the map $\Delta^{0}\to\mathbf{Fin}^{\simeq}$ that corresponds to the object $\left[n\right]\in\mathbf{Fin}^{\simeq}$, we obtain a space $S\left(n\right)$ that is the underlying space of the $\Sigma_{n}$-space on the right hand-side of the above equivalence.

2. 2.

   In relating $\infty$-operads to symmetric sequences it is useful to note that the functor $\mathbf{Fin}\to\mathbf{Fin}_{*}$, which adds a base point, induces an isomorphism of groupoids $\mathbf{Fin}^{\simeq}\overset{\sim}{\longrightarrow}\mathbf{Fin}_{*}^{\simeq}$. Moreover, $\mathbf{Fin}_{*}^{\simeq}$ is isomorphic to $\mathbf{Triv}_{\operatorname{\scriptsize{act}}}^{\otimes}$ (see the notation in T.3.1.1.1).