Subsection 2.3 (0M39) · ScalingStacks

2.3 Symmetric Sequences[0M39]

There is another perspective on reduced $\infty$ -operads provided by the notion of a symmetric sequence. Roughly speaking, a symmetric sequence is a sequence of $\Sigma_{n}$ -spaces $X_{n}$ for $n\geq 0$ , where $\Sigma_{n}$ is the symmetric group on $n$ elements. From an $\infty$ -operad $\mathcal{O}$ with an object $X\in\underline{\mathcal{O}}$ one can construct a symmetric sequence of spaces by

\mathcal{O}\left(n\right)=\operatorname{Mul}_{\mathcal{O}}(X^{(n)};X),

where the action of $\Sigma_{n}$ comes from permuting the inputs. For our purposes it is convenient to use the following model:

[0M2R]

Definition 2.3.1. Let $\mathbf{Fin}$ denote the skeletal version of the category of finite sets, ie the full subcategory of $\mathbf{Set}$ spanned by the objects $\left[n\right]=\left\{1,\dots,n\right\}$ for each integer $n$ . We define the $\infty$ -category of symmetric sequences (in spaces), denoted by $\mathbf{SSeq}$ , to be $\mathcal{S}_{/\mathbf{Fin}^{\simeq}}$ .

[05XF]

Remark 2.3.2. We note two things about this definition:

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.

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).

We next define the underlying symmetric sequence of a pointed $\infty$ -operad $\mathcal{O}_{X}$ , which is given by a map $\mathbf{Triv}\to\mathcal{O}$ , such that $X$ is the image of $\left\langle 1\right\rangle\in\mathbf{Triv}$ (see 2.2.7).

[0M2S]

Definition 2.3.3. Given a pointed $\infty$ -operad $\mathcal{O}_{X}$ , we define its underlying symmetric sequence to be

p\colon\mathbf{Triv}_{\operatorname{\scriptsize{act}}}^{\otimes}\times_{\mathcal{O}_{\operatorname{\scriptsize{act}}}^{\otimes}}\left(\mathcal{O}_{\operatorname{\scriptsize{act}}}^{\otimes}\right)_{/X}\to\mathbf{Triv}_{\operatorname{\scriptsize{act}}}^{\otimes}\simeq\mathbf{Fin}^{\simeq}

and denote it by $\mathcal{O}_{X,\mathbf{SSeq}}$ . By analogy with $\infty$ -operads, we denote by $\mathcal{O}_{X,\mathbf{SSeq}}^{\otimes}$ the source of $p$ .

[05XG]

Lemma 2.3.4. Given a pointed $\infty$ -operad $\mathcal{O}_{X}$ , the map

p\colon\mathbf{Triv}_{\operatorname{\scriptsize{act}}}^{\otimes}\times_{\mathcal{O}_{\operatorname{\scriptsize{act}}}^{\otimes}}\left(\mathcal{O}_{\operatorname{\scriptsize{act}}}^{\otimes}\right)_{/X}\to\mathbf{Triv}_{\operatorname{\scriptsize{act}}}^{\otimes}\simeq\mathbf{Fin}^{\simeq}

is a Kan fibration.

[05XH]

Proof. Since $p$ is a pullback of the right fibration $\left(\mathcal{O}_{\operatorname{\scriptsize{act}}}^{\otimes}\right)_{/X}\to\mathcal{O}_{\operatorname{\scriptsize{act}}}^{\otimes}$ it is itself a right fibration. The simplicial set $\mathbf{Triv}_{\operatorname{\scriptsize{act}}}^{\otimes}$ is isomorphic to $\mathbf{Fin}^{\simeq}$ and is in particular a Kan complex. By T.2.1.3.3 the map $p$ is a Kan fibration. ∎

2.3.3 relates to the informal description at the beginning of the subsection by

[05XI]

Lemma 2.3.5. Given a pointed $\infty$ -operad $\mathcal{O}_{X}$ , there is a homotopy equivalence

\mathcal{O}_{X,\mathbf{SSeq}}(n)\simeq\operatorname{Mul}_{\mathcal{O}}(X^{(n)};X),

which is natural in $\mathcal{O}_{X}$ .

[05XJ]

Proof. This follows directly from unwinding 2.3.3. ∎

Let $f\colon\mathbf{Triv}\to\mathcal{O}$ be a pointed $\infty$ -operad and let $p\colon\mathcal{O}\to\mathcal{U}$ be a map of $\infty$ -operads. Consider $\mathcal{U}$ as pointed by the composition $p\circ f$ . Let $X=f\left(\left\langle 1\right\rangle\right)$ and $Y=p\left(f\left(\left\langle 1\right\rangle\right)\right)$ . The (1-categorical) functoriality of the formula in 2.3.4 induces a map of symmetric sequences

\mathbf{Triv}_{\operatorname{\scriptsize{act}}}^{\otimes}\times_{\mathcal{O}_{\operatorname{\scriptsize{act}}}^{\otimes}}\left(\mathcal{O}_{\operatorname{\scriptsize{act}}}^{\otimes}\right)_{/X}\to\mathbf{Triv}_{\operatorname{\scriptsize{act}}}^{\otimes}\times_{\mathcal{U}_{\operatorname{\scriptsize{act}}}^{\otimes}}\left(\mathcal{U}_{\operatorname{\scriptsize{act}}}^{\otimes}\right)_{/Y}.

One can verify that this yields a functor on the level of homotopy categories

\left(-\right)_{\mathbf{SSeq}}\colon h\left(\mathbf{Op}_{\infty}\right)_{\mathbf{Triv}/}\to h\mathbf{SSeq}.

It will be important in what follows to know the following:

[05XK]

Proposition 2.3.6. The functor $\left(-\right)_{\mathbf{SSeq}}$ is conservative.

[05XL]

Proof. Let $g\colon\mathcal{P}\to\mathcal{Q}$ be a map of reduced $\infty$ -operads such that $g_{\mathbf{SSeq}}$ is an equivalence. The map $g$ is defined by a commutative triangle

To show that $g$ is an equivalence of $\infty$ -operads, we need to show that $g^{\otimes}$ is an equivalence of $\infty$ -categories. Since $\mathcal{P}$ and $\mathcal{Q}$ are reduced, it is clear that $g^{\otimes}$ is essentially surjective. To show that $g^{\otimes}$ is fully faithful, we can use the Segal conditions to reduce this to showing that the map

\mathcal{P}\left(n\right)=\operatorname{Mul}_{\mathcal{P}}(*^{(n)},*)\to\operatorname{Mul}_{\mathcal{Q}}(*^{(n)},*)=\mathcal{Q}\left(n\right)

is a homotopy equivalence for all $n$ . By 2.3.5, those maps are induced by the equivalence $g_{\mathbf{SSeq}}$ and therefore are equivalences. ∎

[05XM]

Remark 2.3.7. It is possible to lift $\left(-\right)_{\mathbf{SSeq}}$ to a functor of $\infty$ -categories, but a bit tedious to do so. We shall be content with the above weaker version as it will suffice for our applications.

Tomer Schlank, Lior Yanovski

Original source: arXiv:1808.06006v3

Original source · 1808.06006v3