Section 10. (0MTN) · ScalingStacks

10. Segal space model category structure[0MTN]

In this section we prove (7.1).

The Segal space closed model category structure on $s{\operatorname{\mathcal{S}}}$ is defined using (9.1) to be the localization of simplicial spaces with respect to the map $\varphi=\coprod_{i\geq 0}\varphi^{i}$ , where $\varphi^{n}\colon G(n)\rightarrow F(n)$ is the map defined in (4.1). Parts (1)-(4) of (7.1) follow immediately from (9.1). The only thing left to prove is the compatibility of this model category structure with the cartesian closure.

To prove this, we need the notion of a cover of $F(n)$ . Let $\alpha^{i}\colon[k]\rightarrow[n]$ for $i=0,\dots,n-k$ denote the maps defined by $\alpha^{i}(j)=i+j$ ; we also write $\alpha^{i}\colon F(k)\rightarrow F(n)$ for the corresponding map of simplicial spaces. We say that a subobject $G\subseteq F(n)$ is a cover of $F(n)$ if

(1)

$G$ and $F(n)$ have the same $0$ -space, i.e., $G_{0}=F(n)_{0}$ , and
(2)

$G$ has the form

$G=\bigcup_{\lambda}\alpha^{i_{\lambda}}F(k_{\lambda})$

where $k_{\lambda}\geq 1$ and $i_{\lambda}=0,\dots,k_{\lambda}-1$ .

In particular, $F(n)$ covers itself, and $G(n)\subset F(n)$ is the smallest cover of $F(n)$ .

[05VH]

Lemma 10.1. Let $G\subset F(n)$ be a cover. Then the inclusion maps $G(n)\xrightarrow{i}G\xrightarrow{j}F(n)$ are weak equivalences in the Segal space model category structure.

[05VI]

Proof. In this proof, weak equivalence will mean weak equivalence in the Segal space model category structure. The composite map $ji$ is a weak equivalence by construction, so it suffices to show that $i$ is also a weak equivalence. Given any $\alpha^{i_{1}}F(k_{1}),\alpha^{i_{2}}F(k_{2})\subset F(n)$ , we see that the intersection $\alpha^{i_{1}}F(k_{1})\cap\alpha^{i_{2}}F(k_{2})$ is either empty, or is equal to $\alpha^{i_{3}}F(k_{3})$ for some $i_{3}$ and $k_{3}$ . Thus $G$ can be written as a colimit over a partially ordered set of subcomplexes of the form $\alpha^{i}F(k)$ . Since $G(n)\cap\alpha^{i}F(k)=\alpha^{i}G(k)$ , we see that $G(n)$ is obtained as a colimit over the same indexing category of subobjects of the form $\alpha^{i}G(k)$ . Since by hypothesis the map $\Map_{s{\operatorname{\mathcal{S}}}}(\alpha^{i}F(k),W)\rightarrow\Map_{s{\operatorname{\mathcal{S}}}}(\alpha^{i}G(k),W)$ is a weak equivalence for any Segal space $W$ , we conclude that $\Map_{s{\operatorname{\mathcal{S}}}}(G,W)\rightarrow\Map_{s{\operatorname{\mathcal{S}}}}(G(n),W)$ is also a weak equivalence for any Segal space $W$ , and hence $i$ is a weak equivalence in the Segal space model category, as desired. ∎

[05VJ]

Remark 10.2. The class of subobjects which are weakly equivalent to $F(n)$ is not exhausted by the coverings. For example, one can show that for $0<i<n$ the subobject $\dot{F}(n)\setminus d^{i}F(n-1)$ (the “boundary” of $F(n)$ with a “face” removed which is neither the first nor the last face) is weakly equivalent to $F(n)$ in the Segal space model category structure, but is not a cover.

To finish the proof of (7.1), we note that by (9.2) it suffices to show that for a Segal space $W$ , the simplicial space $W^{F(1)}$ is also a Segal space; i.e., that the induced maps $\varphi_{k}\colon(W^{F(1)})_{k}\approx\Map_{s{\operatorname{\mathcal{S}}}}(F(k),W^{F(1)})\rightarrow\Map_{s{\operatorname{\mathcal{S}}}}(G(k),W^{F(1)})$ are weak equivalences. This follows immediately from (10.3) below.

[05VK]

Lemma 10.3. The inclusion $F(1)\times G(n)\rightarrow F(1)\times F(n)$ is a weak equivalence in the Segal space model category structure.

[05VL]

Proof. Let $\gamma^{i}\colon[n+1]\rightarrow[1]\times[n]$ denote the map defined by

\gamma^{i}(j)=\begin{cases}(0,j)&\text{if $j\leq i$,}\\ (1,j-1)&\text{if $j>i$.}\end{cases}

Likewise, let $\delta^{i}\colon[n]\rightarrow[1]\times[n]$ denote the map defined by

\delta^{i}(j)=\begin{cases}(0,j)&\text{if $j\leq i$,}\\ (1,j)&\text{if $j>i$.}\end{cases}

Then one can write $F(1)\times F(n)$ as a colimit of the diagram

\gamma^{0}F(n+1)\leftarrow\delta^{0}F(n)\rightarrow\gamma^{1}F(n+1)\leftarrow\delta^{1}F(n)\rightarrow\dots\rightarrow\gamma^{n}F(n+1)

(10.4)

of subobjects. (This is analogous to the decomposition of the simplicial set $\Delta[1]\times\Delta[n]$ into a union of $(n+1)$ copies of $\Delta[n+1]$ , attached along faces.) A straightforward computation shows that the maps $\gamma^{i}F(n+1)\cap(F(1)\times G(n))\rightarrow\gamma^{i}F(n+1)$ and $\delta^{i}F(n)\cap(F(1)\times G(n))\rightarrow\delta^{i}F(n)$ are covers, and hence by (10.1) are weak equivalences. Thus the result follows by comparing diagram (10.4) with the diagram obtained by intersecting each object of (10.4) with $F(1)\times G(n)$ . ∎

Charles Rezk

Original source: arXiv:math/9811037v3

Original source · math/9811037v3