Subsection 11.7. (0MTR) · ScalingStacks

11.7. Proof of (11.1)[0MTR]

For $k\geq 2$ there are push-out diagrams

(11.8)

where $\sigma_{k}$ is the map corresponding to the word $xyx\cdots$ of length $(k+1)$ , and where $H(k)$ denotes the largest subobject of $F(k)$ not containing $d_{0}\iota$ .

We next note that $H(k)$ can itself be decomposed. Thus let $C(k)\subseteq F(k)$ denote the largest subobject of $F(k)$ not containing $d_{0}d_{0}\iota$ . If we let $d^{1}\colon F(k-1)\rightarrow F(k)$ denote the inclusion of the “face” $d_{1}\iota$ , then we have that $d^{1}F(k-1)\cap C(k)=d^{1}H(k-1)$ , and thus an isomorphism

H(k)\approx C(k)\cup_{d^{1}H(k-1)}d^{1}F(k-1).

(11.9)

Let $X$ be a simplicial space and $W$ a Segal space. Then each map $\gamma\colon F(1)\rightarrow X$ induces a map

\gamma^{*}\colon\Map_{s{\operatorname{\mathcal{S}}}}(X,W)\rightarrow\Map_{s{\operatorname{\mathcal{S}}}}(F(1),W)\approx W_{1}

of spaces. We introduce the following notation. Let $\Map_{s{\operatorname{\mathcal{S}}}}(X,W)_{\hoequiv}$ denote the subspace of $\Map_{s{\operatorname{\mathcal{S}}}}(X,W)$ consisting of all simplices $x$ such that $\gamma^{*}(x)\in W_{\hoequiv}\subset W_{1}$ for all $\gamma\colon F(1)\rightarrow X$ . Then $\Map_{s{\operatorname{\mathcal{S}}}}(X,W)_{\hoequiv}$ is isomorphic to a union of some of the path components of $\Map_{s{\operatorname{\mathcal{S}}}}(X,W)$ . In particular, $\Map_{s{\operatorname{\mathcal{S}}}}(F(1),W)_{\hoequiv}\approx W_{\hoequiv}$ by definition, and so $\Map_{s{\operatorname{\mathcal{S}}}}(F(k),W)_{\hoequiv}\approx W_{\hoequiv}\times_{W_{0}}\dots\times_{W_{0}}W_{\hoequiv}$ .

[05VW]

Lemma 11.10. Let $W$ be a Segal space. Then for $k\geq 2$ the induced map

\Map_{s{\operatorname{\mathcal{S}}}}(F(k),W)_{\hoequiv}\rightarrow\Map_{s{\operatorname{\mathcal{S}}}}(H(k),W)_{\hoequiv}

is a weak equivalence.

[05VX]

Proof. The proof is by induction on $k$ . The case $k=2$ is immediate from (11.6).

Now suppose the lemma is proved for the map $\Map_{s{\operatorname{\mathcal{S}}}}(F(k-1),W)_{\hoequiv}\rightarrow\Map_{s{\operatorname{\mathcal{S}}}}(H(k-1),W)_{\hoequiv}$ . From (11.9) we get a commutative square

This square would be a pullback square if we left off the “ $\hoequiv$ ” decorations. Even with these decorations the square is a pullback (and hence a homotopy pullback), as can be seen by recalling that $H(k)_{1}=C(k)_{1}\cup d^{1}F(k-1)_{1}$ .

Thus by induction we see that the map

a\colon\Map_{s{\operatorname{\mathcal{S}}}}(H(k),W)_{\hoequiv}\rightarrow\Map_{s{\operatorname{\mathcal{S}}}}(C(k),W)_{\hoequiv}

is a weak equivalence. The proof now follows from (11.11) and the fact that the map

W_{k}\approx W_{k-1}\times_{W_{0}}W_{1}\xrightarrow{a\times_{W_{0}}1}\Map_{s{\operatorname{\mathcal{S}}}}(C(k),W)\approx\Map_{s{\operatorname{\mathcal{S}}}}(d^{1}H(k-1),W)\times_{W_{0}}W_{1}

is a weak equivalence after restricting to the “ $\hoequiv$ ” components. ∎

[05VY]

Lemma 11.11. There is a natural weak equivalence

\Map_{s{\operatorname{\mathcal{S}}}}(C(k),W)\approx\Map(d^{1}H(k-1),W)\times_{W_{0}}W_{1}.

[05VZ]

Proof. Let $d^{0}H(k-1)\subset C(k)$ denote the image of $H(k-1)$ in $C(k)$ induced by the map $d^{0}\colon F(k-1)\rightarrow F(k)$ . There is a square

of subobjects of $C(k)$ ; we need to show that the inclusion map $d^{0}H(k-1)\cup\alpha^{0}F(1)\rightarrow C(k)$ of the union of these subobjects is a weak equivalence in the Segal space model category structure.

Now $C(k)$ can be written as a colimit of the poset of subcomplexes each of which

(1)

are isomorphic to $F(\ell)$ for some $\ell<k$ , and
(2)

include $0,1\in F(k)_{0}$ .

Straightforward calculation shows that the intersection of $d^{0}H(k-1)\cup\alpha^{0}F(1)$ with each of the objects $F(\ell)$ in the above diagram is a cover of $F(\ell)$ . ∎

[05W0]

Proof of (11.1). It is clear that for $k\geq 3$ every map

\Map_{s{\operatorname{\mathcal{S}}}}(E^{(k)},W)\rightarrow\Map_{s{\operatorname{\mathcal{S}}}}(F(1),W)\approx W_{1}

induced by an inclusion $F(1)\rightarrow E^{(k)}$ must factor through $W_{\hoequiv}\subseteq W_{1}$ , since each point of the mapping space maps to a homotopy equivalence in the sense of (5.5). Let $r_{k}$ denote the map $\Map_{s{\operatorname{\mathcal{S}}}}(E^{(k)},W)\rightarrow W_{1}$ associated to the inclusion $F(1)\rightarrow E^{(k)}$ classifying the point $xy\in E^{(k)}_{1}$ . We have that $\Map_{s{\operatorname{\mathcal{S}}}}(E^{(k)},W)=\Map_{s{\operatorname{\mathcal{S}}}}(E^{(k)},W)_{\hoequiv}$ for $k\geq 3$ , and even when $k=2$ we have that

\Map_{s{\operatorname{\mathcal{S}}}}(E^{(2)},W)_{\hoequiv}\approx\Map_{s{\operatorname{\mathcal{S}}}}(E^{(2)},W)\times_{W_{1}}W_{\hoequiv}.

Then we must show that for each $k\geq 2$ the fiber of $r_{k}$ over any point in the subspace $W_{\hoequiv}\subset W_{1}$ is contractible. The result now follows from (11.10) applied to the pushout diagrams (11.8). ∎

Charles Rezk

Original source: arXiv:math/9811037v3

Original source · math/9811037v3