Section 14. (0MTZ) · ScalingStacks

14. A completion functor[0MTZ]

In this section we prove (7.7). We do this by constructing functorially for each Segal space $W$ a map $i_{W}\colon W\rightarrow\widehat{W}$ called the completion map, such that

(1)

the completion $\widehat{W}$ is a complete Segal space,
(2)

the completion map $i_{W}$ is a weak equivalence in the complete Segal space model category, and
(3)

the completion map $i_{W}$ is a Dwyer-Kan equivalence.

Statement (2) implies that a map $f\colon U\rightarrow V$ between Segal spaces is a weak equivalence in the complete Segal space model category structure if and only if $\widehat{f}$ is. Likewise, statement (3) together with (7.5) imply that $f$ is a Dwyer-Kan equivalence if and only if $\widehat{f}$ is. Thus (7.7) will follow from statement (1) together with (7.6), which shows that the Dwyer-Kan equivalences between complete Segal spaces are precisely the Reedy weak equivalences between such, which are precisely the weak equivalences between fibrant objects in the complete Segal space model category structure.

We should note that it is easy to demonstrate statements (1) and (2) alone. In fact, (7.2) implies that there exists for each simplicial space $W$ a fibrant replacement map $i\colon W\rightarrow W^{f}$ , in which $i$ is a weak equivalence in the complete Segal space model category structure, and $W^{f}$ is a complete Segal space. However, we need a different construction to prove all three statements.

Suppose $W$ is a Segal space. Let $E(m)=\discnerve I[m]$ . For each $n\geq 0$ we can define a simplicial space by $[m]\mapsto\Map_{s{\operatorname{\mathcal{S}}}}(E(m),W^{F(n)})$ . Let

\widetilde{W}_{n}=\diag\left([m]\mapsto\Map_{s{\operatorname{\mathcal{S}}}}(E(m),W^{F(n)})\right)=\diag\left([m]\mapsto(W^{E(m)})_{n}\right).

Then the spaces $\widetilde{W}_{n}$ taken together form a simplicial space $\widetilde{W}$ , and there is a natural map $W\rightarrow\widetilde{W}$ . Since $\Map_{s{\operatorname{\mathcal{S}}}}(E(m),W^{F(n)})=(W^{E(m)})_{n}$ , we can write $\widetilde{W}=\diag^{\prime}([m]\mapsto W^{E(m)})$ , where $\diag^{\prime}\colon s(s{\operatorname{\mathcal{S}}})\rightarrow s{\operatorname{\mathcal{S}}}$ denotes the prolongation of the $\diag$ functor to simplicial objects in $s{\operatorname{\mathcal{S}}}$ .

Let $\widetilde{W}\rightarrow\widehat{W}$ denote the functorial Reedy fibrant replacement of $\widetilde{W}$ . The composite map $i_{W}\colon W\rightarrow\widehat{W}$ is called the completion map of $W$ , and the functor which sends $W$ to $\widehat{W}$ is called the completion functor.

[05WG]

Remark 14.1. This completion is a generalization of the classifying space construction. In fact, suppose $W$ is a Segal space such that $\ho W$ is a groupoid; equivalently, that $W_{1}=W_{\hoequiv}$ . Then the arguments below show that $\widehat{W}$ is weakly equivalent to a constant simplicial space, which in each degree is the realization $\diag W$ . For instance, if $W$ is a “ $\Delta$ -space” (i.e., $W_{0}=*$ ) and thus a model for a loop space with underlying space equivalent to $W_{1}$ , then $\widehat{W}$ is equivalent to the constant object which is $BW_{1}$ , the classifying space of the “loop space” $W_{1}$ , in each degree.

[05WH]

Lemma 14.2. If $C$ is a category, then $\widetilde{\discnerve C}$ is isomorphic to $NC$ . In particular, $NE\approx\widetilde{E}$ and $\widehat{E}$ are weakly equivalent to the terminal object in $s{\operatorname{\mathcal{S}}}$ .

[05WI]

Proof. The first statement is straightforward from the definitions. Since $E$ is equivalent to the terminal object in ${\operatorname{\mathcal{C}at}}$ , and $N$ takes equivalences to weak equivalences by (3.7), the second statement follows. ∎

[05WJ]

Lemma 14.3. If $U\rightarrow V$ is a categorical equivalence between Segal spaces, then $\widehat{U}\rightarrow\widehat{V}$ is a Reedy weak equivalence.

[05WK]

Proof. It is clear from the definition that $\widetilde{U\times E}\approx\widetilde{U}\times\widetilde{E}$ , and $\widetilde{E}$ is contractible by (14.2). Thus categorically homotopic maps are taken to homotopic maps by the completion operator, and hence completion takes categorical equivalences to homotopy equivalences. ∎

[05WL]

Proof of statement (2). By (13.5) the natural maps $W\rightarrow W^{E(m)}$ are categorical equivalences, and hence weak equivalences in the complete Segal space model category structure by (13.6). Thus the induced map on homotopy colimits

W=\diag^{\prime}([m]\mapsto W)\rightarrow\diag^{\prime}([m]\mapsto W^{E(m)})=\widetilde{W}

is a weak equivalence in the complete Segal space model category structure. ∎

[05WM]

Proof of statements (1) and (3). For each simplicial map $\delta\colon[n]\rightarrow[m]\in\boldsymbol{\Delta}$ there is a diagram

By (13.8) the maps $W^{E(m)}\rightarrow W^{E(n)}$ are Dwyer-Kan equivalences, so for each set of objects $x_{0},\dots,x_{k}\in{\operatorname{ob}}W$ the morphism

\map_{W^{E(m)}}(x_{0},\dots,x_{k})\rightarrow\map_{W^{E(n)}}(\delta x_{0},\dots,\delta x_{k})

between the fibers of the vertical maps in the above diagram is a weak equivalence. Thus, the square is a homotopy pullback, with fibers which are weakly equivalent to the products of mapping spaces.

Thus, the induced map of realizations $\diag^{\prime}(W^{E(-)})_{k}\rightarrow\diag^{\prime}(W^{E(-)})_{0}^{\times k}$ has its homotopy fibers weakly equivalent to a $k$ -fold product of mapping spaces, and thus we have shown that $\widehat{W}$ is a Segal space, and that $\map_{W}(x,y)\rightarrow\map_{\widehat{W}}(i(x),i(y))$ are weak equivalences for all $x,y\in{\operatorname{ob}}W$ .

By construction the map $\pi_{0}W_{0}\rightarrow\pi_{0}\widehat{W}_{0}$ is surjective; it follows that $\ho W\rightarrow\ho\widehat{W}$ is surjective on isomorphism classes of objects. Therefore we have shown that $W\rightarrow\widehat{W}$ is a Dwyer-Kan equivalence, proving statement (3).

It remains to show that $\widehat{W}$ is a complete Segal space. Consider the square

induced by a map $j\colon E(n)\rightarrow E(m)$ . Since $W^{E(m)}\rightarrow W^{E(n)}$ is a categorical equivalence by (13.5) and thus a Dwyer-Kan equivalence by (13.8), we may conclude that the induced map $\hoequiv_{W^{E(m)}}(x,y)\rightarrow\hoequiv_{W^{E(n)}}(j(x),j(y))$ is a weak equivalence for each pair $(x,y)\in(W^{E(m)})_{0}\times(W^{E(m)})_{0}$ . Thus the above square is a homotopy pullback, and so the induced map $\diag^{\prime}(W^{E(-)})_{\hoequiv}\rightarrow\diag^{\prime}(W^{E(-)})_{0}^{\times 2}$ has its homotopy fibers weakly equivalent to the spaces $\hoequiv_{W}(x,y)$ . That is,

(\widehat{W})_{\hoequiv}\approx\diag([m]\mapsto(W^{E(m)})_{\hoequiv}).

Since $(W^{E(m)})_{\hoequiv}\approx(W^{E(m)\times E(1)})_{0}$ by (6.2), the above really says that there is an equivalence $(\widehat{W})_{\hoequiv}\approx(\widehat{W^{E(1)}})_{0}$ . Now (14.3) shows that since $W^{E(1)}$ is categorically equivalent to $W$ , we have that $(\widehat{W})_{\hoequiv}\approx(\widehat{W^{E(1)}})_{0}\approx(\widehat{W})_{0}$ ; in other words, $\widehat{W}$ is a complete Segal space. This proves statement (1), and completes the proof. ∎

Charles Rezk

Original source: arXiv:math/9811037v3

previous section

Original source · math/9811037v3