Section 6. (0MTE) · ScalingStacks

6. Complete Segal spaces[0MTE]

A complete Segal space is defined to be a Segal space $W$ for which the map $s_{0}\colon W_{0}\rightarrow W_{\hoequiv}$ is a weak equivalence, where $W_{\hoequiv}$ is the space of homotopy equivalences defined in (5.7).

[05UE]

Proposition 6.1. If $C$ is a small category, then the classifying diagram $NC$ of (3.5) is a complete Segal space.

[05UF]

Proof. This follows from (3.6) and (3.9), together with the fact that $(NC)_{\hoequiv}$ is isomorphic to $\nerve\iso(C^{I[1]})$ and the fact that the natural inclusion $\iso C\rightarrow\iso(C^{I[1]})$ is an equivalence of categories. ∎

Note that the discrete nerve of $C$ is not in general a complete Segal space.

Let $E$ denote the Segal space which is the discrete nerve of the category $I[1]$ which consists of exactly two objects $x$ and $y$ , and two non-identity maps $x\rightarrow y$ and $y\rightarrow x$ which are inverses of each other.

There is an inclusion $i\colon F(1)\rightarrow E$ associated to the arrow $x\rightarrow y$ , inducing a map $\Map_{s{\operatorname{\mathcal{S}}}}(E,W)\rightarrow\Map_{s{\operatorname{\mathcal{S}}}}(F(1),W)\approx W_{1}$ . The following is crucial.

[05UG]

Theorem 6.2. If $W$ is a Segal space, then $\Map_{s{\operatorname{\mathcal{S}}}}(E,W)\rightarrow W_{1}$ factors through $W_{\hoequiv}\subseteq W_{1}$ , and induces a weak equivalence $\Map_{s{\operatorname{\mathcal{S}}}}(E,W)\rightarrow W_{\hoequiv}$ .

The proof of (6.2) is technical, and we defer it to Section 11.

Suppose that $W$ is a Segal space, and let $x,y\in W_{0}$ be objects, and consider the diagram

(6.3)

Here $\hoequiv(x,y)\subseteq\map(x,y)$ denotes the subspace of $\map(x,y)$ consisting of those components which contain homotopy equivalences. This square is a pullback square, and also is a homotopy pullback since $(d_{1},d_{0})$ is a fibration. We have the following result.

[05UH]

Proposition 6.4. Let $W$ be a Segal space. The following are equivalent.

(1)

$W$ is a complete Segal space.
(2)

The map $W_{0}\rightarrow\Map_{s{\operatorname{\mathcal{S}}}}(E,W)$ induced by $E\rightarrow F(0)$ is a weak equivalence.
(3)

Either of the maps $\Map_{s{\operatorname{\mathcal{S}}}}(E,W)\rightarrow W_{0}$ induced by a map $F(0)\rightarrow E$ is a weak equivalence.
(4)

For each pair $x,y\in{\operatorname{ob}}W$ , the space $\hoequiv(x,y)$ is naturally weakly equivalent to the space of paths in $W_{0}$ from $x$ to $y$ .

[05UI]

Proof.

$(1)\Rightarrow(2)$ :: This follows from (6.2).
$(2)\Leftrightarrow(3)$ :: Straightforward.
$(2)\Rightarrow(4)$ :: Part (2) and (6.2) imply that $W_{0}\rightarrow W_{\hoequiv}$ is a weak equivalence, whence the result follows from the fact that the space of paths in $W_{0}$ with endpoints $x$ and $y$ is equivalent to the homotopy fiber of the map $\Delta$ in (6.3).
$(4)\Rightarrow(1)$ :: Immediate from the diagram (6.3).

∎

[05UJ]

Corollary 6.5. Let ${\operatorname{ob}}W/{\sim}$ denote the set of homotopy equivalence classes of objects in $\ho W$ . If $W$ is a complete Segal space, then $\pi_{0}W_{0}\approx{\operatorname{ob}}W/{\sim}$ .

[05UK]

Proof. This is immediate from (6.4, (4)). ∎

[05UL]

Corollary 6.6. Let $W$ be a complete Segal space. Then $\ho W$ is a groupoid if and only if $W$ is Reedy weakly equivalent to a constant simplicial space.

[05UM]

Proof. The category $\ho W$ is a groupoid if and only if $\hoequiv(x,y)=\map(x,y)$ for all $x,y\in{\operatorname{ob}}W$ , if and only if $W_{\hoequiv}=W_{1}$ , if and only if $s_{0}\colon W_{0}\rightarrow W_{1}$ is a weak equivalence (since $W$ is complete). A simplicial space $W$ is weakly equivalent to a constant simplicial space if and only if $s_{0}\colon W_{0}\rightarrow W_{1}$ is a weak equivalence. ∎

Charles Rezk

Original source: arXiv:math/9811037v3

Original source · math/9811037v3