Subsection 5.5. (0MTC) · ScalingStacks

5.5. Homotopy category and homotopy equivalences[0MTC]

In view of (5.4) we define the homotopy category of a Segal space $W$ , denoted by $\ho W$ , to be the category having as objects ${\operatorname{ob}}W$ , and having as maps $\hom_{\ho W}(x,y)=\pi_{0}\map_{W}(x,y)$ . For any $f\in\map_{W}(x,y)$ we can write $[f]\in\hom_{\ho W}(x,y)$ for its associated equivalence class.

[05UB]

Remark 5.6. Recall (3.5) in which we defined an embedding $N\colon{\operatorname{\mathcal{C}at}}\rightarrow s{\operatorname{\mathcal{S}}}$ via the classifying diagram construction (which by (4.4) in fact lands in the subcategory of Segal spaces). By (5.2) we see that $\ho NC\approx C$ . It is possible to show that the functor $N$ admits a left adjoint $L\colon s{\operatorname{\mathcal{S}}}\rightarrow{\operatorname{\mathcal{C}at}}$ , and that $L(W)\approx\ho W$ whenever $W$ is a Segal space.

A homotopy equivalence $g\in\map(x,y)$ is a point for which $[g]$ admits an inverse on each side in $\ho W$ . That is, there exist points $f,h\in\map(y,x)$ such that $g\circ f\sim\id_{y}$ and $h\circ g\sim\id_{x}$ . Note that this implies by (5.4) that $h\sim h\circ g\circ f\sim f$ . Furthermore, for each $x\in{\operatorname{ob}}W$ the map $\id_{x}\in\map(x,x)$ is a homotopy equivalence by (5.4).

We give another characterization of homotopy equivalences in a Segal space. Let $Z(3)=\discnerve(0\rightarrow 2\leftarrow 1\rightarrow 3)\subset F(3)$ be the discrete nerve of a “zig-zag” category; it follows that there is a fibration $W_{3}=\Map_{s{\operatorname{\mathcal{S}}}}(F(3),W)\rightarrow\Map_{s{\operatorname{\mathcal{S}}}}(Z(3),W)$ , and an isomorphism

\Map_{s{\operatorname{\mathcal{S}}}}(Z(3),W)\approx\lim(W_{1}\xrightarrow{d_{1}}W_{0}\xleftarrow{d_{1}}W_{1}\xrightarrow{d_{0}}W_{0}\xleftarrow{d_{0}}W_{1})\approx W_{1}\underset{W_{0}}{\times}W_{1}\underset{W_{0}}{\times}W_{1}.

(We can thus write simplices of $\Map_{s{\operatorname{\mathcal{S}}}}(Z(3),W)$ as certain ordered triples of simplices of $W_{1}$ .) Then a point $g\in\map(x,y)\subset W_{1}$ is a homotopy equivalence if and only if the element $(\id_{x},g,\id_{y})\in\Map_{s{\operatorname{\mathcal{S}}}}(Z(3),W)$ admits a lift to an element $H\in W_{3}$ ; note that if $g\in\map(x,y)$ then $s_{0}d_{1}g=\id_{x}$ and $s_{0}d_{0}g=\id_{y}$ .

Charles Rezk

Original source: arXiv:math/9811037v3

Original source · math/9811037v3