Subsection 5.7. (0MTD) · ScalingStacks

5.7. The space of homotopy equivalences[0MTD]

Clearly, any point in $\map(x,y)$ which is homotopic to a homotopy equivalence is itself a homotopy equivalence. More generally, we have the following.

[05UC]

Lemma 5.8. If $g\in W_{1}$ is a vertex which can be connected by a path in $W_{1}$ to a homotopy equivalence $g^{\prime}\in W_{1}$ , then $g$ is itself a homotopy equivalence.

[05UD]

Proof. Let $G\colon\Delta[1]\rightarrow W_{1}$ denote the path connecting $g$ and $g^{\prime}$ . Then it suffices to note that a dotted arrow exists in

where $H$ is a lift of $(s_{0}d_{1}g^{\prime},g^{\prime},s_{0}d_{0}g^{\prime})=(id_{x^{\prime}},g^{\prime},id_{y^{\prime}})$ to $W_{3}$ , since the right-hand vertical map is a fibration. ∎

Thus, we define the space of homotopy equivalences of $W$ to be the subspace $W_{\hoequiv}\subseteq W_{1}$ consisting of exactly those components whose points are homotopy equivalences. Note that the map $s_{0}\colon W_{0}\rightarrow W_{1}$ necessarily factors through $W_{\hoequiv}$ , since $s_{0}x=\id_{x}$ is a homotopy equivalence for any vertex $x\in W_{0}$ .

Charles Rezk

Original source: arXiv:math/9811037v3

Original source · math/9811037v3