ScalingStacks

Fix a Segal space $W$. We define the set of objects of a Segal space $W$ to be the set of $0$-simplices of $W_{0}$, and we denote the set of objects by ${\operatorname{ob}}W$.

Given two objects $x,y\in{\operatorname{ob}}W$ we define the mapping space $\map_{W}(x,y)$ between them to be the fiber of the morphism $(d_{1},d_{0})\colon W_{1}\rightarrow W_{0}\times W_{0}$ over the point $(x,y)\in W_{0}\times W_{0}$. Note that since $W$ is Reedy fibrant the map $(d_{1},d_{0})$ is a fibration, and thus the homotopy type of $\map_{W}(x,y)$ depends only on the equivalence classes of $x$ and $y$ in $\pi_{0}W_{0}$. We will sometimes write $\map(x,y)$ when $W$ is clear from the context.

Given a vertex $x\in W_{0}$ we have that $d_{0}s_{0}x=d_{1}s_{0}x=x$. Thus for each object $x\in{\operatorname{ob}}W$ the point $s_{0}x\in W_{1}$ defines a point in $\map_{W}(x,x)$, called the identity map of $x$, and denoted $\id_{x}$.

Given $(n+1)$ objects $x_{0},\dots,x_{n}$ in ${\operatorname{ob}}W$ we write $\map_{W}(x_{0},x_{1},\dots,x_{n})$ for the fiber of the map $(\alpha_{0},\dots,\alpha_{n})\colon W_{n}\rightarrow{W_{0}}^{n+1}$ over $(x_{0},\dots,x_{n})\in{W_{0}}^{n+1}$. The commutative triangle

induces trivial fibrations

between the fibers of the slanted maps over $(x_{0},\dots,x_{n})$.

Let $W$ be a Segal space, and suppose $x,y\in{\operatorname{ob}}W$. Given points $f,g\in\map(x,y)$, we say that $f$ and $g$ are homotopic if they lie in the same component of $\map(x,y)$. We write $f\sim g$ if $f$ and $g$ are homotopic.

A Segal space is not a category, so we cannot compose maps in the usual way. Nonetheless, given $f\in\map(x,y)$ and $g\in\map(y,z)$, we define a composition to be a lift of $(g,f)\in\map(y,z)\times\map(x,y)$ along $\varphi_{2}$ to a point $k\in\map(x,y,z)$. The result of the composition $k$ is the point $d_{1}(k)\in\map(x,z)$. Since $\varphi_{2}$ is a trivial fibration the results of any two compositions of $f$ and $g$ are homotopic. Sometimes we write $g\circ f\in\map(x,z)$ to represent the result of some composition of $f$ and $g$.

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.

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

(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 $s_{0}d_{0}g=\id_{y}$.

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.

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 $x\in W_{0}$.