Subsection 5.1. (0MTA) · ScalingStacks

5.1. “Objects” and “mapping spaces”[0MTA]

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

\varphi_{k}\colon\map(x_{0},x_{1},\dots,x_{n})\xrightarrow{\sim}\map(x_{n-1},x_{n})\times\dots\times\map(x_{0},x_{1})

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

[05U8]

Remark 5.2. As an example, if $C$ is a category and either $W=\discnerve C$ or $W=NC$ , then ${\operatorname{ob}}W\approx{\operatorname{ob}}C$ and $\map_{W}(x,y)\approx\hom_{C}(x,y)$ .

Charles Rezk

Original source: arXiv:math/9811037v3

Original source · math/9811037v3