ScalingStacks

Let $G(k)\subseteq F(k)$ denote the smallest subobject having $G(k)_{0}=F(k)_{0}$ and such that $G(k)_{1}$ contains the elements $\alpha^{i}\in F(k)_{1}=\boldsymbol{\Delta}([1],[k])$ defined in (2.2). In other words,

$$G(k)=\bigcup_{i=0}^{k-1}\alpha^{i}F(1)\subset F(k),$$

where $\alpha^{i}F(1)$ denotes the image of the inclusion map $\alpha^{i}\colon F(1)\rightarrow F(k)$. Let $\varphi^{k}\colon G(k)\rightarrow F(k)$ denote the inclusion map. It is straight-forward to check that

$$\Map_{s{\operatorname{\mathcal{S}}}}(G(k),X)\approx X_{1}\times_{X_{0}}\dots\times_{X_{0}}X_{1},$$

where the right-hand side denotes the limit of a diagram

$$X_{1}\xrightarrow{d_{0}}X_{0}\xleftarrow{d_{1}}X_{1}\xrightarrow{d_{0}}X_{0}\xleftarrow{d_{1}}\cdots\xrightarrow{d_{0}}X_{0}\xleftarrow{d_{1}}X_{1}$$

(4.2)

with $k$ copies of $X_{1}$.

We define a Segal space to be a simplicial space $W$ which is Reedy fibrant, and such that the map $\varphi_{k}=\Map_{s{\operatorname{\mathcal{S}}}}(\varphi^{k},W)\colon\Map_{s{\operatorname{\mathcal{S}}}}(F(k),W)\rightarrow\Map_{s{\operatorname{\mathcal{S}}}}(G(k),W)$ is a weak equivalence. In plain language, this means that $W$ is a Reedy fibrant simplicial space such that the maps

$$\varphi_{k}\colon W_{k}\rightarrow W_{1}\times_{W_{0}}\cdots\times_{W_{0}}W_{1}$$

(4.3)

are weak equivalences for $k\geq 2$. Because the maps $\varphi^{k}$ are inclusions and $W$ is Reedy fibrant, the maps $\varphi_{k}$ acting on a Segal space are trivial fibrations. Note also that the maps $d_{0},d_{1}\colon W_{1}\rightarrow W_{0}$ are fibrations as well, so that the fiber-product of (4.2) is in fact a homotopy fiber-product.

Recall that every discrete simplicial space is Reedy fibrant. A discrete simplicial space $W$ is a Segal space if and only if the maps in (4.3) are isomorphisms. Thus, $W$ is a discrete Segal space if and only if it is isomorphic to the discrete nerve of some small category. In particular, the objects $F(k)$ are Segal spaces.

If $C$ is a category, then its classifying diagram $NC$ (as defined in (3.5)) is a Segal space, by (3.6) and (3.9).