ScalingStacks

In this paper we will consider several distinct closed model category structures on $s{\operatorname{\mathcal{S}}}$. If the model category structure is not named in a discussion, assume that the Reedy model category structure is intended.

The Reedy model category structure [Ree], [DKS93, 2.4–6] on $s{\operatorname{\mathcal{S}}}$ has as its weak equivalences maps which are degree-wise weak equivalences. A fibration (resp. trivial fibration) in $s{\operatorname{\mathcal{S}}}$ is a map $X\rightarrow Y$ such that each $k\geq 0$ the induced map

is a fibration (resp. trivial fibration) of simplicial sets, where $\dot{F}(k)$ denotes the largest subobject of $F(k)$ which does not contain $\iota\colon[k]\rightarrow[k]\in\boldsymbol{\Delta}([k],[k])$. It follows that the cofibrations are exactly the inclusions.

With the above definitions, all objects are cofibrant, and the fibrant objects are precisely those $X$ for which each map $\ell_{k}\colon\Map_{s{\operatorname{\mathcal{S}}}}(F(k),X)\rightarrow\Map_{s{\operatorname{\mathcal{S}}}}(\dot{F}(k),X)$ is a fibration of spaces. We note here the fact that discrete simplicial spaces (i.e., simplicial spaces $X$ such that each $X_{n}$ is a discrete space) are Reedy fibrant.

This Reedy model category structure is cofibrantly generated [DHK]; i.e., there exist sets of generating cofibrations and generating trivial cofibrations which have small domains, and trivial fibrations (resp. fibrations) are characterized as having the right lifting property with respect to the generating cofibrations (resp. generating trivial cofibrations). The generating cofibrations are the maps

and the generating trivial cofibrations are the maps