ScalingStacks

By space we always mean “simplicial set” unless otherwise indicated; the category of spaces is denoted by ${\operatorname{\mathcal{S}}}$. Particular examples of spaces which we shall need are $\Delta[n]$, the standard $n$-simplex, $\dot{\Delta}[n]$, the boundary of the standard $n$-simplex, and $\Lambda^{k}[n]$, the boundary of the standard $n$-simplex with the $k$-th face removed. If $X$ and $Y$ are spaces we write $\Map_{{\operatorname{\mathcal{S}}}}(X,Y)$ for the space of maps from $X$ to $Y$; the $n$-simplices of $\Map_{{\operatorname{\mathcal{S}}}}(X,Y)$ correspond to maps $X\times\Delta[n]\rightarrow Y$.

We will sometimes speak of a “point” in a space, by which is meant a $0$-simplex, or of a “path” in a space, by which is meant a $1$-simplex.

For $n\geq 0$ let $[n]$ denote the category consisting of $n+1$ objects and a sequence of $n$ composable arrows: $\{0\rightarrow 1\rightarrow\dots\rightarrow n\}$. Let $\boldsymbol{\Delta}$ denote the full subcategory of the category of categories consisting of the objects $[n]$. We write $\iota\colon[n]\rightarrow[n]$ for the identity map in this category.

As is customary, we let $d^{i}\colon[n]\rightarrow[n+1]$ for $i=0,\dots,n$ denote the injective functor which omits the $i$th object, and we let $s^{i}\colon[n]\rightarrow[n-1]$ for $i=0,\dots,n-1$ denote the surjective functor which maps the $i$th and $(i+1)$st objects to the same object. Additionally, we introduce the following notation: let $\alpha^{i}\colon[m]\rightarrow[n]$ for $i=0,\dots,n-m$ denote the functor defined on on objects by $\alpha^{i}(k)=k+i$.

Let $s{\operatorname{\mathcal{S}}}$ denote the category of simplicial spaces. An object in this category is a functor $X\colon\boldsymbol{\Delta}^{\operatorname{op}}\rightarrow{\operatorname{\mathcal{S}}}$, sending $[n]\mapsto X_{n}$. We write $d_{i}\colon X_{n}\rightarrow X_{n-1}$, $s_{i}\colon X_{n}\rightarrow X_{n+1}$ and $\alpha_{i}\colon X_{n}\rightarrow X_{m}$ for the maps corresponding respectively to the morphisms $d^{i}\colon[n+1]\rightarrow[n]$, $s^{i}\colon[n-1]\rightarrow[n]$, and $\alpha^{i}\colon[m]\rightarrow[n]$ in $\boldsymbol{\Delta}$.

The category $s{\operatorname{\mathcal{S}}}$ is enriched over spaces. We denote the mapping space by $\Map_{s{\operatorname{\mathcal{S}}}}(X,Y)\in{\operatorname{\mathcal{S}}}$. It is convenient to identify ${\operatorname{\mathcal{S}}}$ with the full subcategory of $s{\operatorname{\mathcal{S}}}$ consisting of constant simplicial objects (i.e., those $K\in s{\operatorname{\mathcal{S}}}$ such that $K_{n}=K_{0}$ for all $n$), whence for a space $K$ and simplicial spaces $X$ and $Y$,

In particular, the $n$-simplices of $\Map_{s{\operatorname{\mathcal{S}}}}(X,Y)$ correspond precisely to the set of maps $X\times\Delta[n]\rightarrow Y$ of simplicial spaces.

Let $F(k)\in s{\operatorname{\mathcal{S}}}$ denote the simplicial space defined by

where the set $\boldsymbol{\Delta}([n],[k])$ is regarded as a discrete space. The $F(k)$’s represent the $k$-th space functor, i.e.,

We write $d^{i}\colon F(n)\rightarrow F(n+1)$, $s^{i}\colon F(n)\rightarrow F(n-1)$, and $\alpha^{i}\colon F(m)\rightarrow F(n)$ for the maps of simplicial spaces corresponding to the maps $d^{i}$, $s^{i}$, and $\alpha^{i}$ in $\boldsymbol{\Delta}$.

The category of simplicial spaces is cartesian closed; for $X,Y\in s{\operatorname{\mathcal{S}}}$ there is an internal hom-object $Y^{X}\in s{\operatorname{\mathcal{S}}}$ characterized by the natural isomorphism

In particular, $(Y^{X})_{0}\approx\Map_{s{\operatorname{\mathcal{S}}}}(X,Y)$, and

Furthermore, if $K\in{\operatorname{\mathcal{S}}}$ is regarded as a constant simplicial space, then $(X^{K})_{n}\approx\Map_{s{\operatorname{\mathcal{S}}}}(K,X_{n})$.

Finally, we note the existence of a diagonal functor $\diag\colon s{\operatorname{\mathcal{S}}}\rightarrow{\operatorname{\mathcal{S}}}$, defined so that the $n$-simplices of $\diag X$ are the $n$-simplices of $X_{n}$.

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

Given a model category structure on $s{\operatorname{\mathcal{S}}}$, we say that it is compatible with the cartesian closure if for any cofibrations $i\colon A\rightarrow B$ and $j\colon C\rightarrow D$ and any fibration $k\colon X\rightarrow Y$, either (and hence both) of the following two equivalent assertions hold:

1. (1)

   The induced map $A\times D\amalg_{A\times C}B\times C\rightarrow B\times D$ is a cofibration, and additionally is a weak equivalence if either $i$ or $j$ is.

2. (2)

   The induced map $Y^{B}\rightarrow Y^{A}\times_{X^{A}}X^{B}$ is a fibration, and additionally is a weak equivalence if either $i$ or $k$ is.

(A closed symmetric monoidal category together with a Quillen closed model category structure which satisfies the above properties is sometimes also called a “Quillen ring”.) Assuming (as will always be the case for us) that a weak equivalence or a fibration $X\rightarrow Y$ in our model category structure induces a weak equivalence or a fibration $X_{0}\rightarrow Y_{0}$ on the degree $0$ spaces, then it follows that such a model category structure makes $s{\operatorname{\mathcal{S}}}$ into a simplicial model category in the sense of [Qui67], since $\Map_{s{\operatorname{\mathcal{S}}}}(X,Y)\approx(Y^{X})_{0}$ for any simplicial spaces $X$ and $Y$.

The Reedy model category structure on $s{\operatorname{\mathcal{S}}}$ is compatible with the cartesian closure; to prove (1) in this case, it suffices to recall that cofibrations are exactly inclusions, and that weak equivalences are degree-wise.

A closed model category is said to be proper if

1. (1)

   the pushout of a weak equivalence along a cofibration is a weak equivalence, and

2. (2)

   the pullback of a weak equivalence along a fibration is a weak equivalence.

The Reedy model category structure is proper, because cofibrations and fibrations are in particular cofibrations and fibrations in each degree, and ${\operatorname{\mathcal{S}}}$ is proper.