ScalingStacks

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 $X_{n}$.