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.