ScalingStacks

Consider a pair $(C,W)$ consisting of a category $C$ together with a subcategory $W$ such that ${\operatorname{ob}}W={\operatorname{ob}}C$; we refer to a morphism of $C$ as a weak equivalence if it is contained in $W$. More generally, given a natural transformation $\alpha\colon f\rightarrow g$ of functors $f,g\colon D\rightarrow C$, we say that $\alpha$ is a weak equivalence if $\alpha d\in W$ for each $d\in{\operatorname{ob}}D$, and write $\we(C^{D})$ for the category consisting of all functors from $D$ to $C$ and all weak equivalences between them; thus $\we(C)=W$.

For any such pair $(C,W)$ of categories we define a simplicial space $N(C,W)$, called the classification diagram of $(C,W)$, by setting

If we view the category $[m]\times[n]$ as an $m$-by-$n$ grid of objects with rows of $m$ composable horizontal arrows and columns of $n$ composable vertical arrows, then the set of $n$-simplices of the $m$th space of $N(C,W)$ corresponds to the set of functors $[m]\times[n]\rightarrow C$ in which the vertical arrows are sent into $W\subset C$.

We consider several special cases of this construction.