ScalingStacks

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 $\alpha^{i}(k)=k+i$.