ScalingStacks

Definition 7.5. Write $\Ar_{n_{1},\dotsc,n_{q}}$ for $\Ar[n_{1}]\times\dotsb\times\Ar[n_{q}]$. For a functor

$$A\colon\mathrm{N}(\Ar_{n_{1},\dotsc,n_{q}})=\mathrm{N}(\Ar[n_{1}]\times\dotsb\times\Ar[n_{q}])\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}},$$

we write $A_{i_{1},j_{1};\dotsc;i_{q},j_{q}}$ for the value of $A$ on the object $((i_{1},j_{1}),\dotsc,(i_{q},j_{q}))$. Let $\Gap(([n_{1}],\dotsc,[n_{q}]),{\mathcal{C}})$ be the full subcategory of $\mathrm{Fun}(\mathrm{N}(\Ar_{n_{1},\dotsc,n_{q}}),{\mathcal{C}})$ spanned by the functors such that

• •

  $A_{i_{1},j_{1};\dotsc;i_{q},j_{q}}\simeq*$ whenever $i_{k}=j_{k}$ for some $k$.

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  For every object $(i_{1},j_{1};\dotsc;i_{q},j_{q})$ in $\Ar[n_{1}]\times\dotsb\times\Ar[n_{q}]$, every $1\leq r\leq q$, and every $j_{r}\leq k\leq n_{r}$, the square

  is a cocartesian square.