ScalingStacks

Notation 12.1. Objects of $\Delta^{\!\times n}$ will be denoted $\mathbf{m}=([m_{k}])_{k=1,\dots,n}$. Let

$$j:\Delta^{\!\times n}\to\Fun((\Delta^{\!\times n})^{\mathrm{op}},\set)$$

denote the Yoneda embedding. Let

$$\boxtimes\colon\Fun(\Delta^{\mathrm{op}},\set)\times\Fun((\Delta^{\!\times n-1})^{\mathrm{op}},\set)\to\Fun((\Delta^{\!\times n})^{\mathrm{op}},\set)$$

be the essentially unique functor that preserves colimits separately in each variable and sends $(j[k],j(\mathbf{m}))$ to $j([k],\mathbf{m})$. Let $\mathrm{Segal}_{\Delta}$ denote the collection of maps that corepresent the Segal squares:

$$\mathrm{Segal}_{\Delta}=\{j{\{0,1,\dots,k\}}\cup^{j{\{k\}}}j{\{k,k+1,\dots,m\}}\to j[m]\ |\ 1\leq k\leq m-1\}$$

and inductively define

$$\mathrm{Segal}_{\Delta^{\!\times n}}=\{\mathrm{Segal}_{\Delta}\boxtimes j(\mathbf{m})\ |\ \mathbf{m}\in\Delta^{\!\times n-1}\}\cup\{j[k]\boxtimes\mathrm{Segal}_{\Delta^{\!\times n-1}}\ |\ [k]\in\Delta\}.$$

Moreover for each $\mathbf{m}\in\Delta^{\times n}$, let $\widehat{\mathbf{m}}=([\widehat{m}_{j}])_{1\leq j\leq n}$ be defined by the formula

$$[\widehat{m}_{j}]=\begin{cases}[0]&\textrm{if there exists }i\leq j\textrm{ with }[m_{i}]=[0],\textrm{ and}\\ [m_{j}]&\textrm{else}\end{cases},$$

and let

$$\mathrm{Glob}_{\Delta^{\times n}}=\{j(\mathbf{m})\to j(\widehat{\mathbf{m}})\;|\;\mathbf{m}\in\Delta^{\times n}\}.$$

The presheaf underlying a strict $n$-category $C$ will be called its nerve $\nu C$.