ScalingStacks

Notation 12.2. Let $K$ denote the simplicial set

$$\Delta^{3}\cup^{(\Delta^{\{0,2\}}\sqcup\Delta^{\{1,3\}})}(\Delta^{0}\sqcup\Delta^{0})$$

obtained by contracting two edges in the three simplex.

Rezk observed [34, § 10] that $K$ detects equivalences in nerves of categories, and consequently it may be used to formulate his completeness criterion. We shall use it to identify the gaunt $n$-categories. To this end set

	$\displaystyle\mathrm{Comp}_{\Delta}$	$\displaystyle=\{K\to j[0]\}$
	$\displaystyle\mathrm{Comp}_{\Delta^{\!\times n}}$	$\displaystyle=\{\mathrm{Comp}_{\Delta}\boxtimes j(\mathbf{0})\}\cup\{j[k]\boxtimes\mathrm{Comp}_{\Delta^{\!\times n-1}}\}$
	$\displaystyle\mathrm{Comp}_{\Theta_{n}}$	$\displaystyle=\iota_{!}\mathrm{Comp}_{\Delta}\cup\sigma_{!}\mathrm{Comp}_{\Theta_{n-1}}.$

where

$$\iota_{!}\colon\Fun(\Delta^{\textrm{op}},\set)\to\Fun(\Theta_{n}^{\textrm{op}},\set)\textrm{\quad and\quad}\sigma_{!}\colon\Fun(\Theta_{n-1}^{\textrm{op}},\set)\to\Fun(\Theta_{n}^{\textrm{op}},\set)$$

are given by left Kan extension along $\iota$ and $\sigma$, respectively.