Section 12. (0MM0) · ScalingStacks

12. Strict $n$ -categories as presheaves of sets[0MM0]

A category internal to an ordinary category $\mathcal{D}$ may be described as a simplicial object in $\mathcal{D}$ , that is a $\mathcal{D}$ -valued presheaf $C\colon\Delta^{\textrm{op}}\to\mathcal{D}$ , which satisfies the following strong Segal conditions. For any nonnegative integer $m$ and any integer $1\leq k\leq m-1$ , the following square is a pullback square:

Thus a strict $n$ -category consists of a presheaf of sets on the category $\Delta^{\!\times n}$ which satisfies the Segal condition in each factor and further satisfies a globularity condition. Equivalently a strict $n$ -category is a presheaf of sets on $\Delta^{\!\times n}$ which is local with respect to the classes of maps $\mathrm{Segal}_{\Delta^{\!\times n}}$ and $\mathrm{Glob}_{\Delta^{\!\times n}}$ defined below.

[0MJX]

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 $\nu C$ .

Strict $n$ -categories may also be described as certain presheaves on the category $\Theta_{n}$ , the opposite of Joyal’s category of $n$ -disks (Definition 6.1).

As $i\colon\Theta_{n}\to\cat_{n}$ is a dense functor, the corresponding nerve functor $\nu:\cat_{n}\to\Fun(\Theta_{n}^{\text{op}},\set)$ is fully-faithful. The essential image consists of precisely those presheaves which are local with respect to the class of maps $\mathrm{Segal}_{\Theta_{n}}$ defined inductively to be the union of $\sigma\mathrm{Segal}_{\Theta_{n-1}}$ and the following:

\displaystyle\left\{\begin{aligned} &j({\{0,\dots,k\}};o_{1},\dots,o_{k})\cup^{j({\{k\}})}j({\{k,\dots,m\}};o_{k+1},\dots,o_{m})\to j({[m]};o_{1},\dots,o_{m})\\ &0\leq k\leq m,\quad o_{i}\in\Theta_{n-1}\end{aligned}\right\}.

We will call this latter class $\mathrm{Se}_{\Theta_{n}}$ for later reference.

[0MJY]

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.

[0MJZ]

Corollary 12.3. A presheaf of sets on $\Delta^{\!\times n}$ is isomorphic to the nerve of a gaunt $n$ -category if and only if it is local with respect to the classes $\mathrm{Segal}_{\Delta^{\!\times n}}$ , $\mathrm{Glob}_{\Delta^{\!\times n}}$ , and $\mathrm{Comp}_{\Delta^{\!\times n}}$ . A presheaf of sets on $\Theta_{n}$ is isomorphic to the nerve of a gaunt $n$ -category if and only if it is local with respect to the classes $\mathrm{Segal}_{\Theta_{n}}$ and $\mathrm{Comp}_{\Theta_{n}}$ .

[0MK0]

Proof. Being local with respect $\mathrm{Segal}_{\Delta^{\!\times n}}$ and $\mathrm{Glob}_{\Delta^{\!\times n}}$ (or to $\mathrm{Segal}_{\Theta_{n}}$ for $\Theta_{n}$ -presheaves) implies that the presheaf is the nerve of a strict $n$ -category. Such an $n$ -category is gaunt if and only if it is local with respect to the morphisms $\sigma^{k}(E)\to\sigma^{k}(C_{0})$ . This last follows from locality with respect to $\mathrm{Comp}_{\Delta^{\!\times n}}$ (or, respectively, with respect to $\mathrm{Comp}_{\Theta_{n}}$ ) because the square

is a pushout square of strict $n$ -categories. ∎

The description of $\Theta_{n}$ as an iterated wreath product gives rise to a canonical functor $\delta_{n}\colon\Delta^{\!\times n}\to\Theta_{n}$ , described in [9, Definition 3.8], which sends $[k_{1}]\times[k_{2}]\times\cdots\times[k_{n}]$ to the object

([k_{n}];\underbrace{i_{\Delta^{\times(n-1)}}([k_{1}]\times\cdots\times[k_{n-1}]),\dots,i_{\Delta^{\times(n-1)}}([k_{1}]\times\cdots\times[k_{n-1}])}_{k_{n}\textrm{ times}}).

This object may be thought of as generated by a $k_{1}\times k_{2}\times\cdots\times k_{n}$ grid of cells. If $X$ is a strict $n$ -category then its nerve $\nu_{\Delta^{\times n}}X$ in $\Fun((\Delta^{\times n})^{\mathrm{op}},\set)$ is obtained by the formula $\delta_{n}^{*}\nu_{\Theta_{n}}X$ , where $\nu_{\Theta_{n}}$ is the nerve induced from the inclusion $\Theta_{n}\to\cat_{n}$ [9, Proposition 3.9, Rk. 3.12].

[0MK1]

Proposition 12.4. Joyal’s category $\Theta_{n}$ is the smallest full subcategory of $\gaunt_{n}$ containing the grids (the full subcategory of $\Theta_{n}$ spanned by the image of $\delta_{n}$ ) and closed under retracts. Furthermore, the morphisms in the set $\mathrm{Segal}_{\Theta_{n}}$ may be obtained as retracts of the set $(\delta_{n})_{!}(\mathrm{Segal}_{\Delta^{\times n}})$ .

[0MK2]

Proof. Both statements follow by induction. First note that $\Theta_{n}$ itself is closed under retracts [9, Proposition 3.14]. In the base case, the category $\Theta_{1}=\Delta$ consists of precisely the grids, and the sets of morphisms agree $\mathrm{Segal}_{\Theta_{1}}=\mathrm{Segal}_{\Delta}$ . Now assume, by induction, that every object of $o\in\Theta_{n-1}$ is the retract of a grid $\delta_{n}(\mathbf{m}^{o})$ , for some object $\mathbf{m}^{o}=[m_{1}^{o}]\times\cdots\times[m_{n-1}^{o}]\in\Delta^{\times n}$ . In fact, given any finite collection of objects $\{o_{i}\in\Theta_{n-1}\}$ they may be obtained as the retract of a single grid. This grid may be obtained as the image of $\mathbf{k}=[k_{1}]\times\cdots\times[k_{n-1}]$ , where $k_{j}$ is the maximum of the collection $\{m_{j}^{o_{i}}\}$ . It now follows easily that the object $([n];o_{1},\dots,o_{i})\in\Theta_{n}$ is a retract of the grid coming from the object $[n]\times\mathbf{k}$ .

To prove the second statement we note that there are two types of maps in $\mathrm{Segal}_{\Theta_{n}}$ , those in $\sigma_{!}(\mathrm{Segal}_{\Theta_{n-1}})$ and the maps

j({\{0,\dots,k\}};o_{1},\dots,o_{k})\cup^{j({\{k\}})}j({\{k,\dots,m\}};o_{k+1},\dots,o_{m})\to j({[m]};o_{1},\dots,o_{m})

for $0\leq k\leq m$ and $o_{i}\in\Theta_{n-1}$ . This later map is a retract of the image under $(\delta_{n})_{!}$ of the map

\left(j{\{0,1,\dots,k\}}\cup^{j{\{k\}}}j{\{k,\dots,m\}}\to j[m]\right)\boxtimes j(\mathbf{m})

which is a map in $\mathrm{Segal}_{\Delta^{\times n}}$ . Here $\mathbf{m}$ is such that $({\{0,\dots,k\}};o_{1},\dots,o_{k})$ is the retract of the grid corresponding to $[k]\times\mathbf{m}$ .

The former class of morphisms in $\mathrm{Segal}_{\Theta_{n}}$ , those in $\sigma_{!}(\mathrm{Segal}_{\Theta_{n-1}})$ , are also retracts on elements in $\mathrm{Segal}_{\Delta^{\times n}}$ . Specifically, if $\sigma_{!}(f)\in\sigma_{!}(\mathrm{Segal}_{\Theta_{n-1}})$ , then by induction $f$ is the retract of $(\delta_{n-1})_{!}(g)$ for some $g\in\mathrm{Segal}_{\Delta^{\times n-1}}$ . One may then readily check that $\sigma(f)$ is the retract of $j[1]\boxtimes g\in\mathrm{Segal}_{\Delta^{\times n}}$ . ∎

Clark Barwick, Christopher Schommer-Pries

Original source: arXiv:1112.0040v6

Original source · 1112.0040v6

12. Strict nn-categories as presheaves of sets[0MM0]

12. Strict $n$ -categories as presheaves of sets[0MM0]