Section 6. (0MLU) · ScalingStacks

6. The categories $\Theta_{n}$ and $\Upsilon_{n}$ [0MLU]

For many purposes, it is unwieldy to contemplate all gaunt $n$ -categories (or even all compact gaunt $n$ -categories). The critical structural features of $n$ -categories are already captured by far smaller categories. One such smaller category is Joyal’s category $\Theta_{n}$ of ‘ $n$ -disks’ (Definition 6.1):

[0MIH]

Definition 6.1 ([9, Definition 3.1]). Let $C$ be a small category. The wreath product $\Delta\wr C$ is the category

•

whose objects consist of tuples $([n];c_{1},\dots,c_{n})$ where $[n]\in\Delta$ and $c_{i}\in C$ , and
•

whose morphisms from $([m];a_{1},\dots,a_{m})$ to $([n];b_{1},\dots,b_{n})$ consist of tuples $(\phi;\phi_{ij})$ , where $\phi:[m]\to[n]$ , and $\phi_{ij}:a_{i}\to b_{j}$ where $0<i\leq m$ , and $\phi(i-1)<j\leq\phi(i)$ .

The category $\Theta_{n}$ is now defined inductively as a wreath product: $\Theta_{1}=\Delta$ , and $\Theta_{n}=\Delta\wr\Theta_{n-1}$ . In particular this gives rise to embeddings $\sigma:\Theta_{n-1}\to\Theta_{n}$ , given by $\sigma(o)=([1];o)$ , and $\iota:\Delta\to\Theta_{n}$ given by $\iota([n])=([n];([0]),\dots,([0]))$ .

There is a fully-faithful embedding $i\colon\Theta_{n}\hookrightarrow\gaunt_{n}$ as a dense subcategory [9, Th. 3.7]. The image under $i$ of $([m];a_{1},\dots,a_{m})$ may be described inductively as the following colimit:

i([m];o_{1},\dots,o_{m})=\sigma(i(o_{1}))\cup^{C_{0}}\sigma(i(o_{2}))\cup^{C_{0}}\cdots\cup^{C_{0}}\sigma(i(o_{m})).

This colimit, taken in $\gaunt_{n}$ , is a series of pushouts in which $C_{0}$ is embedded into $\sigma(i(o_{k}))$ via $\top$ and into $\sigma(i(o_{k+1}))$ via $\bot$ as described after Ex. 2.5. There is no possible confusion by the meaning of $\sigma$ , as $i(\sigma(o))=\sigma(i(o))$ for all $o\in\Theta_{n-1}$ .

Since we will be concerned with the study of correspondences, it is convenient to enlarge $\Theta_{n}$ to contain products of correspondences:

[0MII]

Definition 6.2. The category $\Upsilon_{n}$ is the smallest full subcategory of $\gaunt_{n}$ that contains $\Theta_{n}$ and is closed under products of correspondences, $(M,N)\mapsto M\times_{C_{k}}N$ .

[0MIJ]

Remark 6.3. We now examine the fiber products of cells in detail. We aim to express these fiber products as simple iterated colimits of cells. Let $\varphi:C_{i}\to C_{j}$ and $\psi:C_{k}\to C_{j}$ be a pair of functors ( $i,j,k\geq 0$ ). A map of cells $\varphi:C_{i}\to C_{j}$ either factors as a composite $C_{i}\to C_{0}\to C_{j}$ or is a suspension $\varphi=\sigma(\xi)$ of some map $\xi:C_{i-1}\to C_{j-1}$ .

We thus begin by contemplating the case in which $\varphi$ is not the suspension of a map of lower dimensional cells. In this case we have a diagram of pullback squares

Here $F$ is the fiber of $\psi:C_{k}\to C_{j}$ over the unique object in the image of $\varphi$ . There are four possibilities:

(A)

The image of $\psi$ may be disjoint from the image of $\varphi$ , in which case $F=\partial C_{0}=\emptyset$ . Hence $F$ and also $C_{i}\times F$ are the empty colimit of cells.
(B)

The fiber may be a zero cell, $F=C_{0}$ , in which case $C_{i}\times F\cong C_{i}$ is trivially a colimit of cells.
(C)

The fiber may be the $k$ -cell $F\cong C_{k}$ , but we have $i=0$ . In this case $C_{i}\times F\cong F\cong C_{k}$ is again trivially a colimit of cells.
(D)

The fiber may be an $k$ -cell $F\cong C_{k}$ , and we have $i\geq 1$ . In this case we have (cf. [34, Proposition 4.9])

$C_{i}\times C_{k}\cong(C_{i}\cup^{C_{0}}C_{k})\cup^{\sigma(C_{i-1}\times C_{k-1})}(C_{k}\cup^{C_{0}}C_{i})$

where for each pushout $C_{x}\cup^{C_{0}}C_{y}$ , the object $C_{0}$ is included into the final object of $C_{x}$ and the initial object of $C_{y}$ .

As the suspension functor $\sigma$ commutes with pullback squares, a general pullback of cells is the suspension of one of the types just considered. Moreover, as the suspension functor also commutes with pushout squares, the above considerations give a recipe for writing any fiber product of cells as an iterated pushout of cells. This will be made precise in Lemma 6.7.

[0MIK]

Notation 6.4. The inclusion $\Upsilon_{n}\hookrightarrow\gaunt_{n}$ induces a fully faithful nerve functor

\nu:\gaunt_{n}\hookrightarrow\Fun(\Upsilon_{n}^{\mathrm{op}},\set).

In particular, we may regard gaunt $n$ -categories as particular presheaves of sets on the category $\Upsilon_{n}$ (precisely which presheaves will be determined in Corollary 10.2). Note that the nerve functor commutes with all limits, hence in particular fiber products.

[0MIL]

Notation 6.5. Let $S_{00}$ consist of the union $A\cup B\cup C\cup D$ of the following four finite sets of maps of presheaves on $\Upsilon_{n}$ :

A\mathrel{\mathop{:}}=\left\{\nu C_{i-1}\cup^{\nu(\partial C_{i-1})}\nu C_{i-1}\to\nu(\partial C_{i})\;|\;0\leq i\leq n-1\right\}

(when $i=0$ , we interpret this as the empty presheaf mapping to the nerve of the empty $n$ -category),

B\mathrel{\mathop{:}}=\left\{\nu C_{j}\cup^{\nu C_{i}}\nu C_{j}\to\nu(C_{j}\cup^{C_{i}}C_{j})\;|\;0\leq i<j\leq n\right\},

\begin{split}C\mathrel{\mathop{:}}=\Big\{\nu(C_{i+j}\cup^{C_{i}}C_{i+k})\cup^{\nu\sigma^{i+1}(C_{j-1}\times C_{k-1})}\nu(C_{i+k}&\cup^{C_{i}}C_{i+j})\to\nu(C_{i+j}\times_{C_{i}}C_{i+k})\\ &\Big|\;0\leq i\leq n\textrm{, }0<j,k\leq n-i\Big\},\end{split}

and, lastly,

D\mathrel{\mathop{:}}=\left\{\nu\sigma^{k}(\Delta^{3})\cup^{\nu\sigma^{k}(\Delta^{\{0,2\}}\sqcup\Delta^{\{1,3\}})}\nu\sigma^{k}(\Delta^{0}\sqcup\Delta^{0})\to\nu C_{k}\;|\;0\leq k\leq n\right\}.

Now let $S_{0}$ be the smallest class of morphisms $U\to V$ in $\Fun(\Upsilon_{n}^{\mathrm{op}},\set)$ that (a) is closed under isomorphism, (b) contains $S_{00}$ , and (c) is closed under the operation $-\times_{C_{k}}N$ for any functor $V\to C_{k}$ and any $k$ -correspondence $N\to C_{k}$ with $N\in\Upsilon_{n}$ .

[0MIM]

Lemma 6.6. Suppose $X$ a gaunt $n$ -category. Then the presheaf $\nu X:\Upsilon_{n}^{\mathrm{op}}\to\set$ is local with respect to the morphisms of $S_{0}$ .

[0MIN]

Proof. Forming each of the pushouts of $S_{00}$ in $\gaunt_{n}$ yields an equivalence, so $X$ is local with respect to $S_{00}$ .

Now let $S_{0}^{\prime}\subseteq S_{0}$ denote the class of morphisms $f:U\to V$ in $S_{0}$ such that $\nu X$ is local with respect to $f$ for any gaunt $X$ . We have observed that $S_{0}^{\prime}$ contains $S_{00}$ . It is also visibly closed under isomorphism.

We complete the proof by showing that $S^{\prime}_{0}$ is closed under the operation $-\times_{C_{i}}N$ for any $N\in\Upsilon_{n}$ . Indeed, suppose $U\to V$ a morphism of $S_{0}^{\prime}$ . We claim that for any morphism $V\to C_{k}$ and any functor $N\to C_{k}$ , the map

\Upsilon_{n}(V\times_{C_{k}}N,X)\to\Upsilon_{n}(U\times_{C_{k}}N,X)

is a bijection. For each $W\in\Upsilon_{n}$ , we have

	$\displaystyle\gaunt_{n}(W\times_{C_{i}}N,X)$	$\displaystyle\cong(\gaunt_{n}/C_{k})(W\times_{C_{k}}N,X\times C_{k})$
		$\displaystyle\cong(\gaunt_{n}/C_{k})(W,\uHom_{C_{k}}(N,X\times C_{k})),$

where $\uHom_{C_{k}}(N,-)$ denotes the right adjoint of Lemma 5.6. The claim now follows from the observation that as $\uHom_{C_{i}}(N,X\times C_{i})$ is gaunt, it is local with respect to $U\to V$ . ∎

[0MIP]

Lemma 6.7. Let

\mathcal{C}\mathrel{\mathop{:}}=S_{00}^{-1}\Fun(\Upsilon_{n}^{\mathrm{op}},\set)

denote the full subcategory of presheaves of sets which are local with respect to the the morphisms of $S_{00}$ . Let $\varphi:C_{i}\to C_{j}$ and $\psi:C_{k}\to C_{j}$ be an arbitrary pair of maps ( $i,j,k\geq 0$ ). Then $\nu(C_{i}\times_{C_{j}}C_{k})$ is contained in the smallest full subcategory of $\mathcal{C}$ that contains the nerves of cells and is closed under the formation of colimits.

[0MIQ]

Proof. Recall that $\nu$ commutes with limits. Let $m$ ( $\leq i,j,k$ ) be the largest integer such that $\varphi=\sigma^{m}(g)$ and $\psi=\sigma^{m}(f)$ are both $m$ -fold suspensions of maps, $g:C_{i-m}\to C_{j-m}$ and $f:C_{k-m}\to C_{j-m}$ .

Suppose, without loss of generality, that $\varphi$ is not an $(m+1)$ -fold suspension of a map. We thus have an $m$ -suspension of the situation considered in Rk. 6.3; that is, we have a diagram of pullback squares

where as above $F$ denotes the fiber of $f:C_{k-m}\to C_{j-m}$ over the image of $g$ . So let us consider each of the cases A-D of Rk. 6.3 in turn.

(A)

If $F=\emptyset$ , then

$C_{i}\times_{C_{j}}C_{k}\cong\sigma^{m}(C_{i-m}\times_{C_{0}}F)\cong\sigma^{m}(\emptyset)\cong\partial C_{m}.$

In this case, the morphisms of $A\subset S_{00}$ provide an iterative construction of $\nu\partial C_{m}$ as a colimit in $S_{00}^{-1}\Fun(\Upsilon_{n}^{\mathrm{op}},\set)$ of cells.
(B)

Next, if $F\cong C_{0}$ , then

$C_{i}\times_{C_{j}}C_{k}\cong\sigma^{m}(C_{i-m}\times_{C_{0}}F)\cong\sigma^{m}(C_{i-m})\cong C_{i}$

is already a cell.
(C)

Similarly, if $F\cong C_{k-m}$ , but $i=m$ , then

$C_{i}\times_{C_{j}}C_{k}\cong\sigma^{m}(C_{0}\times_{C_{0}}F)\cong\sigma^{m}(C_{k-m})\cong C_{k}$

is again already a cell.
(D)

Finally, let us suppose that $F\cong C_{\ell}$ with $i=m+p$ and $k=m+\ell$ for $p>0$ . In this case we have,

$C_{i}\times_{C_{j}}C_{k}\cong C_{m+p}\times_{C_{m}}C_{m+\ell}$

is precisely the fiber product considered in the set $C\subset S_{00}$ . One readily observes that morphisms of $B$ and $C$ provide an inductive construction of this fiber product as an iterated colimit of cells in $\mathcal{C}$ .∎

Clark Barwick, Christopher Schommer-Pries

Original source: arXiv:1112.0040v6

Original source · 1112.0040v6

6. The categories Θn\Theta_{n} and Υn\Upsilon_{n}[0MLU]

6. The categories $\Theta_{n}$ and $\Upsilon_{n}$ [0MLU]