# A "join" of ω-categorical simplices

Recall that the category of level trees $\mathcal{T}$ is defined to be the category $[\mathbb{N}^{op},\Delta_a]$, where $\Delta_a$ is the skeleton of the category of finite possibly empty linearly ordered sets. The category of finite rooted level trees is the full subcategory $\mathcal{T}_f$ spanned by those level trees $X$ such that $X(0)$ is a singleton and $X(n)=\emptyset$ for some $n>0$.

Let $X$ be a finite rooted level tree. We call a vertex $x$ of $X$ a source vertex if there are no vertices lying above it, that is, its fibre (the set of vertices of height $n+1$ lying over $x$) is empty. We say that a subtree of $X_0\subseteq X$ is a full subtree if for any vertex $x$ of $X_0$, the fibre of $x$ in $X_0$ is a (possibly empty) interval in the fibre of $x$ in $X$.

We define a sector of a vertex $x$ in $X$ to be a choice of a partition of its fibre into two disjoint, possibly empty intervals.

Then $\Theta_0$ is defined to be the category whose objects are finite level trees and whose morphisms $s\to t$ are given by pairs $(f:s\overset{full}{\hookrightarrow} t, (\varepsilon_i)_{i\in I})$ where $f$ is the inclusion of a full subtree and $(\varepsilon_i)_{i\in I})$ is the choice of a sector of $f(x)$ for each source vertex of $x_i$ of $s$. Define $D_n$ to be the unique finite rooted level tree where $D_n(i)={*}$ for $0\leq i\leq n$ and $D_n(i)=\emptyset$ for $i>n$. Then we see that

$$Hom_{\Theta_0}(D_n,D_m)=\begin{cases}\emptyset \quad \text{if n>m}\\ \{s,t\} \quad \text{if n<m}\\ \{\operatorname{id} \quad \text{for m=n}\}\end{cases}$$

This is a full embedding of the globe category $\mathbb{G}$, and this gives a functor $\eta:\Theta_0\hookrightarrow \operatorname{Psh}(\mathbb{G})$ (which turns out to be full and faithful). There is an adjoint pair $$F:\operatorname{Psh}(\mathbb{G})\leftrightarrows \operatorname{Str-\omega-cat}:U$$

Then define $\Theta$ to be the full subcategory of $\operatorname{Str-\omega-cat}$ spanned by the images of the objects of $\Theta_0$ under the composite functor $F\circ \eta$.

The category $\Theta$ contains $\Delta$ as the full subcategory spanned by the trees of height $1$ and is in many ways the $\omega$-categorical analogue of $\Delta$ (see The petit topos of globular sets by Ross Street).

However, $\Delta$ is equipped with a functorial ordinal sum $[m],[n]\mapsto [m+1+n]$ arising ultimately from the monoidal product on $\Delta_a$, the augmented simplex category. This gives an operation (by Day convolution) on the category of simplicial sets, which is known as the join. The join is central to a lot of the theory of quasicategories, since it is used to define categories of cones and comma categories, which are used to define universal constructions like the limit and colimit of a diagram in a quasicategory.

## Question

Is there any sort of generalization of the ordinal sum to $\Theta$ leading to a similar kind of join operation that might allow us to define higher cones and higher overcategories (to formalize universal constructions in strict $\omega$-categories and the higher analogues of quasicategories?

• Just happened across this old question and thought it might be worth noting that in the meantime an extensive study of this join operation in the strict setting has been undertaken by Ara and Maltsiniotis. Jan 30, 2022 at 20:27

Months later, I come to my own rescue with the following answer:

In what follows, let dComp denote the category of directed complexes in the sense of [1].

There is no monoidal product on θ itself that induces the join, but we may perform the following construction:

We define a functor $$\Theta_0\to dComp$$ sending the globular set $$t$$ that generates $$[t]\in \Theta$$ as a strict ω-category to the directed complex whose underlying set is the set $$\coprod_{n\in \mathbf{n}} t_n$$, whose dimension map is defined componentwise, and whose $$\partial^-(x)$$ and $$\partial^+(x)$$ are defined as $$\{s(x)\}$$ $$\{t(x)\}$$ respectively for each element $$x$$.

It is not hard to see using the theory in [1] that this defines a totally loop-free directed complex. Then the trick is to use the join of directed complexes, realize to a strict ω-category, and then show that this determines a pro-monoidal structure on $$\Theta$$ in the sense of Day. This induces a join on the category $$\widehat{\Theta}$$ of cellular sets with the correct join property. A similar construction can be used to obtain a version of the lax Gray tensor product of cellular sets.

Edit: Probably should clear things up: You have much less than a monoidal product on Θ-sets, but at least conjecturally (and proven by Yuki Maehara in the case of Θ_2-sets), you have a kind of colax monoidal structure in which the structure maps are natural weak equivalences, and whose binary part is a left-Quillen bifunctor.

To elaborate: Given a finite family of objects $$\{[t_1],\dots,[t_n]\}$$ of $$\Theta$$, we can take the nerve $$N([t_1]\otimes\dots\otimes[t_n])$$ of their lax Gray tensor product in $$\mathbf{Cat}_\omega$$. For each family of $$n$$ cellular sets $$X_1,\dots,X_n$$, we can define an $$n$$-ary tensor product by forming a coend $$\int^{t_1,\dots,t_n}(X_1([t_1])\times\dots\times X_n([t_n])) \cdot N([t_1]\otimes\dots\otimes[t_n])$$. These are the functors in the colax monoidal structure: $$\boxtimes_n:(\widehat{\Theta})^n \to \widehat{\Theta}$$. While there are natural maps $$\boxtimes_n\circ (\boxtimes_{m_1},\dots,\boxtimes_{m_n})\to \boxtimes_{\sum_{i=1}^n m_i},$$ and these maps all cohere, they are not invertible. However, they are inner anodyne (at least in the case $$n=2$$), and moreover, the binary part $$\boxtimes_2$$ is a left-Quillen bifunctor. Taking these facts together, we learn that the $$\infty$$-category obtained by localization carries the desired biclosed monoidal structure.

• The join produced by this procedure is lax, as is aforementioned Gray tensor product. Dec 11, 2011 at 1:55
• 1. Link [1] is broken for me. Is it Ara-Maltsiniotis? 2. I'm really curious whether there's a "fat-join" way to construct the join as asked about here. I haven't (yet?) found this formula in Ara-Maltsiniotis, but I'm really hoping it's there somewhere... Feb 7, 2022 at 19:57
• 3. I'm a little unclear about the extension of the join to $\widehat \Theta$ which you're discussing. Normally, the Day convolution of a monoidal structure is used to produce a biclosed monoidal structure, but the join should only preserve connected colimits in each variable. Are you doing something like Ara-Maltsiniotis do when extending the join to strict $\omega$-categories, where they modify Day convolution to work for such not-quite-biclosed monoidal structures? 4. Are you claiming that this gives a monoidal structure on weak $\omega$-categories? Quillen bifunctor seems hard to check. Feb 7, 2022 at 20:00
• @TimCampion Yeah, I dunno. I think in the strict ω-categorical case, they will actually be isomorphic. In Ara-Maltsiniotis, take a look near the end at the note about suspensions and oplax transformations. I expect you have to do something like this, but more generally (I am only ever interested in forming cones, so in that case, it works out fine). Feb 8, 2022 at 1:01