# 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?

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  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.