# Extending the vertex-facet correspondence from Δ to Θ

Recall that in the $n$-simplex $\Delta[n]$, we have a combinatorially crucial bijection between facets, (codimension $1$ faces) and vertices, where the $i$th face of a simplex corresponds to the full subcomplex $[0,\dots,\hat{\imath},\dots,n]$ on the vertices not equal to $i$ (the facet opposite the $i$th vertex). What this allows us to do is control the simplex with data readily available in the quiver that freely generates it as a free category.

Recall that Joyal's category $\Theta$ is the full subcategory of the category of strict $\omega$-categories $\omega\operatorname{-}\mathbf{Cat}$ spanned by the free strict $\omega$-categories associated with the globular sets ($\omega$-quivers or $\omega$-graphs) obtained by taking sequential pushouts of globes (representable disks $D^n$) along their various sources and targets.

We can describe the globular sets freely generating the objects of $\Theta$ simply by finite double-sequences of natural numbers $$i_0>j_0<i_1>\dots <i_{n-1}>j_{n-1}<i_n,$$ where the generating diagram is the colimit of the diagram $$D_{i_0}\xleftarrow{\sigma^{i_0}_{j_0}}D_{j_0}\xrightarrow{\tau^{i_1}_{j_0}}D_{i_1}\xleftarrow{\sigma^{i_1}_{j_1}} \dots \xrightarrow{\tau^{i_{n-1}}_{j_{n-2}}} D_{i_{n-1}}\xleftarrow{\sigma^{i_{n-1}}_{j_{n-1}}}D_{j_{n-1}}\xrightarrow{\tau^{i_n}_{j_{n-1}}} D_{i_n},$$ where $\sigma^i_j$ and $\tau^i_j$ are the unique cosource and cotarget maps $D_j\to D_i$.

We define the dimension of an object $$[t]=[i_0>j_0<i_1>\dots <i_{n-1}>j_{n-1}<i_n]$$ of $\Theta$ to be $$i_0 - j_0 + i_1 - \cdots + i_{n-1} - j_{n-1} + i_n.$$ The morphisms of $\Theta$ are then defined using the free-strict $\omega$-category functor or using a more sophisticated (but also more computable) theory of pasting diagrams (for instance, [Steiner, 2006]). We say a map between two objects of $\Theta$ is a coface (the inclusion of a face) if it is an injective map. We call such a map a cofacet (or the inclusion of a facet) if its source has codimension $1$ in its target.

Since facets play such a major role in the combinatorics of simplicial sets, and since they can easily be read off the generating diagram of an $n$-simplex using the duality between vertices and facets, it seems reasonable to ask if we might be able to "get a handle" on the facets of an object $[t]$ of $\Theta$ in terms of the generating diagram or the double sequence corresponding to its generating diagram. Moreover, the rule for this association should reduce to the classical case of facet-vertex duality when we are given a double sequence such that each of the $i$ terms is $1$ and each of the $j$ terms is zero, since $\Delta$ embeds isomorphically onto this full subcategory of $\Theta$.

What I've been able to work out is as follows:

There are facets corresponding to certain kinds of "peaks" in the double sequence. For instance, if $i_k$ is a peak such that $j_{k-1}+1< i_k >j_k+1$ (with $j_{-1}$ and $j_n$ taken to be zero), it corresponds to a pair (left and right) of "outer" facets obtained by omitting the cells corresponding respectively to the source and target of the corresponding disk. However, there are also special inner facets corresponding (non-bijectively) to the omission of a valley element $j$. However, it appears that simply choosing such a $j$ does not determine a facet, but merely a family of facets living opposite it. There are also special cases involving valleys $j$ when one or both of its surrounding peaks is equal to $j+1$. Indeed, in the case of an n-simplex, all of the "inner vertices" belong to this special case.

However, this still doesn't accomplish the goal of giving a combinatorially useful description of the facets in terms of the generating information, first because in the case of removing a valley, this doesn't determine a single facet; it only determines a family of facets not containing that cell, each of which then corresponds to a choice of a kind of shuffle of nearby data. Moreover, we then need to deal with the two or three special cases where a valley is directly adjacent to a peak of height $j+1$.

Then the question: Is there a way to further refine this partial classification of the facets in terms of generating data such that we can work combinatorially with them instead of having to deal with the underlying categorical picture? That is, can we find data that can be read directly off from the corresponding double sequence that can be thought of as "dual" to or "opposite" from a particular facet?

Edit: There is an obvious bijective correspondence between the double sequences that we are interested in and the continuously-graded linear intervals, which are given by the data of an even-dimensional simplex $[2n]=[0<\dots<2n]$ and a grading function function $f:[2n]\to \mathbf{N}$ such that $f(0)=f(2n)=0$ and such that $f(i)=f(i-1)\pm 1$ for all $i>0$. To obtain such a continuously-graded linear interval from a double-sequence $$i_0>j_0<i_1>\dots <i_{m-1}>j_{n-1}<i_m,$$ we let $j_{-1}=j_m=0$, and interpolate the sequence by adding a strict monotone continuous grading between each peak and valley. To go in the other direction, we simply take the alternating sequence of peaks and valleys (ignoring the leftmost and rightmost zero valleys) with respect to the grading. It is easy to see by counting the up-and-down steps that if we are given a double sequence $$i_0>j_0<i_1>\dots <i_{m-1}>j_{n-1}<i_m$$ corresponding to a continuously graded linear interval $([2n],f)$, the dimension of the associated object in $\Theta$ is equal to $n$. Since this "interpolated interval" is readily available from the generating diagram, we might be able to crush a little bit more information out of the diagram if the double sequence itself doesn't tell the whole story in an obvious way.

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Here's the sort of trouble we run into with omitting a valley: Consider the object associated with $$2>1<2>\dots<2>1<2>0<2>1<2>\dots<2>1<2$$ where the lefthand side has $m-2$ valleys equal to $1$ and where the righthand side has $n-2$ valleys equal to $1$. Omitting any valleys with height $1$, we have no strange choices, but if we omit the valley in the middle (of height $0$), we obtain a suspended copy of $[m]\times [n]$, and the representables making up this object are given by suspended shuffles of an $[m]$ and an $[n]$, each of which is a facet. These shuffles come from the exchange law. – Harry Gindi May 21 '12 at 21:14
You can see what's going on if you take the graded interval of the peak/valley sequence and use it to trace out the sectors of a planar tree. Omitting the valley of height zero corresponds to gluing the two branches between which the sector of height zero lives and then choosing a new ordering on the branches living above those two branches that is compatible with the original orders. Omitting a peak and choosing a left or right element corresponds to embedding a subtree with one input sector removed and then choosing a sector beneath it. The problem with this is that it's quite nasty. – Harry Gindi May 21 '12 at 21:34
This is a very nice question! – David Roberts May 22 '12 at 0:09