# $\omega$-nerve versus $\Theta$-nerve

To which extent the adjunction $F\dashv N_\omega$ generated by the $\omega$-nerve described at $n$Lab - oriental (obtained as a particular instance of the nerve-realization paradigm) is linked to the adjunction generated by the functor $O_{[\Theta]}\colon \Theta\to \textbf{Str}\text{-}\omega\text{-}\mathbf{Cat}$ (Joyal's $\Theta$-category), $$\text{Lan}_y(O_{[\Theta]})\dashv N_{[\Theta]},$$ where the functor $N_{[\Theta]}\colon \textbf{Str}\text{-}\omega\text{-}\mathbf{Cat} \to [\Theta^{op}, \mathbf{Sets}]$ sends $C\in \textbf{Str}\text{-}\omega\text{-}\mathbf{Cat}$ to the presheaf $\textbf{Str}\text{-}\omega\text{-}\mathbf{Cat}(O(-), C)$? Is there any reference to learn about, and quote properly, affinities and differences between the two?

As you remark, the cellular and simplicial nerves arise from the functors $J \colon \Theta \longrightarrow \omega\text{-Cat}$ and $O \colon \Delta \longrightarrow \omega\text{-Cat}$ respectively, where $J$ is the full inclusion of Joyal's cell category and $O$ is Street's orientals functor.

A significant difference between these two is that $J$ is dense, and hence the nerve functor $\omega\text{-Cat} \longrightarrow [\Theta^{op},\text{Set}]$ is fully faithful, whereas this is not true of $O$ and the simplicial nerve induced by the orientals. (For a reference, this comment is made in the introduction to Section 1 of Berger's 'A cellular nerve for higher categories'). Note however that the modification of the simplicial nerve which lands in stratified simplicial sets is fully faithful (see Verity's 'Complicial sets...', whose introduction contains an account of the history of this nerve).

The cellular nerve has these nice properties because it arises from general theory; I mean the nerve theorem (see for instance the Introduction and Theorem 1.10 of Berger, Melliès & Weber's 'Monads with arities...'), which applies upon recognising the objects of $\Theta$ as the free $\omega$-categories on the canonical arities for the free $\omega$-category monad on globular sets.

Sadly, there is no such nice theoretical description of the orientals functor; the best we can give is an explicit construction using parity complexes or similar. Yet there is still another description of the simplicial nerve of a 2-category (see section 10 on nerves in Lack's 'A 2-categories companion'). The composite functor $\Delta \longrightarrow \text{Cat} \longrightarrow 2\text{-Cat}_{\text{nlax}}$, where $2\text{-Cat}_{\text{nlax}}$ is the category of 2-categories and normal lax 2-functors, is dense and so gives a fully faithful nerve functor $2\text{-Cat}_{\text{nlax}} \longrightarrow [\Delta^{op},\text{Set}]$ which agrees (up to duality) on $2\text{-Cat}$ with the nerve induced by the orientals. So we could hope that, if we had a category $\omega\text{-Cat}_{\text{nlax}}$ whose morphisms are "normal lax $\omega$-functors", the composite functor $\Delta \longrightarrow \text{Cat} \longrightarrow \omega\text{-Cat}_{\text{nlax}}$ would be dense and give a fully faithful nerve functor $\omega\text{-Cat}_{\text{nlax}} \longrightarrow [\Delta^{op},\text{Set}]$ extending the simplicial nerve given by the orientals. This suggests the tentative theoretical description of the $n$-th oriental as the "normal lax $\omega$-functor classifier" of $[n]$. However, I do not know of any definition of normal lax $\omega$-functor, which would likely make use of the orientals anyway.

With regard to their affinities, I am not aware of any study in the literature. At the heart of the comparison is the module $\omega\text{-Cat}(J,O) \colon \Theta \nrightarrow \Delta$, corresponding to the functor $\Delta \longrightarrow [\Theta^{op},\text{Set}]$ that gives the cellular nerves of the orientals. One could hope that this module has a nice combinatorial description. Perhaps someone out there has considered this, maybe in the comparison of the models for higher categories based on weak complicial sets and cellular sets.

Important edit: A few years ago, Dimitri Ara mentioned to me in private correspondence that he had written a bruteforce program to find objecta in $\Theta$ that do not admit an orientation that can be coherently pasted together. He told me way back then that they found an example in low dimensions very quickly. I do not know if this ended up in any of his published articles yet, but definitively, there is no such extension. He and Maltsiniotis ended up defining a non-full subcategory of $\Theta$ that was coherently orientable, but I don't know the definition.

It turns out, due to a series of papers by Richard Steiner and then a later paper by Dimitri Ara and Georges Maltsiniotis, that there doesn't appear to be a meaningful definition for an 'oriental' for any given object of the category $\Theta$. What Ara and Maltsiniotis did figure out was the notion of a lax join of strict $\omega$-categories using Steiner's representation of pasting diagrams as special chain complexes with some additional structure.

Essentially the construction is given on "loop-free pasting diagrams", that is to say, directed complexes in the sense of Steiner (which happen to include the pasting diagrams that define $\Theta$ and also the simplicial orientals), then extend it to the category of strict $\omega$-categories by merit of a specialized version of the Day Convolution theorem. On the chain complexes underlying the directed complexes, this is given by the desuspension of the tensor product of the suspensions of two directed complexes, that is,

$$A\star B = \Sigma^{-1}(\Sigma A \otimes \Sigma B)$$.

In particular, the orientals are obtained by the formula $$\mathcal{O}[n+1] = \mathcal{O}[n] \star [0]$$.

This is all contained in the paper of Ara and Maltsiniotis (link)

Their definition of the join can also be extended easily to the category of cellular sets using the much simpler (weighted) Day Convolution construction for presheaves.

That is, for cellular sets $S,T \in \operatorname{Psh}(\Theta)$, the join

$$S\star T (-) = \int^{[a],[b]\in \Theta^+} S^+_a\times T^+_b \times \operatorname{Hom}_{\operatorname{DirComp}^+}( - , [a]\star[b])$$

where the $+$ means that we're looking at augmented Directed Complexes, augmented cellular sets, and the augmented Disk category $\Theta$. This essentially means attaching an initial object to the categories.

There are several other nerves that one can take with respect to different cocellular objects $\Theta\to \operatorname{Str-\omega-Cat}$, but it is important to note that the naive version induced by the fully faithful embedding of $\Theta$ in $\operatorname{Str-\omega-Cat}$ does not induce a Quillen adjunction between strict $\omega$-categories and the various models for weak $\omega$-categories. It turns out that the 'correct' nerve to induce the Quillen adjunction is something a bit more complicated and involves first applying Metayer's resolution by polygraphs to the objects of $\Theta$.

Edit: Andrea Gagna informed me by personal correspondence that it appears to be nontrivial and thusfar unproven that the lax join as described above for cellular sets will be associative (and similarly the lax tensor product's extension to cellular sets). The extension of the join is locally biclosed (similarly the extension of the tensor product is truly biclosed), but without further work, the associativity and coherence axioms for monoidal catagories remain to be demonstrated.

For example, Andrea gave an interesting example that Dendroidal sets equipped with the Day Convolution-induced tensor product from the tensor product of operads fails to be associative (see the erratum to this paper of Cisinski and Moerdijk), so it's not just free.

I have geometric/combinatorial reasons to expect that both will be associative for cellular sets, but it is, as far as I know, an open, if peripheral, question.

• I'm not so sure about this edit now. I was going by what Andrea said, but the erratum doesn't appear to show a failure of associativity on the category of dendroidal sets, but rather on its homotopy category (the model structure rather than the category fails to be monoidal). Aug 2 '17 at 9:28
• An explicit counter-example appears in full detail in the paper On the equivalence between Lurie’s model and the dendroidal model for $\infty$-operads, by Heuts, Hinich, Moerdijk. You can also check this other question of mine, with a nice answer by Gijs. Aug 2 '17 at 10:58
• I guess to prove it, you would want to show that the coend decomposes in a sane way (i.e. that tensor products/joins of $\Theta$-pasting schemes are given by a union of same-geometric-dimension $\Theta$-subobjects (which ought to be true because they're free)) Aug 2 '17 at 11:19