This is a very simple question coming from the observation that every (pre)sheaf category has the maximal Cisinski model structure on it. This is the Cisinski model structure with the smallest class of weak equivalences possible. 


Now, it is natural to ask: what is the maximal Cisinski model structure on the most canonical category in this setting, namely the category of simplicial sets? Is it larger than the Joyal model structure? Assuming that the answer is "yes", can we give an explicit description of its weak equivalences and fibrant objects?



Maximal Cisinski model structure on simplicial sets
(Edit: after 2.5 years, I've finally typed up the details as a paper! <https://arxiv.org/abs/2201.13400> )


I think I've worked out a relatively nice description of the fibrant objects. I won't be able to include the full proofs here, but I will give the description and explain a bit about why it works. (I'm planning to include the details in a longer note, to which I'll post a link here once I make it available, but that will probably take at least a couple more months.)


Let me first recap some of Andrea's answer and the comments following it: the fibrant objects in the minimal model structure are those with lifts against $\textbf{An}\_{\mathfrak{L}}$. The class $\textbf{An}\_{\mathfrak{L}}$ is generated by the pushout-products of the boundary inclusions $\partial\Delta[n]\hookrightarrow\Delta[n]$ with the maps $\{\varepsilon\}\hookrightarrow L$ for $\varepsilon=0,1$, where $L$ is the subobject classifier in simplicial sets.


The first thing to notice is that we can actually use $J$, the nerve of the groupoid with two objects and an isomorphism between them, instead of $L$. One way to see why is to work out what $L$ is as a simplicial set, and see that $J$ is actually a retract of $L$. I talked to Alex Campbell, however, and he told me that if you trace through Cisinski's arguments in his book, you can actually see that everything he says about $L$ would apply to a separating cylinder which is trivially fibrant, such as $J$.


So, at any rate, now we know the fibrant objects are those with lifts against $\textbf{An}\_{J}$, which is generated by the pushout-products of the boundary inclusions $\partial\Delta[n]\hookrightarrow\Delta[n]$ with the map $\{0\}\hookrightarrow J$. With this description already, there is some nice intuition: having lifts against the pushout-product


$(\partial\Delta[n]\times J) \cup (\Delta[n]\times \{0\})\hookrightarrow \Delta[n]\times J$


tells you that any time you have an $n$-simplex and an "isomorphism of boundaries" which starts at the boundary of that $n$-simplex, then it extends to an "isomorphism of $n$-simplices" which starts at that $n$-simplex.


The description so far, which is not new, was not satisfying to me. I wanted something which felt more like the horn extension definition of Kan complexes and quasi-categories, so I played around a bit and found the following:


Define an "isoplex," denoted $\mathfrak{D}\_i[n]$, to be the nerve of the category


$c\_0\rightarrow c\_1 \rightarrow \ldots c\_{i-1} \rightarrow c\_i \leftrightarrow c\_{i+1} \rightarrow c\_{i+2} \rightarrow \ldots \rightarrow c\_n$,


i.e., where the morphism $c\_i\rightarrow c\_{i+1}$ is an isomorphism. Think of this as analogous to $\Delta[n]$, which is the nerve of the category $c\_0\rightarrow c\_1 \rightarrow \ldots \rightarrow c\_n$. In particular, we can define faces of $\mathfrak{D}\_i[n]$ in a similar manner, where $d\_j\mathfrak{D}\_i[n]$ is the maximal subcomplex not containing the $j$ vertex. Then we can define the iso-horn of $\mathfrak{D}\_i[n]$, denoted $\mathbb{V}\_i[n]$, to be the union of all of its faces except the $i$th face. Let $\text{IsoHorn}$ be the set of all iso-horn inclusions $\mathbb{V}\_i[n]\hookrightarrow \mathfrak{D}\_i[n]$.


It turns out that $\text{IsoHorn}$ generates the class $\textbf{An}\_J$. In particular, the fibrant objects in the minimal model structure are the simplicial sets with lifts against $\text{IsoHorn}$.


That $\overline{\text{IsoHorn}}\subseteq \textbf{An}\_J$ is not so bad to check, because each iso-horn extension is a retract of a generator of $\textbf{An}\_J$. The other direction follows because each of the generators of $\textbf{An}\_J$ can be built out of iso-horn extensions (by transfinite composition of pushouts), which I think is pretty intuitive, but which did take a couple pages of combinatorics for me to check carefully.


As for the minimal model structure being different from Joyal's, in addition to Andrea's properness argument, we can also just see directly that the horn $\Lambda^1[2]$ is fibrant in the minimal model structure.


But not only is the minimal model structure different from Joyal's, I'm confident there will be multiple interesting Cisinski model structures between them. In particular, my thesis project is to find a Cisinski model structure where the fibrant objects model up-to-homotopy versions of 2-Segal sets (which is why I was thinking about the minimal model structure in the first place).


