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Recall that $\Theta_2$ is the full subcategory of $Str2-cat$ consisting of “free 2-categories on pasting diagrams”. Consider the category $tSet:=Psh(\Theta_2)$ (I’m calling this $tSet$ to avoid having to write out $\Theta_2$ every time).
Consider the presheaf $X$ represented by the object $[2]([2],[2])$. There exists a subpresheaf $\Lambda\subset X$, which is, roughly geometrically, $\Lambda^2_1(\Lambda^2_1,\Lambda^2_1)$. Another way to think about this geometrically is simply as the pasting diagram that generates $[2]([2],[2])$.
Now, we may apply Cisinski’s machinery in the following way: Let $L$ be the subobject classifier of $tSet$, which, being injective, is fibrant and contractible for every Cisinski-type model structure. The object $L$ defines a separated interval object in the following canonical way: Since the terminal object, $\ast$ has exactly two subobjects $\emptyset \hookrightarrow \ast$ and $\ast \hookrightarrow \ast$, we have exactly two morphisms $\{0\},\{1\}:\ast \hookrightarrow L$, and their intersection is empty. Also, $L\to \ast$ is the unique morphism to the terminal object, so this determines a separated interval. We let $\partial L$ be the subobject of $L$ induced from the coproduct $\{0\}\coprod \{1\}\to L$.
We choose the following homotopy data: Let $M$ be a small set of generating monomorphisms (in the sense that $llp(rlp(M))$ is the class of monomorphisms), which exists by virtue of $tSet$ being a presheaf category. Let $S$ be the set containing just the morphism $\iota:\Lambda\hookrightarrow X$ Then the data of $(L,M,S)$ may be completed to a homotopy structure on $tSet$ by setting $An_L:=llp(rlp(C_L(S,M)))$, where $C_L(S,M)$ is the set of morphisms defined recursively as follows:
$C_0=\{A\times L \coprod_{A\times \{\varepsilon\}} B\times \{\varepsilon\}:A\hookrightarrow B \in M, \varepsilon=0,1\} \cup S$For $Y$ any set of morphisms of $tSet$, let
$C(Y)=\{A\times L \coprod_{A\times \partial L} B\times \partial L:A\to B \in Y\}$Define $C_n=C(C_{n-1})$ for $n\geq 1$, and let $C_L(S,M)=\bigcup_{n\geq 0} C_n$.
It is a lemma of Cisinski that the class $An_L=llp(rlp(C_L(S,M)))$ is a class of anodynes for the cylinder $L$. Further, Cisinski’s lemma states that $An_L$ is the absolute smallest weakly saturated class of morphisms containing $S$ that is a compatible class of anodynes for the cylinder $L$.
By Cisinski’s big theorem in chapter 1.3 of Astérisque, we see that the homotopy structure $(L,An_L)$ generates a combinatorial model structure on $tSet$, and that this is the minimal Cisinski model structure for which $\iota:\Lambda\hookrightarrow X$ is anodyne.
The resulting model structure is a model for $(oo,2)$-categories and is on the nose equal to Joyal’s conjectural model structure for quasi-2-categories as presheaves on $\Theta_2$.
If we rerun the argument by replacing $\Theta_2$ with $\Delta$, $X$ with $\Delta^2$, and $\Lambda$ with $\Lambda^2_1$, it is known that we recover the Joyal model structure. If further, we add the outer 2-horn inclusions to $S$, we recover the theory of Kan complexes. This, among other things, leads me to believe that we can think of the set $S$ above as encoding “operations” satisfying certain “identities” on fibrant objects. The inclusion $\Lambda\hookrightarrow X$ that I chose above seems to encode composition of 1-morphisms, composition of 2-morphisms, and the interchange law.
At least what I’m hoping is that the same magic that happens for quasicategories and Kan complexes works for quasi-2-categories. If it works (pretty big “if”!), I think it should be fairly straightforward to generalize the idea to $(\infty,n)$-categories modeled by $\Theta_n$ for finite $n$ by induction.
I’m posting this here to ask you guys if it sounds plausible. Does it?
Hi Harry,
I can’t make a useful technical comment right now, but have a general comment, concerning for instance
Joyal’s conjectural model structure for quasi-2-categories as presheaves on $\Theta_2$.
There has been a tremendous amount of activity recently by Julie Bergner, Charles Rezk and others on relating Theta-space-models for $(\infty,n)$-categories in all kinds of ways to all kinds of other models. I am not up to speed with this, though, only know that Julie and others told me about exciting developoments.
I expect you can get a really good answer by emailing her or Charles Rezk (if he doesn’t see you on MO, already ;-)
Alright, I sent Julie an e-mail.
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