Proof of existence of Joyal model structure via Cisinski theory?

Question

I'm looking for a proof of the existence of the Joyal model structure -- with its usual description -- which uses Cisinski theory directly. The closest thing I know of is Theorem 5.26 of Ara's Higher quasi-categories vs higher Rezk spaces, but even there it seems he needs to assume the existence of the Joyal model structure before he can compare it to a certain model structure obtained from Cisinski theory.

More precisely, let

• $\mathsf{IH} = \{\Lambda^k[n] \to \Delta[n] \mid 0 < k < n\}$ be the set of inner horns;

• $\mathsf{J} = \{\Delta[0] \to J\}$ where $J$ is the walking isomorphism.

Then we have the following

Theorem: (Existence of the Joyal model structure) There exists a Cisinski model structure on $sSet$ such that

• $(\ast(\mathsf{IH} \cup \mathsf{J})):$ The fibrant objects (resp. fibrations between fibrant objects) are those objects (resp. morphisms between fibrant objects) which have the right lifting property with respect to $\mathsf{IH} \cup \mathsf{J}$.

What I'm looking for is a write-up of the following proof outline of the above theorem:

Proof Sketch: By Cisinski theory, there exists a Cisinski model structure on $sSet$ such that $\ast(\Lambda(\mathsf{IH} \cup \mathsf{J}))$ holds, where for any $S$ we define

• $\Lambda^0(S) = S \cup \{\Delta[n] \cup \partial \Delta[n] \times J \to \Delta[n] \times J\}$

• $\Lambda^{n+1}(S) = \{B \times \partial \Delta[1] \cup A \times J \to B \times J \mid A \to B \in \Lambda^n(S)\}$

• $\Lambda(S) = \cup_n \Lambda^n(S)$

We now verify (and these verifications are what I'd like to see!) that the morphisms of $\Lambda(\mathsf{IH} \cup \mathsf{J})$ can be constructed as retracts of transfinite composites of cobase changes of morphisms of $\mathsf{IH} \cup \mathsf{J}$. Therefore $\ast(\mathsf{IH} \cup \mathsf{J})$ and $\ast(\Lambda(\mathsf{IH} \cup \mathsf{J}))$ are equivalent, and we are done.

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I'm pretty sure that all the necessary combinatorics has been done somewhere, but I'd like to see it strung together.

I'd also be interested in an analogous approach to the Kan-Quillen model structure.

Answer 1

Such a proof is given in Chapter 3 of Cisinski's book Higher categories and homotopical algebra, see Definition 3.3.7 and Theorem 3.6.1. (Note that Cisinski's proof uses as the interval object not the nerve of the free-living isomorphism, but the simplicial set freely generated by a "left and right invertible" 1-simplex, see Definition 3.3.3.)

I attempted to give a geodesic such proof in the Appendix of my paper Joyal's cylinder conjecture, see Theorem A.7 (take $B = \Delta[0]$). I take the nerve $J$ of the free-living isomorphism as the interval object. This choice has the advantage that the necessary combinatorics can be boiled down to the following lemma (and the standard fact that the pushout-product of an inner horn inclusion with a monomorphism is inner anodyne), see Lemma A.5:

Lemma. Let $j \colon S \to T$ be a bijective-on-$0$-simplices monomorphism of simplicial sets, and suppose that $T$ is a quasi-category. Then the pushout-product of $j \colon S \to T$ with the inclusion $\{0\} \to J$ is inner anodyne.

For the Kan--Quillen model structure, see Section 3.1 of Cisinski op. cit.