# Proof of existence of Joyal model structure via Cisinski theory?

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;

• $$\mathsf{J} = \{\Delta \to J\}$$ 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

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

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

• $$\Lambda(S) = \cup_n \Lambda^n(S)$$

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.

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.

## 1 Answer

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 $$B = \Delta$$). 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.

• Thanks! I probably should have guessed I could find this in Cisinski, but I'm glad I asked because I probably wouldn't have guessed to look in your appendix for the nicer result with the obvious cylinder! – Tim Campion Oct 1 '20 at 19:28