Joyal's CatLab The model structure for quasi-categories

• Recall that the category of simplicial sets $SS=[\Delta^o, Set]$ admits a Quillen model structure? in which the cofibrations? are the monomorphisms, the weak equivalences are the weak homotopy equivalences? and the fibrations? are the Kan fibrations?. We shall say that it is the Kan-Gabriel-Zisman-Quillen model structure.

• Recall that a model structure on a category is determined by its cofibrations together with its fibrant objects. When the class of cofibrations is the class of monomorphisms, the model structure is determined by its class of fibrant objects. In particular, the Kan-Gabriel-Zisman-Quillen model structure is determined by the Kan complexes; we shall say that it is the model structure for Kan complexes.

Theorem

The category of simplicial sets $SS=[\Delta^o, Set]$ admits a Quillen model structure? in which the cofibrations? are the monomorphisms and the fibrant objects are the quasi-categories?. The weak equivalences are called the weak categorical equivalences? and the fibrations are called the isofibrations.
The model structure is cofibrantly generated and cartesian closed; we shall say that it is the model structure for quasi-categories.

Definition

Recall that a cofibrantly generated model structure on a (Grothendieck) topos $E$ is said to be a [Cisinski model structure] if its cofibrations are the monomorphisms. Recall that a class $W$ of maps in a Grothendiect topos $E$ is said to be a [localiser] if $W$ is the class of weak equivalences of a Cisinski model structure on $E$. Recall that every set of map $\Sigma \subseteq E$ is contained in a smallest localiser $W(\Sigma)$ by a theorem of Cisinski. We say that $W(\Sigma)$ is the localiser generated by $\Sigma$.

Theorem

In the category $SS$, the class of weak categorical equivalences is the localiser generated by the set of inner horn inclusions $\Lambda^k[n]\subset \Delta[n]$ (for $0\lt k\lt n$).

Definition

If $n \gt 0$, the spine $I[n]$ of a simplex $\Delta[n]$ is defined to be the union of the edges $(i,i+1):\Delta[1]\to \Delta[n]$ for $0\leq i \lt n$. We shall put $I[0]=\Delta[0]$.

Theorem

In the category $SS$, the class of weak categorical equivalences is the localiser generated by the set of spine inclusions $I[n]\subset \Delta[n]$ (for $n\geq 0$).