Joyal's CatLab
The model structure for quasi-categories

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Theorem

The category of simplicial sets SS=[Δ o,Set]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 EE is said to be a [Cisinski model structure] if its cofibrations are the monomorphisms. Recall that a class WW of maps in a Grothendiect topos EE is said to be a [localiser] if WW is the class of weak equivalences of a Cisinski model structure on EE. Recall that every set of map ΣE\Sigma \subseteq E is contained in a smallest localiser W(Σ)W(\Sigma) by a theorem of Cisinski. We say that W(Σ)W(\Sigma) is the localiser generated by Σ\Sigma.

Theorem

In the category SSSS, the class of weak categorical equivalences is the localiser generated by the set of inner horn inclusions Λ k[n]Δ[n]\Lambda^k[n]\subset \Delta[n] (for 0<k<n0\lt k\lt n).

Definition

If n>0n \gt 0, the spine I[n]I[n] of a simplex Δ[n]\Delta[n] is defined to be the union of the edges (i,i+1):Δ[1]Δ[n](i,i+1):\Delta[1]\to \Delta[n] for 0i<n0\leq i \lt n. We shall put I[0]=Δ[0]I[0]=\Delta[0].

Theorem

In the category SSSS, the class of weak categorical equivalences is the localiser generated by the set of spine inclusions I[n]Δ[n]I[n]\subset \Delta[n] (for n0n\geq 0).

Revised on June 12, 2012 at 11:10:00 by Andrew Stacey