Kan complexes and quasicategories

Definition 1.6.7. Let \(X\) be a simplicial set.

(1)

We say that \(X\) is a Kan complex if for every \(n \geq 1\) and \(0 \leq k \leq n\), every morphism of simplicial sets \(\Lambda ^n_k \to X\) extends to a morphism \(\Delta ^n \to X\):

(2)

We say that \(X\) is a quasicategory if the above condition holds for all \(n \geq 1\) but only for \(0 < k < n\).

We denote by \[ \Kan \quad \subseteq \quad \qCat \quad \subseteq \quad \sSet \] the full subcategories on the Kan complexes and the quasicategories, respectively.