Fibration categories are fibrant relative categories (Q728314)

A relative category is a pair \((\mathcal{C},\mathcal{W})\) where \(\mathcal{C}\) is a category and \(\mathcal{W}\) a subcategory containing all objects. In [\textit{C. Barwick} and \textit{D. M. Kan}, Indag. Math., New Ser. 23, No. 1--2, 42--68 (2012; Zbl 1245.18006)], it is proved that there exists on the category of small relative categories a model category structure which is Quillen equivalent to the Joyal model structure on simplicial sets and to the Rezk model structure on simplicial spaces. In this paper, the author proves that the underlying relative category of a model category or even a fibration category is fibrant in the Barwick-Kan model structure. The even stronger theorem is proved: every homotopically full subcategory of a fibration category is fibrant in the Barwick-Kan model structure.