Fibration Categories are Fibrant Relative Categories

TL;DR

This paper proves that the underlying relative category of a model or fibration category is fibrant within the Barwick--Kan model structure, linking model categories to fibrant objects in this framework.

Contribution

It establishes that model categories and fibration categories are fibrant in the Barwick--Kan model structure, enhancing the understanding of their homotopical properties.

Findings

Model categories are fibrant in the Barwick--Kan structure.

Fibration categories are fibrant in the Barwick--Kan structure.

Connections between different model structures are clarified.

Abstract

A relative category is a category with a chosen class of weak equivalences. Barwick and Kan produced a model structure on the category of all relative categories, which is Quillen equivalent to the Joyal model structure on simplicial sets and the Rezk model structure on simplicial spaces. We will prove that the underlying relative category of a model category or even a fibration category is fibrant in the Barwick--Kan model structure.