Fibrancy of partial model categories (Q5963044)

The main result of this paper is that the category of weak equivalences of a partial model category is fibrant in the Thomason model structure. By partial model category, it is meant a relative category equipped with two additional class of maps playing the role of trivial cofibrations and trivial fibrations respectively such that: 1) the class of weak equivalences has the \(2\)-out-of-\(6\) property, 2) the class of trivial cofibrations (of trivial fibrations resp.) is closed under pushout (under pullback resp.), 3) every weak equivalence factors functorially as a composite a trivial cofibration followed by a trivial fibration. In other terms, a partial model category is a homotopical category equipped with a \(3\)-arrow calculus. The authors also prove that this fibrancy condition is not necessary.