Weak cofibrations in categories of cofibrant objects (Q1770316)

A category of cofibrant objects is a triple (\({\mathcal C},\;cof, \;we)\) with usual axioms of model categories (\(cof\) is closed under composition or change of cobase and contains the isos, \(we\) contains the isos and if two of \(f\), \(g\), \(f\circ g\) are in \(we\) so is the third) and each object has a cylinder object and is cofibrant. This paper introduces and studies a fibre homotopy relation and the weak fibrations (morphisms having the weak right lifting property with respect to \(we\)) in such a category equipped with a choice of cylinder objects. Comparison with the model situation, dualization and given examples explain the interest of these new notions.