A cofibration is a member of a distinguished class of cofibrations in one of the several setups in homotopy theory:
Quillen categories of models for homotopy theory
the categories with cofibrations? of Baues
In category theory and descent theory, Grothendieck however introduced a notion of cofibered category or cofibration, whose definition is categorically dual to that of a fibered category and based on the generalization of the universal property of coCartesian squares; however it does not correspond to the extension property in topology, but to a lifting property, like for fibrations. For that reason, Gray suggested (and this is to some extent adopted in lab) to call such categories opfibration?s. In the quasicategorical setup, the generalizations of fibered categories are called Cartesian fibrations, and the generalizations of op/co-fibered categories are called coCartesian fibrations. In the -categorical setup, the coCartesian arrows in op/co-fibrations are indeed the ones which complete coCartesian squares in the special and most important case of domain opfibrations.