homotopy theory, (∞,1)-category theory, homotopy type theory flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed… models: topological, simplicial, localic, … see also algebraic topology Introductions Introduction to Basic Homotopy Theory Introduction to Abstract Homotopy Theory geometry of physics – homotopy types Definitions Paths and cylinders Homotopy groups Basic facts Theorems The notion of cofibration is dual to that of fibration. See there for more details. 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 Waldhausen categories In traditional topology, one usually means a Hurewicz cofibration. 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 opfibrations. 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. relative cell complex inclusions are the cofibrations in the classical model structure on topological spaces Closed Hurewicz cofibration are the cofibrations in in the Strøm's model category on Top. monomorphisms are the cofibrations in Cisinski model structures such as the classical model structure on simplicial...