nLab cofibration



Homotopy theory

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



Paths and cylinders

Homotopy groups

Basic facts


Category theory



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:

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 nnlab) 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 11-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.


Stability properties

Cofibrations are usually defined in such a way that they are stable at least under the following operations in the category under consideration

  • composition
  • pushouts of spans at least one of whose legs is a cofibration

(Please mind the precise definitions of the category you are using. Also compare the stability properties of the dual notion fibration.)

Last revised on May 31, 2017 at 19:05:21. See the history of this page for a list of all contributions to it.