nLab h-cofibration

The term “h-cofibration” can refer to two closely related, but different notions:

In this article, we concentrate on the latter.


A map f:XYf\colon X\to Y in a relative category CC is an h-cofibration if the cobase change functor X/CY/CX/C\to Y/C is a relative functor, i.e., preserves weak equivalences.


Another name for such morphisms is “flat map”, used by Hill–Hopkins–Ravenel (Appendix B.2). This choice of terminology conflicts with flat maps used to define flat monoidal model categories.


A model category is left proper if and only if all cofibrations are h-cofibrations.

In a left proper model category, cobase changes along h-cofibrations are homotopy cobase changes.

The notion of h-cofibrations is most useful in the left proper case, and one can argue that in the non-left proper case, the above property should be taken as the definition instead.

  • The dual notion is known as a “sharp map” or “h-fibration”.


Created on May 31, 2022 at 13:53:56. See the history of this page for a list of all contributions to it.