# nLab h-cofibration

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

## Definition

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

## Terminology

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.

## Properties

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”.

## References

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