nLab h-cofibration

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

• Hurewicz cofibrations;

• the dual notion to sharp maps.

In this article, we concentrate on the latter.

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.

Related concepts

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

References

• Michael Batanin, Clemens Berger, Homotopy theory for algebras over polynomial monads, Theory and Applications of Categories 32:6 (2017), 148–253. arXiv.