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The term “h-cofibration” can refer to two closely related, but different notions:
the dual notion to sharp maps.
In this article, we concentrate on the latter.
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.
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.
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