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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 31st 2022

    Created:

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

    In this article, we concentrate on the latter.

    Definition

    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.

    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.

    References

    !redirects h-cofibrations

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