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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeMay 31st 2022
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: 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. ## 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 * [[Michael Batanin]], [[Clemens Berger]], _Homotopy theory for algebras over polynomial monads_, Theory and Applications of Categories 32:6 (2017), 148–253. [arXiv](http://arxiv.org/abs/1305.0086). [[!redirects h-cofibrations]] <a href="https://ncatlab.org/nlab/revision/h-cofibration/1">v1</a>, <a href="https://ncatlab.org/nlab/show/h-cofibration">current</a>

   Created:

   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.

   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

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

   !redirects h-cofibrations

   v1, current