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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMar 7th 2012
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added a little bit of discussion to the Idea-section at _[[anodyne morphism]]_, added references, added a mentioning of the dendroidal case, and made _[[anodyne extension]]_ redirect to it.

   I have added a little bit of discussion to the Idea-section at anodyne morphism, added references, added a mentioning of the dendroidal case, and made anodyne extension redirect to it.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeApr 19th 2016
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   Author: Urs
   Format: MarkdownItexI have added the original reference (ch. IV.2 of Grabriel-Zisman) to _[[anodyne extension]]_

   I have added the original reference (ch. IV.2 of Grabriel-Zisman) to anodyne extension

3. • CommentRowNumber3.
   • CommentAuthorDmitri Pavlov
   • CommentTimeJul 22nd 2022
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   Author: Dmitri Pavlov
   Format: MarkdownItexExpanded the Idea section: The original definition by Gabriel–Zisman (Definition IV.2.1.4 \cite{GabrielZisman67}) defined __anodyne extensions__ as the [[weak saturation]] of [[simplicial horn]] inclusions. More generally, the same definition can be used to talk about the [[weak saturation]] of any set $S$ of morphisms in any category. One also talks about _anodyne maps_ or _anodyne morphisms_. If the [[small object argument]] is applicable, anodyne maps are precisely maps with a [[left lifting property]] with respect to all [[fibrations]], where the latter is defined as morphisms with a [[right lifting property]] with respect to $S$. In particular, if $S$ is a set of generating [[acyclic cofibrations]] in a [[model category]] with applicable [[small object argument]], then anodyne maps are precisely [[acyclic cofibrations]]. <a href="https://ncatlab.org/nlab/revision/diff/anodyne+morphism/14">diff</a>, <a href="https://ncatlab.org/nlab/revision/anodyne+morphism/14">v14</a>, <a href="https://ncatlab.org/nlab/show/anodyne+morphism">current</a>

   Expanded the Idea section:

   The original definition by Gabriel–Zisman (Definition IV.2.1.4 \cite{GabrielZisman67}) defined anodyne extensions as the weak saturation of simplicial horn inclusions.

   More generally, the same definition can be used to talk about the weak saturation of any set of morphisms in any category. One also talks about anodyne maps or anodyne morphisms.

   If the small object argument is applicable, anodyne maps are precisely maps with a left lifting property with respect to all fibrations, where the latter is defined as morphisms with a right lifting property with respect to .

   In particular, if is a set of generating acyclic cofibrations in a model category with applicable small object argument, then anodyne maps are precisely acyclic cofibrations.

   diff, v14, current