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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.
I have added the original reference (ch. IV.2 of Grabriel-Zisman) to anodyne extension
Expanded the Idea section:
The original definition by Gabrielâ€“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.
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