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The original definition by Gabriel–Zisman (Definition IV.2.1.4) 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.
The standard example, often taken to be the default, is that of morphisms in the category sSet of simplicial sets which have the left lifting property against all Kan fibrations. In this case, anodyne morphisms (Gabriel-Zisman 67, chapter IV.2) are equivalent to acyclic cofibrations in the classical model structure on simplicial sets.
So in the standard example of left lifting against Kan fibrations, one typically speaks of anodyne extensions if one produces morphisms by these operations from the set of horn inclusions. (see for instance (Jardine)).
A morphism $f : A \to B$ of simplicial sets is called anodyne if it has the left lifting property with respect to all Kan fibrations.
So $f$ is anodyne if for every Kan fibration $X \to Y$ and every commuting diagram
there exists a lift
See for instance (Jardine) for details.
Similarly a morphism is called
left anodyne if it has the left lifting property with respect to all left Kan fibrations
right anodyne if it has the left lifting property with respect to all right Kan fibrations
inner anodyne if it has the left lifting property with respect to all inner Kan fibrations
See (Lurie) (following Joyal).
In the category of dendroidal sets there is a notion of horn inclusions that generazies that of simplicial sets. The corresponding saturated class of morphisms is called that of dendroidal inner anodyne morphisms.
See (Cisinski-Moerdijk 09).
The pushout product $f \Box g$ of two monomorphisms $f,g$ in sSet is again a monomorphism, which is anodyne (a weak homotopy equivalence) if $f$ or $g$ is so.
This is due to (Gabriel-Zisman 67, IV.2, prop. 2.2). The argument is somewhat more streamlined form is also in Joyal-Tierney 05, theorem 3.2.2
Prop. is the key lemma which implies (is effectively equivalent to) the statement that the classical model structure on simplicial sets is an enirched model category? over itself.
Kan fibration, anodyne morphism
The original concept of anodyne extensions as morphisms in the saturation class of the simplicial horn inclusions originates in
Review includes
John Frederick Jardine, Homotopy theory, lecture 5 (pdf)
André Joyal, Myles Tierney, An introduction to simplicial homotopy theory, 2005 (chapter I, more notes pdf)
Left/right and inner anodyne morphisms of simplicial sets are discussed in section 2 of
Inner anodyne morphisms of dendroidal sets are discussed in
Last revised on October 12, 2022 at 12:02:06. See the history of this page for a list of all contributions to it.