nLab
anodyne morphism

Idea

In the presence of a notion of fibration and a given class of such, a morphism is called anodyne if it has the left lifting property against all these.

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 are equivalent to acyclic cofibration in the standard model structure on simplicial sets.

Typically one says anodyne morphism or, actually, anodyne extension if one thinks of these morphisms produced by retracts and transfinite composition of pushouts of a certain generating set of morphisms.

So in the standard example of left lifting against Kan fibrations, one typically speaks of anodyne extension if one produces morphisms by these operations from the set of horn inclusions. (see for instance (Jardine)).

Definition

Relative to Kan fibrations of simplicial sets

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

$$\begin{array}{ccc}A& \to & X\\ {\downarrow }^f& & \downarrow \\ B& \to & Y\end{array}$$\array{ A &\to& X \\ \downarrow^f && \downarrow \\ B &\to& Y }

there exists a lift

$$\begin{array}{ccc}A& \to & X\\ {\downarrow }^f& \nearrow & \downarrow \\ B& \to & Y\end{array}.$$\array{ A &\to& X \\ \downarrow^f &\nearrow& \downarrow \\ B &\to& Y } \,.

See for instance (Jardine) for details.

Relative to left/right inner Kan fibrations of simplicial sets

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).

Relative to inner Kan fibrations of dendroidal sets

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).

Related concepts

• factorization system, small object argument

• Kan fibration, anodyne morphism

• right/left Kan fibration, right/left anodyne map

• inner fibration

• Cartesian fibration

References

The standard anodyne extensions as morphisms in the saturation class of the simplicial horn inclusions is discussed in some detail in

• Rick Jardine, Homotopy theory, lecture 5 (pdf)

Left/right and inner anodyne morphisms of simplicial sets are discussed in section 2 of

• Jacob Lurie, Higher Topos Theory

Inner anodyne morphisms of dendroidal sets are discussed in

• Denis-Charles Cisinski, Ieke Moerdijk, Dendroidal sets as models for homotopy operads (arXiv:0902.1954)