nLab anodyne morphism

Idea

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)).

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

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

there exists a lift

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

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).

Properties

Pushout-products with inclusions

Proposition

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

Remark

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.

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 original concept of anodyne extensions as morphisms in the saturation class of the simplicial horn inclusions originates in

• Pierre Gabriel, Michel Zisman, chapter IV.2 of Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, Springer (1967) (pdf)

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

• 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)