nLab anodyne morphism



(,1)(\infty,1)-Category theory

Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




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 SS 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 SS.

In particular, if SS 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)).


Relative to Kan fibrations of simplicial sets

A morphism f:ABf : A \to B of simplicial sets is called anodyne if it has the left lifting property with respect to all Kan fibrations.

So ff is anodyne if for every Kan fibration XYX \to Y and every commuting diagram

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

there exists a lift

A X f B Y. \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

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).


Pushout-products with inclusions


The pushout product fgf \Box g of two monomorphisms f,gf,g in sSet is again a monomorphism, which is anodyne (a weak homotopy equivalence) if ff or gg 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.


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

Review includes

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.