# nLab anodyne morphism

Contents

### Context

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

(∞,1)-category theory

# Contents

## 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 (Gabriel-Zisman 67, chapter IV.2) are equivalent to acyclic cofibrations in the classical model structure on simplicial sets.

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

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

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.

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