Fibration of simplicial sets

In mathematics, especially in homotopy theory, a left fibration of simplicial sets is a map that has the right lifting property with respect to the horn inclusions

Λ

i

n

⊂

Δ

n

,

0

≤

i

<

n

{\displaystyle \Lambda _{i}^{n}\subset \Delta ^{n},0\leq i<n}

. A right fibration is defined similarly with the condition

0

<

i

≤

n

{\displaystyle 0<i\leq n}

. A Kan fibration is one with the right lifting property with respect to every horn inclusion; hence, a Kan fibration is exactly a map that is both a left and right fibration.

Examples

A right fibration is a cartesian fibration such that each fiber is a Kan complex.

In particular, a category fibered in groupoids over another category is a special case of a right fibration of simplicial sets in the ∞-category setup.

Anodyne extensions

A left anodyne extension is a map in the saturation of the set of the horn inclusions

Λ

k

n

→

Δ

n

{\displaystyle \Lambda _{k}^{n}\to \Delta ^{n}}

for

n

≥

1

,

0

≤

k

<

n

{\displaystyle n\geq 1,0\leq k<n}

in the category of simplicial sets, where the saturation of a class is the smallest class that contains the class and is stable under pushouts, retracts and transfinite compositions (compositions of infinitely many maps). A right anodyne extension is defined by replacing the condition

0

≤

k

<

n

{\displaystyle 0\leq k<n}

with

0

<

k

≤

n

{\displaystyle 0<k\leq n}

. The notions are originally due to Gabriel–Zisman and are used to study fibrations for simplicial sets.

A left (or right) anodyne extension is a monomorphism (since the class of monomorphisms is saturated, the saturation lies in the class of monomorphisms).

Given a class

F

{\displaystyle F}

of maps, let

r

(

F

)

{\displaystyle r(F)}

denote the class of maps satisfying the right lifting property with respect to

F

{\displaystyle F}

.