nLab
homotopy Kan fibration

Context

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

Model category theory

model category, model \infty -category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

for (,1)(\infty,1)-sheaves / \infty-stacks

Contents

Idea

The notion of homotopy Kan fibration is the evident homotopy-theoretic generalization ( ( , 1 ) (\infty,1) -categorification) of the notion of Kan fibration: where a Kan fibration is a morphism of simplicial sets (hence: simplicial objects internal to Sets) which satisfies the right lifting property against horn inclusions, so a homotopy Kan fibration is a morphism of simplicial \infty -groupoids (“simplicial spaces”, bisimplicial sets) which satisfies a homotopy-lifting property along all horn inclusions.

Just like Kan fibrations serve as the fibrations in the classical model structure on simplicial sets, so homotopy Kan fibrations are the fibrations in the archetypical model \infty -category-structure on simplicial \infty -groupoids.

Definition

Definition

A morphism f :X Y f_\bullet \,\colon\, X_\bullet \xrightarrow{\;} Y_\bullet is a homotopy Kan fibration if for all nn \in \mathbb{N} and 0kn0 \leq k \leq n the induced map

(1)X(Δ n)X(Λ k n)× hY(Λ k n)Y(Δ n) X(\Delta^n) \xrightarrow{\;\;} X(\Lambda^n_k) \underset { Y(\Lambda^n_k) } {\times^h} Y(\Delta^n)

(into the homotopy fiber product of the space of space of ( n , k ) (n,k) -horns in XX with that of n n -simplices in YY)

is surjective on connected components,

hence in that for all solid homotopy-commutative squares as follows, there exists a dashed lift up to homotopy (with the left morphism being the ( n , k ) (n,k) -horn-inclusion in simplicial sets regarded as degree-wise discrete topological spaces):

(Lurie 2011, Def. 3; Mazel-Gee 2014, p. 2)

Examples

For GGrp(Grpd )G \,\in\, Grp(Grpd_\infty) an \infty -group, consider a homomorphism of GG- \infty -actions

(…)

Properties

Relation to homotopy colimit (geometric realization)

See at geometric realization of simplicial topological spaces the section Preservation of homotopy limits.

References