# nLab homotopy Kan fibration

Contents

### Context

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

(∞,1)-category theory

Background

Basic concepts

Universal constructions

Local presentation

Theorems

Extra stuff, structure, properties

Models

#### Model category theory

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for $\infty$-groupoids

for ∞-groupoids

for equivariant $\infty$-groupoids

for rational $\infty$-groupoids

for rational equivariant $\infty$-groupoids

for $n$-groupoids

for $\infty$-groups

for $\infty$-algebras

general $\infty$-algebras

specific $\infty$-algebras

for stable/spectrum objects

for $(\infty,1)$-categories

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

for $(\infty,1)$-operads

for $(n,r)$-categories

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

# Contents

## Idea

The notion of homotopy Kan fibration is the evident homotopy-theoretic generalization ($(\infty,1)$-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 $f_\bullet \,\colon\, X_\bullet \xrightarrow{\;} Y_\bullet$ is a homotopy Kan fibration if for all positive numbers $n \in \mathbb{N}_+$ and all $0 \leq k \leq n$ the induced map

(1)$X(\Delta^n) \xrightarrow{\;\;} X(\Lambda^n_k) \underset { Y(\Lambda^n_k) } {\times^h} Y(\Delta^n)$

(into the homotopy fiber product of the space of space of $(n,k)$-horns in $X$ with that of $n$-simplices in $Y$)

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 $(n,k)$-horn-inclusion in simplicial sets regarded as degree-wise discrete topological spaces):

## Examples

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

(…)

## References

Last revised on July 7, 2022 at 12:42:03. See the history of this page for a list of all contributions to it.