nLab homotopy Kan fibration

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 $Y$)

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

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

Examples

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

(…)

Properties

General

• Homotopy Kan fibrations participate in a pullback-power axiom with (acyclic) homotopy cofibrations as expected in a monoidal model category (Mazel-Gee 2015, Def. 4.3 with Exp. 5.3)

Relation to homotopy colimits (geometric realization)

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

Related concepts

• Kan fibration

• model $\infty$-category

References

• Jacob Lurie, Simplicial spaces, Def. 3 in: Algebraic L-theory and Surgery, 2011 (pdf)

• Aaron Mazel-Gee, Model $\infty$-categories I: some pleasant properties of the $\infty$-category of simplicial spaces [arXiv:1412.8411]

• Aaron Mazel-Gee, Model ∞-categories II: Quillen adjunctions, New York Journal of Mathematics 27 (2021) 508-550. [arXiv:1510.04392, nyjm:27-21]