nLab homotopy Kan fibration



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

Model category theory

model category, model \infty -category



Universal constructions


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 \infty-algebras

specific \infty-algebras

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



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.



A morphism f :X Y f_\bullet \,\colon\, X_\bullet \xrightarrow{\;} Y_\bullet is a homotopy Kan fibration if for all positive numbers n +n \in \mathbb{N}_+ and all 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)


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




Relation to homotopy colimits (geometric realization)

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


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