equivalences in/of $(\infty,1)$-categories
model category, model $\infty$-category
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
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.
A morphism $f_\bullet \,\colon\, X_\bullet \xrightarrow{\;} Y_\bullet$ is a homotopy Kan fibration if for all $n \in \mathbb{N}$ and $0 \leq k \leq n$ the induced map
(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)
For $G \,\in\, Grp(Grpd_\infty)$ an $\infty$-group, consider a homomorphism of $G$-$\infty$-actions
(…)
See at geometric realization of simplicial topological spaces the section Preservation of homotopy limits.
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)