Realizability of Univalence: Modest Kan complexes
  A modest Kan complex is a modest simplicial set which has a right lifting property with respect to horn inclusions $\Lambda_k[n] \to \Delta[n]$. This paper develops the categorical logical that is required to show that there is a univalent universe of modest Kan complexes among simplicial assemblies. 
Wouter Pieter Stekelenburg
math.LO math.CT