  We provide a partial solution to the problem of defining a constructive
version of Voevodsky's simplicial model of univalent foundations. For this, we
prove constructive counterparts of the necessary results of simplicial homotopy
theory, building on the constructive version of the Kan-Quillen model structure
established by the second-named author. In particular, we show that dependent
products along fibrations with cofibrant domains preserve fibrations, establish
the weak equivalence extension property for weak equivalences between
fibrations with cofibrant domain and define a univalent classifying fibration
for small fibrations between bifibrant objects. These results allow us to
define a comprehension category supporting identity types, $\Sigma$-types,
$\Pi$-types and a univalent universe, leaving only a coherence question to be
addressed.
