  We show that Martin Hyland's effective topos can be exhibited as the homotopy
category of a path category $\mathbb{EFF}$. Path categories are categories of
fibrant objects in the sense of Brown satisfying two additional properties and
as such provide a context in which one can interpret many notions from homotopy
theory and Homotopy Type Theory. Within the path category $\mathbb{EFF}$ one
can identify a class of discrete fibrations which is closed under push forward
along arbitrary fibrations (in other words, this class is polymorphic or closed
under impredicative quantification) and satisfies propositional resizing. This
class does not have a univalent representation, but if one restricts to those
discrete fibrations whose fibres are propositions in the sense of Homotopy Type
Theory, then it does. This means that, modulo the usual coherence problems, it
can be seen as a model of the Calculus of Constructions with a univalent type
of propositions. We will also build a more complicated path category in which
the class of discrete fibrations whose fibres are sets in the sense of Homotopy
Type Theory has a univalent representation, which means that this will be a
model of the Calculus of Constructions with a univalent type of sets.
