  Postulating an impredicative universe in dependent type theory allows System
F style encodings of finitary inductive types, but these fail to satisfy the
relevant {\eta}-equalities and consequently do not admit dependent eliminators.
To recover {\eta} and dependent elimination, we present a method to construct
refinements of these impredicative encodings, using ideas from homotopy type
theory. We then extend our method to construct impredicative encodings of some
higher inductive types, such as 1-truncation and the unit circle S1.
