  We describe a homotopical version of the relational and gluing models of type
theory, and generalize it to inverse diagrams and oplax limits. Our method uses
the Reedy homotopy theory on inverse diagrams, and relies on the fact that
Reedy fibrant diagrams correspond to contexts of a certain shape in type
theory. This has two main applications. First, by considering inverse diagrams
in Voevodsky's univalent model in simplicial sets, we obtain new models of
univalence in a number of (infinity,1)-toposes; this answers a question raised
at the Oberwolfach workshop on homotopical type theory. Second, by gluing the
syntactic category of univalent type theory along its global sections functor
to groupoids, we obtain a partial answer to Voevodsky's homotopy-canonicity
conjecture: in 1-truncated type theory with one univalent universe of sets, any
closed term of natural number type is homotopic to a numeral.
