  We define and develop the infrastructure of homotopical inverse diagrams in
categories with attributes.
  Specifically, given a category with attributes $C$ and an ordered homotopical
inverse category $I$, we construct the category with attributes $C^I$ of
homotopical diagrams of shape $I$ in $C$ and Reedy types over these; and we
show how various logical structure ($\Pi$-types, identity types, and so on)
lifts from $C$ to $C^I$. This may be seen as providing a general class of
diagram models of type theory.
  In a companion paper "The homotopy theory of type theories"
(arXiv:1610.00037), we apply the present results to construct semi-model
structures on categories of contextual categories.
