Homotopical inverse diagrams in categories with attributes.

We define and develop the infrastructure of homotopical inverse diagrams in categories with attributes (CwA's). 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 the original CwA to the diagram CwA. This may be seen as providing a general class of diagram models of type theory, and forms a companion paper to arXiv:1610.00037, "The homotopy theory of type theories" , which applies the present results in constructing semi-model structures on categories of contextual categories.