Awesome

Cartesian Cubical Type Theory

by Carlo Angiuli, Guillaume Brunerie, Thierry Coquand, Kuen-Bang Hou (Favonia), Robert Harper, Daniel R. Licata

  cart-cube.pdf    -- current version of paper
  cart-cube-v2.pdf -- second version paper
                      (lacking some presentational improvements)
  cart-cube-v1.pdf -- orignal version of paper
                      (with a different composition for Glue types)

The formalization that corresponds exactly to the paper is browsable and downloadable.

The code in this repository includes that, as well as some other, related material:

  agda/ -- full Agda formalization
           (compile ABCFHL.agda to check everything)

The formalizations were checked by Agda 2.6.1.2, which includes the 'flat' modality that was previously in a special agda-flat branch.

A Model of Type Theory with Directed Univalence in Bicubical Sets

by Matthew Z. Weaver and Daniel R. Licata

  agda/directed/ -- Agda formalization
                    (compile All.agda to check everything)