A Model of Type Theory in Groupoid Assemblies

Abstract: We consider the category $\GrpA{A}$ of groupoids defined internally to the category of assemblies on a partial combinatory algebra $A$. In this thesis we exhibit the structure of a $\pi$-tribe on $\GrpA{A}$ showing the category to be a model of type theory. We also show that $\GrpA{A}$ has $W$-types with reductions and univalent object classifier for assemblies and modest assemblies, where the latter is an impredicative object classifier. Using the $W$-types with reductions, we show that $\GrpA{A}$ has a model structure. Finally, we construct $\pGrA{A}$, the full subcategory of partitioned groupoid assemblies, and show that $\pGrA{A}$ has finite bilimits and bicolimits as well as showing that the homotopy category of the full subcategory of the $0$-types of $\pGrA{A}$ is $\RT{A}$, the realizability topos of $A$.