A Model of Type Theory in Groupoid Assemblies

Abstract

We consider the category Grpd(Asm$(A)$) 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 Grpd(Asm$(A)$) showing the category to be a model of type theory. We also show that Grpd(Asm$(A)$) 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 Grpd(Asm$(A)$) has a model structure. Finally, we construct pGrpd(Asm$(A)$), the full subcategory of partitioned groupoid assemblies, and show that pGrpd(Asm$(A)$) has finite bilimits and bicolimits as well as showing that the homotopy category of the full subcategory of the $0$-types of pGrpd(Asm$(A)$) is RT$[A]$, the realizability topos of $A$.