Localization Equivalence of Realizability Groupoid Models

Determine whether realizability models of homotopy type theory constructed using internal groupoids in a realizability topos or in the category of assemblies (as in Hughes 2025 and Agwu 2025) admit a localization whose resulting model is equivalent to the model based on coherent groupoids CohGpd(P), i.e., the full subcategory of coherent presheaves of groupoids in the presheaf category [P^op, Gpd] built over the category of partitioned assemblies P.

Background

The paper proposes a candidate effective (2,1)-topos by working with coherent groupoids in the presheaf category over partitioned assemblies, and shows that the fibrant 0-types in this setting recover the effective topos Eff. In contrast, some existing realizability models of homotopy type theory based on internal groupoids in realizability toposes or assemblies are known to add further 0-types beyond Eff.

To reconcile these approaches, the authors raise the question of whether those alternative groupoid-based realizability models can be localized—intuitively, adjusted by inverting certain morphisms or imposing a descent condition—so that the resulting semantics become equivalent to the coherent groupoid model CohGpd(P) introduced in the paper.