Since homotopy colimits are preserved by inverse images of geometric morphisms, this type theory is likely to include at least non-recursive higher inductive types, and furthermore perhaps free models for every (finite) essentially algebraic (infinity,1)-theory. (Arbitrary recursive higher inductive types only make sense with reference to Pi-types, which are not geometric.)

Traditionally the types in geometric homotopy type theory, hence the geometric homotopy types, are known as ∞-stacks and maybe better as (∞,1)-sheaves, notably as moduli ∞-stacks.