Validity of the plus principle in general Grothendieck (∞,1)-topoi

Determine whether the plus principle—asserting that every acyclic and simply connected object is contractible—holds in all Grothendieck (∞,1)-topoi. Establish this general validity without assuming additional principles such as Whitehead’s principle (hypercompleteness), which implies the plus principle but may fail in some settings.

Background

The paper introduces the plus principle (PP), defined as the assertion that every acyclic and simply connected type is contractible. This principle plays a key role in several results, including characterizations of extensions along acyclic maps and links between acyclic and balanced maps.

The authors note that PP follows from Whitehead’s principle (hypercompleteness), which asserts that infinitely connected types are contractible. However, Whitehead’s principle does not hold in all higher topos-theoretic contexts (e.g., parametrized spectra), while PP can still hold there, suggesting PP may be strictly weaker than hypercompleteness.

Hoyois highlighted PP in the context of Grothendieck (∞,1)-topoi and raised the question of its general validity. The authors record that it remains open whether PP holds universally in that setting, making this a central unresolved question relevant to establishing orthogonal factorization systems for acyclic maps in univalent mathematics.