Classification Aggregation with Multiple Memberships

People face the problem of classifying objects into categories while aggregating their opinions. We axiomatically study such a classification aggregation problem with multiple memberships, meaning that each individual assigns one category to every object, whereas the resulting classification may assign more than one category. Under the natural settings and axioms (i.e., Independence and Supportiveness), each of our new axioms -- Responsibility, Split Stability and Rejection Stability -- characterises what we call the junta rules. The rules assign a category to an object exactly when some member of a fixed coalition does so. The coalitions that can secure an assignment form a grill, a topological structure implied by an ultrafilter. Hence, the study parallels the literature on Arrow's theorem characterised by the decisive coalition structures. An infinite population setting is also investigated to clarify the differences among the axioms.