We define strict and weak duality involutions on 2-categories, and prove a coherence theorem that every bicategory with a weak duality involution is biequivalent to a 2-category with a strict duality involution. For this purpose we introduce "2-categories with contravariance", a sort of enhanced 2-category with a basic notion of "contravariant morphism", which can be regarded either as generalized multicategories or as enriched categories. This enables a universal characterization of duality involutions using absolute weighted colimits, leading to a conceptual proof of the coherence theorem.
Keywords: opposite category, contravariant functor, generalized multicategory, enriched category, coherence theorem
2010 MSC: 18D20, 18D05
Theory and Applications of Categories, Vol. 33, 2018, No. 5, pp 95-130.