adjunctionina

Adjunctionina is a proposed generalization of the classical notion of adjunction between functors in category theory. It is intended for use in contexts where the ambient setting is a bicategory, a 2-category, or an enriched category, and where the standard triangle identities of an adjunction hold only up to specified coherence data.

In a bicategory, an adjunctionina between F: C → D and G: D → C consists of a unit η: Id_C ⇒ G F and a counit ε: F G ⇒ Id_D, together with a family of coherence 2-cells that mediate the triangle identities. These coherence cells provide a generalized form of associativity and unitality. When all coherence 2-cells are identities, adjunctionina reduces to a traditional adjunction.

Variants of adjunctionina arise depending on the chosen coherence framework. In a lax adjunctionina, the triangle identities hold with non-invertible 2-cells; in a pseudo adjunctionina, the coherence cells are invertible. This mirrors the standard lax and pseudo notions in higher category theory and allows adjunction-like behavior in settings where strict equalities are unavailable.

Properties and consequences include the way adjunctioninas induce adjoint-like correspondences on hom-categories, how they give rise to associated monads and comonads under suitable conditions, and their compatibility with enrichment and monoidal structures. They provide a flexible framework for universal properties that are weaker than strict adjunctions yet still support constructions such as factorization, limits, and colimits in higher-categorical contexts.

Examples often cited include classical adjunctions in Set or Cat viewed as strict adjunctioninas, and biadjunctions in bicategories of profunctors, where coherence data encodes higher-dimensional relations.