Extranatural transformations, adjunctions of two variables and conjugation
Simon Willerton
TL;DR
This paper provides a formal, diagrammatic framework for conjugation of adjunctions of two variables, extending the classical mates/conjugation between left and right adjoints to the setting of two-variable adjunctions. It recasts closed monoidal structure through extranatural transformations and introduces left and right adjunctions of two variables, together with explicit conjugation maps that interrelate natural transformations across the two-variable adjunctions. By embedding these concepts in the monoidal double category of functors and profunctors, the work connects to Grothendieck’s six operations and clarifies constructions such as the projection formula and internal adjunctions, including a suite of concrete examples. The results yield a unified, diagrammatic method for tracking conjugate relationships in complex adjunction strings, with potential applications to Hopf monads, surface-diagram bookkeeping, and formal category theory beyond standard two-variable adjunctions. Overall, the approach provides a rigorous, visually intuitive account of how conjugation governs the interplay between tensor-like operations and internalHom-like structures in a broad, non-symmetric setting, illuminating foundational aspects of formal category theory and its link to established operational formalisms.
Abstract
Adjunctions of two variables generalize the relationship between tensor product and the internal hom functor in a closed monoidal category. For a pair of ordinary adjunctions $(F\dashv U, F'\dashv U')$ conjugation relates natural transformations of the form $F\Rightarrow F'$ with natural transformations of the form $U' \Rightarrow U$. We look at conjugation for general two variable adjunctions. It is useful in the context of Grothendieck's six operations as we will show that this is an appropriate way to view the constructions of Fausk, Hu and May where they discuss things like the projection formula and internal adjunctions. Extensive use is made of surface diagram notation as this is a helpful way to keep track of the three dimensions of composition. This also places the work in the context of formal category theory as, for instance, closed monoidal categories are defined without reference to objects or morphisms inside them. In an appendix it is explained how an appropriate setting for this perspective is the monoidal double category of functors and profunctors, using the fact that the bicategory of profunctors has duals.