The nucleus of an adjunction and the Street monad on monads

An adjunction is a pair of functors related by a pair of natural transformations, and relating a pair of categories. It displays how a structure, or a concept, projects from each category to the other, and back. Adjunctions are the common denominator of Galois connections, representation theories, spectra, and generalized quantifiers. We call an adjunction nuclear when its categories determine each other. We show that every adjunction can be resolved into a nuclear adjunction. This resolution is idempotent in a strong sense. The nucleus of an adjunction displays its conceptual core, just as the singular value decomposition of an adjoint pair of linear operators displays their canonical bases. The two composites of an adjoint pair of functors induce a monad and a comonad. Monads and comonads generalize the closure and the interior operators from topology, or modalities from logic, while providing a saturated view of algebraic structures and compositions on one side, and of coalgebraic dynamics and decompositions on the other. They are resolved back into adjunctions over the induced categories of algebras and of coalgebras. The nucleus of an adjunction is an adjunction between the induced categories of algebras and coalgebras. It provides new presentations for both, revealing the meaning of constructing algebras for a comonad and coalgebras for a monad. In his seminal early work, Ross Street described an adjunction between monads and comonads in 2-categories. Lifting the nucleus construction, we show that the resulting Street monad on monads is strongly idempotent, and extracts the nucleus of a monad. A dual treatment achieves the same for comonads. Applying a notable fragment of pure 2-category theory on an acute practical problem of data analysis thus led to new theoretical result.