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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. The resolution is idempotent in a strict sense. The resulting nucleus displays the concept that was implicit in the original adjunction, just as the singular value decomposition of an adjoint pair of linear operators displays their canonical bases. [snip] 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 strictly idempotent, and extracts the nucleus of a monad. A dual treatment achieves the same for comonads. This uncovers remarkably concrete applications behind a notable fragment of pure 2-category theory. The other way around, driven by the tasks and methods of machine learning and data analysis, the nucleus construction also seems to uncover remarkably pure and general mathematical content lurking behind the daily practices of network computation and data analysis.
The Vietoris monad on the category of compact Hausdorff spaces is a topological analogue of the power-set monad on the category of sets. Exploiting Manes characterisation of the compact Hausdorff spaces as algebras for the ultrafilter monad on sets,
We provide new categorical perspectives on Jacobs notion of hypernormalisation of sub-probability distributions. In particular, we show that a suitable general framework for notions of hypernormalisation is that of a symmetric monoidal category endow
We introduce a general construction on 2-monads. We develop background on maps of 2-monads, their left semi-algebras, and colimits in 2-category. Then, we introduce the construction of a colimit induced by a map of 2-monads, show that we obtain the s
Let $U$ be a strong monoidal functor between monoidal categories. If it has both a left adjoint $L$ and a right adjoint $R$, we show that the pair $(R,L)$ is a linearly distributive functor and $(U,U)dashv (R,L)$ is a linearly distributive adjunction
We characterize the category of monads on $Set$ and the category of Lawvere theories that are equivalent to the category of regular equational theories.