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1. The nucleus of an adjunction and the Street monad on monads

   Pavlovic, Dusko (April 2020, Journal of Computer Research Repository)

   null (Ed.)

   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. 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. 

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2. ∞-Operads as Analytic Monads

   https://doi.org/10.1093/imrn/rnaa332  

   Gepner, David; Haugseng, Rune; Kock, Joachim (April 2021, International Mathematics Research Notices)

   null (Ed.)

   Abstract We develop an $$\infty $$-categorical version of the classical theory of polynomial and analytic functors, initial algebras, and free monads. Using this machinery, we provide a new model for $$\infty $$-operads, namely $$\infty $$-operads as analytic monads. We justify this definition by proving that the $$\infty $$-category of analytic monads is equivalent to that of dendroidal Segal spaces, known to be equivalent to the other existing models for $$\infty $$-operads. 

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3. Unitary Anchored Planar Algebras

   https://doi.org/10.1007/s00220-024-04985-w  

   Henriques, André; Penneys, David; Tener, James (May 2024, Communications in Mathematical Physics)

   Abstract In our previous article (http://arxiv.org/abs/1607.06041), we established an equivalence between pointed pivotal module tensor categories and anchored planar algebras. This article introduces the notion of unitarity for both module tensor categories and anchored planar algebras, and establishes the unitary analog of the above equivalence. Our constructions use Baez’s 2-Hilbert spaces (i.e., semisimple$$\textrm{C}^*$$ ${\text{C}}^{\ast }$-categories equipped with unitary traces), the unitary Yoneda embedding, and the notion of unitary adjunction for dagger functors between 2-Hilbert spaces. 

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4. An analogue of adjoint ideals and PLT singularities in mixed characteristic

   https://doi.org/10.1090/jag/797  

   Ma, Linquan; Schwede, Karl; Tucker, Kevin; Waldron, Joe; Witaszek, Jakub (July 2022, Journal of Algebraic Geometry)

   We use the framework of perfectoid big Cohen-Macaulay (BCM) algebras to define a class of singularities for pairs in mixed characteristic, which we call purely BCM-regular singularities, and a corresponding adjoint ideal. We prove that these satisfy adjunction and inversion of adjunction with respect to the notion of BCM-regularity and the BCM test ideal defined by the first two authors. We compare them with the existing equal characteristic purely log terminal (PLT) and purely F F -regular singularities and adjoint ideals. As an application, we obtain a uniform version of the Briançon-Skoda theorem in mixed characteristic. We also use our theory to prove that two-dimensional Kawamata log terminal singularities are BCM-regular if the residue characteristic p > 5 p>5 , which implies an inversion of adjunction for three-dimensional PLT pairs of residue characteristic p > 5 p>5 . In particular, divisorial centers of PLT pairs in dimension three are normal when p > 5 p > 5 . Furthermore, in Appendix A we provide a streamlined construction of perfectoid big Cohen-Macaulay algebras and show new functoriality properties for them using the perfectoidization functor of Bhatt and Scholze. 

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5. CATEGORICAL COMPLEXITY

   https://doi.org/10.1017/fms.2020.26  

   BASU, SAUGATA; ISIK, UMUT (January 2020, Forum of Mathematics, Sigma)

   We introduce a notion of complexity of diagrams (and, in particular, of objects and morphisms) in an arbitrary category, as well as a notion of complexity of functors between categories equipped with complexity functions. We discuss several examples of this new definition in categories of wide common interest such as finite sets, Boolean functions, topological spaces, vector spaces, semilinear and semialgebraic sets, graded algebras, affine and projective varieties and schemes, and modules over polynomial rings. We show that on one hand categorical complexity recovers in several settings classical notions of nonuniform computational complexity (such as circuit complexity), while on the other hand it has features that make it mathematically more natural. We also postulate that studying functor complexity is the categorical analog of classical questions in complexity theory about separating different complexity classes. 

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