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

   Pavlovic, Dusko; Hughes, Dominic (January 2021, Theory and applications of categories)

   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. 

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2. Formalizing Colimits in 𝒞at

   https://doi.org/10.4230/lipics.itp.2025.20  

   Carneiro, Mario; Riehl, Emily (September 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Forster, Yannick; Keller, Chantal (Ed.)

   {"Abstract":["Certain results involving "higher structures" are not currently accessible to computer formalization because the prerequisite ∞-category theory has not been formalized. To support future work on formalizing ∞-category theory in Lean’s mathematics library, we formalize some fundamental constructions involving the 1-category of categories. Specifically, we construct the left adjoint to the nerve embedding of categories into simplicial sets, defining the homotopy category functor. We prove further that this adjunction is reflective, which allows us to conclude that 𝒞at has colimits. To our knowledge this is the first formalized proof that the nerve functor is a fully faithful right adjoint and that the category of categories is cocomplete."]} 

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3. ∞-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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4. SYMMETRIC MONOIDAL G-CATEGORIES AND THEIR STRICTIFICATION

   https://doi.org/10.1093/qmathj/haz034  

   Guillou, Bertrand J.; May, J. Peter; Merling, Mona; Osorno, Angélica M. (December 2019, The Quarterly Journal of Mathematics)

   Abstract We give an operadic definition of a genuine symmetric monoidal $$G$$-category, and we prove that its classifying space is a genuine $$E_\infty $$G$-space. We do this by developing some very general categorical coherence theory. We combine results of Corner and Gurski, Power and Lack to develop a strictification theory for pseudoalgebras over operads and monads. It specializes to strictify genuine symmetric monoidal $$G$$-categories to genuine permutative $$G$$-categories. All of our work takes place in a general internal categorical framework that has many quite different specializations. When $$G$$ is a finite group, the theory here combines with previous work to generalize equivariant infinite loop space theory from strict space level input to considerably more general category level input. It takes genuine symmetric monoidal $$G$$-categories as input to an equivariant infinite loop space machine that gives genuine $$\Omega $$-$$G$-spectra as output. 

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5. Rear cortex contraction aids in nuclear transit during confined migration by increasing pressure in the cell posterior

   https://doi.org/10.1242/jcs.260623  

   Keys, Jeremy; Cheung, Brian_C H; Elpers, Margaret A; Wu, Mingming; Lammerding, Jan (June 2024, Journal of Cell Science)

   ABSTRACT As cells migrate through biological tissues, they must frequently squeeze through micron-sized constrictions in the form of interstitial pores between extracellular matrix fibers and/or other cells. Although it is now well recognized that such confined migration is limited by the nucleus, which is the largest and stiffest organelle, it remains incompletely understood how cells apply sufficient force to move their nucleus through small constrictions. Here, we report a mechanism by which contraction of the cell rear cortex pushes the nucleus forward to mediate nuclear transit through constrictions. Laser ablation of the rear cortex reveals that pushing forces behind the nucleus are the result of increased intracellular pressure in the rear compartment of the cell. The pushing forces behind the nucleus depend on accumulation of actomyosin in the rear cortex and require Rho kinase (ROCK) activity. Collectively, our results suggest a mechanism by which cells generate elevated intracellular pressure in the posterior compartment to facilitate nuclear transit through three-dimensional (3D) constrictions. This mechanism might supplement or even substitute for other mechanisms supporting nuclear transit, ensuring robust cell migrations in confined 3D environments. 

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