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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. 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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3. Flipper Games for Monadically Stable Graph Classes

   https://doi.org/10.4230/LIPIcs.ICALP.2023.128  

   Gajarský, Jakub; Mählmann, Nikolas; McCarty, Rose; Ohlmann, Pierre; Pilipczuk, Michał; Przybyszewski, Wojciech; Siebertz, Sebastian; Sokołowski, Marek; Toruńczyk, Szymon (January 2023, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Etessami, Kousha; Feige, Uriel; Puppis, Gabriele (Ed.)

   A class of graphs C is monadically stable if for every unary expansion Ĉ of C, one cannot encode - using first-order transductions - arbitrarily long linear orders in graphs from C. It is known that nowhere dense graph classes are monadically stable; these include classes of bounded maximum degree and classes that exclude a fixed topological minor. On the other hand, monadic stability is a property expressed in purely model-theoretic terms that is also suited for capturing structure in dense graphs. In this work we provide a characterization of monadic stability in terms of the Flipper game: a game on a graph played by Flipper, who in each round can complement the edge relation between any pair of vertex subsets, and Localizer, who in each round is forced to restrict the game to a ball of bounded radius. This is an analog of the Splitter game, which characterizes nowhere dense classes of graphs (Grohe, Kreutzer, and Siebertz, J. ACM '17). We give two different proofs of our main result. The first proof is based on tools borrowed from model theory, and it exposes an additional property of monadically stable graph classes that is close in spirit to definability of types. Also, as a byproduct, we show that monadic stability for graph classes coincides with monadic stability of existential formulas with two free variables, and we provide another combinatorial characterization of monadic stability via forbidden patterns. The second proof relies on the recently introduced notion of flip-flatness (Dreier, Mählmann, Siebertz, and Toruńczyk, arXiv 2206.13765) and provides an efficient algorithm to compute Flipper’s moves in a winning strategy. 

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4. Closed-Form Minkowski Sum Approximations for Efficient Optimization-Based Collision Avoidance

   Guthrie, James; Kobilarov, Marin; Mallada, Enrique (January 2022, Proceedings of the American Control Conference)

   Motion planning methods for autonomous systems based on nonlinear programming offer great flexibility in incorporating various dynamics, objectives, and constraints. One limitation of such tools is the difficulty of efficiently representing obstacle avoidance conditions for non-trivial shapes. For example, it is possible to define collision avoidance constraints suitable for nonlinear programming solvers in the canonical setting of a circular robot navigating around M convex polytopes over N time steps. However, it requires introducing (2+L)MN additional constraints and LMN additional variables, with L being the number of halfplanes per polytope, leading to larger nonlinear programs with slower and less reliable solving time. In this paper, we overcome this issue by building closed-form representations of the collision avoidance conditions by outerapproximating the Minkowski sum conditions for collision. Our solution requires only MN constraints (and no additional variables), leading to a smaller nonlinear program. On motion planning problems for an autonomous car and quadcopter in cluttered environments, we achieve speedups of 4.0x and 10x respectively with significantly less variance in solve times and negligible impact on performance arising from the use of outer approximations. 

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5. Closed-Form Minkowski Sum Approximations for Efficient Optimization-Based Collision Avoidance

   https://doi.org/10.23919/ACC53348.2022.9867524  

   Guthrie, James; Kobilarov, Marin; Mallada, Enrique (January 2022, Proceedings of the American Control Conference)

   Motion planning methods for autonomous systems based on nonlinear programming offer great flexibility in incorporating various dynamics, objectives, and constraints. One limitation of such tools is the difficulty of efficiently representing obstacle avoidance conditions for non-trivial shapes. For example, it is possible to define collision avoidance constraints suitable for nonlinear programming solvers in the canonical setting of a circular robot navigating around $$M$$ convex polytopes over $$N$$ time steps. However, it requires introducing $(2+L)MN$ additional constraints and $LMN$ additional variables, with $$L$$ being the number of halfplanes per polytope, leading to larger nonlinear programs with slower and less reliable solving time. In this paper, we overcome this issue by building closed-form representations of the collision avoidance conditions by outer-approximating the Minkowski sum conditions for collision. Our solution requires only $MN$ constraints (and no additional variables), leading to a smaller nonlinear program. On motion planning problems for an autonomous car and quadcopter in cluttered environments, we achieve speedups of 4.0x and 10x respectively with significantly less variance in solve times and negligible impact on performance arising from the use of outer approximations. 

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