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1. ∞-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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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. 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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4. Effects and Coeffects in Call-by-Push-Value

   https://doi.org/10.1145/3689750  

   Torczon, Cassia; Suárez_Acevedo, Emmanuel; Agrawal, Shubh; Velez-Ginorio, Joey; Weirich, Stephanie (October 2024, Proceedings of the ACM on Programming Languages)

   Effect and coeffect tracking integrate many types of compile-time analysis, such as cost, liveness, or dataflow, directly into a language's type system. In this paper, we investigate the addition of effect and coeffect tracking to the type system of call-by-push-value (CBPV), a computational model useful in compilation for its isolation of effects and for its ability to cleanly express both call-by-name and call-by-value computations. Our main result is effect-and-coeffect soundness, which asserts that the type system accurately bounds the effects that the program may trigger during execution and accurately tracks the demands that the program may make on its environment. This result holds for two different dynamic semantics: a generic one that can be adapted for different coeffects and one that is adapted for reasoning about resource usage. In particular, the second semantics discards the evaluation of unused values and pure computations while ensuring that effectful computations are always evaluated, even if their results are not required. Our results have been mechanized using the Coq proof assistant. 

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5. Parsing randomness

   https://doi.org/10.1145/3563291  

   Goldstein, Harrison; Pierce, Benjamin_C (October 2022, Proceedings of the ACM on Programming Languages)

   Random data generators can be thought of as parsers of streams of randomness. This perspective on generators for random data structures is established folklore in the programming languages community, but it has never been formalized, nor have its consequences been deeply explored. We build on the idea of freer monads to develop free generators, which unify parsing and generation using a common structure that makes the relationship between the two concepts precise. Free generators lead naturally to a proof that a monadic generator can be factored into a parser plus a distribution over choice sequences. Free generators also support a notion of derivative, analogous to the familiar Brzozowski derivatives of formal languages, allowing analysis tools to preview the effect of a particular generator choice. This gives rise to a novel algorithm for generating data structures satisfying user-specified preconditions. 

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