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1. 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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2. The combinatorics of $$N_\infty$$ operads for $$C_{qp^n}$$ and $$D_{p^n}$$

   https://doi.org/10.1017/S0017089524000211  

   Balchin, Scott; MacBrough, Ethan; Ormsby, Kyle (January 2025, Glasgow Mathematical Journal)

   We provide a general recursive method for constructing transfer systems on finite lattices. Using this, we calculate the number of homotopically distinct $$N_{\infty} $$ operads for dihedral groups $$D_{p^n}$$, p>2 prime, and cyclic groups $$C_{qp^n}$$, $$p \neq q$$ prime. We then further display some of the beautiful combinatorics obtained by restricting to certain homotopically meaningful $$N_\infty$$ operads for these groups. 

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3. Model structures on finite total orders

   https://doi.org/10.1007/s00209-023-03287-6  

   Balchin, Scott; Ormsby, Kyle; Osorno, Angélica M.; Roitzheim, Constanze (July 2023, Mathematische Zeitschrift)

   Abstract We initiate the study of model structures on (categories induced by) lattice posets, a subject we dub homotopical combinatorics . In the case of a finite total order [ n ], we enumerate all model structures, exhibiting a rich combinatorial structure encoded by Shapiro’s Catalan triangle. This is an application of previous work of the authors on the theory of $$N_\infty $$ N ∞ -operads for cyclic groups of prime power order, along with new structural insights concerning extending choices of certain model structures on subcategories of [ n ]. 

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4. A simple proof of reflexivity and separability of N^{1,p} Sobolev spaces

   https://doi.org/10.54330/afm.127419  

   Alvarado, Ryan; Hajłasz, Piotr; Malý, Lukáš (October 2022, Annales Fennici Mathematici)

   We present an elementary proof of a well-known theorem of Cheeger which states that if a metric-measure space \(X\) supports a \(p\)-Poincaré inequality, then the \(N^{1,p}(X)\) Sobolev space is reflexive and separable whenever \(p\in (1,\infty)\). We also prove separability of the space when \(p=1\). Our proof is based on a straightforward construction of an equivalent norm on \(N^{1,p}(X)\), \(p\in [1,\infty)\), that is uniformly convex when \(p\in (1,\infty)\). Finally, we explicitly construct a functional that is pointwise comparable to the minimal \(p\)-weak upper gradient, when \(p\in (1,\infty)\). 

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5. Program adverbs and Tlön embeddings

   https://doi.org/10.1145/3547632  

   Li, Yao; Weirich, Stephanie (August 2022, Proceedings of the ACM on Programming Languages)

   Free monads (and their variants) have become a popular general-purpose tool for representing the semantics of effectful programs in proof assistants. These data structures support the compositional definition of semantics parameterized by uninterpreted events, while admitting a rich equational theory of equivalence. But monads are not the only way to structure effectful computation, why should we limit ourselves? In this paper, inspired by applicative functors, selective functors, and other structures, we define a collection of data structures and theories, which we call program adverbs, that capture a variety of computational patterns. Program adverbs are themselves composable, allowing them to be used to specify the semantics of languages with multiple computation patterns. We use program adverbs as the basis for a new class of semantic embeddings called Tlön embeddings. Compared with embeddings based on free monads, Tlön embeddings allow more flexibility in computational modeling of effects, while retaining more information about the program's syntactic structure. 

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