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1. Unitary braided-enriched monoidal categories

   https://doi.org/10.4171/QT/226  

   Dell, Zachary; Huston, Peter; Penneys, David (October 2024, Quantum Topology)

   Braided-enriched monoidal categories were introduced in the work of Morrison–Penneys, where they were characterized using braided central functors. The recent work of Kong–Yuan–Zhang–Zheng and Dell extended this characterization to an equivalence of 2-categories. Since their introduction, braided-enriched fusion categories have been used to describe certain phenomena in topologically ordered systems in theoretical condensed matter physics. While these systems are unitary, there was previously no general notion of unitarity for enriched categories in the literature. We supply the notion of unitarity for enriched categories and braided-enriched monoidal categories and extend the above 2-equivalence to the unitary setting. 

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2. Monoidal categories, representation gap and cryptography

   https://doi.org/10.1090/btran/151  

   Khovanov, Mikhail; Sitaraman, Maithreya; Tubbenhauer, Daniel (January 2024, Transactions of the American Mathematical Society, Series B)

   The linear decomposition attack provides a serious obstacle to direct applications of noncommutative groups and monoids (or semigroups) in cryptography. To overcome this issue we propose to look at monoids with only big representations, in the sense made precise in the paper, and undertake a systematic study of such monoids. One of our main tools is Green’s theory of cells (Green’s relations). A large supply of monoids is delivered by monoidal categories. We consider simple examples of monoidal categories of diagrammatic origin, including the Temperley–Lieb, the Brauer and partition categories, and discuss lower bounds for their representations. 

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3. Coalgebras in symmetric monoidal categories of spectra

   https://doi.org/10.4310/HHA.2019.v21.n1.a1  

   Péroux, Maximilien; Shipley, Brooke (January 2019, Homology, Homotopy and Applications)
4. Riemann–Roch theorems in monoidal 2-categories

   https://doi.org/10.1093/qmath/haad003  

   Campbell, Jonathan A; Ponto,  Kate (April 2023, The Quarterly Journal of Mathematics)

   Smooth and proper dg-algebras have an Euler class valued in the Hochschild homology of the algebra. This Euler class is worthy of this name since it satisfies many familiar properties including compatibility with the pairing on the Hochschild homology of the algebra and that of its opposite. This compatibility is the Riemann–Roch theorems of [21, 14]. In this paper, we prove a broad generalization of these Riemann–Roch theorems. We generalize from the bicategory of dg-algebras and their bimodules to symmetric monoidal bicategories and from the Euler class to traces of non-identity maps. Our generalization also implies spectral Riemann–Roch theorems. We regard this result as an instantiation of a 2-dimensional generalized cobordism hypothesis. This perspective draws the result close to many others that generalize results about Euler characteristics and classes to bicategorical traces. 

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5. 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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