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1. A Shadow Perspective on Equivariant Hochschild Homologies

   https://doi.org/10.1093/imrn/rnac250  

   Adamyk, Katharine; Gerhardt, Teena; Hess, Kathryn; Klang, Inbar; Kong, Hana Jia (September 2022, International Mathematics Research Notices)

   Abstract Shadows for bicategories, defined by Ponto, provide a useful framework that generalizes classical and topological Hochschild homology. In this paper, we define Hochschild-type invariants for monoids in a symmetric monoidal, simplicial model category $$\mathsf V$$, as well as for small $$\mathsf V$$-categories. We show that each of these constructions extends to a shadow on an appropriate bicategory, which implies in particular that they are Morita invariant. We also define a generalized theory of Hochschild homology twisted by an automorphism and show that it is Morita invariant. Hochschild homology of Green functors and $$C_n$$-twisted topological Hochschild homology fit into this framework, which allows us to conclude that these theories are Morita invariant. We also study linearization maps relating the topological and algebraic theories, proving that the linearization map for topological Hochschild homology arises as a lax shadow functor, and constructing a new linearization map relating topological restriction homology and algebraic restriction homology. Finally, we construct a twisted Dennis trace map from the fixed points of equivariant algebraic $$K$$-theory to twisted topological Hochschild homology. 

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2. 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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3. Coherence for bicategories, lax functors, and shadows

   Cary Malkiewich; Kate Ponto (January 2022, Theory and applications of categories)

   Coherence theorems are fundamental to how we think about monoidal categories and their generalizations. In this paper we revisit Mac Lane's original proof of coherence for monoidal categories using the Grothendieck construction. This perspective makes the approach of Mac Lane's proof very amenable to generalization. We use the technique to give efficient proofs of many standard coherence theorems and new coherence results for bicategories with shadow and for their functors. 

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4. Coherence for bicategories, lax functors, and shadows

   Malkiewich, Cary; Ponto, Kate (February 2022, Theory and applications of categories)

   Shulman, Michael (Ed.)

   Coherence theorems are fundamental to how we think about monoidal categories and their generalizations. In this paper we revisit Mac Lane's original proof of coherence for monoidal categories using the Grothendieck construction. This perspective makes the approach of Mac Lane's proof very amenable to generalization. We use the technique to give efficient proofs of many standard coherence theorems and new coherence results for bicategories with shadow and for their functors. 

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5. Algebraic Systems Motivated by DNA Origami

   Garrett, J.; Jonoska, N.; Saito, M (January 2019, Algebraic Informatics)

   Ćirić, M.; Droste, M.; Pin, JÉ. (Ed.)

   We initiate an algebraic approach to study DNA origami structures. We identify two types of basic building blocks and describe a DNA origami structure by their composition. These building blocks are taken as generators of a monoid, called the origami monoid, and motivated by the well studied Temperley-Lieb algebras, we identify a set of relations that characterize the origami monoid. We present several observations about Green’s relations for the origami monoid and study the relations to a direct product of Jones monoids, which is a morphic image of an origami monoid. 

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