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1. On dualizability of braided tensor categories

   https://doi.org/10.1112/S0010437X20007630  

   Brochier, Adrien; Jordan, David; Snyder, Noah (March 2021, Compositio Mathematica)

   null (Ed.)

   We study the question of dualizability in higher Morita categories of locally presentable tensor categories and braided tensor categories. Our main results are that the 3-category of rigid tensor categories with enough compact projectives is 2-dualizable, that the 4-category of rigid braided tensor categories with enough compact projectives is 3-dualizable, and that (in characteristic zero) the 4-category of braided multi-fusion categories is 4-dualizable. Via the cobordism hypothesis, this produces respectively two-, three- and four-dimensional framed local topological field theories. In particular, we produce a framed three-dimensional local topological field theory attached to the category of representations of a quantum group at any value of $$q$$ . 

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2. Support varieties without the tensor product property

   Bergh, Petter A; Plavnik, Julia Y; Witherspoon, Sarah J (April 2024, Bulletin of the London Mathematical Society)

   Iyengar, Srikanth (Ed.)

   We show that over a perfect field, every non-semisimple finite tensor category with finitely generated cohomology embeds into a larger such category where the tensor product property does not hold for support varieties. 

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3. Stratification and the comparison between homological and tensor triangular support

   https://doi.org/10.1093/qmath/haac040  

   Barthel, Tobias; Heard, Drew; Sanders, Beren (December 2022, The Quarterly Journal of Mathematics)

   Abstract We compare the homological support and tensor triangular support for ‘big’ objects in a rigidly-compactly generated tensor triangulated category. We prove that the comparison map from the homological spectrum to the tensor triangular spectrum is a bijection and that the two notions of support coincide whenever the category is stratified, extending the work of Balmer. Moreover, we clarify the relations between salient properties of support functions and exhibit counter-examples highlighting the differences between homological and tensor triangular support. 

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4. Tensor category describing anyons in the quantum Hall effect and quantization of conductance

   https://doi.org/10.1142/S0129055X24610075  

   Bachmann, Sven; Corbelli, Matthew; Fraas, Martin; Ogata, Yoshiko (May 2025, Reviews in Mathematical Physics)

   In this study, we examine the quantization of Hall conductance in an infinite plane geometry. We consider a microscopic charge-conserving system with a pure, gapped infinite-volume ground state. While Hall conductance is well-defined in this scenario, existing proofs of its quantization have relied on assumptions of either weak interactions, or properties of finite volume ground state spaces, or invertibility. Here, we assume that the conditions necessary to construct the braided [Formula: see text]-tensor category which describes anyonic excitations are satisfied, and we demonstrate that the Hall conductance is rational if the tensor category is finite. 

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5. A Formal Logic for Formal Category Theory

   https://doi.org/10.1007/978-3-031-30829-1_6  

   New, Max S; Licata, Daniel R (January 2023, Springer Nature Switzerland)

   Abstract We present a domain-specific type theory for constructions and proofs in category theory. The type theory axiomatizes notions of category, functor, profunctor and a generalized form of natural transformations. The type theory imposes an ordered linear restriction on standard predicate logic, which guarantees that all functions between categories are functorial, all relations are profunctorial, and all transformations are natural by construction, with no separate proofs necessary. Important category-theoretic proofs such as the Yoneda lemma and Co-yoneda lemma become simple type-theoretic proofs about the relationship between unit, tensor and (ordered) function types, and can be seen to be ordered refinements of theorems in predicate logic. The type theory is sound and complete for a categorical model invirtual equipments, which model both internal and enriched category theory. While the proofs in our type theory look like standard set-based arguments, the syntactic discipline ensure that all proofs and constructions carry over to enriched and internal settings as well. 

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