Advances in Mathematics 102, 20–78. Morita 4-category of braided tensor categories, and I'll report on joint work with Brochier and Snyder which identifies two natural subcategories therein which are 3- and 4-dualizable. Adv. Check out our Code of Conduct. Geoff C is a new contributor to this site. Theorem 4.2 will be proved by using a formula of the Frobenius-Perron dimension of the tensor full subcategory generated by two tensor full subcategories . Share. In mathematics, a monoidal category (or tensor category) is a category equipped with a bifunctor ⊗: × → that is ... "Braided Tensor Categories". Finally, we indicate how these considerations generalize to arbitrary finitely generated rigid braided C*-tensor categories. MTS. Lecture III: The Morita 4-category of braided tensor categories and the cobordism hypothesis 17 1. Cite. compute it, one must work with a braid group representation. Rowell and Wang, Mathematics of topological quantum computing. These categories are braided, rigid and non semi-simple tensor categories. Section: 1. J. I define them generally and naturally using a braided tensor category, a topological structure familiar from the study of knot invariants, anyons and conformal field theory. We list their simple and projective objects, their tensor products and their Hopf links. Follow edited Jul 9 '12 at 21:00. We study the question of dualizability in higher Morita categories of locally presentable tensor categories and braided tensor categories. This leads us to the definition of theta-, product-, subgrading and orbitcategories in order to construct new families of BTC's from given ones. over braided tensor categories Andr e Henriques, David Penneys, and James Tener September 11, 2015 Abstract Given a braided pivotal category Cand a pivotal module tensor category M, we de ne a functor Tr C: M!C, called the associated categori ed trace. (Precise de nitions will be given later.) Braided Tensor Categories Related to \({\mathcal {B}}_{p}\) Vertex Algebras. The present text provides an introduction and overview of [I]–[IV] and [C] and places these works into context. I define them generally and naturally using a braided tensor category, a topological structure familiar from the study of knot invariants, anyons and conformal field theory. I'll also explain work in progress by Follow asked 1 min ago. quantum-groups braided-tensor-categories. 15 (1948), 271 – 292. While giving a detailed overview of general tensor categories, it focuses especially on the theory of finite tensor categories and fusion categories (in particular, braided and modular ones), and discusses the main results about them with proofs. The concept of a braided tensor category has its roots in Mac Lane's work [18] on the category-theoretic foundations of associativity and commutativity, and a certain class, the Tannakian categories, have long found wide application in number theory and other subjects [5, 27]. Joyal, André; Street, Ross (1988). Metrics details. 24 Moore, G. and Seiberg, N.. Duke Math. We realize this physical picture in terms of braided tensor categories and their braid group representations. In particular, the category of V-modules has a natural structure of modular tensor category. Dimensions of tensor full subcategories. braided tensor categories—that is, the properties of rigidity and of nondegeneracy—were proved by the first author in [H8] by the use of the Moore–Seiberg equations, which the first author had proved for suitable classes of representations of vertex operator algebras in [H7]. On dualizability of braided tensor categories w/ A. Brochier and N. Snyder. Via the cobordism hypothesis, this produces respectively 2, 3 and 4-dimensional framed local topological field theories. arXiv:1804.07538. 22 Karpilovsky, G.. Projective representations of finite groups, vol. The cohomology theory of group extensions. REFERENCES. Improve this question. Jean Auger 1 nAff2, Thomas Creutzig 1, Shashank Kanade 3 & Matthew Rupert 1 Communications in Mathematical Physics volume 378, pages 219 – 260 (2020)Cite this article. We first clarify what we mean by a tensor full subcategory: Definition 4.3. Authors: Thomas Kerler (Submitted on 3 Feb 1994) Abstract: We investigate invertible elements and gradings in braided tensor categories. ACKNOWLEDGMENTS. A braided finite tensor category is factorizable if and only if its Müger center is trivial. Modular categories, as well as the (possibly) more general non-degenerate braided fusion categories, are braided tensor categories that are linear over a eld and satisfy some natural additional axioms, like existence of duals, semisimplicity, niteness, and an important non-degeneracy condition. asked Apr 8 '10 at 2:30. Abstract. Abstract. A coboundary or “cactus” monoidal category is a monoidal category (, ⊗,). Title: On braided tensor categories. We investigate invertible elements and gradings in braided tensor categories. We study categories related to the $\mathcal{B}_p$ vertex operator algebras using their conjectural relation to unrolled restricted quantum groups of $\mathfrak{sl}_2$. 14 – 18 and 22 is rigid, has a natural structure of ribbon category, and has the nondegeneracy property. In all known examples, these weights also satisfy the Yang-Baxter equation. Presentable braided tensor categories and their factorization homology 16 3. Bull. While giving a detailed overview of general tensor categories, it focuses especially on the theory of finite tensor categories and fusion categories (in particular, braided and modular ones), and discusses the main results about them with proofs. 177 Accesses. David Jordan David Jordan. Abstract. Math. of rigid braided tensor categories with enough compact projectives is 3-dualizable, and that (in characteristic zero) the 4-category of braided fusion categories is 4-dualizable. Take care in asking for clarification, commenting, and answering. ct.category-theory type-theory braided-tensor-categories linear-logic. We investigate invertible elements and gradings in braided tensor categories. 7,889 2 2 gold badges 28 28 silver badges 60 60 bronze badges. Ribbon categories. 11701293). A braided monoidal (weak) 2-category is a monoidal 2-category with a categorified version of a braiding.. That is, it is a 2-category C C equipped with a tensor product ⊗: C × C → C \otimes : C \times C \to C 2-functor which satisfies the first in a hierarchy of conditions for being commutative up to equivalence. Baxterization via fractional-spin conserved currents Will give a simple formula for the Boltzmann weights in terms of category data that guarantees the existence of conserved currents. Braided tensor categories and the cobordism hypothesis. Braided tensor categories. Such a current amounts to terminating a lattice topological defect, and I will touch on related work on such done with Aasen and Mong. 102 (1993), 20 – 78. Kelly, G. Max (1964). In particular, it shows how the main properties of finite-dimensional Hopf algebras may be derived from the theory of 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 fusion categories is 4-dualizable. fold tensor product factors through the symmetric group. K-theory of AF-algebras from braided C*-tensor categories Aaserud, Andreas and Evans, David 2020. In particular, it shows how the main properties of finite-dimensional Hopf algebras may be derived from the theory of tensor categories. Cite. 23 Lyndon, R. C.. I define them generally and naturally using a braided tensor category, a topological structure familiar from the study of knot invariants, anyons and conformal field theory. 94, Monographs and Textbooks in Pure and Applied Mathematics (New York: Marcel Dekker Inc, 1985). The author acknowledges the support from the NSFC (Grant No. We will explain how the fusion code can compute these. Lecture I: Overview The goal of these lecture notes is to give an informal exposition of a number of recent applications of \higher Morita theory" { i.e. For an abelian group $ A $, we study a close connection between braided crossed $ A $-categories with a trivialization of the $ A $-action and $ A $-graded braided tensor categories. Reviews in Mathematical Physics 32 (08) , 2030005. A braided tensor category (or braided tensor category, BTC) $\cat{C}$ is a tensor category with a natural isomorphism \[\sigma_{x,y} \colon x\otimes y \overset\sim\to y\otimes x, \quad \forall x,y \in \cat{C}\] such that the following diagrams (hexagon equations) commute: In the graphical notation, $\sigma$ and its inverse can be visualized as . A braided monoidal category is a ribbon category if it is rigid, and it may preserve quantum trace and co-quantum trace.Ribbon categories are particularly useful in constructing knot invariants.. Coboundary monoidal categories. "On MacLane's Conditions for Coherence of Natural Associativities, Commutativities, etc." New contributor. Then the braided tensor category structure on the category of V-modules constructed in refs. Geoff C Geoff C. 1. I derive simple constraints on the Boltzmann weights necessary and sufficient for such a current to exist, generalizing those found using quantum-group algebras. "Planar diagrams and tensor algebra". Idea. We show that TLJcat(delta) is equivalent, as a braided C*-tensor category, to the full subcategory of ModB whose objects are those modules which admit a finite orthonormal basis. K-theory of AF-algebras from braided C*-tensor categories. Braided tensor categories 5. 2 Altmetric. 1. Sketch of the proof: Note that condition 1 implies that V' is equivalent to V as a V-module. Such a current amounts to terminating a lattice topological defect, and I will touch on related work on such done with Aasen and Mong. Fusion Rings Quantum Methods F-matrices Some references Bakalov and Kirillov, Lectures on tensor categories and modular functors, AMS (2001). Let V be a vertex operator algebra satisfying suitable conditions such that in particular its module category has a natural vertex tensor category structure, and consequently, a natural braided tensor category structure. Share. The cobordism hypothesis gives a functorial bijection between oriented n-dimensional fully local topological field theories, valued in some higher category C, and the fully dualizable object of C equipped with the structure of SO(n)-fixed point. These are the rigid braided tensor categories with enough compact projectives, and the braided fusion categories, respectively. 5,803 23 23 silver badges 39 39 bronze badges $\endgroup$ 3 $\begingroup$ This is a neat construction. This leads us to the definition of theta-, product-, subgrading and orbitcategories in order to construct new families of BTC's from given ones. 4.2. braided tensor categories are investigated in depths. 10.1142/S0129055X20300058 Item availability restricted. AMS 55 (2018).arXiv:1705.06206 Di Francesco, Mathieu and …
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