When Cis in fact a modular tensor category, the theory Z Creduces to that of Reshetikhin and Turaev. The corresponding rules are interpreted by precomposing the interpretation of a sequent with one of these maps. Idea. arXiv:1412.0212v1 [math.CT] 30 Nov 2014 INTERCATEGORIES II: EXAMPLES MARCO GRANDIS AND ROBERT PARE Abstract. A biclosed monoidal category is a monoidal category that is both left and right closed. Category theory has come to occupy a central position in contemporary mathematics and theoretical computer science, and is also applied to mathematical physics. braided It is a kind of twice-categorified version of an abelian group. braided monoidal functor. It may mean any : any monoidal category,. arxiv-export1.library.cornell.edu Internal monoids. Internal monoids. We would like to show you a description here but the site wont allow us. Quantum spacetime This documentation was automatically generated using doc-gen on the following source commits: PDF | On Feb 1, 2019, Daniel Bulacu and others published Monoidal and Braided Categories | Find, read and cite all the research you need on ResearchGate MSC Classification Codes a symmetric monoidal category (and then a quasitensor category is a braided monoidal category),. The braiding isomorphism in Rep(G) is a unitary operator. The hom-category on that 0-cell is the original monoidal category and compositition of 1-cells is the monoidal operation. Examples. If one has the category Et (X) Et(X) of covering spaces of a (nice enough) topological space X X, then after Theorems. The momentum space for the theory is another copy of the same algebra and there is a certain 'braided addition' of momentum on it expressed as the structure of a braided Hopf algebra or quantum group in a certain braided monoidal category). monoidal coherence theorem for monoidal categories. Hopf algebra closed monoidal structure on presheaves. Examples. arXiv:2210.04286v1 [math.QA] 9 Oct 2022 Our coherence theorem, as well as the construction of the We classify the braided monoidal structures of a class of linear Gr-categories via explicit computations of the normalized 3-cocycles and the quasi-bicharacters of finite abelian groups which are direct product of two cyclic groups. Idea. Welcome to mathlib's documentation page. nLab Then we have the category A braided monoidal We develop the Witt group for certain braided monoidal categories with duality. The partition function is . We give examples of intercategories, a special kind of lax triple ca Being braided means that this bicategory is itself (bi? Day convolution in nLab : Day convolution. Frontiers of Mathematics in China 137.Springer-Verlag, 1970, pp 1-38 (),as well as in Days thesis. Monoidal maps. 13.1, Shulman 12, theorem 2.14). Z = i X e E i / kT Z = \sum_{i \in X} e^{- E_i/kT} . An elementary example of a mapping describable as a tensor is the dot product, which maps two vectors to a scalar.A more complex example is the Cauchy stress tensor T, which takes a directional unit vector v as input and maps it to the stress vector T (v), which is the force (per unit area) exerted by material on the negative side of the plane orthogonal to v against the material In the sumless Sweedler notation, this property can also be expressed as (()) = (()) = ().As for algebras, one can replace the underlying field K with a commutative ring R in the above definition.. symmetric monoidal functor. The precise definition associated with the term tensor category varies somewhat in the literature. Definition 2.3.A (strict) V-monoidal category is a V-category Cequipped with an associative monoid structure on objects, denoted for , C, whose unit object is denoted by 1 coherence theorem for monoidal categories. Monoidal category theory played a central role in the discovery of new invariants of knots and links and in the development of the theory of quantum groups. C!VectZ-gr, constructed by De Renzi [DR22], is de ned on a category of admissible decorated three dimensional cobordisms and takes values in a braided monoidal category of Z-graded vector spaces. Download to read the full article text References Andruskiewitsch, N., Schneider, H.-J. monoidal topos; References. A different notion that is also sometimes called an enriched bicategory is that of a bicategory enriched over a monoidal 1-category V V (which must be at least braided) at the level of 2-cells only. arXiv:math/9902030v1 [math.QA] 4 Feb 1999 In case of a braided fusion category over an algebraically closed field of characteristic zero, we explicitly describe this structure. module over a monoid. Symmetric Monoidal Bicategories Crossed Modules in Braided Monoidal Categories See E- algebra. The braided monoidal structures on the category of vector spaces graded by the Klein group - Volume 54 Issue 3 tensor product. BRAIDED Brian Day, Construction of Biclosed Categories, PhD thesis.School of Mathematics of the University of New South Wales, Braided Access Denied - LiveJournal Brian Day, On closed categories of functors, Reports of the Midwest Category Seminar IV, Lecture Notes in Mathematics Vol. Closed monoidal category The (braided) monoidal equivalence of both categories is proved provided B is a Hopf algebra (with invertible antipode). A V-category Cis called tensored if every V-representable functor C( ): CVadmits a left V-adjoint. Monoidal symmetric monoidal functor. closed monoidal structure on presheaves. braided Monoidal category If all the fibers are not just monoidal but closed monoidal categories and the base change morphisms are not just strong monoidal but also strong closed monoidal functors, then the indexed monoidal category is an indexed closed monoidal category (Shulman 08, def. Category A forgetful functor from a category of actions/representations to the underlying sets/spaces is often called a fiber functor, notably in the context of Tannaka duality and Galois theory.. monoidal category The geometric nerve of a tricategory - Higher categorical structures We then use this description to prove that this tool provides an invariant for finite isocategorical groups. These braidings are shown to arise from, and classify, cobraidings (also known as coquasitriangular structures) on bialgebras. This is the API reference for mathlib, the library of mathematics being developed in Lean. is cartesian closed, and the product there coincides with the product in the base category.The exponential (unsurprisingly for a Kleisli category) is B A ! Group cohomology commutative monoid in a symmetric monoidal category. )monoidal so that it can be delooped again to Thus we may safely speak of a 'symmetric monoidal closed category' without specifying whether it is Related concepts. Examples. This fact was first noticed by J. See also algebra in an (,1)-category. an Ab-enriched or Vect-enriched Nevertheless, due to the switch of source and target, the opposite category C op C^{op} is usually far from being equivalent to C C. See the examples below. A monoidal category is a category equipped with a bifunctor called the tensor product or monoidal product, an object called the unit object or identity object, three natural isomorphisms subject to certain coherence conditions expressing the fact that the tensor operation Theorems. nLab monoid in a monoidal category. linear logic Central charge and Central extensions fiber functor in nLab For more information about this format, please see the Archive Torrents collection. Closely related is the statement that the braided monoidal structure on the modular representation category of the corresponding vertex operator algebra is fully determined by the genus=0 conformal blocks, a statement that seems to be folklore (highlighted in EGNO 15, p. 266, Runkel, Sec. Tensor Day convolution. braided monoidal 18D20: Enriched categories (over closed or monoidal categories) 18D25: Strong functors, strong adjunctions; 18D30: Fibered categories The concept originates in. category Particular monoidal and * *-autonomous you deloop a monoidal category more than once If you need information about installing Lean or mathlib, or getting started with a project, please visit our community website.. The homology of an E 2 E_2-algebra in chain complexes is a Gerstenhaber algebra. opposite category 18D10: Monoidal categories (= multiplicative categories), symmetric monoidal categories, braided categories; 18D15: Closed categories (closed monoidal and Cartesian closed categories, etc.) In this paper, we study the category of corepresentations of a monoidal comonad. Examples of such monoidal categories include symmetric, braided, ribbon, and coboundary monoidal categories, which naturally arise in the representation theory of quantum groups and of coboundary Hopf algebras and in the theory of crystals of A tensor category is a category equipped with an operation similar to the tensor product in Ab.. The braided monoidal structure on the category of Hom-type Doi-Hopf modules Daowei Lu Abstract Let $ (H,\a_H)$ be a Hom-Hopf algebra, $ (A,\a_A)$ a right H -comodule algebra and $ (C,\a_C)$ a left H -module coalgebra. Braided Monoidal Corollary 2.1 The classifying space of a braided monoidal category is the classifying space of its double suspension tricategory, that is, B 3 (C, , c) = B 3 2 (C, , c) . We now assume Vis a braided monoidal category. It is known that the group completion of the classifying space BC of a braided monoidal category (C, , c) is a double loop space. Braided Monoidal An E 1 E_1 algebra in the symmetric monoidal (,1)-category Spec of spectra is a ring spectrum. Monoidal category
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