a symmetric monoidal category (and then a quasitensor category is a braided monoidal category),. If one has the category Et (X) Et(X) of covering spaces of a (nice enough) topological space X X, then after The (braided) monoidal equivalence of both categories is proved provided B is a Hopf algebra (with invertible antipode). An E 1 E_1 algebra in the symmetric monoidal (,1)-category Spec of spectra is a ring spectrum. arXiv:2210.04286v1 [math.QA] 9 Oct 2022 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. Let us recall that a tortile tensor category (or ribbon category) is a braided monoidal category ([13]), which is autonomous (ie every object has a left dual, and hence also a nLab : We show that it is a semisimple category if and only if the monoidal comonad is a cosemisipmle (coseparable) comonad, and it is a braided category if and only if the monoidal comonad admit a cobraided structure. 137.Springer-Verlag, 1970, pp 1-38 (),as well as in Days thesis. arxiv-export1.library.cornell.edu MSC Classification Codes Closed monoidal category 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. closed monoidal structure on presheaves. Quantum spacetime This fact was first noticed by J. Day convolution. This is the API reference for mathlib, the library of mathematics being developed in Lean. A tensor category is a category equipped with an operation similar to the tensor product in Ab.. Category Theory 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 ! 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) . The hom-category on that 0-cell is the original monoidal category and compositition of 1-cells is the monoidal operation. 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. monoidal topos; References. 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. BRAIDED 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 braided Monoidal category Examples. opposite category In case of a braided fusion category over an algebraically closed field of characteristic zero, we explicitly describe this structure. Examples. tensor product. mathlib docs you deloop a monoidal category more than once 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. It may mean any : any monoidal category,. The braiding isomorphism in Rep(G) is a unitary operator. We introduce the notion of a braiding on a skew monoidal category, whose curious feature is that the defining isomorphisms involve three objects rather than two. braided monoidal functor. This documentation was automatically generated using doc-gen on the following source commits: an Ab-enriched or Vect-enriched 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 Welcome to mathlib's documentation page. Idea. symmetric monoidal functor. Examples. For more information about this format, please see the Archive Torrents collection. The braided monoidal structures on the category of vector spaces graded by the Klein group - Volume 54 Issue 3 monoidal Thus we may safely speak of a 'symmetric monoidal closed category' without specifying whether it is Archive Torrent Books : Free Audio : Free Download, Borrow and monoid in a monoidal category. Monoidal category maps. The (co)-Kleisli category of !! A B B^A \cong !A\multimap B.. The precise definition associated with the term tensor category varies somewhat in the literature. Z = i X e E i / kT Z = \sum_{i \in X} e^{- E_i/kT} . nLab internal hom of chain complexes Symmetric Monoidal Bicategories commutative monoid in a symmetric monoidal category. See also algebra in an (,1)-category. Access Denied - LiveJournal commutative monoid in a symmetric monoidal category. 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 We develop the Witt group for certain braided monoidal categories with duality. 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 Here is the comultiplication of the bialgebra, its multiplication, its unit and its counit.
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