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Topic: Braided monoidal category

	NationMaster - Encyclopedia: Monoidal category
		In mathematics, a monoidal category (or tensor category) is a 2-category with one object (a 2-monoid).
		A monoidal category may be regarded as a bicategory with one object.
		Braided monoidal category is a mathematical concept in terms of category theory and is, as its name suggests, a monoidal category with braiding.
	www.nationmaster.com /encyclopedia/Monoidal-category (1358 words)

	qg13.1
		dimension 1 -- category dimension 2 -- monoidal category dimension 3 -- braided monoidal category dimension 4 -- symmetric monoidal category dimension 5 -- symmetric monoidal category dimension 6 -- symmetric monoidal category.
		An n-dimensional TQFT is a symmetric monoidal functor from nCob to Hilb.
		A unitary n-dimensional TQFT is a symmetric monoidal *-functor from nCob to Hilb.
	math.ucr.edu /home/baez/qg-winter2001/qg13.1.html (1318 words)

	Monoidal category - Wikipedia, the free encyclopedia
		In mathematics, a monoidal category (or tensor category) is a category
		A monoidal category may be regarded as a bicategory with one object.
		Monoidal categories are used to define models for linear logic.
	en.wikipedia.org /wiki/Monoidal_category (404 words)

	[No title]
		A monoidal category is a bicategory with one object.
		A braided monoidal category is a tri-category with one object and one map.
		This is a belated reply to the query of John Baez regarding the nerve of a braided monoidal category.
	www.mta.ca /~cat-dist/catlist/1999/braided (1985 words)

	Monoidal category - TheBestLinks.com - Category theory, Field (mathematics), Identity element, Mathematics, ... (Site not responding. Last check: )
		In mathematics, a strict monoidal category is a category with a product operation × on objects that has properties analogous to those of the tensor product.
		Any category with standard categorical products and a terminal object is a strict monoidal category, with the categorical product as tensor product and the terminal object as identity.
		Also, any category with coproducts and an initial object is a strict monoidal category - with the coproduct as tensor product and the initial object as identity.
	www.thebestlinks.com /Monoidal_category.html (337 words)

	Monoidal category - Definition, explanation
		In mathematics, a monoidal category (or tensor category) is a category
		Many monoidal categories have additional structure such as braiding or symmetry: the references describe this in detail.
		Monoidal categories are used to define models for linear logic.
	www.calsky.com /lexikon/en/txt/m/mo/monoidal_category.php (370 words)

	The Maseeh Mathematics & Statistics Colloquium Series (Site not responding. Last check: )
		In particular, a strict monoidal category, braiding, and pivotal category are defined.
		The language of strict monoidal categories is then constructed using the concept of a graph with relations.
		This category is in fact a free braided pivotal category.
	www.mth.pdx.edu /Events/colloquium.asp?id=61 (165 words)

	This Week's Finds in Mathematical Physics (Week 137)
		A braided monoidal category is simple algebraic gadget that captures a bit of the essence of 3-dimensionality in its rawest form.
		The center of a braided monoidal category is obviously a symmetric monoidal category.
		For example, the center of a monoidal category is a braided monoidal category.
	www.lns.cornell.edu /spr/1999-09/msg0017904.html (1673 words)

	Quantum Gravity Seminar: Week 13: Track 1
		A strict braided monoidal category is a strict monoidal category C equipped with: 1) for every pair of objects u and v, an isomorphism B_{u,v}: u (x) v -> v (x) u, called the "braiding".
		Similarly, a weak braided monoidal category is a weak monoidal category with a braiding satisfying all the above laws...
		Luckily, Mac Lane proved that weak monoidal categories are all equivalent (in a certain precise sense) to strict ones.
	www.lns.cornell.edu /spr/2001-02/msg0031289.html (1439 words)

	[No title]
		In the case of tensor categories and functors on $\M$ we obtain as diagonal on $A$ an entwined double measuring $\widetilde \Delta_A: C \tensor C \tensor A \to A \tensor A$ and as multiplication on $C$ an entwined double comeasuring $\widetilde \nabla_C: C \tensor C \to A \tensor A \tensor C$ satisfying certain compatibility conditions.
		We generalize this construction to the category $\Mp$ of entwined modules, that is $A$-modules and $C$-comodules over Hopf algebras $A$ and $C$ where the structures are only related by an entwining map $\psi: C \tensor A \to A \tensor C$.
		Braided monoidal categories have important applications in knot theory, algebraic quantum field theory, and the theory of quantum groups and Hopf algebras.
	www.mathematik.uni-muenchen.de /~pareigis/pa_schft.html (2950 words)

	Citations: Higher-dimensional algebra II: 2-Hilbert spaces - Baez (ResearchIndex)
		Moreover, the 2 category of representations of any Hopf category is a monoidal 2 category [23] and when unitary representations can be be defined, the 2 category of unitary representations should be a monoidal 2 category with duals.
		Finally, there are several examples of braided monoidal 2 categories which have been constructed from solutions of the Zamolodchikov tetrahedron equations [11, 7] and it is possible that one of these may give a braided monoidal 2 category with duals.
		, her result is that isotopy classes of framed oriented tangles in 3 dimensions are the morphisms of the free braided monoidal category with duals on one object.
	citeseer.ist.psu.edu /context/279240/627007 (2094 words)

	Categories
		A functor F from a category C to a category D is a map from the set of objects of C to the set of objects of D together with a map from the set Hom(X,Y) for any objects X,Y of C to Hom(F(X),F(Y)).
		A representation of a group, if we think of a group as a category as Sibley suggests, is just a functor from that category to the category Vect of vector spaces.
		The objects in the category Tang are {0,1,2,...} and the morphisms in Hom(m,n) are (isotopy classes of) tangles with m strands going in and n strands coming out.
	math.ucr.edu /home/baez/categories.html (2616 words)

	[No title] (Site not responding. Last check: )
		Say we do it to the category C of all representations of a finite group G. This is in fact a monoidal category, so the result C' is a braided monoidal category.
		The relationship between Hopf algebras and monoidal categories is given by "Tannaka-Krein reconstruction theorems", which give conditions under which a monoidal category is equivalent to the category of representations of a Hopf algebra, and actually constructs the Hopf algebra for you.
		Suppose C is actually braided monoidal and f preserves the braiding and monoidal structure.
	www.infomag.ru:8083 /dbase/B003E/971008-067.txt (1321 words)

	LMS Proceedings Abstract, paper PLMS 1418 (Site not responding. Last check: )
		For a braided tensor category ${\cal C}$ and a subcategory ${\cal K}$ there is a notion of a centralizer $C_{\cal C}({\cal K})$, which is a full tensor subcategory of ${\cal C}$.
		Let ${\cal C}$ be a modular category and ${\cal K}$ a full tensor subcategory closed with respect to direct sums, subobjects and duals.
		We study the prime factorizations of the categories $D(G)$-Mod, where $G$ is a finite abelian group.
	www.lms.ac.uk /publications/proceedings/abstracts/p1418a.html (250 words)

	The monoidal centre as a limit (Site not responding. Last check: )
		The centre of a monoidal category is a braided monoidal category.
		Monoidal categories are monoidal objects (or pseudomonoids) in the monoidal bicategory of categories.
		This paper provides a universal construction in a braided monoidal bicategory that produces a braided monoidal object from any monoidal object.
	www.tac.mta.ca /tac/volumes/13/13/13-13abs.html (57 words)

	Category of graded vector spaces (Site not responding. Last check: )
		In this sense, it is a symmetric monoidal category.
		There is also a parity reversing functor from this category to itself which interchanges the even and odd subspaces.
		This category is the foundation for the study of "superobjects" like supervector spaces, superalgebras, Lie superalgebras, supergroups, supermanifolds, superspace etc.
	www.worldhistory.com /wiki/C/Category-of-graded-vector-spaces.htm (310 words)

	Citebase - Higher-Dimensional Algebra I: Braided Monoidal 2-Categories
		We begin with a brief sketch of what is known and conjectured concerning braided monoidal 2-categories and their applications to 4d topological quantum field theories and 2-tangles (surfaces embedded in 4-dimensional space).
		Finally, we describe how to construct a semistrict braided monoidal 2-category Z(C) as the `center' of a semistrict monoidal category C. This is analogous to the construction of a braided monoidal category as the center, or `quantum double', of a monoidal category.
		The categories with noninvertible morphisms are studied analogously to the semisupermanifolds with noninvertible transition functions.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:q-alg/9511013 (715 words)

	Centre of Australian Category Theory, Macquarie University :: Projects
		Higher-dimensional categories are complex structures that are currently gaining a lot of attention from mathematicians, physicists and computer scientists because of developing applications in those fields.
		Apart from their appearance in mathematics, star-autonomous categories have been known for over a decade to model the linear logic of Girard used extensively in computer science as a way of coping with resources and resource control.
		He furthermore constructed an action of a categorical analogue of the little n-cubes operad on the category of extensions in a monoidal abelian category; this is an interesting weak form of the generalised Deligne hypothesis.
	www.ics.mq.edu.au /CoACT/projects (1438 words)

	GT Monographs 2 (1999) Paper 21 (Abstract) (Site not responding. Last check: )
		A group-category is an additively semisimple category with a monoidal product structure in which the simple objects are invertible.
		For example in the category of representations of a group, 1-dimensional representations are the invertible simple objects.
		Braided-commutative categories most naturally give theories on 4-manifold thickenings of 2-complexes; the usual 3-manifold theories are obtained from these by normalizing them (using results of Kirby) to depend mostly on the boundary of the thickening.
	www.maths.warwick.ac.uk /gt/GTMon2/paper21.abs.html (176 words)

	PlanetMath: almost cocommutative bialgebra
		-modules into a quasi-tensor or braided monoidal category.
		is not necessarily the identity (this is the braiding of the category).
		Cross-references: identity, category, representations, natural isomorphism, tensor product, map, comultiplication, opposite, unit, cocommutative, bialgebra
	www.planetmath.org /encyclopedia/AlmostCocommutativeBialgebra.html (121 words)

	Math Forum Discussions
		monoidal categories with duals, and these are fairly unfamiliar -
		In general, whenever we have a monoidal category, we say an object
		a monoidal category where all the objects have duals.
	www.mathforum.com /kb/message.jspa?messageID=3652524&tstart=0 (902 words)

	Math Forum Discussions
		In the definition of a weak monoidal category we impose a coherence
		monoidal n-category with k >= n+2 a `stable n-category'.
		monoid on one generator; the integers are the free group on one
	www.mathforum.org /kb/thread.jspa?messageID=127152&tstart=0 (2796 words)

	polyhedra
		When working with permutative operads, it is often convenient to work with functors on the category of finite sets and bijections, rather than with functors on the permutation category (which is a skeleton).
		Of course, this is an incarnation of the free monoid construction, although in this case, R* = 1 + R + R^2 +...
		In other words, it is a structure cell which obtains in a braided monoidal 2-category.
	www.math.ucr.edu /home/baez/trimble/polyhedra.html (2342 words)

	Atlas: Vassiliev Theory as Deformation Theory by David N. Yetter (Site not responding. Last check: )
		We describe the construction of cochain complexes associated to monoidal categories, monoidal functors, and braided monoidal categories, and theorems relating the cohomology of the category (functor) to infinitesimal deformations of its structure maps.
		When a symmetric monoidal category with duals is deformed to give rise to a ribbon category, the k
		Ëxtrinsic deformations" in which a braided monoidal category is deformed a subcategory of a larger category are shown to provide a setting for the consideration of universal Vassiliev invariants over general coefficient rings.
	atlas-conferences.com /c/a/e/a/30.htm (191 words)

	Book \| Higher Operads, Higher Categories (London Mathematical Society Lecture Note Series) (Site not responding. Last check: )
		Structures such as braided monoidal categories, operads, and Hopf algebras are familiar to those who have studied topological quantum field theory, knot theory, string theory, and the renormalization procedure in quantum field theory.
		The braided monoidal category that arises in knot theory is a perfect example of this.
		Higher-dimensional category theory or `n-category theory,' is viewed as a generalization of the notion of category.
	store.worldsearch.com /higher_operads%2c_higher_categories-amco-0521532159.htm (2007 words)

	ATCAT 1997-1998
		ABSTRACT: A semi-quantaloid is a semicategory enriched in the monoidal category SUP of complete lattices and supremum preserving functions.
		In the second, an adjunction is established for monoidal bicategories, and then applied to the bicategory of V-categories and bimodules.
		Another motivating example is cat, the (mere) category of categories, together with profunctors which specializes somewhat to ord, the category of ordered sets, together with ordered ideals.
	www.mscs.dal.ca /~pare/Sem97-98.html (808 words)

	Talk:Monoidal category - InformationBlast
		1) Strict monoidal category = a category C with a specified object I (unit object) and a bifunctor m: C x C -> C satisfying the axioms of monoid.
		By canonical I mean that the natural equivalences are part of the structure and that however you combine them to obtain new equivalences, you can obtain at most one equivalence between any two functors (this is the content of the coherence axioms).
		In particular, a braided monoidal category is NOT a relaxed version of a monoidal category.
	www.informationblast.com /Talk:Monoidal_category.html (269 words)

	Science Fair Projects - Monoidal category
		Or else, you can start by choosing any of the categories below.
		The tensor operation must be associative in the sense that there is a natural isomorphism α with components
		Given a field (or commutative ring) K, the category K-Vect is a symmetric monoidal category with product ⊗ and identity K.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Monoidal_category (533 words)

	IRMA Strasbourg - Publication 2001 (Site not responding. Last check: )
		A braided monoidal category $\Cal G_{\Lambda,\theta}$ of $\Lambda$-graded associative algebras over a field $k$ is established.
		The structural feature (including its PBW-basis) of the braided universal enveloping algebra $\Cal U(L)$ of a $\theta$-Lie algebra $L$ is investigated as an object in $\Cal G_{\Lambda,\theta}$ and a class of quantum groups arising from $\Cal U(L)$ is constructed.
		The quantum affine space $k[A_q^{n0}]$, as the braided universal enveloping algebra of an abelian $\theta$-Lie algebra, is a braided Hopf algebra.
	www-irma.u-strasbg.fr /irma/publications/2001/01026.shtml (179 words)

	[No title] (Site not responding. Last check: )
		Enrichment as Categorical Delooping I: Enrichment Over Iterated Monoidal Categories The 2-category V-Cat of categories enriched over a braided monoidal category V is not itself braided in any way that is based upon the braiding of V. The exception is the case in which V is symmetric, which leads to V-Cat being symmetric as well.
		It seems that the analogy of loop spaces is a good guide for how to define the concept of enrichment over various types of monoidal objects, including k-fold monoidal categories and their higher dimensional counterparts.
		The main result is that for V a k-fold monoidal category, V-Cat becomes a (k-1)-fold monoidal 2-category in a canonical way.
	www.lehigh.edu /~dmd1/sf620.txt (291 words)

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