en>Ad Homonym: Fixed a restrictive clause which/that problem to standard American English. Also slightly reordered some words for easier reading.

2014-11-23T21:38:57Z

Fixed a restrictive clause which/that problem to standard American English. Also slightly reordered some words for easier reading.

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	~~Greetings~~. ~~Allow me begin by telling you~~ the ~~author's name~~ - ~~Phebe~~. ~~In her professional life she~~ is ~~a payroll clerk but she's usually needed her~~ [~~http~~://~~www~~.~~bestdiettips~~.~~com~~/~~diet~~-~~meal-plans-delivery personal business~~]. ~~One~~ of the ~~things she enjoys meal delivery service most~~ is ~~to do aerobics~~ and ~~now she~~ is ~~attempting to make money with~~ it. ~~Years ago we moved~~ to ~~[http~~://~~Www~~.~~Freedieting.com~~/~~diet_delivery_reviews~~.~~htm Puerto Rico~~] ~~and my family members loves it.~~<br><br>~~healthy food delivery Also visit my web page~~ ... ~~meal delivery service~~ ([http://~~Mmservice~~.dk/~~weightlossfoodprograms67803 recommended you read~~])		In [[mathematics]], a '''commutativity constraint''' <math> \gamma </math> on a [[monoidal category]] ''<math>\mathcal{C}</math>'' is a choice of isomorphism
			<math>\gamma_{A,B}:A\otimes B \rightarrow B\otimes A</math>
			for each pair of objects A and B which form a "natural family." In particular, to have a commutativity constraint, one must have <math> A \otimes B \cong B \otimes A </math> for all pairs of objects <math> A,B \in \mathcal{C}</math>.

			A '''braided monoidal category''' is a monoidal category <math>\mathcal{C}</math> equipped with a commutativity constraint <math> \gamma </math> which satisfies the hexagon axiom (see below). The term braided comes from the fact that the [[braid group]] plays an important role in the theory of braided monoidal categories. Partly for this reason, braided monoidal categories and various related notions are important in the theory of [[knot invariants]].

			Alternatively, a braided monoidal category can be seen as a [[tricategory]] with one 0-cell and one 1-cell.

			==The hexagon axiom==
			For <math>\mathcal{C}</math> along with the commutativity constraint <math>\gamma</math> to be called a braided monoidal category, the
			following hexagonal diagrams must commute for all objects <math>A,B,C \in \mathcal{C}</math>. Here <math>\alpha</math> is the associativity isomorphism coming from the [[monoidal category\|monoidal structure]] on <math>\mathcal{C}</math>:
			{\| align=center
			\|-
			\| [[Image:CategoryBraiding-02.png]]
			\| width=40 \|
			\| [[Image:CategoryBraiding-03.png]]
			\|}

			==Properties==

			===Coherence===

			It can be shown that the natural isomorphism <math>\gamma</math> along with the maps <math> \alpha, \lambda, \rho </math> coming from the monoidal structure on the category <math>\mathcal{C}</math>, satisfy various [[coherence condition]]s which state that various compositions of structure maps are equal. In particular:

			* The braiding commutes with the units. That is, the following diagram commutes:
			[[Image:CategoryBraiding-01.png\|center]]

			* The action of <math>\gamma</math> on an <math>N</math>-fold tensor product factors through the [[braid group]]. In particular,

			<math> (\gamma_{B,C} \otimes \text{Id}) \circ (\text{Id} \otimes \gamma_{A, C}) \circ (\gamma_{A,B} \otimes \text{Id}) =
			(\text{Id} \otimes \gamma_{A,B}) \circ (\gamma_{A,C} \otimes \text{Id}) \circ (\text{Id} \otimes \gamma_{B, C})
			</math>

			as maps <math> A \otimes B \otimes C \rightarrow C \otimes B \otimes A</math>. Here we have left out the associator maps.

			==Variations==

			There are several variants of braided monoidal categories that are used in various contexts. See, for example, the expository paper of Savage (2009) for an explanation of symmetric and coboundary monoidal categories, and the book by Chari and Pressley (1995) for ribbon categories.

			===Symmetric monoidal categories===
			{{main\|Symmetric monoidal category}}

			A braided monoidal category is called symmetric if <math>\gamma</math> also satisfies <math> \gamma_{B,A} \circ \gamma_{A,B} = Id</math> for all pairs of objects <math>A</math> and <math>B</math>. In this case the action of <math>\gamma</math> on an <math>N</math>-fold tensor product factors through the [[symmetric group]]

			===Ribbon categories===
			A braided monoidal category is a ''[[ribbon category]]'' if it is [[rigid category\|rigid]], and it has a good notion of [[quantum trace]] and co-quantum trace. Ribbon categories are particularly useful in constructing [[knot invariants]].

			===Coboundary monoidal categories===

			Sometimes categories are assumed to have n-ary monoidal products for all finite n (in particular n>2), diminishing the role of associator morphisms. In such categories, the following variant is used, where the hexagon axiom is replaced by the two conditions:

			* <math> \gamma_{B,A} \circ \gamma_{A,B} = \text{Id}</math> for all pairs of objects <math>A</math> and <math>B</math>.

			*<math> \gamma_{B \otimes A, C} \circ (\gamma_{A,B} \otimes \text{Id}) = \gamma_{A, C \otimes B} \circ (\text{Id} \otimes \gamma_{B,C})</math>

			==Examples==

			* The category of representations of a group (or a lie algebra) is a symmetric monoidal category where <math> \gamma (v \otimes w) = w \otimes v </math>.

			* The category of representations of a [[quantum group\|quantized universal enveloping algebra]] <math> U_q(\mathfrak{g})</math> is a braided monoidal category, where <math> \gamma </math> is constructed using the [[Quasitriangular Hopf algebra\|Universal R-matrix]]. In fact, this example is a ribbon category as well.

			==Applications==

			* [[knot invariants]].
			* Symmetric [[closed monoidal category\|closed monoidal categories]] are used in denotational models of [[linear logic]] and [[linear types]].

			==References==

			*Chari, Vyjayanthi; Pressley, Andrew. "A guide to quantum groups". Cambridge University Press. 1995.

			* Joyal, André; Street, Ross (1993). "Braided Tensor Categories". ''Advances in Mathematics'' ''102'', 20–78.

			*Savage, Alistair. Braided and coboundary monoidal categories. Algebras, representations and applications, 229–251, Contemp. Math., 483, Amer. Math. Soc., Providence, RI, 2009. [http://arxiv.org/abs/0804.4688 Available on the arXiv]

			==External links==
			*[[John Baez]] (1999), [http://math.ucr.edu/home/baez/week137.html An introduction to braided monoidal categories], ''This week's finds in mathematical physics'' 137.

			[[Category:Braids]]
			[[Category:Monoidal categories]]

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Greetings. Allow me begin by telling you the author's name - Phebe. In her professional life she is a payroll clerk but she's usually needed her [http://www.bestdiettips.com/diet-meal-plans-delivery personal business]. One of the things she enjoys meal delivery service most is to do aerobics and now she is attempting to make money with it. Years ago we moved to [http://Www.Freedieting.com/diet_delivery_reviews.htm Puerto Rico] and my family members loves it.<br><br>healthy food delivery Also visit my web page ... meal delivery service ([http://Mmservice.dk/weightlossfoodprograms67803 recommended you read])

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en>Ad Homonym: Fixed a restrictive clause which/that problem to standard American English. Also slightly reordered some words for easier reading.

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