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Topic: Tensor category

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	Monoidal category - Wikipedia, the free encyclopedia
		In mathematics, a monoidal category (or tensor category) is a bicategory with one object.
		Monoidal categories are used to define models for the multiplicative fragment of intuitionistic linear logic.
		There is a general notion of monoid object in a monoidal category, which generalizes the ordinary notion of monoid.
	en.wikipedia.org /wiki/Monoidal_category (608 words)

	[No title]
		A monoidal category is a bicategory with one object.
		A monoid in monoidal categories is a braided monoidal category.
		Strict tensor functors between tensor categories become strict 2-functors between 2-categories with a single object, and it is from this entirely equivalent point of view that we wish to pursue the subject of this talk.
	www.mta.ca /~cat-dist/catlist/1999/braided (1985 words)

	ESI program 2004 "tensor categories"
		These categories may be described either as categories of representations of affine Kac-Moody algebras (the central extensions of loop Lie algebras) or as categories of D-modules on the flag varieties of loop groups.
		For an Abelian category A we would like to define the quotient of the category of complexes in A over the subcategory of acyclic complexes as a unital A_infinity-category, whose 0-th cohomology is the usual derived category D(A).
		This category is a braided crossed G-category in the sense of Turaev.
	www.ingvet.kau.se /teofys/conf/esi04/abs-esi04.html (2737 words)

	index
		We introduced the notion of vertex tensor category which is a sophisticated analogue, involving geometry and not just topology, of the notion of symmetric tensor category.
		We proved that the category of modules for a suitable vertex operator algebra is a vertex tensor category.
		In particular, we proved that the category generated by the standard modules of a fixed positive integral level for an affine Lie algebra and certain categories of modules for the Virasoro algebra are braided tensor categories.
	www.rci.rutgers.edu /~yzhuang/page25.html (369 words)

	[No title]
		The mentioned result, in particular, implies that the space of derived transformations of the identity endofunctor (of any DG category) is an algebra over a contractible 2-operad, which, according to Batanin, means that this space is an algebra over the chain operad of litlle disks, thus yielding another proof of Deligne's conjecture on Hochschild cochains.
		Categories of correspondences Abstract Let C be a 2-category such that for any two objects X,Y the category Hom(X,Y) is in fact a triangulated category (enhanced by spectra, or by complexes).
		Objects of C are finite-dimensional supervector spaces, Hom(U,V) is the derived category of finite-dimensional representations of the free algebra generated by the usual hom-space Hom(V,U).
	www.math.uchicago.edu /seminars/geometric_langlands.html (5218 words)

	[No title]
		That is, we con- struct the derived category Stable() of (A,) as the homotopy category* * of a Quillen model structure on Ch(), the category of unbounded chain com- plexes of -comodules.
		The category -comod is a symmetric monoidal category.
		6.The stable category We define the homotopy category of the homotopy model structure on Ch() to be the stable homotopy category of (A,), and we denote it by Stable(), follo* *w- ing Palmieri [Pal01] in the case of the Steenrod algebra.
	hopf.math.purdue.edu /Hovey/comodule.txt (11965 words)

	[No title]
		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.
		Anyway, Mueger's elegant characterization of a modular tensor category amounts to this: it's a braided 2-H*-algebra whose center is "trivial".
	math.ucr.edu /home/baez/twf_ascii/week137 (1601 words)

	[No title]
		For Adams graded objects X and Y in a tensored category, we define X (X Y)(r) = X(s) Y (r - s); and the category of Adams graded objects is again tensored; its unit is the uni* t of the given category* viewed as concentrated in Adams grading zero.
		Since we have tensored the interval coordinate on the right, t* he differential on X is the same as the differential on X, without the introductio *n of a sign.
		Since we defined cofiber sequenc* es in terms of tensoring* with k-modules, the cofiber sequence generated by a map of A-modules is clearly a sequence of A-modules.
	hopf.math.purdue.edu /Kriz-May/operads_motives.txt (13330 words)

	Teaching at I
		The isotopy class of framed oriented tangles in 3-dimensional space forms a "free braided tensor category with dual of one object".
		This allows to form a functor from this category to the other braided tensor categories with dual, such as the category of representations of a quantum group.
		Tensor categories also play a role in generalization of subfactors, this generalization gives a useful tool for the proof of new results in categorical algebra and low dimensional topology.
	www.math.iitb.ac.in /~vishvajit (562 words)

	Abstract (Site not responding. Last check: 2007-10-31)
		Galois Theory for Braided Tensor Categories and the Modular Closure
		Given a braided tensor -category C with conjugate (dual) objects and irreducible unit together with a full symmetric subcategory S we define a crossed product C\rtimes S. This construction yields a tensor* *-category with conjugates and an irreducible unit.
		The category C\rtimes D is called the modular closure of C since in the rational case it is modular, i.e.
	www.lqp.uni-goettingen.de /papers/04/04/04041503.html (205 words)

	week169
		Part of the fun of category theory is that it lets you take mathematical arguments and generalize them to their full extent by finding the proper context for them: that is, by figuring out in exactly what sort of category you can carry out the argument.
		So, one job of a category theorist is to figure out what features are actually needed in a given situation, and isolate the kind of category that has those features.
		In a monoidal category we can't "duplicate" or "delete" arguments: if X is an object in a monoidal category, there's no god-given map from X to X tensor X, or from X to 1.
	math.ucr.edu /home/baez/week169.html (2717 words)

	LMS Proceedings Abstract, paper PLMS 1418 (Site not responding. Last check: 2007-10-31)
		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)

	Quantum Math Seminar
		Abstract We introduce the notion of a supercategory as a generalization of the tensor category of vector superspaces.
		Then, by using certain generalized Knizhnik-Zamolodchikov equations, we prove the ``convergence and expansion properties'' for this category and obtain a new construction of the braided tensor category structure.
		Compared to the original algebraic-geometric method, the vertex algebraic approach further establishes a vertex tensor category structure on this category.
	www.math.rutgers.edu /~seminars/QuantumMath.html (794 words)

	PlanetMath: tensor algebra
		From the point of view of category theory, one can describe the tensor algebra construction as a functor
		Cross-references: homomorphism, algebras, forgetful functor, category, functor, category theory, point, right, bimodule, module, non-commutative, ground ring, cover, tensor product, power, tensor, component, commutative ring
		This is version 7 of tensor algebra, born on 2002-12-18, modified 2005-09-30.
	planetmath.org /encyclopedia/TensorAlgebra.html (95 words)

	Tensor Pond Vacuum in category accessory \| Bradshaws Direct UK
		Bradshaws Direct based in York welcome you to view this product "Tensor - Pond Vacuum in category accessory" at their York showroom or order direct from the website.
		Pond accessory skimmers and vacs is one of numerous categories stocked by Bradshaws who at times provide special offers on top selling products within the wide range of pond products sold.
		You can choose to pick up Tensor - Pond Vacuum or depending what your urgency is the Tensor - Pond Vacuum can be delivered using various transport systems.
	www.realponds.com /tensorpond.htm (259 words)

	Amazon.com: Quantum Groups (Graduate Texts in Mathematics): Books: Christian Kassel (Site not responding. Last check: 2007-10-31)
		A certain strict tensor category is built out of tangles, and shown to give isotopy invariants of links.
		Braiding in the tensor category is used to formalize the notion of crossing in link and tangle diagrams.
		Tensor categories modeled on framed tangles or "ribbons" are introduced to illustrate duality.
	www.amazon.com /exec/obidos/tg/detail/-/0387943706?v=glance (1789 words)

	Previous Rutgers Algebra Seminars (Since 1995)
		We introduce the notion of a supercategory as a generalization of the tensor category of vector superspaces.
		There is a well-developed tensor category theory for certain modules of a fixed positive integral level for an affine Lie algebra, important in the study of conformal field theory.
		Immediate applications of these equations are a construction of braided tensor categories on the category of modules for the vertex operator algebra and a construction of intertwining operator algebras (or chiral genus-zero conformal field theories) from irreducible modules for the vertex operator algebra.
	math.rutgers.edu /~weibel/oldalgebra.sem.html (8477 words)

	Frobenius Algebras and 2D QFT \| The String Coffee Table
		It is that ambient tensor category which ‘knows’ if the Frobenius algebra describes a topological or a conformal field theory (in 2D) - and which one.
		The main result is, roughly, that given any modular tensor category with certain properties, and given any (symmetric and special) Frobenius algebra object internal to that category, one can construct functions on surfaces that satisfy all the properties that one would demand of an
		This procedure is deeply rooted in well-known relations between 3-(!)-dimensional topological field theory, modular functors and modular tensor categories and may seem very natural to people who have thought long enough about it.
	golem.ph.utexas.edu /string/archives/000699.html (1123 words)

	Müger on Doplicher-Roberts \| The String Coffee Table
		As far as I understood, the main point is that once you use nowadays obvious category-theoretic reasoning and building on ideas by Deligne concerning this problem, the problem becomes pretty easy.
		In case there is anyone out there who knows what a category is but not what the Doplicher-Roberts theorem says, here is a brief outline.
		-category is equivalent to the representation category of the vertex group of this groupoid.
	golem.ph.utexas.edu /string/archives/000711.html (642 words)

	[No title]
		A Tannakian category is, roughly, a suitably small k-linear abelian category wi* th tensor* product and duality - such as the category of finite-dimensional linear * *rep- resentations of a proalgebraic group over a field k.
		However, we're not as lucky here as in the pre* ced- ing case: the stable homotopy category* (at p) is not the same as the category of modules over the Steenrod algebra.
		There is evidence [2, 4 x6, 22, 42 x4] that * some generalization of his theory to a derived category* context could accomodate a v* *ery general form of the theory of vanishing cycles [17, 9].
	hopf.math.purdue.edu /Morava/Rosendal.txt (6109 words)

	"Area Metric" Manifolds \| The String Coffee Table
		Normally our category should be “monoidal” - it should have “tensor products”, as evident from page 2 of my talk.
		This is a monoidal category under disjoint union.
		category whose morphisms are spin networks to the category whose morphisms are intertwining operators between representations of
	golem.ph.utexas.edu /string/archives/000687.html (2801 words)

	Ketan Mehta's Research Blog
		Tensor field understanding is enhanced by using a new glyph (NLCGlyph) based on a new design metric which is closely related to the underlying physical properties of an NLC, described using the Q-tensor.
		The first level uses a timeline based visualization technique for providing an overall view of the dataset and selected timestep's detail are shown in the second level.
		Third level visualizes tensor glyphs and other details using region of interest selected in the second level.
	vis.cse.msstate.edu /weblog/km223/TensorVisualization (2742 words)

	Good Math, Bad Math
		A zero morphism in a category C is part of a set of arrows, called the zero family of morphisms of C, where composing any morphism with a member of the zero family results in a morphism in the zero family.
		As I keep complaining, the problem with category theory is that anytime you want to do something interesting, you have to wade through a bunch of definitions to set up your structures.
		A monoidal category C (with terminal object t, tensor ⊗, and natural isomorphisms λ, ρ) is symmetric if/f for all objects a and b in C, there is a natural isomorphism γ
	www.scienceblogs.com /goodmath (7840 words)

	NSDL Metadata Record -- $II_1$-Subfactors associated with the $C^*$-Tensor Category of a Finite Group
		NSDL Metadata Record -- $II_1$-Subfactors associated with the $C^$-Tensor Category* of a Finite Group
		$II_1$-Subfactors associated with the $C^$-Tensor Category* of a Finite Group
		We determine the subfactors $N\subset R$ of the hyperfinite $II_1$-factor R with finite index for which the $C^$-tensor category* of the associated $(N,N)$-bimodules is equivalent to the $C^$-tensor category* $\C{U}_G$ of all unitary finite dimensional representations of a given finite group G. description:
	nsdl.org /mr/1212728 (77 words)

	Homology of schemes, I, by Vladimir Voevodsky (Site not responding. Last check: 2007-10-31)
		The first one contains a construction which produces a tensor triangulated category (called the homological category) out of a site with some additional structure.
		In the second one we define two new Grothendieck topologies on the category of Noetherian schemes called h- and qfh-topology and prove some of their basic properties including the comparison theorems for cohomology.
		Finally in the third part we apply the construction of homological category to the category of schemes over a base S considered as a site with respect to h- or qfh-topology.
	www.math.uiuc.edu /K-theory/0031 (106 words)

	Citebase - A construction of a quotient tensor category (Site not responding. Last check: 2007-10-31)
		Citebase - A construction of a quotient tensor category
		Let f:G→ A be a surjective homomorphism of transitive groupoid schemes and let L denote the kernel of f.
		We also generalize this setting to the framework where the tensor categories are not necessarily Tannaka categories (i.e.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0603279 (145 words)

	Motivic cohomology and algebraic cycles a categorical approach, by Marc Levine
		For each scheme S, we construct a triangulated tensor category DM(S), functorially in S, which we propose as a candidate for the derived category of the conjectural category of mixed motives over S. The resulting cohomology theory has all the properties of a Bloch-Ogus cohomology theory, including cycle classes and Chern classes for higher K-theory.
		For S a field of characteristic zero, or a smooth curve over a field of characteristic zero, the motivic cohomology agrees with Bloch's higher Chow groups; the same is true in characteristic p>0 if one uses Q-coefficients.
		In addition, each reasonable graded cohomology theory Gamma() on the category* of smooth, quasi-projective schemes over a fixed base S gives rise to a realization functor Re_Gamma for DM(S); for example, we have the Betti, e'tale and Hodge realizations of DM(S).
	www.math.uiuc.edu /K-theory/0107 (202 words)

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