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Topics referred to by the same term
Tensor category (within the subfield category theory of mathematics) may refer to: General monoidal categories; or More specifically symmetric monoidal
Tensor_category
Type of monoidal category
A modular tensor category (or modular fusion category) is a type of monoidal category that plays a role in the areas of topological quantum field theory
Modular_tensor_category
mathematics, a unitary modular tensor category is a certain type of algebraic structure, defined by equipping a modular tensor category with additional data that
Unitary modular tensor category
Unitary_modular_tensor_category
specifically category theory, a pseudo-tensor category is a generalization of a symmetric monoidal category (also known as a tensor category) introduced
Pseudo-tensor_category
Algebraic object with geometric applications
(electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, etc.), and general relativity (stress–energy tensor, curvature tensor, etc.). In
Tensor
Category admitting tensor products
In mathematics, a monoidal category (or tensor category) is a category 𝐂 equipped with a bifunctor ⊗:𝐂×𝐂→𝐂 that is associative up to a natural isomorphism
Monoidal_category
Object in category theory
action of γ on an N-fold tensor product factors through the symmetric group. A braided monoidal category is a ribbon category if it is rigid, and it may
Braided_monoidal_category
Mathematical operation on vector spaces
called the tensor product of v and w. An element of V⊗W is a tensor, and the tensor product of two vectors is sometimes called an elementary tensor or a decomposable
Tensor_product
Topics referred to by the same term
classes/categories Category of being Categories (Aristotle) Categoriae Decem Category (Kant) Categories (Peirce) Category (Vaisheshika) Stoic categories Category
Category
Concept in mathematical category theory
In category theory, a branch of mathematics, a symmetric monoidal category is a monoidal category (i.e. a category in which a "tensor product" ⊗ {\displaystyle
Symmetric_monoidal_category
Tensor describing energy momentum density in spacetime
stress-energy tensor The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor field quantity
Stress–energy_tensor
Universal construction in multilinear algebra
define the kth tensor power of V to be the tensor product of V with itself k times: TkV=V⊗k=V⊗V⊗⋯⊗V. That is, TkV consists of all tensors on V of order
Tensor_algebra
additive category is the smallest pseudo-abelian category containing the given category. pseudo-tensor category pseudo-tensor category. Q Q-category. Quillen
Glossary_of_category_theory
asserts that the Drinfeld center of every spherical fusion category is a modular tensor category. Müger's theorem was introduced in 2003 by mathematician
Müger's_theorem
Algebra used in certain conformal field theories
A , B , C ∈ C {\displaystyle A,B,C\in {\mathcal {C}}} in a modular tensor category, the Verlinde formula relates the fusion coefficient N C A , B {\displaystyle
Verlinde_algebra
Categorical procedure
relation The pivotal structure on a modular tensor category (or more generally a pre-modular tensor category) can be encoded skeletally using θ {\displaystyle
Skeletonization of fusion categories
Skeletonization_of_fusion_categories
Monoidal category
the Galois theory is replaced by an exact and faithful tensor functor F from C to the category of finite-dimensional vector spaces over K. The group of
Tannakian_formalism
Tensor field in Riemannian geometry
mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the
Riemann_curvature_tensor
Muscle of the thigh
The tensor fasciae latae (or tensor fasciæ latæ or, formerly, tensor vaginae femoris) is a muscle of the thigh. Together with the gluteus maximus, it acts
Tensor_fasciae_latae_muscle
Expression of time reference in grammar
In grammar, tense is a category that expresses time reference. Tenses are usually manifested by the use of specific forms of verbs, particularly in their
Grammatical_tense
Introduced by Gray, a Gray tensor product is a replacement of a product of 2-categories that is more convenient for higher category theory. Precisely, given
3-category
Graphical representation of a morphism
the monoidal category of finite-dimensional vector spaces and linear maps with the tensor product, string diagrams are called tensor networks or Penrose
String_diagram
Tensor used in general relativity
differential geometry, the Einstein tensor (named after Albert Einstein; also known as the trace-reversed Ricci tensor) is used to express the curvature
Einstein_tensor
theorem about modular tensor categories. It asserts that two different formulations of the modularity condition of a modular tensor category are equivalent.
Bruguières_modularity_theorem
Operation that pairs a left and a right R-module into an abelian group
universal property of the tensor product of vector spaces extends to more general situations in abstract algebra. The tensor product of an algebra and
Tensor_product_of_modules
Tensor product space endowed with a special inner product
analysis, the tensor product of Hilbert spaces is a way to extend the tensor product construction so that the result of taking a tensor product of two
Tensor product of Hilbert spaces
Tensor_product_of_Hilbert_spaces
Concept from category theory
identity" diagram commutes. A tensor category,[full citation needed] or monoidal category, is a semi-groupal category with an identity object, I {\displaystyle
Associativity_isomorphism
of tensor theory. For expositions of tensor theory from different points of view, see: Tensor Tensor (intrinsic definition) Application of tensor theory
Glossary_of_tensor_theory
Particle
by the fact that they are described by the Fibonacci category, a certain modular tensor category. Due to its connections with quantum field theory and
Fibonacci_anyons
Representation of a modular tensor category
group representation (or simply modular representation) of a modular tensor category 𝒞 is a representation of the modular group SL2(ℤ) associated to 𝒞
Modular_group_representation
introduced the notion of chiral algebra, which based on the pseudo-tensor category of D-modules. They give a 'coordinate independent' notion of vertex
Chiral_algebra
Type of category in mathematics
non-cartesian example is the category of vector spaces, K-Vect, over a field K. Here the monoidal product is the usual tensor product of vector spaces, and
Closed_monoidal_category
Generalization of tensor fields
differential geometry, a tensor density or relative tensor is a generalization of the tensor field concept. A tensor density transforms as a tensor field when passing
Tensor_density
Structure in algebraic geometry
SmProj(k), the category Corr(k) has direct sums (X ⊕ Y := X ∐ Y) and tensor products (X ⊗ Y := X × Y). It is a preadditive category. The sum of morphisms
Motive_(algebraic_geometry)
Field-equations in general relativity
Einstein in 1915 in the form of a tensor equation which related the local spacetime curvature (expressed by the Einstein tensor) with the local energy, momentum
Einstein_field_equations
Mathematical category
4007/annals.2005.162.581. ISSN 0003-486X. Etingof, Pavel; Nikshych, Dmitri; Ostrik, Viktor (2005). Tensor Categories. ISSN 0885-4653. Portal: Mathematics v t e
Fusion_category
Tensor that describes the 4D geometry of spacetime
four-dimensional differentiable manifold M and the metric tensor is given as a covariant, second-degree, symmetric tensor on M, conventionally denoted by g. Moreover
Metric tensor (general relativity)
Metric_tensor_(general_relativity)
Symmetric monoidal closed category equipped with a dualizing object
{\displaystyle \gamma } is the symmetry of the tensor product. An object ⊥ {\displaystyle \bot } of the category C {\displaystyle {\mathcal {C}}} is called
*-autonomous_category
Operation in algebra and mathematics
endofunctor on the category of vector spaces which maps a vector space V to its tensor algebra T(V), and which maps linear maps to their tensor product. We then
Monad_(category_theory)
Graphical notation for multilinear algebra calculations
essentially the composition of functions. In the language of tensor algebra, a particular tensor is associated with a particular shape with many lines projecting
Penrose_graphical_notation
Generalization of bialgebra
{B'_{A'}}}} . Then the induced tensor functor ( α ∗ , i d , ϕ 2 F ) {\displaystyle (\alpha ^{*},id,\phi _{2}^{F})} is a tensor category equivalence between A ′
Quasi-bialgebra
Generalization of category theory
Set is the one given by the cartesian product as tensor and a singleton as unit. In fact any category with finite products can be given a monoidal structure
Higher_category_theory
Theorem in category theory
namely that every monoidal category is monoidally equivalent to a strict monoidal category. In a monoidal category, the tensor product is associative and
Mac_Lane's_coherence_theorem
Generalization of a Lie algebra
algebra in that it too is a "Lie algebra," though in a different pseudo-tensor category. Lie conformal algebras are very closely related to vertex algebras
Lie_conformal_algebra
Tensor product of algebras over a field; itself another algebra
commutative, then the tensor product is commutative as well. The tensor product turns the category of R-algebras into a symmetric monoidal category.[citation needed]
Tensor_product_of_algebras
Topics referred to by the same term
product Seven-dimensional cross product Triple product, in vector calculus Tensor product Product topology Cap product Cup product Slant product Smash product
Product
Mathematical form
of a tensor product; it captures exactly the notion of why it is that tensor products behave the way they do. More precisely, a monoidal category is the
Product_(mathematics)
Possible grammatical tense distinctions
Relative tense and absolute tense are distinct possible uses of the grammatical category of tense. Absolute tense means the grammatical expression of time
Relative_and_absolute_tense
AI accelerator ASIC by Google
The Tensor Processing Unit (TPU) is a neural processing unit (NPU) application-specific integrated circuit (ASIC) developed by Google for neural network
Tensor_Processing_Unit
Graph representation in quantum mechanics
compression and contraction are both efficient. Tensor network states are the special case in which a tensor network represents a variational wave function
Tensor_network
Word classes, largely corresponding to traditional parts of speech
features), which are properties such as tense, gender, etc. At least three criteria are used in defining syntactic categories: The type of meaning it expresses
Syntactic_category
Generalization of the concept of category that allows morphisms of multiple arity
knew what both a category and a multilinear map were". pseudo-tensor category Tom Leinster (2004). Higher Operads, Higher Categories. Cambridge University
Multicategory
Algebraic theory
statistics) naturally fit together into the structure of a unitary modular tensor category. This data is called the anyon content of the topological phase. Anyon
Algebraic theory of topological quantum information
Algebraic_theory_of_topological_quantum_information
Concept in mathematics
In mathematics, the tensor-hom adjunction is the statement that the tensor product − ⊗ X {\displaystyle -\otimes X} and hom-functor Hom ( X , − ) {\displaystyle
Tensor–hom_adjunction
the derived tensor product of M and N. In particular, π 0 ( M ⊗ R L N ) {\displaystyle \pi _{0}(M\otimes _{R}^{L}N)} is the usual tensor product of modules
Derived_tensor_product
Most general completion of a commutative square given two morphisms with same domain
below). In CRing, the category of commutative rings (a full subcategory of the category of rings), the pushout is given by the tensor product of rings A⊗CB
Pushout_(category_theory)
Topics referred to by the same term
low-voltage desk lamp Tensor muscle (disambiguation) Tensor type, in tensor analysis Category: Tensors Glossary of tensor theory Curvature tensor (disambiguation)
Tensor_(disambiguation)
Convolution
convolution acts as a tensor product for a monoidal category structure on the category of functors [𝐂,V] over some monoidal category V. Given F,G:𝐂→𝐃
Day_convolution
Vector behavior under coordinate changes
consequently a vector is called a contravariant tensor. A vector, which is an example of a contravariant tensor, has components that transform inversely to
Covariance and contravariance of vectors
Covariance_and_contravariance_of_vectors
Zero in algebraic structures
Taking a tensor product of any tensor with any zero tensor results in another zero tensor. Among tensors of a given type, the zero tensor of that type
Zero_element
Operation in graph theory
chromatic number of a tensor product, was disproved by Yaroslav Shitov (2019). The tensor product of graphs equips the category of graphs and graph homomorphisms
Tensor_product_of_graphs
category of small categories 𝐂𝐚𝐭 is a closed monoidal category in exactly two ways: with the usual categorical product and with the funny tensor product
Premonoidal_category
Type of category in category theory
as category theory, a monoidal category where the monoidal ("tensor") product is the categorical product is called a cartesian monoidal category. Any
Cartesian_monoidal_category
Ring that is also a vector space or a module
coproduct in the category of R-algebras. Tensor products The tensor product of two R-algebras is also an R-algebra in a natural way. See tensor product of algebras
Associative_algebra
Category of finite-dimension vector spaces
categorical product and a coproduct, the tensor product, which makes FinVect a compact closed category. Tensor networks are string diagrams interpreted
FinVect
the rank-finiteness theorem for modular tensor categories. Here and throughout, the rank of a fusion category refers to the number of isomorphism classes
Rank-finiteness
Topics referred to by the same term
equivalence). It may refer to: Tensor product of vector spaces, an operation on vector spaces (the original tensor product) Tensor product of modules, the same
Tensor product (disambiguation)
Tensor_product_(disambiguation)
Abbreviation in the fields of special and general relativity
relativity, a four-tensor is an abbreviation for a tensor in a four-dimensional spacetime. General four-tensors are usually written in tensor index notation
Four-tensor
Sums in quantum mathematics
symbols are matrix coefficients of the associator isomorphism in a tensor category. For example, if we are given three representations Vi, Vj, Vk of a
6-j_symbol
Grammatical category of tense
Periodic tense is a subtype of the grammatical category of tense, which encodes that the event expressed by the verb occurs within a particular period
Periodic_tense
Algebra used in 2D conformal field theories and string theory
vertex operator algebra admit a fusion tensor product operation, and form a braided tensor category. When the category of V-modules is semisimple with finitely
Vertex_operator_algebra
Property of items within the grammar of a language
grammatical elements corresponding to each of the three categories; see Tense–aspect–mood. Categories may be marked on words by means of inflection. In English
Grammatical_category
Method of utilizing water in magnetic resonance imaging
more gradient directions, sufficient to compute the diffusion tensor. The diffusion tensor model is a rather simple model of the diffusion process, assuming
Diffusion-weighted magnetic resonance imaging
Diffusion-weighted_magnetic_resonance_imaging
Scalar measure of the rotational inertia with respect to a fixed axis of rotation
point about which the tensor is calculated and δij is the Kronecker delta. Note that, by the definition, 𝐈 is a symmetric tensor. The diagonal elements
Moment_of_inertia
Mathematical operation
\mathbf {R} } be a multilinear form on W (also known as a tensor – not to be confused with a tensor field – of rank (0, s), where s is the number of factors
Pullback (differential geometry)
Pullback_(differential_geometry)
Ontology for descriptive linguistics
include phrasal categories Grammatical category, a grammatical feature such as tense, gender, etc. The definition of linguistic categories is a major concern
Linguistic_categories
contravariant (ai) indices on the tensor, r+s (a number called the rank of the tensor). Some physical quantities are represented by tensors not all of whose components
Mathematics of general relativity
Mathematics_of_general_relativity
Category whose objects are rings and whose morphisms are ring homomorphisms
single object—is a preadditive category). The category of rings is a symmetric monoidal category with the tensor product of rings ⊗Z as the monoidal product
Category_of_rings
Canadian mathematician
tensor calculus, I". Advances in Mathematics. 88: 55–112. doi:10.1016/0001-8708(91)90003-P.; Joyal, A.; Street, R. (1993). "Braided Tensor Categories"
André_Joyal
Class of mathematical software
similar to MATLAB and GNU Octave, but designed specifically for tensors. Tensor is a tensor package written for the Mathematica system. It provides many
Tensor_software
Type of order at absolute zero
by unitary modular tensor categories. 2+1D bosonic topological orders with symmetry G are classified by G-crossed tensor categories. 2+1D bosonic/fermionic
Topological_order
"Smallest" commutative algebra that contains a vector space
the tensor algebra, an element of the symmetric algebra is not a tensor, and, in particular, is not a symmetric tensor. However, symmetric tensors are
Symmetric_algebra
Topics referred to by the same term
The Memory Test Computer built as part of Project Whirlwind Modular tensor category, an algebraic structure appearing in mathematical physics Meerut City
MTC
Variant of the notion of the center of a monoid, group, or ring to a category
given by the tensor product. The category Z ( C ) {\displaystyle {\mathcal {Z(C)}}} becomes a braided monoidal category with the tensor product on objects
Center_(category_theory)
History of maths
This is a timeline of category theory and related mathematics. Its scope ("related mathematics") is taken as: Categories of abstract algebraic structures
Timeline of category theory and related mathematics
Timeline_of_category_theory_and_related_mathematics
Topic in mathematics
the tensor product of V with the complex numbers (thought of as a 2-dimensional vector space over the reals): Vℂ=V⊗ℝℂ. The subscript, ℝ, on the tensor product
Complexification
field. Tensors also have extensive applications in physics: Electromagnetic tensor (or Faraday's tensor) in electromagnetism Finite deformation tensors for
Introduction to the mathematics of general relativity
Introduction_to_the_mathematics_of_general_relativity
Mathematical operation on matrices
specialization of the tensor product (which is denoted by the same symbol) from vectors to matrices and gives the matrix of the tensor product linear map
Kronecker_product
Hypothetical elementary particle that mediates gravity
stress–energy tensor, a second-order tensor (compared with electromagnetism's spin-1 photon, the source of which is the four-current, a first-order tensor). Additionally
Graviton
Symmetric monoidal category with a special involution
converse. The category FdHilb of finite-dimensional Hilbert spaces is a dagger symmetric monoidal category where the tensor is the usual tensor product of
Dagger symmetric monoidal category
Dagger_symmetric_monoidal_category
Collection of knots that do not intersect, but may be linked
algebraic structure. Isotopy classes of tangles form a tensor category, where for the category structure, one can compose two tangles if the bottom end
Link_(knot_theory)
Algebraic operation on coordinate vectors
between a tensor of order n and a tensor of order m is a tensor of order n+m−2 (more generally, each dot reduces the order by 2), see Tensor contraction
Dot_product
Conserved physical quantity; rotational analogue of linear momentum
as an anti-symmetric second order tensor, with components ωij. The relation between the two anti-symmetric tensors is given by the moment of inertia which
Angular_momentum
θC=(−1)n+1idC. The category of projective modules over a commutative ring. In this category, the monoidal structure is the tensor product, the dual object
Ribbon_category
Duality between a group and its representations
the tensor product of representations. However, irreducible representations of G in general fail to form a group, or even a monoid, because a tensor product
Tannaka–Krein_duality
Differentiable manifold with nondegenerate metric tensor
Y∈TpM. Given a metric tensor g on an n-dimensional real manifold, the quadratic form q(x) = g(x, x) associated with the metric tensor applied to each vector
Pseudo-Riemannian_manifold
Literary technique used in religious texts
to happen that they are referred to in the past tense as if they had already happened. The category of "prophetic perfect" was already suggested by medieval
Prophetic_perfect_tense
Dravidian language
suffixes are used to mark noun class, number, and case, verb tense and other grammatical categories. Tamil's standard metalinguistic terminology and scholarly
Tamil_language
Generalization of associativity properties
P(n)⊗P(k1)⊗⋯⊗P(kn)→P(k1+⋯+kn) in C (where ⊗ denotes the tensor product of the monoidal category), and the actions of the symmetric group elements are given
Operad
Algebra associated to any vector space
the r-homogeneous alternating tensor subspace. On the other hand, the image 𝒜(T(V)) is always the alternating tensor graded subspace (not yet an algebra
Exterior_algebra
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TENSOR CATEGORY
TENSOR CATEGORY
Surname or Lastname
English
English : variant of Tennyson.
Surname or Lastname
English
English : variant spelling of Ensor.
Boy/Male
Muslim
Winner
Surname or Lastname
English
English : patronymic from Penn 3 or Paine 1.English : habitational name from Penson in Devon.
Boy/Male
Polish Spanish
Surname or Lastname
English
English : patronymic from a reduced form of the personal name Steven.English : habitational name from a place in Derbyshire, recorded in Domesday Book as Steintune, later as Steineston, from the Old Norse personal name Steinn (meaning ‘stone’) + Old English tūn ‘enclosure’, ‘settlement’.Variant of Steenson 2.
Male
Greek
(ÎœÎντωÏ) Greek name derived from the word menos, MENTOR means "spirit." In mythology, this is the name of the son of Ãlkimos.
Boy/Male
French
Works in iron.
Surname or Lastname
German
German : variant of Tanner 2.English : from Old French teneor, teneur, tenor, ‘holder of a tenement’, hence an equivalent of Tennant.
Surname or Lastname
English
English : habitational name for someone from Edensor in Derbyshire, which derives its name from the genitive case of the Old English personal name Ēadhūn (see Eden 1) + Old English ofer ‘ridge’.
Surname or Lastname
French
French : unexplained.English : unexplained.Possibly a respelling of Menter, an unexplained name of German origin.
Surname or Lastname
English
English : patronymic from the personal name Henn(e), a short form of Henry 1, Hayne (see Hain 2), or Hendy.Irish : Anglicized form of Gaelic Ó hAmhsaigh (see Hampson 2).
Male
English
English surname transferred to forename use, BENSON means "son of Ben."
Surname or Lastname
English
English : variant of Windsor. This is the spelling used for places so named in Devon and Hampshire.Perhaps also an Americanized spelling of German Winzer.
Male
Scandinavian
Scandinavian form of Latin Theodorus, TEODOR means "gift of God."
Surname or Lastname
English
English : probably a variant of Manser.
Surname or Lastname
English
English : unexplained.
Surname or Lastname
English (mainly Yorkshire)
English (mainly Yorkshire) : nickname for a peasant who gave himself airs and graces, from Anglo-Norman French segneur ‘lord’ (Latin senior ‘elder’).English and Dutch : distinguishing nickname for the elder of two bearers of the same personal name (for example, a father and son or two brothers), from Latin senior ‘elder’.
Surname or Lastname
English
English : patronymic from the medieval personal name Benne, a pet form of Benedict (see Benn).English : habitational name from a place in Oxfordshire named Benson, from Old English Benesingtūn ‘settlement (Old English tūn) associated with Benesa’, a personal name of obscure origin, perhaps a derivative of Bana meaning ‘slayer’.Jewish (Ashkenazic) : patronymic composed of a pet form of the personal name Beniamin (see Bien, Benjamin) + German Sohn ‘son’.Scandinavian : altered form of such names as Bengtsson, Bendtsen, patronymics from Bengt, Bendt, etc., Scandinavian forms of Benedict.
Surname or Lastname
English
English : perhaps an altered spelling of Janson.Respelling of Danish, Norwegian, and North German Jensen.
TENSOR CATEGORY
TENSOR CATEGORY
TENSOR CATEGORY
TENSOR CATEGORY
TENSOR CATEGORY
TENSOR CATEGORY
TENSOR CATEGORY
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