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Spherical Category
In category theory, a branch of mathematics, a spherical category is a pivotal category (a monoidal category with traces) in which left and right traces coincide. Spherical fusion categories give rise to a family of three-dimensional topological state sum models (a particular formulation of a topological quantum field theory In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants. Although TQFTs were invented by physicists, they are also of mathem ...), the Turaev-Viro model, or rather Turaev-Viro-Barrett-Westbury model. References Category theory {{categorytheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, category theory is used in almost all areas of mathematics, and in some areas of computer science. In particular, many constructions of new mathematical objects from previous ones, that appear similarly in several contexts are conveniently expressed and unified in terms of categories. Examples include quotient spaces, direct products, completion, and duality. A category is formed by two sorts of objects: the objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. One often says that a morphism is an ''arrow'' that ''maps'' its source to its target. Morphisms can be ''composed'' if the target of the first morphism equals the source of the second one, and morphism com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pivotal Category
In category theory, a branch of mathematics, a dual object is an analogue of a dual vector space from linear algebra for objects in arbitrary monoidal categories. It is only a partial generalization, based upon the categorical properties of duality for finite-dimensional vector spaces. An object admitting a dual is called a dualizable object. In this formalism, infinite-dimensional vector spaces are not dualizable, since the dual vector space ''V''∗ doesn't satisfy the axioms. Often, an object is dualizable only when it satisfies some finiteness or compactness property. A category in which each object has a dual is called autonomous or rigid. The category of finite-dimensional vector spaces with the standard tensor product is rigid, while the category of all vector spaces is not. Motivation Let ''V'' be a finite-dimensional vector space over some field ''K''. The standard notion of a dual vector space ''V''∗ has the following property: for any ''K''-vector spaces ''U'' an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoidal Category
In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor :\otimes : \mathbf \times \mathbf \to \mathbf that is associative up to a natural isomorphism, and an object ''I'' that is both a left and right identity for ⊗, again up to a natural isomorphism. The associated natural isomorphisms are subject to certain coherence conditions, which ensure that all the relevant diagrams commute. The ordinary tensor product makes vector spaces, abelian groups, ''R''-modules, or ''R''-algebras into monoidal categories. Monoidal categories can be seen as a generalization of these and other examples. Every ( small) monoidal category may also be viewed as a "categorification" of an underlying monoid, namely the monoid whose elements are the isomorphism classes of the category's objects and whose binary operation is given by the category's tensor product. A rather different application, of which monoidal categories can be considered an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Traced Monoidal Category
In category theory, a traced monoidal category is a category with some extra structure which gives a reasonable notion of feedback. A traced symmetric monoidal category is a symmetric monoidal category C together with a family of functions :\mathrm^U_:\mathbf(X\otimes U,Y\otimes U)\to\mathbf(X,Y) called a ''trace'', satisfying the following conditions: * naturality in X: for every f:X\otimes U\to Y\otimes U and g:X'\to X, ::\mathrm^U_(f \circ (g\otimes \mathrm_U)) = \mathrm^U_(f) \circ g * naturality in Y: for every f:X\otimes U\to Y\otimes U and g:Y\to Y', ::\mathrm^U_((g\otimes \mathrm_U) \circ f) = g \circ \mathrm^U_(f) * dinaturality in U: for every f:X\otimes U\to Y\otimes U' and g:U'\to U ::\mathrm^U_((\mathrm_Y\otimes g) \circ f)=\mathrm^_(f \circ (\mathrm_X\otimes g)) * vanishing I: for every f:X \otimes I \to Y \otimes I, (with \rho_X \colon X\otimes I\cong X being the right unitor), ::\mathrm^I_(f)=\rho_Y \circ f \circ \rho_X^ * vanishing II: for every f:X\oti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Advances In Mathematics
''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes. At the origin, the journal aimed at publishing articles addressed to a broader "mathematical community", and not only to mathematicians in the author's field. Herbert Busemann writes, in the preface of the first issue, "The need for expository articles addressing either all mathematicians or only those in somewhat related fields has long been felt, but little has been done outside of the USSR. The serial publication ''Advances in Mathematics'' was created in response to this demand." Abstracting and indexing The journal is abstracted and indexed in: * [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fusion Category
In mathematics, a fusion category is a category that is rigid, semisimple, k-linear, monoidal and has only finitely many isomorphism classes of simple objects, such that the monoidal unit is simple. If the ground field k is algebraically closed, then the latter is equivalent to \mathrm(1,1)\cong k by Schur's lemma. Examples * Representation Category of a finite group Reconstruction Under Tannaka–Krein duality In mathematics, Tannaka–Krein duality theory concerns the interaction of a compact topological group and its category of linear representations. It is a natural extension of Pontryagin duality, between compact and discrete commutative topologi ..., every fusion category arises as the representations of a weak Hopf algebra. References Category theory {{categorytheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Quantum Field Theory
In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants. Although TQFTs were invented by physicists, they are also of mathematical interest, being related to, among other things, knot theory and the theory of four-manifolds in algebraic topology, and to the theory of moduli spaces in algebraic geometry. Donaldson, Jones, Witten, and Kontsevich have all won Fields Medals for mathematical work related to topological field theory. In condensed matter physics, topological quantum field theories are the low-energy effective theories of topologically ordered states, such as fractional quantum Hall states, string-net condensed states, and other strongly correlated quantum liquid states. Overview In a topological field theory, correlation functions do not depend on the metric of spacetime. This means that the theory is not sensitive to changes in the shape ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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