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Internal Bialgebroid
In mathematics, an internal bialgebroid is a structure which generalizes the notion of an associative bialgebroid to the setup where the ambient symmetric monoidal category of vector spaces is replaced by any abstract symmetric monoidal category (''C'', \otimes, ''I'',''s'') admitting coequalizers commuting with the monoidal product \otimes. It consists of two monoids in the monoidal category (''C'', \otimes, ''I''), namely the base monoid A and the total monoid H, and several structure morphisms involving A and H as first axiomatized by G. Böhm.Gabriella Böhm, Internal bialgebroids, entwining structures and corings, in: Algebraic structures and their representations, 207–226, Contemp. Math. 376, Amer. Math. Soc. 2005Cornell University Library, retrieved 11 September, 2017/ref> The coequalizers are needed to introduce the tensor product \otimes_A of (internal) bimodules over the base monoid; this tensor product is consequently (a part of) a monoidal structure on the category ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associative Bialgebroid
In mathematics, if L is an associative algebra over some ground field ''k'', then a left associative L-bialgebroid is another associative ''k''-algebra H together with the following additional maps: an algebra map \alpha:L\to H called the source map, an algebra map \beta:L^\to H called the target map, so that the elements of the images of \alpha and \beta commute in H, therefore inducing an L-bimodule structure on H via the rule a.h.b = \alpha(a)\beta(b) h for a,b\in L, h\in H; an L-bimodule morphism \Delta:H\to H\otimes_L H which is required to be a counital coassociative comultiplication on H in the monoidal category of L-bimodules with monoidal product \otimes_L. The corresponding counit \varepsilon:H\to L is required to be a left character (equivalently, the map H\otimes L\ni h\otimes \ell\mapsto \varepsilon(h\alpha(\ell))\in L must be a left action extending the multiplication L\otimes L\to L along \alpha\otimes\mathrm_L). Furthermore, a compatibility between the comultiplic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Spaces
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space. Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems of linear equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Monoidal Category
In category theory, a branch of mathematics, a symmetric monoidal category is a monoidal category (i.e. a category in which a "tensor product" \otimes is defined) such that the tensor product is symmetric (i.e. A\otimes B is, in a certain strict sense, naturally isomorphic to B\otimes A for all objects A and B of the category). One of the prototypical examples of a symmetric monoidal category is the category of vector spaces over some fixed field ''k,'' using the ordinary tensor product of vector spaces. Definition A symmetric monoidal category is a monoidal category (''C'', ⊗, ''I'') such that, for every pair ''A'', ''B'' of objects in ''C'', there is an isomorphism s_: A \otimes B \to B \otimes A that is natural in both ''A'' and ''B'' and such that the following diagrams commute: *The unit coherence: *: *The associativity coherence: *: *The inverse law: *: In the diagrams above, ''a'', ''l'' , ''r'' are the associativity isomorphism, the left unit isomorphism, and the right un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coequalizer
In category theory, a coequalizer (or coequaliser) is a generalization of a quotient by an equivalence relation to objects in an arbitrary category. It is the categorical construction dual to the equalizer. Definition A coequalizer is a colimit of the diagram consisting of two objects ''X'' and ''Y'' and two parallel morphisms ''f'', ''g'' : ''X'' → ''Y''. More explicitly, a coequalizer can be defined as an object ''Q'' together with a morphism ''q'' : ''Y'' → ''Q'' such that ''q'' ∘ ''f'' = ''q'' ∘ ''g''. Moreover, the pair (''Q'', ''q'') must be universal in the sense that given any other such pair (''Q''′, ''q''′) there exists a unique morphism ''u'' : ''Q'' → ''Q''′ such that ''u'' ∘ ''q'' = ''q''′. This information can be captured by the following commutative diagram: As with all universal constructions, a coequalizer, if it exists, is unique up to a unique isomorphism (this is why, by abuse of language, one sometimes speaks of " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monoid (category Theory)
In category theory, a branch of mathematics, a monoid (or monoid object, or internal monoid, or algebra) in a monoidal category is an object ''M'' together with two morphisms * ''μ'': ''M'' ⊗ ''M'' → ''M'' called ''multiplication'', * ''η'': ''I'' → ''M'' called ''unit'', such that the pentagon diagram : and the unitor diagram : commute. In the above notation, is the identity morphism of , is the unit element and α, λ and ρ are respectively the associativity, the left identity and the right identity of the monoidal category C. Dually, a comonoid in a monoidal category C is a monoid in the dual category Cop. Suppose that the monoidal category C has a symmetry ''γ''. A monoid ''M'' in C is commutative when . Examples * A monoid object in Set, the category of sets (with the monoidal structure induced by the Cartesian product), is a monoid in the usual sense. * A monoid object in Top, the category of topological spaces (with the monoidal structure induced by the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bialgebra
In mathematics, a bialgebra over a field ''K'' is a vector space over ''K'' which is both a unital associative algebra and a counital coassociative coalgebra. The algebraic and coalgebraic structures are made compatible with a few more axioms. Specifically, the comultiplication and the counit are both unital algebra homomorphisms, or equivalently, the multiplication and the unit of the algebra both are coalgebra morphisms. (These statements are equivalent since they are expressed by the same commutative diagrams.) Similar bialgebras are related by bialgebra homomorphisms. A bialgebra homomorphism is a linear map that is both an algebra and a coalgebra homomorphism. As reflected in the symmetry of the commutative diagrams, the definition of bialgebra is self-dual, so if one can define a dual of ''B'' (which is always possible if ''B'' is finite-dimensional), then it is automatically a bialgebra. Formal definition (''B'', ∇, η, Δ, ε) is a bialgebra over ''K'' if it h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
