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Distributive Law Between Monads
In category theory, an abstract branch of mathematics, distributive laws between monads are a way to express abstractly that two algebraic structures distribute one over the other one. Suppose that (S,\mu^S,\eta^S) and (T,\mu^T,\eta^T) are two monads on a category C. In general, there is no natural monad structure on the composite functor ''ST''. However, there is a natural monad structure on the functor ''ST'' if there is a distributive law of the monad ''S'' over the monad ''T''. Formally, a distributive law of the monad ''S'' over the monad ''T'' is a natural transformation :l:TS\to ST such that the diagrams : : commute. This law induces a composite monad ''ST'' with * as multiplication: STST\xrightarrowSSTT\xrightarrowST, * as unit: 1\xrightarrowST. See also * distributive law In mathematics, the distributive property of binary operations generalizes the distributive ... [...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 compos ... [...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] |
Distributive Law
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality x \cdot (y + z) = x \cdot y + x \cdot z is always true in elementary algebra. For example, in elementary arithmetic, one has 2 \cdot (1 + 3) = (2 \cdot 1) + (2 \cdot 3). One says that multiplication ''distributes'' over addition. This basic property of numbers is part of the definition of most algebraic structures that have two operations called addition and multiplication, such as complex numbers, polynomials, matrices, rings, and fields. It is also encountered in Boolean algebra and mathematical logic, where each of the logical and (denoted \,\land\,) and the logical or (denoted \,\lor\,) distributes over the other. Definition Given a set S and two binary operators \,*\, and \,+\, on S, *the operation \,*\, is over (or with respect to) \,+\, if, given any elements x, y, \text z of S, x * (y + z) = (x * y) + (x * z); *the operation \,*\, is over ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monad (category Theory)
In category theory, a branch of mathematics, a monad (also triple, triad, standard construction and fundamental construction) is a monoid in the category of endofunctors. An endofunctor is a functor mapping a category to itself, and a monad is an endofunctor together with two natural transformations required to fulfill certain coherence conditions. Monads are used in the theory of pairs of adjoint functors, and they generalize closure operators on partially ordered sets to arbitrary categories. Monads are also useful in the theory of datatypes and in functional programming languages, allowing languages with non-mutable states to do things such as simulate for-loops; see Monad (functional programming). Introduction and definition A monad is a certain type of endofunctor. For example, if F and G are a pair of adjoint functors, with F left adjoint to G, then the composition G \circ F is a monad. If F and G are inverse functors, the corresponding monad is the identity functor. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Informally, the notion of a natural transformation states that a particular map between functors can be done consistently over an entire category. Indeed, this intuition can be formalized to define so-called functor categories. Natural transformations are, after categories and functors, one of the most fundamental notions of category theory and consequently appear in the majority of its applications. Definition If F and G are functors between the categories C and D , then a natural transformation \eta from F to G is a family of morphisms that satisfies two requirements. # The natural transformation must associate, to every object X in C, a morphism \eta_X : F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distributive Law Monads Mult1
Distributive may refer to: *Distributive property, in algebra, logic and mathematics *Distributive pronoun and distributive adjective (determiner), in linguistics *Distributive case The distributive case (abbreviated ) is used on nouns for the meanings of ''per'' or ''each.'' In Hungarian it is ''-nként'' and expresses the manner when something happens to each member of a set one by one (e.g., ''fejenként'' "per head", ''e ..., in linguistics * Distributive numeral, in linguistics {{Disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distributive Law Monads Unit1
Distributive may refer to: *Distributive property, in algebra, logic and mathematics *Distributive pronoun and distributive adjective (determiner), in linguistics *Distributive case The distributive case (abbreviated ) is used on nouns for the meanings of ''per'' or ''each.'' In Hungarian it is ''-nként'' and expresses the manner when something happens to each member of a set one by one (e.g., ''fejenként'' "per head", ''e ..., in linguistics * Distributive numeral, in linguistics {{Disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distributive Law Monads Mult2
Distributive may refer to: *Distributive property, in algebra, logic and mathematics *Distributive pronoun and distributive adjective (determiner), in linguistics *Distributive case The distributive case (abbreviated ) is used on nouns for the meanings of ''per'' or ''each.'' In Hungarian it is ''-nként'' and expresses the manner when something happens to each member of a set one by one (e.g., ''fejenként'' "per head", ''e ..., in linguistics * Distributive numeral, in linguistics {{Disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distributive Law Monads Unit2
Distributive may refer to: *Distributive property, in algebra, logic and mathematics *Distributive pronoun and distributive adjective (determiner), in linguistics *Distributive case The distributive case (abbreviated ) is used on nouns for the meanings of ''per'' or ''each.'' In Hungarian it is ''-nként'' and expresses the manner when something happens to each member of a set one by one (e.g., ''fejenként'' "per head", ''e ..., in linguistics * Distributive numeral, in linguistics {{Disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
