|
Multicategory
In mathematics (especially category theory), a multicategory is a generalization of the concept of category that allows morphisms of multiple arity. If morphisms in a category are viewed as analogous to functions, then morphisms in a multicategory are analogous to functions of several variables. Multicategories are also sometimes called operads, or colored operads. Definition A (non-symmetric) multicategory consists of * a collection (often a proper class) of ''objects''; * for every finite sequence (X_i)_ of objects (for von Neumann ordinal n \in \mathbb) and object ''Y'', a set of ''morphisms'' from (X_i)_ to ''Y''; and * for every object ''X'', a special identity morphism (with ''n'' = 1) from ''X'' to ''X''. Additionally, there are composition operations: Given a sequence of sequences ((X_)_)_ of objects, a sequence (Y_j)_ of objects, and an object ''Z'': if * for each j \in m, ''f''''j'' is a morphism from (X_)_ to ''Y''''j''; and * ''g'' is a morphism from (Y_j)_ to ''Z'': t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operad
In mathematics, an operad is a structure that consists of abstract operations, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose these operations. Given an operad O, one defines an ''algebra over O'' to be a set together with concrete operations on this set which behave just like the abstract operations of O. For instance, there is a Lie operad L such that the algebras over L are precisely the Lie algebras; in a sense L abstractly encodes the operations that are common to all Lie algebras. An operad is to its algebras as a group is to its group representations. History Operads originate in algebraic topology; they were introduced to characterize iterated loop spaces by J. Michael Boardman and Rainer M. Vogt in 1969 and by J. Peter May in 1970. The word "operad" was created by May as a portmanteau of "operations" and "monad" (and also because his mother was an opera singer). Interest in operads was consid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operad Theory
In mathematics, an operad is a structure that consists of abstract Operation (mathematics), operations, each one having a fixed finite number of inputs (arguments) and one output, as well as a specification of how to compose these operations. Given an operad O, one defines an ''algebra over O'' to be a set together with concrete operations on this set which behave just like the abstract operations of O. For instance, there is a Lie operad L such that the algebras over L are precisely the Lie algebras; in a sense L abstractly encodes the operations that are common to all Lie algebras. An operad is to its algebras as a Group (mathematics), group is to its group representations. History Operads originate in algebraic topology; they were introduced to characterize iterated loop spaces by Michael Boardman, J. Michael Boardman and Rainer M. Vogt in 1969 and by J. Peter May in 1970. The word "operad" was created by May as a portmanteau of "operations" and "monad (category theory), monad ... [...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] |
Multilinear Operator
In mathematics, a tensor is an mathematical object, algebraic object that describes a Multilinear map, multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as Vector (mathematics and physics), vectors, Scalar (mathematics), scalars, and even other tensors. There are many types of tensors, including Scalar (mathematics), scalars and Vector (mathematics and physics), vectors (which are the simplest tensors), dual vectors, multilinear map, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined Tensor (intrinsic definition), independent of any Basis (linear algebra), basis, although they are often referred to by their components in a basis related to a particular coordinate system. Tensors have become important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics (St ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean Bénabou
Jean Bénabou (1932 – 11 February 2022) was a Moroccan-born French mathematician, known for his contributions to category theory. He directed the Research Seminar in Category Theory at the Institut Henri Poincaré and Institut de mathématiques de Jussieu – Paris Rive Gauche, Institut de mathématiques de Jussieu from 1969 to 2001. Bénabou died on 11 February 2022. Partial bibliography * * * * * * See also * References 1932 births 2022 deaths 20th-century French mathematicians Category theorists Moroccan emigrants to France People from Rabat External links * {{France-mathematician-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jim Lambek
Joachim "Jim" Lambek (5 December 1922 – 23 June 2014) was a German-born Canadian mathematician. He was Peter Redpath Emeritus Professor of Pure Mathematics at McGill University, where he earned his PhD degree in 1950 with Hans Zassenhaus as advisor. Biography Lambek was born in Leipzig, Germany, where he attended a Gymnasium. He came to England in 1938 as a refugee on the ''Kindertransport''. From there he was interned as an enemy alien and deported to a prison work camp in New Brunswick, Canada. There, he began in his spare time a mathematical apprenticeship with Fritz Rothberger, also interned, and wrote the McGill Junior Matriculation in fall of 1941. In the spring of 1942, he was released and settled in Montreal, where he entered studies at McGill University, graduating with an honours mathematics degree in 1945 and an MSc a year later. In 1950, he completed his doctorate under Hans Zassenhaus becoming McGill's first PhD in mathematics. Lambek became assistant profes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Associative Algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplication operations together give ''A'' the structure of a ring; the addition and scalar multiplication operations together give ''A'' the structure of a vector space over ''K''. In this article we will also use the term ''K''-algebra to mean an associative algebra over the field ''K''. A standard first example of a ''K''-algebra is a ring of square matrices over a field ''K'', with the usual matrix multiplication. A commutative algebra is an associative algebra that has a commutative multiplication, or, equivalently, an associative algebra that is also a commutative ring. In this article associative algebras are assumed to have a multiplicative identity, denoted 1; they are sometimes called unital associative algebras for clarification. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics given the need to associate pictures with coordinates (e.g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Theory
In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topology, the theory has also been used in other areas of mathematics such as algebraic geometry (e.g., A1 homotopy theory) and category theory (specifically the study of higher categories). Concepts Spaces and maps In homotopy theory and algebraic topology, the word "space" denotes a topological space. In order to avoid pathologies, one rarely works with arbitrary spaces; instead, one requires spaces to meet extra constraints, such as being compactly generated, or Hausdorff, or a CW complex. In the same vein as above, a "map" is a continuous function, possibly with some extra constraints. Often, one works with a pointed space -- that is, a space with a "distinguished point", called a basepoint. A pointed map is then a map which preserv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up to homeomorphism, though usually most classify up to Homotopy#Homotopy equivalence and null-homotopy, homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Main branches of algebraic topology Below are some of the main areas studied in algebraic topology: Homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy gro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Higher Category Theory
In mathematics, higher category theory is the part of category theory at a ''higher order'', which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities. Higher category theory is often applied in algebraic topology (especially in homotopy theory), where one studies algebraic invariants of spaces, such as their fundamental weak ∞-groupoid. Strict higher categories An ordinary category has objects and morphisms, which are called 1-morphisms in the context of higher category theory. A 2-category generalizes this by also including 2-morphisms between the 1-morphisms. Continuing this up to ''n''-morphisms between (''n'' − 1)-morphisms gives an ''n''-category. Just as the category known as Cat, which is the category of small categories and functors is actually a 2-category with natural transformations as its 2-morphisms, the category ''n''-Cat of (small) ''n''-categories is actually a ... [...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 abstractio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
