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Opetope
In category theory, a branch of mathematics, an opetope, a portmanteau of "operation" and "polytope", is a shape that captures higher-dimensional substitutions. It was introduced by John C. Baez and James Dolan so that they could define a weak ''n''-category as a certain presheaf In mathematics, a sheaf (: sheaves) is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the d ... on the category of opetopes. See also * higher-order operad References * External links * Category theory {{categorytheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weak N-category
In category theory in mathematics, a weak ''n''-category is a generalization of the notion of strict ''n''-category where composition and identities are not strictly associative and unital, but only associative and unital up to coherent equivalence or coherent isomorphism. A weak 0-category is just a set, and a weak 1-category is a ordinarily category. This generalisation only becomes noticeable at dimensions two and above where weak 2-, 3- and 4-categories are typically referred to as bicategories, tricategories, and tetracategories. History There is much work to determine what the coherence laws for weak ''n''-categories should be. Weak ''n''-categories have become the main object of study in higher category theory. There are basically two classes of theories: those in which the higher cells and higher compositions are realized algebraically (most remarkably Michael Batanin's theory of weak higher categories) and those in which more topological models are used (e.g. a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polytope
In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an -dimensional polytope or -polytope. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a -polytope consist of -polytopes that may have -polytopes in common. Some theories further generalize the idea to include such objects as unbounded apeirotopes and tessellations, decompositions or tilings of curved manifolds including spherical polyhedra, and set-theoretic abstract polytopes. Polytopes of more than three dimensions were first discovered by Ludwig Schläfli before 1853, who called such a figure a polyschem. The German term ''Polytop'' was coined by the mathematician Reinhold Hoppe, and was introduced to English mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Higher-order 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 1968 and by J. Peter May in 1972. Martin Markl, Steve Shnider, and Jim Stasheff write in their book on operads:"Operads in Algebra, Topology and Physics": Martin Markl, Steve Shnider, Jim Stasheff, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Category Theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory is used in most areas of mathematics. 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 space (other), quotient spaces, direct products, completion, and duality (mathematics), duality. Many areas of computer science also rely on category theory, such as functional programming and Semantics (computer science), semantics. A category (mathematics), category is formed by two sorts of mathematical object, objects: the object (category theory), objects of the category, and the morphisms, which relate two objects called the ''source'' and the ''target'' of the morphism. Metapho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operation (algebra)
In mathematics, a basic algebraic operation is any one of the common operations of elementary algebra, which include addition, subtraction, multiplication, division, raising to a whole number power, and taking roots (fractional power). These operations may be performed on numbers, in which case they are often called ''arithmetic operations''. They may also be performed, in a similar way, on variables, algebraic expressions, and more generally, on elements of algebraic structures, such as groups and fields. An algebraic operation may also be defined more generally as a function from a Cartesian power of a given set to the same set. The term ''algebraic operation'' may also be used for operations that may be defined by compounding basic algebraic operations, such as the dot product. In calculus and mathematical analysis, ''algebraic operation'' is also used for the operations that may be defined by purely algebraic methods. For example, exponentiation with an integer or rational e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shape
A shape is a graphics, graphical representation of an object's form or its external boundary, outline, or external Surface (mathematics), surface. It is distinct from other object properties, such as color, Surface texture, texture, or material type. In geometry, ''shape'' excludes information about the object's Position (geometry), position, size, Orientation (geometry), orientation and chirality. A ''figure'' is a representation including both shape and size (as in, e.g., figure of the Earth). A plane shape or plane figure is constrained to lie on a ''plane (geometry), plane'', in contrast to ''solid figure, solid'' 3D shapes. A two-dimensional shape or two-dimensional figure (also: 2D shape or 2D figure) may lie on a more general curved ''surface (mathematics), surface'' (a two-dimensional space). Classification of simple shapes Some simple shapes can be put into broad categories. For instance, polygons are classified according to their number of edges as triangles, qua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Substitution (algebra)
A substitution is a syntactic transformation on formal expressions. To ''apply'' a substitution to an expression means to consistently replace its variable, or placeholder, symbols with other expressions. The resulting expression is called a ''substitution instance'', or ''instance'' for short, of the original expression. Propositional logic Definition Where ''ψ'' and ''φ'' represent formulas of propositional logic, ''ψ'' is a ''substitution instance'' of ''φ'' if and only if ''ψ'' may be obtained from ''φ'' by substituting formulas for propositional variables in ''φ'', replacing each occurrence of the same variable by an occurrence of the same formula. For example: ::''ψ:'' (R → S) & (T → S) is a substitution instance of ::''φ:'' P & Q That is, ''ψ'' can be obtained by replacing P and Q in ''φ'' with (R → S) and (T → S) respectively. Similarly: ::''ψ:'' (A ↔ A) ↔ (A ↔ A) is a substitution instance of: ::''φ:'' (A ↔ A) since ''ψ'' can be obta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John C
John is a common English name and surname: * John (given name) * John (surname) John may also refer to: New Testament Works * Gospel of John, a title often shortened to John * First Epistle of John, often shortened to 1 John * Second Epistle of John, often shortened to 2 John * Third Epistle of John, often shortened to 3 John People * John the Baptist (died ), regarded as a prophet and the forerunner of Jesus Christ * John the Apostle (died ), one of the twelve apostles of Jesus Christ * John the Evangelist, assigned author of the Fourth Gospel, once identified with the Apostle * John of Patmos, also known as John the Divine or John the Revelator, the author of the Book of Revelation, once identified with the Apostle * John the Presbyter, a figure either identified with or distinguished from the Apostle, the Evangelist and John of Patmos Other people with the given name Religious figures * John, father of Andrew the Apostle and Saint Peter * Pope John ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Presheaf (category Theory)
In category theory, a branch of mathematics, a presheaf on a category C is a functor F\colon C^\mathrm\to\mathbf. If C is the poset of open sets in a topological space, interpreted as a category, then one recovers the usual notion of presheaf on a topological space. A morphism of presheaves is defined to be a natural transformation of functors. This makes the collection of all presheaves on C into a category, and is an example of a functor category. It is often written as \widehat = \mathbf^ and it is called the category of presheaves on C. A functor into \widehat is sometimes called a profunctor. A presheaf that is naturally isomorphic to the contravariant hom-functor Hom(–, ''A'') for some object ''A'' of C is called a representable presheaf. Some authors refer to a functor F\colon C^\mathrm\to\mathbf as a \mathbf-valued presheaf. Examples * A simplicial set is a Set-valued presheaf on the simplex category C=\Delta. * A directed multigraph is a presheaf on the catego ... [...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] |
