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Tricategory
In mathematics, a tricategory is a kind of structure of category theory studied in higher-dimensional category theory. Whereas a weak 2-category is said to be a ''bicategory'', a weak 3-category is said to be a ''tricategory'' (Gordon, Power & Street 1995; Baez & Dolan 1996; Leinster 1998). Tetracategories are the corresponding notion in dimension four. Dimensions beyond three are seen as increasingly significant to the relationship between knot theory and physics. John Baez John Carlos Baez (; born June 12, 1961) is an American mathematical physicist and a professor of mathematics at the University of California, Riverside (UCR) in Riverside, California. He has worked on spin foams in loop quantum gravity, appl ..., R. Gordon, A. J. Power and Ross Street have done much of the significant work with categories beyond bicategories thus far. See also * Weak ''n''-category References External linksThe Dimensional Ladder [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Higher-dimensional 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] |
Bicategory
In mathematics, a bicategory (or a weak 2-category) is a concept in category theory used to extend the notion of category to handle the cases where the composition of morphisms is not (strictly) associative, but only associative ''up to'' an isomorphism. The notion was introduced in 1967 by Jean Bénabou. Bicategories may be considered as a weakening of the definition of 2-categories. A similar process for 3-categories leads to tricategories, and more generally to weak ''n''-categories for ''n''-categories. Definition Formally, a bicategory B consists of: * objects ''a'', ''b'', ... called 0-''cells''; * morphisms ''f'', ''g'', ... with fixed source and target objects called 1-''cells''; * "morphisms between morphisms" ρ, σ, ... with fixed source and target morphisms (which should have themselves the same source and the same target), called 2-''cells''; with some more structure: * given two objects ''a'' and ''b'' there is a category B(''a'', ''b'') whose objects are the 1- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3-category
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] |
Weak N-category
In category theory, 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. 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. The subject of weak ''n''-categories is an area of ongoing research. History There is currently 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 higher category as a simplicial set satisfy ... [...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] |
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] |
2-category
In category theory, a strict 2-category is a category with "morphisms between morphisms", that is, where each hom-set itself carries the structure of a category. It can be formally defined as a category enriched over Cat (the category of categories and functors, with the monoidal structure given by product of categories). The concept of 2-category was first introduced by Charles Ehresmann in his work on enriched categories in 1965. The more general concept of bicategory (or ''weak'' 2-''category''), where composition of morphisms is associative only up to a 2-isomorphism, was introduced in 1968 by Jean Bénabou.Jean Bénabou, Introduction to bicategories, in Reports of the Midwest Category Seminar, Springer, Berlin, 1967, pp. 1--77. Definition A 2-category C consists of: * A class of 0-''cells'' (or ''objects'') , , .... * For all objects and , a category \mathbf(A,B). The objects f,g: A \to B of this category are called 1-''cells'' and its morphisms \alpha: f \Ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ross Street
Ross Howard Street (born 29 September 1945, Sydney) is an Australian mathematician specialising in category theory.Street, Ross Howard (1945 - ) ''Biographical entry'', Encyclopaedia of Australian ScienceStreet, Ross Howard, FAA (1945-) trove.nla.gov.au Biography Street completed his undergraduate and postgraduate study at the , where his dissertation advisor was Max Kelly. He is a ...[...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 c. AD 30), regarded as a prophet and the forerunner of Jesus Christ * John the Apostle (lived c. AD 30), one of the twelve apostles of Jesus * 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 Jo ... [...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: * |
Tetracategory
In category theory, a tetracategory is a weakened definition of a 4-category. See also * Weak ''n''-category * infinity category In mathematics, more specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a generalization of the notion of a category. T ... External links Notes on tetracategoriesby Todd Trimble. Higher category theory {{categorytheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Knot Theory
In the mathematical field of topology, knot theory is the study of knot (mathematics), mathematical knots. While inspired by knots which appear in daily life, such as those in shoelaces and rope, a mathematical knot differs in that the ends are joined so it cannot be undone, Unknot, the simplest knot being a ring (or "unknot"). In mathematical language, a knot is an embedding of a circle in 3-dimensional Euclidean space, \mathbb^3 (in topology, a circle is not bound to the classical geometric concept, but to all of its homeomorphisms). Two mathematical knots are equivalent if one can be transformed into the other via a deformation of \mathbb^3 upon itself (known as an ambient isotopy); these transformations correspond to manipulations of a knotted string that do not involve cutting it or passing through itself. Knots can be described in various ways. Using different description methods, there may be more than one description of the same knot. For example, a common method of descr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
