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2-group
In mathematics, a 2-group, or 2-dimensional higher group, is a certain combination of group and groupoid. The 2-groups are part of a larger hierarchy of ''n''-groups. In some of the literature, 2-groups are also called gr-categories or groupal groupoids. Definition A 2-group is a monoidal category ''G'' in which every morphism is invertible and every object has a weak inverse. (Here, a ''weak inverse'' of an object ''x'' is an object ''y'' such that ''xy'' and ''yx'' are both isomorphic to the unit object.) Strict 2-groups Much of the literature focuses on ''strict 2-groups''. A strict 2-group is a ''strict'' monoidal category in which every morphism is invertible and every object has a strict inverse (so that ''xy'' and ''yx'' are actually equal to the unit object). A strict 2-group is a group object in a category of categories; as such, they are also called ''groupal categories''. Conversely, a strict 2-group is a category object in the category of groups; as such, t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N-group (category Theory)
In mathematics, an ''n''-group, or ''n''-dimensional higher group, is a special kind of ''n''-category that generalises the concept of group to higher-dimensional algebra. Here, n may be any natural number or infinity. The thesis of Alexander Grothendieck's student Hoàng Xuân Sính was an in-depth study of 2-groups under the moniker 'gr-category'. The general definition of n-group is a matter of ongoing research. However, it is expected that every topological space will have a ''homotopy n-group'' at every point, which will encapsulate the Postnikov tower of the space up to the homotopy group \pi_n, or the entire Postnikov tower for n=\infty. Examples Eilenberg-Maclane spaces One of the principal examples of higher groups come from the homotopy types of Eilenberg–MacLane spaces K(A,n) since they are the fundamental building blocks for constructing higher groups, and homotopy types in general. For instance, every group G can be turned into an Eilenberg-Maclane space K ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crossed Module
In mathematics, and especially in homotopy theory, a crossed module consists of groups G and H, where G acts on H by automorphisms (which we will write on the left, (g,h) \mapsto g \cdot h , and a homomorphism of groups : d\colon H \longrightarrow G, that is equivariant with respect to the conjugation action of G on itself: : d(g \cdot h) = gd(h)g^ and also satisfies the so-called Peiffer identity: : d(h_) \cdot h_ = h_h_h_^ Origin The first mention of the second identity for a crossed module seems to be in footnote 25 on p. 422 of J. H. C. Whitehead's 1941 paper cited below, while the term 'crossed module' is introduced in his 1946 paper cited below. These ideas were well worked up in his 1949 paper 'Combinatorial homotopy II', which also introduced the important idea of a free crossed module. Whitehead's ideas on crossed modules and their applications are developed and explained in the book by Brown, Higgins, Sivera listed below. Some generalisations of the id ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Crossed Modules
In mathematics, and especially in homotopy theory, a crossed module consists of groups G and H, where G acts on H by automorphisms (which we will write on the left, (g,h) \mapsto g \cdot h , and a homomorphism of groups : d\colon H \longrightarrow G, that is equivariant with respect to the conjugation action of G on itself: : d(g \cdot h) = gd(h)g^ and also satisfies the so-called Peiffer identity: : d(h_) \cdot h_ = h_h_h_^ Origin The first mention of the second identity for a crossed module seems to be in footnote 25 on p. 422 of J. H. C. Whitehead's 1941 paper cited below, while the term 'crossed module' is introduced in his 1946 paper cited below. These ideas were well worked up in his 1949 paper 'Combinatorial homotopy II', which also introduced the important idea of a free crossed module. Whitehead's ideas on crossed modules and their applications are developed and explained in the book by Brown, Higgins, Sivera listed below. Some generalisations of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hoàng Xuân Sính
Hoàng Xuân Sính (born September 8, 1933) is a Vietnamese mathematician, a student of Grothendieck, the first female mathematician in Vietnam, the founder of Thang Long University, and the recipient of the ''Ordre des Palmes Académiques''. Early life and career Hoàng was born in Cót, in the Từ Liêm District of Vietnam, one of seven children of fabric merchant Hoàng Thuc Tan. Her mother died when she was eight years old, and she was raised by a stepmother. She has also frequently been said to be the granddaughter of Vietnamese mathematician Hoàng Xuân Hãn. She completed a bachelor's degree in 1951 in Hanoi, studying English and French, and then traveled to Paris for a second baccalaureate in mathematics. She stayed in France to study for an '' agrégation'' at the University of Toulouse, which she completed in 1959, before returning to Vietnam to become a mathematics teacher at the Hanoi National University of Education. Hoàng became the first female mathematician in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Groupoid
In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: *''Group'' with a partial function replacing the binary operation; *''Category'' in which every morphism is invertible. A category of this sort can be viewed as augmented with a unary operation on the morphisms, called ''inverse'' by analogy with group theory. A groupoid where there is only one object is a usual group. In the presence of dependent typing, a category in general can be viewed as a typed monoid, and similarly, a groupoid can be viewed as simply a typed group. The morphisms take one from one object to another, and form a dependent family of types, thus morphisms might be typed g:A \rightarrow B, h:B \rightarrow C, say. Composition is then a total function: \circ : (B \rightarrow C) \rightarrow (A \rightarrow B) \rightarrow A \rightarrow C , so that h \circ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Action (mathematics)
In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group ''acts'' on the space or structure. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it. For example, it acts on the set of all triangles. Similarly, the group of symmetries of a polyhedron acts on the vertices, the edges, and the faces of the polyhedron. A group action on a vector space is called a representation of the group. In the case of a finite-dimensional vector space, it allows one to identify many groups with subgroups of , the group of the invertible matrices of dimension over a field . The symmetric group acts on any set wit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Object
In category theory, a branch of mathematics, group objects are certain generalizations of groups that are built on more complicated structures than sets. A typical example of a group object is a topological group, a group whose underlying set is a topological space such that the group operations are continuous. Definition Formally, we start with a category ''C'' with finite products (i.e. ''C'' has a terminal object 1 and any two objects of ''C'' have a product). A group object in ''C'' is an object ''G'' of ''C'' together with morphisms *''m'' : ''G'' × ''G'' → ''G'' (thought of as the "group multiplication") *''e'' : 1 → ''G'' (thought of as the "inclusion of the identity element") *''inv'' : ''G'' → ''G'' (thought of as the "inversion operation") such that the following properties (modeled on the group axioms – more precisely, on the definition of a group used in universal algebra) are satisfied * ''m'' is associative, i.e. ''m'' (''m'' × id''G'') = ''m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Cohomology
In mathematics (more specifically, in homological algebra), group cohomology is a set of mathematical tools used to study groups using cohomology theory, a technique from algebraic topology. Analogous to group representations, group cohomology looks at the group actions of a group ''G'' in an associated ''G''-module ''M'' to elucidate the properties of the group. By treating the ''G''-module as a kind of topological space with elements of G^n representing ''n''-simplices, topological properties of the space may be computed, such as the set of cohomology groups H^n(G,M). The cohomology groups in turn provide insight into the structure of the group ''G'' and ''G''-module ''M'' themselves. Group cohomology plays a role in the investigation of fixed points of a group action in a module or space and the quotient module or space with respect to a group action. Group cohomology is used in the fields of abstract algebra, homological algebra, algebraic topology and algebraic number theory, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classification Theorem
In mathematics, a classification theorem answers the classification problem "What are the objects of a given type, up to some equivalence?". It gives a non-redundant enumeration: each object is equivalent to exactly one class. A few issues related to classification are the following. *The equivalence problem is "given two objects, determine if they are equivalent". *A complete set of invariants, together with which invariants are solves the classification problem, and is often a step in solving it. *A (together with which invariants are realizable) solves both the classification problem and the equivalence problem. * A canonical form solves the classification problem, and is more data: it not only classifies every class, but provides a distinguished (canonical) element of each class. There exist many classification theorems in mathematics, as described below. Geometry * Classification of Euclidean plane isometries * Classification theorems of surfaces ** Classification of two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Up To
Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' are equal. This figure of speech is mostly used in connection with expressions derived from equality, such as uniqueness or count. For example, ''x'' is unique up to ''R'' means that all objects ''x'' under consideration are in the same equivalence class with respect to the relation ''R''. Moreover, the equivalence relation ''R'' is often designated rather implicitly by a generating condition or transformation. For example, the statement "an integer's prime factorization is unique up to ordering" is a concise way to say that any two lists of prime factors of a given integer are equivalent with respect to the relation ''R'' that relates two lists if one can be obtained by reordering (permutation) from the other. As anot ... [...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] |
