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Picard Stack
In mathematics, an Abelian 2-group is a higher dimensional analogue of an Abelian group, in the sense of higher algebra, which were originally introduced by Alexander Grothendieck while studying abstract structures surrounding Abelian varieties and Picard groups. More concretely, they are given by groupoids \mathbb which have a bifunctor +:\mathbb\times\mathbb \to \mathbb which acts formally like the addition an Abelian group. Namely, the bifunctor + has a notion of commutativity, associativity, and an identity structure. Although this seems like a rather lofty and abstract structure, there are several (very concrete) examples of Abelian 2-groups. In fact, some of which provide prototypes for more complex examples of higher algebraic structures, such as Abelian n-groups. Definition An Abelian 2-group is a groupoid \mathbb with a bifunctor +:\mathbb\times\mathbb \to \mathbb and natural transformations\begin \tau: & X+Y \Rightarrow Y + X \\ \sigma: & (X+Y)+Z \Rightarrow X+(Y+Z) \ ... [...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] |
Postnikov System
In homotopy theory, a branch of algebraic topology, a Postnikov system (or Postnikov tower) is a way of decomposing a topological space's homotopy groups using an inverse system of topological spaces whose homotopy type at degree k agrees with the truncated homotopy type of the original space X. Postnikov systems were introduced by, and are named after, Mikhail Postnikov. Definition A Postnikov system of a path-connected space X is an inverse system of spaces :\cdots \to X_n \xrightarrow X_\xrightarrow \cdots \xrightarrow X_2 \xrightarrow X_1 \xrightarrow * with a sequence of maps \phi_n\colon X \to X_n compatible with the inverse system such that # The map \phi_n\colon X \to X_n induces an isomorphism \pi_i(X) \to \pi_i(X_n) for every i\leq n. # \pi_i(X_n) = 0 for i > n. # Each map p_n\colon X_n \to X_ is a fibration, and so the fiber F_n is an Eilenberg–MacLane space, K(\pi_n(X),n). The first two conditions imply that X_1 is also a K(\pi_1(X),1)-space. More generally, if X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Group Theory
Abelian may refer to: Mathematics Group theory * Abelian group, a group in which the binary operation is commutative ** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms * Metabelian group, a group where the commutator subgroup is abelian * Abelianisation Topology and number theory * Abelian variety, a complex torus that can be embedded into projective space * Abelian surface, a two-dimensional abelian variety * Abelian function, a meromorphic function on an abelian variety * Abelian integral, a function related to the indefinite integral of a differential of the first kind Other mathematics * Abelian category, in category theory, a preabelian category in which every monomorphism is a kernel and every epimorphism is a cokernel * Abelian and Tauberian theorems, in real analysis, used in the summation of divergent series * Abelian extension, in Galois theory, a field extension for which the associated Galois group is abelian * Abe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1606
Events January–June * January 24 – Gunpowder Plot: The trial of Guy Fawkes and other conspirators, for plotting against Parliament and James I of England, begins. * January 29 – Pedro Fernandes de Queirós discovers the Pitcairn Islands. * February 9 – Pedro Fernandes de Queirós discovers Mehetia. * February 12 – Pedro Fernandes de Queirós discovers Tauere atoll. * February 26 – Dutch navigator Willem Janszoon makes the first confirmed sighting of Australia by a European. * March – The Duke of York's ship ''Duyfken'', under Captain Willem Janszoon, explores the western coast of Cape York Peninsula. * March 19 – Ferdinando I de' Medici, Grand Duke of Tuscany, in the Fortezza Vecchia Chapel of Saint Francesco, elevates Livorno to the rank of city. * April – Venetian Interdict: Pope Paul V places the Republic of Venice under an interdict. * April 10 – Charter of 1606: The First Charter of Virginia is adopted, by whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1201
Year 1201 ( MCCI) was a common year starting on Monday (link will display the full calendar) of the Julian calendar. Events By place Byzantine Empire * July 31 – John Komnenos the Fat, a Byzantine aristocrat, attempts to usurp the imperial throne; he is proclaimed emperor and crowned by Patriarch John X Kamateros, at Constantinople. Meanwhile, Emperor Alexios III Angelos, who resides in the Palace of Blachernae, dispatches a small force under Alexios Palaiologos, Alexios' son-in-law, who is regarded as his heir-apparent. With support of the Varangian Guard, John is overthrown and decapitated by the end of the day. His head is displayed at the Forum of Constantine, while John's supporters are captured and tortured to extract the names of all the conspirators. * Autumn – Prince Alexios Angelos, son of the deposed, blinded and imprisoned late Emperor Isaac II Angelos, escapes from Constantinople. He makes his way to Sicily and then Rome where he is turned a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1702
In the Swedish calendar it was a common year starting on Wednesday, one day ahead of the Julian and ten days behind the Gregorian calendar. Events January–March * January 2 – A total solar eclipse is visible from the southern Pacific Ocean. * January 12 – In North America, ships from Fort Maurepas arrive at Twenty-Seven Mile Bluff, to build ''Fort Louis de la Mobile'' (future Mobile, Alabama), to become the capital of French Louisiana. * February 1 – The Duc de Villeroy, commander of the French Army, is taken as a prisoner of war by the Austrian Army during the Battle of Cremona * March 3 (February 20 O.S.) – King William III of England is fatally injured in an accident when he is thrown from his horse, "Sorrel", while riding in Hampton Court Park near London. Already in poor health before the accident, he dies from his injuries 16 days later at the age of 51. * March 14 – An earthquake in the middle of the Calore valley in Italy, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1101
Year 1101 (Roman numerals, MCI) was a common year starting on Tuesday (link will display the full calendar) of the Julian calendar. It was the 2nd year of the 1100s decade, and the 1st year of the 12th century. Events By place Byzantine Empire * Crusade of 1101: A second wave of European crusaders attempts to cross Anatolia, to reach the Kingdom of Jerusalem. They are defeated by the Seljuq dynasty, Seljuk troops under Sultan Kilij Arslan I, at Heraclea Cybistra, Heraclea. A handful of crusaders under Raymond IV, Count of Toulouse, Raymond IV (Saint-Gilles) manage to reach the Byzantine port of Bafra, at the mouth of the Kızılırmak River, River Halys. * Summer – The Byzantine fleet under Admiral Eustathios Kymineianos, Eustathios recaptures the ports of western Cilicia, Seleucia and Corycus. Eustathios extends his power over Cilician territory (belonging to Bohemond I of Antioch, Bohemond I) further east – occupying Tarsus, Mersin, Tarsus, Adana and Mopsuest ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1007
Year 1007 ( MVII) was a common year starting on Wednesday (link will display the full calendar) of the Julian calendar. Events By place England * King Æthelred the Unready pays the Danish Vikings a sum of 36,000 pounds of silver (Danegeld) to stop further invasions.John Haywood (1995). ''Historical Atlas of the Vikings'', p. 118. . Ireland * The Book of Kells is stolen from the Abbey of Kells. Japan * January 1 (New Year’s Day) – Imperial Princess Shushi is granted the title Ippon Shinno (first rank princess). * January 29 – Ranking ceremony of Murasaki Shikibu – as a renowned writer and lady-in-waiting, tutor of Empress Shōshi, she is elevated to the highest position in the palace below the empress. * April – Imperial Prince Tomohira receives the title ''nihon'' (second rank prince). By topic Religion * November 1 – King Henry II of Germany founds the Archdiocese of Bamberg during a synod held in Frankfurt. * Ælfheah of Cante ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gerbe
In mathematics, a gerbe (; ) is a construct in homological algebra and topology. Gerbes were introduced by Jean Giraud (mathematician), Jean Giraud following ideas of Alexandre Grothendieck as a tool for non-commutative cohomology in degree 2. They can be seen as an analogue of fibre bundles where the fibre is the classifying stack of a group. Gerbes provide a convenient, if highly abstract, language for dealing with many types of Deformation theory, deformation questions especially in modern algebraic geometry. In addition, special cases of gerbes have been used more recently in differential topology and differential geometry to give alternative descriptions to certain cohomology classes and additional structures attached to them. "Gerbe" is a French (and archaic English) word that literally means wheat sheaf (agriculture), sheaf. Definitions Gerbes on a topological space A gerbe on a topological space S is a stack (mathematics), stack \mathcal of groupoids over S which is ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
∞-groupoid
In category theory, a branch of mathematics, an ∞-groupoid is an abstract homotopical model for topological spaces. One model uses Kan complexes which are fibrant objects in the category of simplicial sets (with the standard model structure). It is an ∞-category generalization of a groupoid, a category in which every morphism is an isomorphism. The homotopy hypothesis states that ∞-groupoids are spaces. Globular Groupoids Alexander Grothendieck suggested in ''Pursuing Stacks'' that there should be an extraordinarily simple model of ∞-groupoids using globular sets, originally called hemispherical complexes. These sets are constructed as presheaves on the globular category \mathbb. This is defined as the category whose objects are finite ordinals /math> and morphisms are given by \begin \sigma_n: \to +1\ \tau_n: \to +1\end such that the globular relations hold \begin \sigma_\circ\sigma_n &= \tau_\circ\sigma_n \\ \sigma_\circ\tau_n &= \tau_\circ\tau_n \end These encod ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphere Spectrum
In stable homotopy theory, a branch of mathematics, the sphere spectrum ''S'' is the monoidal unit in the category of spectra. It is the suspension spectrum of ''S''0, i.e., a set of two points. Explicitly, the ''n''th space in the sphere spectrum is the ''n''-dimensional sphere ''S''''n'', and the structure maps from the suspension of ''S''''n'' to ''S''''n''+1 are the canonical homeomorphisms. The ''k''-th homotopy group of a sphere spectrum is the ''k''-th stable homotopy group of spheres. The localization of the sphere spectrum at a prime number ''p'' is called the local sphere at ''p'' and is denoted by S_. See also * Chromatic homotopy theory * Adams-Novikov spectral sequence *Framed cobordism Framed may refer to: Common meanings *A painting or photograph that has been placed within a picture frame *Someone falsely shown to be guilty of a crime as part of a frameup Film and television * ''Framed'' (1930 film), a pre-code crime action ... References * Algebraic topo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eilenberg–MacLane Space
In mathematics, specifically algebraic topology, an Eilenberg–MacLane spaceSaunders Mac Lane originally spelt his name "MacLane" (without a space), and co-published the papers establishing the notion of Eilenberg–MacLane spaces under this name. (See e.g. ) In this context it is therefore conventional to write the name without a space. is a topological space with a single nontrivial homotopy group. Let ''G'' be a group and ''n'' a positive integer. A connected topological space ''X'' is called an Eilenberg–MacLane space of type K(G,n), if it has ''n''-th homotopy group \pi_n(X) isomorphic to ''G'' and all other homotopy groups trivial. If n > 1 then ''G'' must be abelian. Such a space exists, is a CW-complex, and is unique up to a weak homotopy equivalence, therefore any such space is often just called K(G,n). The name is derived from Samuel Eilenberg and Saunders Mac Lane, who introduced such spaces in the late 1940s. As such, an Eilenberg–MacLane space is a special k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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