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Double Complex
In mathematics, specifically Homological algebra, a double complex is a generalization of a chain complex where instead of having a \mathbb-grading, the objects in the bicomplex have a \mathbb\times\mathbb-grading. The most general definition of a double complex, or a bicomplex, is given with objects in an additive category \mathcal. A bicomplex is a sequence of objects C_ \in \text(\mathcal) with two differentials, the horizontal differentiald^h: C_ \to C_and the vertical differentiald^v:C_ \to C_which have the compatibility relationd_h\circ d_v = d_v\circ d_hHence a double complex is a commutative diagram of the form\begin & & \vdots & & \vdots & & \\ & & \uparrow & & \uparrow & & \\ \cdots & \to & C_ & \to & C_ & \to & \cdots \\ & & \uparrow & & \uparrow & & \\ \cdots & \to & C_ & \to & C_ & \to & \cdots \\ & & \uparrow & & \uparrow & & \\ & & \vdots & & \vdots & & \\ \endwhere the rows and columns form chain complexes. Some authors instead require that the squares anticom ... [...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] |
Homological Algebra
Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of module (mathematics), modules and Syzygy (mathematics), syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. Homological algebra is the study of homological functors and the intricate algebraic structures that they entail; its development was closely intertwined with the emergence of category theory. A central concept is that of chain complexes, which can be studied through both their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariant (mathematics), invariants of ring (mathematics), rings, modules, topological spaces, and other 'tan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chain Complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or module (mathematics), modules) and a sequence of group homomorphism, homomorphisms between consecutive groups such that the image (mathematics), image of each homomorphism is included in the kernel (algebra)#Group homomorphisms, kernel of the next. Associated to a chain complex is its Homology (mathematics), homology, which describes how the images are included in the kernels. A cochain complex is similar to a chain complex, except that its homomorphisms are in the opposite direction. The homology of a cochain complex is called its cohomology. In algebraic topology, the singular chain complex of a topological space X is constructed using continuous function#continuous functions between topological spaces, continuous maps from a simplex to X, and the homomorphisms of the chain complex capture how these maps restrict to the boundary of the simplex. The homology of this chain co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Additive Category
In mathematics, specifically in category theory, an additive category is a preadditive category C admitting all finitary biproducts. Definition A category C is preadditive if all its hom-sets are abelian groups and composition of morphisms is bilinear; in other words, C is enriched over the monoidal category of abelian groups. In a preadditive category, every finitary product (including the empty product, i.e., a final object) is necessarily a coproduct (or initial object in the case of an empty diagram), and hence a biproduct, and conversely every finitary coproduct is necessarily a product (this is a consequence of the definition, not a part of it). Thus an additive category is equivalently described as a preadditive category admitting all finitary products, or a preadditive category admitting all finitary coproducts. Another, yet equivalent, way to define an additive category is a category (not assumed to be preadditive) that has a zero object, finite coprodu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Total Complexes
Total may refer to: Mathematics * Total, the summation of a set of numbers * Total order, a partial order without incomparable pairs * Total relation, which may also mean ** connected relation (a binary relation in which any two elements are comparable). * Total function, a partial function that is also a total relation Business * TotalEnergies, a French petroleum company * Total (cereal), a food brand by General Mills * Total, a brand of strained yogurt made by Fage * Total, a database management system marketed by Cincom Systems * Total Linhas Aéreas - a brazilian airline * Total, a line of dental products by Colgate Music and culture * Total (group), an American R&B girl group * '' Total: From Joy Division to New Order'', a compilation album * ''Total'' (Sebastian album) * ''Total'' (Total album) * ''Total'' (Teenage Bottlerocket album) * ''Total'' (Seigmen album) * ''Total'' (Wanessa album) * ''Total'' (Belinda Peregrín album) * ''Total 1'', an annual compilation alb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Groupoid
In mathematics, a Lie groupoid is a groupoid where the set \operatorname of objects and the set \operatorname of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations :s,t : \operatorname \to \operatorname are submersions. A Lie groupoid can thus be thought of as a "many-object generalization" of a Lie group, just as a groupoid is a many-object generalization of a group. Accordingly, while Lie groups provide a natural model for (classical) continuous symmetries, Lie groupoids are often used as model for (and arise from) generalised, point-dependent symmetries. Extending the correspondence between Lie groups and Lie algebras, Lie groupoids are the global counterparts of Lie algebroids. Lie groupoids were introduced by Charles Ehresmann under the name ''differentiable groupoids''. Definition and basic concepts A Lie groupoid consists of * two smooth ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Rham Cohomology
In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. It is a cohomology theory based on the existence of differential forms with prescribed properties. On any smooth manifold, every exact form is closed, but the converse may fail to hold. Roughly speaking, this failure is related to the possible existence of "holes" in the manifold, and the de Rham cohomology groups comprise a set of topological invariants of smooth manifolds that precisely quantify this relationship. Definition The de Rham complex is the cochain complex of differential forms on some smooth manifold , with the exterior derivative as the differential: :0 \to \Omega^0(M)\ \stackrel\ \Omega^1(M)\ \stackrel\ \Omega^2(M)\ \stackrel\ \Omega^3(M) \to \cd ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hodge Theory
In mathematics, Hodge theory, named after W. V. D. Hodge, is a method for studying the cohomology groups of a smooth manifold ''M'' using partial differential equations. The key observation is that, given a Riemannian metric on ''M'', every cohomology class has a canonical representative, a differential form that vanishes under the Laplacian operator of the metric. Such forms are called harmonic. The theory was developed by Hodge in the 1930s to study algebraic geometry, and it built on the work of Georges de Rham on de Rham cohomology. It has major applications in two settings: Riemannian manifolds and Kähler manifolds. Hodge's primary motivation, the study of complex projective varieties, is encompassed by the latter case. Hodge theory has become an important tool in algebraic geometry, particularly through its connection to the study of algebraic cycles. While Hodge theory is intrinsically dependent upon the real and complex numbers, it can be applied to questions in nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Complex Manifold
In mathematics, an almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost complex manifold, but there are almost complex manifolds that are not complex manifolds. Almost complex structures have important applications in symplectic geometry. The concept is due to Charles Ehresmann and Heinz Hopf in the 1940s. Formal definition Let ''M'' be a smooth manifold. An almost complex structure ''J'' on ''M'' is a linear complex structure (that is, a linear map which squares to −1) on each tangent space of the manifold, which varies smoothly on the manifold. In other words, we have a smooth tensor field ''J'' of degree such that J^2=-1 when regarded as a vector bundle isomorphism J\colon TM\to TM on the tangent bundle. A manifold equipped with an almost complex structure is called an almost complex manifold. If ''M'' admits an almost complex structure, it must be even-dimensional. This ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chain Complex
In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or module (mathematics), modules) and a sequence of group homomorphism, homomorphisms between consecutive groups such that the image (mathematics), image of each homomorphism is included in the kernel (algebra)#Group homomorphisms, kernel of the next. Associated to a chain complex is its Homology (mathematics), homology, which describes how the images are included in the kernels. A cochain complex is similar to a chain complex, except that its homomorphisms are in the opposite direction. The homology of a cochain complex is called its cohomology. In algebraic topology, the singular chain complex of a topological space X is constructed using continuous function#continuous functions between topological spaces, continuous maps from a simplex to X, and the homomorphisms of the chain complex capture how these maps restrict to the boundary of the simplex. The homology of this chain co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derived Algebraic Geometry
Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded algebras (over \mathbb), simplicial commutative rings or E_-ring spectra from algebraic topology, whose higher homotopy groups account for the non-discreteness (e.g., Tor) of the structure sheaf. Grothendieck's scheme theory allows the structure sheaf to carry nilpotent elements. Derived algebraic geometry can be thought of as an extension of this idea, and provides natural settings for intersection theory (or motivic homotopy theory) of singular algebraic varieties and cotangent complexes in deformation theory (cf. J. Francis), among the other applications. Introduction Basic objects of study in the field are derived schemes and derived stacks. The oft-cited motivation is Serre's intersection formula. In the usual formulation, the formula involves the Tor functor and thus, unless h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
