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Verdier Duality
In mathematics, Verdier duality is a cohomological duality in algebraic topology that generalizes Poincaré duality for manifolds. Verdier duality was introduced in 1965 by as an analog for locally compact topological spaces of Alexander Grothendieck's theory of Poincaré duality in étale cohomology for schemes in algebraic geometry. It is thus (together with the said étale theory and for example Grothendieck's coherent duality) one instance of Grothendieck's six operations formalism. Verdier duality generalises the classical Poincaré duality of manifolds in two directions: it applies to continuous maps from one space to another (reducing to the classical case for the unique map from a manifold to a one-point space), and it applies to spaces that fail to be manifolds due to the presence of singularities. It is commonly encountered when studying constructible or perverse sheaves. Verdier duality Verdier duality states that (subject to suitable finiteness conditions disc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adjoint Functors
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems (i.e., constructions of objects having a certain universal property), such as the construction of a free group on a set in algebra, or the construction of the Stone–Čech compactification of a topological space in topology. By definition, an adjunction between categories \mathcal and \mathcal is a pair of functors (assumed to be covariant) :F: \mathcal \rightarrow \mathcal and G: \mathcal \rightarrow \mathcal and, for all objects c in \mathcal and d in \mathcal, a bijection between the respective morphism sets :\ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Société Mathématique De France
Groupe Lactalis S.A. (doing business as Lactalis) is a French multinational dairy products corporation, owned by the Besnier family and based in Laval, Mayenne, France. The company's former name was Besnier S.A. Lactalis is the largest dairy products group in the world, and is the second largest food products group in France, behind Danone. It owns brands such as Parmalat, Président, Kraft Natural Cheese, Siggi's Dairy, Skånemejerier, Rachel's Organic, and Stonyfield Farm. History André Besnier started a small cheesemaking company in 1933 and launched its '' Président'' brand of Camembert in 1968. In 1990, it acquired Group Bridel (2,300 employees, 10 factories, fourth-largest French dairy group) with a presence in 60 countries. In 1992, it acquired United States cheese company Sorrento. In 1999, ''la société Besnier'' became ''le groupe Lactalis'' owned by Belgian holding company BSA International SA. In 2006, they bought Italian group Galbani, and in 2008, bough ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer Science+Business Media
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second-largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, op ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coherent Duality
In mathematics, coherent duality is any of a number of generalisations of Serre duality, applying to coherent sheaves, in algebraic geometry and complex manifold theory, as well as some aspects of commutative algebra that are part of the 'local' theory. The historical roots of the theory lie in the idea of the adjoint linear system of a linear system of divisors in classical algebraic geometry. This was re-expressed, with the advent of sheaf theory, in a way that made an analogy with Poincaré duality more apparent. Then according to a general principle, Grothendieck's relative point of view, the theory of Jean-Pierre Serre was extended to a proper morphism; Serre duality was recovered as the case of the morphism of a non-singular projective variety (or complete variety) to a point. The resulting theory is now sometimes called Serre–Grothendieck–Verdier duality, and is a basic tool in algebraic geometry. A treatment of this theory, ''Residues and Duality'' (1966) by Robin Hart ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Six Operations
In mathematics, Grothendieck's six operations, named after Alexander Grothendieck, is a formalism in homological algebra, also known as the six-functor formalism. It originally sprang from the relations in étale cohomology that arise from a morphism of schemes . The basic insight was that many of the elementary facts relating cohomology on ''X'' and ''Y'' were formal consequences of a small number of axioms. These axioms hold in many cases completely unrelated to the original context, and therefore the formal consequences also hold. The six operations formalism has since been shown to apply to contexts such as ''D''-modules on algebraic varieties, sheaves on locally compact topological spaces, and motives. The operations The operations are six functors. Usually these are functors between derived categories and so are actually left and right derived functors. * the direct image f_* * the inverse image f^* * the proper (or extraordinary) direct image f_! * the proper ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Category Of Chain Complexes
In homological algebra in mathematics, the homotopy category ''K(A)'' of chain complexes in an additive category ''A'' is a framework for working with chain homotopies and homotopy equivalences. It lies intermediate between the category of chain complexes ''Kom(A)'' of ''A'' and the derived category ''D(A)'' of ''A'' when ''A'' is abelian; unlike the former it is a triangulated category, and unlike the latter its formation does not require that ''A'' is abelian. Philosophically, while ''D(A)'' turns into isomorphisms any maps of complexes that are quasi-isomorphisms in ''Kom(A)'', ''K(A)'' does so only for those that are quasi-isomorphisms for a "good reason", namely actually having an inverse up to homotopy equivalence. Thus, ''K(A)'' is more understandable than ''D(A)''. Definitions Let ''A'' be an additive category. The homotopy category ''K(A)'' is based on the following definition: if we have complexes ''A'', ''B'' and maps ''f'', ''g'' from ''A'' to ''B'', a chain homo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constant Sheaf
In mathematics, the constant sheaf on a topological space X associated to a set (mathematics), set A is a Sheaf (mathematics), sheaf of sets on X whose stalk (sheaf), stalks are all equal to A. It is denoted by \underline or A_X. The constant presheaf with value A is the presheaf that assigns to each open set, open subset of X the value A, and all of whose restriction maps are the identity map A\to A. The constant sheaf associated to A is the sheafification of the constant presheaf associated to A. This sheaf may be identified with the sheaf of locally constant A-valued functions on X. In certain cases, the set A may be replaced with an object (category theory), object A in some category (mathematics), category \textbf (e.g. when \textbf is the category of abelian groups, or category of commutative rings, commutative rings). Constant sheaves of abelian groups appear in particular as coefficients in sheaf cohomology. Basics Let X be a topological space, and A a set. The sections o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sheaf Cohomology
In mathematics, sheaf cohomology is the application of homological algebra to analyze the global sections of a sheaf on a topological space. Broadly speaking, sheaf cohomology describes the obstructions (holes) to solving a geometric problem globally when it can be solved locally. The central work for the study of sheaf cohomology is Grothendieck's 1957 Tôhoku paper. Sheaves, sheaf cohomology, and spectral sequences were introduced by Jean Leray at the prisoner-of-war camp Oflag XVII-A in Austria. From 1940 to 1945, Leray and other prisoners organized a "université en captivité" in the camp. Leray's definitions were simplified and clarified in the 1950s. It became clear that sheaf cohomology was not only a new approach to cohomology in algebraic topology, but also a powerful method in complex analytic geometry and algebraic geometry. These subjects often involve constructing global functions with specified local properties, and sheaf cohomology is ideally suited to such ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contravariant Functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied. The words ''category'' and ''functor'' were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively. The latter used ''functor'' in a linguistic context; see function word. Definition Let ''C'' and ''D'' be categories. A functor ''F'' from ''C'' to ''D'' is a mapping that * associates each object X in ''C'' to an object F(X) in ''D'', * associates each morphism f \colon X \to Y in ''C'' to a morphism F(f) \colon F(X) \to F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
CW-complexes
In mathematics, and specifically in topology, a CW complex (also cellular complex or cell complex) is a topological space that is built by gluing together topological balls (so-called ''cells'') of different dimensions in specific ways. It generalizes both manifolds and simplicial complexes and has particular significance for algebraic topology. It was initially introduced by J. H. C. Whitehead to meet the needs of homotopy theory. (open access) CW complexes have better categorical properties than simplicial complexes, but still retain a combinatorial nature that allows for computation (often with a much smaller complex). The C in CW stands for "closure-finite", and the W for "weak" topology. Definition CW complex A CW complex is constructed by taking the union of a sequence of topological spaces \emptyset = X_ \subset X_0 \subset X_1 \subset \cdots such that each X_k is obtained from X_ by gluing copies of k-cells (e^k_\alpha)_\alpha, each homeomorphic to the open k-ball B^k ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cohomological Dimension
In abstract algebra, cohomological dimension is an invariant of a group which measures the homological complexity of its representations. It has important applications in geometric group theory, topology, and algebraic number theory. Cohomological dimension of a group As most cohomological invariants, the cohomological dimension involves a choice of a "ring of coefficients" ''R'', with a prominent special case given by R=\Z, the ring of integers. Let ''G'' be a discrete group, ''R'' a non-zero ring with a unit, and RG the group ring. The group ''G'' has cohomological dimension less than or equal to ''n'', denoted \operatorname_R(G)\le n, if the trivial RG-module ''R'' has a projective resolution of length ''n'', i.e. there are projective RG-modules P_0,\dots , P_n and RG-module homomorphisms d_k\colon P_k\to P_ (k = 1,\dots, n) and d_0\colon P_0\to R, such that the image of d_k coincides with the kernel of d_ for k = 1, \dots, n and the kernel of d_n is trivial. Equivalently, th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
