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Perverse Sheaf
The mathematical term perverse sheaves refers to the objects of certain abelian categories associated to topological spaces, which may be a real or complex manifold, or more general topologically stratified spaces, possibly singular. The concept was introduced in the work of Joseph Bernstein, Alexander Beilinson, and Pierre Deligne and Ofer Gabber (1982) as a consequence of the Riemann-Hilbert correspondence, which establishes a connection between the derived categories regular holonomic D-modules and constructible sheaves. Perverse sheaves are the objects in the latter that correspond to individual D-modules (and not more general complexes thereof); a perverse sheaf ''is'' in general represented by a complex of sheaves. The concept of perverse sheaves is already implicit in a 75's paper of Kashiwara on the constructibility of solutions of holonomic D-modules. A key observation was that the intersection homology of Mark Goresky and Robert MacPherson could be described using ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Abelian Category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of abelian groups, . Abelian categories are very ''stable'' categories; for example they are regular and they satisfy the snake lemma. The class of abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an abelian category, or the category of functors from a small category to an abelian category are abelian as well. These stability properties make them inevitable in homological algebra and beyond; the theory has major applications in algebraic geometry, cohomology and pure category theory. Mac Lane says Alexander Grothendieck defined the abelian category, but there is a reference that says Eilenberg's disciple, Buchsbaum, proposed the concept in his PhD thesis, and Groth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |