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Derivator
In mathematics, derivators are a proposed frameworkpg 190-195 for homological algebra giving a foundation for both abelian and non-abelian homological algebra and various generalizations of it. They were introduced to address the deficiencies of derived categories (such as the non-functoriality of the cone construction) and provide at the same time a language for homotopical algebra. Derivators were first introduced by Alexander Grothendieck in his long unpublished 1983 manuscript '' Pursuing Stacks''. They were then further developed by him in the huge unpublished 1991 manuscript ''Les Dérivateurs'' of almost 2000 pages. Essentially the same concept was introduced (apparently independently) by Alex Heller. The manuscript has been edited for on-line publication by Georges Maltsiniotis. The theory has been further developed by several other people, including Heller, Franke, Keller and Groth. Motivations One of the motivating reasons for considering derivators is the lack of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pursuing Stacks
''Pursuing Stacks'' (french: À la Poursuite des Champs) is an influential 1983 mathematical manuscript by Alexander Grothendieck. It consists of a 12-page letter to Daniel Quillen followed by about 600 pages of research notes. The topic of the work is a generalized homotopy theory using higher category theory. The word "stacks" in the title refers to what are nowadays usually called " ∞-groupoids", one possible definition of which Grothendieck sketches in his manuscript. (The stacks of algebraic geometry, which also go back to Grothendieck, are not the focus of this manuscript.) Among the concepts introduced in the work are derivators and test categories. Some parts of the manuscript were later developed in: * * Overview of manuscript I. The letter to Daniel Quillen Pursuing stacks started out as a letter from Grothendieck to Daniel Quillen. In this letter he discusses Quillen's progress on the foundations for homotopy theory and remarked on the lack of progress since ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangulated Category
In mathematics, a triangulated category is a category with the additional structure of a "translation functor" and a class of "exact triangles". Prominent examples are the derived category of an abelian category, as well as the stable homotopy category. The exact triangles generalize the short exact sequences in an abelian category, as well as fiber sequences and cofiber sequences in topology. Much of homological algebra is clarified and extended by the language of triangulated categories, an important example being the theory of sheaf cohomology. In the 1960s, a typical use of triangulated categories was to extend properties of sheaves on a space ''X'' to complexes of sheaves, viewed as objects of the derived category of sheaves on ''X''. More recently, triangulated categories have become objects of interest in their own right. Many equivalences between triangulated categories of different origins have been proved or conjectured. For example, the homological mirror symmetry conj ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Colimit
In mathematics, especially in algebraic topology, the homotopy limit and colimitpg 52 are variants of the notions of limit and colimit extended to the homotopy category \text(\textbf). The main idea is this: if we have a diagramF: I \to \textbfconsidered as an object in the homotopy category of diagrams F \in \text(\textbf^I), (where the homotopy equivalence of diagrams is considered pointwise), then the homotopy limit and colimits then correspond to the cone and cocone\begin \underset(F)&: * \to \textbf\\ \underset(F)&: * \to \textbf \endwhich are objects in the homotopy category \text(\textbf^*), where * is the category with one object and one morphism. Note this category is equivalent to the standard homotopy category \text(\textbf) since the latter homotopy functor category has functors which picks out an object in \text and a natural transformation corresponds to a continuous function of topological spaces. Note this construction can be generalized to model categories, which g ... [...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] |
Grothendieck Site
In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category ''C'' that makes the objects of ''C'' act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site. Grothendieck topologies axiomatize the notion of an open cover. Using the notion of covering provided by a Grothendieck topology, it becomes possible to define sheaves on a category and their cohomology. This was first done in algebraic geometry and algebraic number theory by Alexander Grothendieck to define the étale cohomology of a scheme. It has been used to define other cohomology theories since then, such as ℓ-adic cohomology, flat cohomology, and crystalline cohomology. While Grothendieck topologies are most often used to define cohomology theories, they have found other applications as well, such as to John Tate's theory of rigid analytic geometry. There is a natural way to associate a site to an ordinary ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Space
In mathematics, algebraic spaces form a generalization of the schemes of algebraic geometry, introduced by Michael Artin for use in deformation theory. Intuitively, schemes are given by gluing together affine schemes using the Zariski topology, while algebraic spaces are given by gluing together affine schemes using the finer étale topology. Alternatively one can think of schemes as being locally isomorphic to affine schemes in the Zariski topology, while algebraic spaces are locally isomorphic to affine schemes in the étale topology. The resulting category of algebraic spaces extends the category of schemes and allows one to carry out several natural constructions that are used in the construction of moduli spaces but are not always possible in the smaller category of schemes, such as taking the quotient of a free action by a finite group (cf. the Keel–Mori theorem). Definition There are two common ways to define algebraic spaces: they can be defined as either quotients of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scheme (mathematics)
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations ''x'' = 0 and ''x''2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers). Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise "Éléments de géométrie algébrique"; one of its aims was developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne). Strongly based on commutative algebra, scheme theory allows a systematic use of methods of topology and homological algebra. Scheme theory also unifies algebraic geometry with much of number theory, which eventually led to Wiles's proof of Fermat's Last Theorem. Formally, a scheme is a topological space together with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topos
In mathematics, a topos (, ; plural topoi or , or toposes) is a category that behaves like the category of sheaves of sets on a topological space (or more generally: on a site). Topoi behave much like the category of sets and possess a notion of localization; they are a direct generalization of point-set topology. The Grothendieck topoi find applications in algebraic geometry; the more general elementary topoi are used in logic. The mathematical field that studies topoi is called topos theory. Grothendieck topos (topos in geometry) Since the introduction of sheaves into mathematics in the 1940s, a major theme has been to study a space by studying sheaves on a space. This idea was expounded by Alexander Grothendieck by introducing the notion of a "topos". The main utility of this notion is in the abundance of situations in mathematics where topological heuristics are very effective, but an honest topological space is lacking; it is sometimes possible to find a topos formaliz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Etale Site
In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category ''C'' that makes the objects of ''C'' act like the open sets of a topological space. A category together with a choice of Grothendieck topology is called a site. Grothendieck topologies axiomatize the notion of an open cover. Using the notion of covering provided by a Grothendieck topology, it becomes possible to define sheaves on a category and their cohomology. This was first done in algebraic geometry and algebraic number theory by Alexander Grothendieck to define the étale cohomology of a scheme. It has been used to define other cohomology theories since then, such as ℓ-adic cohomology, flat cohomology, and crystalline cohomology. While Grothendieck topologies are most often used to define cohomology theories, they have found other applications as well, such as to John Tate's theory of rigid analytic geometry. There is a natural way to associate a site to an ordinary ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zariski Topology
In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is not Hausdorff. This topology was introduced primarily by Oscar Zariski and later generalized for making the set of prime ideals of a commutative ring (called the spectrum of the ring) a topological space. The Zariski topology allows tools from topology to be used to study algebraic varieties, even when the underlying field is not a topological field. This is one of the basic ideas of scheme theory, which allows one to build general algebraic varieties by gluing together affine varieties in a way similar to that in manifold theory, where manifolds are built by gluing together charts, which are open subsets of real affine spaces. The Zariski topology of an algebraic variety is the topology whose closed sets are the algebraic subsets of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derived Category
In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction proceeds on the basis that the objects of ''D''(''A'') should be chain complexes in ''A'', with two such chain complexes considered isomorphic when there is a chain map that induces an isomorphism on the level of homology of the chain complexes. Derived functors can then be defined for chain complexes, refining the concept of hypercohomology. The definitions lead to a significant simplification of formulas otherwise described (not completely faithfully) by complicated spectral sequences. The development of the derived category, by Alexander Grothendieck and his student Jean-Louis Verdier shortly after 1960, now appears as one terminal point in the explosive development of homological algebra in the 1950s, a decade in which it had made remarkab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
