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Test Category
''Pursuing Stacks'' () 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 then. He remarks how some of his fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Grothendieck
Alexander Grothendieck, later Alexandre Grothendieck in French (; ; ; 28 March 1928 – 13 November 2014), was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry. His research extended the scope of the field and added elements of commutative algebra, homological algebra, sheaf theory, and category theory to its foundations, while his so-called Grothendieck's relative point of view, "relative" perspective led to revolutionary advances in many areas of pure mathematics. He is considered by many to be the greatest mathematician of the twentieth century. Grothendieck began his productive and public career as a mathematician in 1949. In 1958, he was appointed a research professor at the Institut des Hautes Études Scientifiques, Institut des hautes études scientifiques (IHÉS) and remained there until 1970, when, driven by personal and political convictions, he left following a dispute over military funding. He receive ... [...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 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 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. Definitions Gerbes on a topological space A gerbe on a topological space S is a stack \mathcal of groupoids over S that is ''locally non-empty'' (each point p \in S has an open neighbourhood U_p over which the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Hypothesis
In category theory, a branch of mathematics, Grothendieck's homotopy hypothesis states, homotopy theory speaking, that the ∞-groupoids are space (mathematics), spaces. One version of the hypothesis was claimed to be proved in the 1991 paper by Mikhail Kapranov, Kapranov and Vladimir Voevodsky, Voevodsky. Their proof turned out to be flawed and their result in the form interpreted by Carlos Simpson is now known as the Simpson conjecture. In higher category theory, one considers a space-valued presheaf instead of a presheaf (category theory), set-valued presheaf in ordinary category theory. In view of homotopy hypothesis, a space here can be taken to an ∞-groupoid. Formulations A precise formulation of the hypothesis very strongly depends on the definition of an ∞-groupoid. One definition is that, mimicking the ordinary category case, an ∞-groupoid is an ∞-category in which each morphism is invertible or equivalently its homotopy category of an ∞-category, homotopy cat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homotopy Category
In mathematics, the homotopy category is a category built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) categories, as discussed below. More generally, instead of starting with the category of topological spaces, one may start with any model category and define its associated homotopy category, with a construction introduced by Quillen in 1967. In this way, homotopy theory can be applied to many other categories in geometry and algebra. The naive homotopy category The category of topological spaces Top has topological spaces as objects and as morphisms the continuous maps between them. The older definition of the homotopy category hTop, called the naive homotopy category for clarity in this article, has the same objects, and a morphism is a homotopy class of continuous maps. That is, two continuous maps ''f'' : ''X'' → ''Y'' are considered the same in the na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cubical Set
In topology, a branch of mathematics, a cubical set is a set-valued contravariant functor on the category of (various) ''n''-cubes. Cubical sets have been often considered as an alternative to simplicial sets in combinatorial topology, including in the early work of Daniel Kan and Jean-Pierre Serre Jean-Pierre Serre (; born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the inau .... They have also been developed in computer science, in particular in concurrency theory and in homotopy type theory. See also * Simplicial presheaf References * nLabCubical set * Rick JardineCubical sets Lecture 12 in "Lectures on simplicial presheaves" https://web.archive.org/web/20110104053206/http://www.math.uwo.ca/~jardine/papers/sPre/index.shtml Topology {{topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annales Scientifiques De L'École Normale Supérieure
''Annales Scientifiques de l'École Normale Supérieure'' is a French scientific journal of mathematics published by the Société Mathématique de France. It was established in 1864 by the French chemist Louis Pasteur and published articles in mathematics, physics, chemistry, biology, and geology. In 1900, it became a purely mathematical journal. It is published with help of the Centre national de la recherche scientifique. Its web site is hosted by the mathematics department of the École Normale Supérieure École or Ecole may refer to: * an elementary school in the French educational stages normally followed by Secondary education in France, secondary education establishments (collège and lycée) * École (river), a tributary of the Seine flowing i .... External links * Archive(1864–2013) Mathematics journals Publications established in 1864 Multidisciplinary scientific journals Société Mathématique de France academic journals {{math-journal-stub English-F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michèle Raynaud
Michèle Raynaud (born Michèle Chaumartin; ) is a French mathematician, who works on algebraic geometry and who worked with Alexandre Grothendieck in Paris in the 1960s at the Institut des hautes études scientifiques (IHÉS). Biography Raynaud was a member of the séminaire de géométrie algébrique du Bois Marie (SGA) 1 and 2 and obtained her doctorate in 1972, supervised by Grothendieck at Paris Diderot University. Her thesis was entitled ''Théorèmes de Lefschetz en cohomologie cohérente et en cohomologie étale''. Grothendieck wrote about her doctoral thesis in ''Récoltes et Semailles'' (p.168 Chapitre 8.1.) describing it as original, entirely independent, and a major work. Michèle Raynaud was married to the mathematician Michel Raynaud Michel Raynaud (; 16 June 1938 – 10 March 2018 Décès de Mich ... [...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] |
Simplicial Group
In mathematics, more precisely, in the theory of simplicial sets, a simplicial group is a simplicial object in the category of groups. Similarly, a simplicial abelian group is a simplicial object in the category of abelian groups. A simplicial group is a Kan complex (in particular, its homotopy groups make sense). The Dold–Kan correspondence says that a simplicial abelian group may be identified with a chain complex. In fact it can be shown that any simplicial abelian group A is non-canonically homotopy equivalent to a product of Eilenberg–MacLane spaces, \prod_ K(\pi_iA,i). A commutative monoid in the category of simplicial abelian groups is a simplicial commutative ring. discusses a simplicial analogue of the fact that a cohomology class on a Kähler manifold has a unique harmonic representative and deduces Kirchhoff's circuit laws from these observations. See also *Simplicial commutative ring References * * * Charles Weibel Charles Alexander Weibel (born October ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after the Norwegian mathematician Niels Henrik Abel. The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified. Definition An abelian group is a set A, together with an operation ･ , that combines any two elements a and b of A to form another element of A, denoted a \cdot b. The sym ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dold–Kan Correspondence
In mathematics, more precisely, in the theory of simplicial sets, the Dold–Kan correspondence (named after Albrecht Dold and Daniel Kan) states that there is an equivalence between the category of (nonnegatively graded) chain complexes and the category of simplicial abelian groups. Moreover, under the equivalence, the nth homology group of a chain complex is the nth homotopy group of the corresponding simplicial abelian group, and a chain homotopy corresponds to a simplicial homotopy. (In fact, the correspondence preserves the respective standard model structures.) The correspondence is an example of the nerve and realization paradigm. There is also an ∞-category-version of the Dold–Kan correspondence. The book "Nonabelian Algebraic Topology" has a Section 14.8 on cubical versions of the Dold–Kan theorem, and relates them to a previous equivalence of categories between cubical omega-groupoids and crossed complexes, which is fundamental to the work of that book. Examp ... [...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 modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is contained in the kernel of the next. Associated to a chain complex is its homology, which is (loosely speaking) a measure of the failure of a chain complex to be exact. 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 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 complex is called the singular homology of X, and is a commonly used invariant of a topological space. Chain complexes are studied in homological algebra, but a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |