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36 relations: Alexander Grothendieck, Algebraic geometry, Algebraic topology, Associated bundle, Atlas (topology), Beck's monadicity theorem, Cambridge University Press, Category theory, Chain rule, Cohomological descent, Descent along torsors, Disjoint union, Equaliser (mathematics), Equivalence relation, Fiber bundle, Fibred category, Fondements de la Géometrie Algébrique, Grothendieck connection, Grothendieck topology, Jacobian matrix and determinant, Mathematics, Moduli space, Monad (category theory), Pullback (category theory), Quotient space (topology), Representable functor, Springer Science+Business Media, Stack (mathematics), Surjective function, Tangent bundle, Tensor field, Topological space, Topology, Topos, Up to, Vector bundle.
Alexander Grothendieck
Alexander Grothendieck (28 March 1928 – 13 November 2014) was a German-born French mathematician who became the leading figure in the creation of modern algebraic geometry.
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Algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials.
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Algebraic topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces.
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Associated bundle
In mathematics, the theory of fiber bundles with a structure group G (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F_1 to F_2, which are both topological spaces with a group action of G. For a fibre bundle F with structure group G, the transition functions of the fibre (i.e., the cocycle) in an overlap of two coordinate systems Uα and Uβ are given as a G-valued function gαβ on Uα∩Uβ.
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Atlas (topology)
In mathematics, particularly topology, one describes a manifold using an atlas.
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Beck's monadicity theorem
In category theory, a branch of mathematics, Beck's monadicity theorem gives a criterion that characterises monadic functors, introduced by in about 1964.
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Cambridge University Press
Cambridge University Press (CUP) is the publishing business of the University of Cambridge.
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Category theory
Category theory formalizes mathematical structure and its concepts in terms of a labeled directed graph called a category, whose nodes are called objects, and whose labelled directed edges are called arrows (or morphisms).
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Chain rule
In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions.
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Cohomological descent
In algebraic geometry, a cohomological descent is, roughly, a "derived" version of a fully faithful descent in the classical descent theory.
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Descent along torsors
In mathematics, given a G-torsor X → Y and a stack F, the descent along torsors says there is a canonical equivalence between F(Y), the category of Y-points and F(X)G, the category of G-equivariant X-points.
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Disjoint union
In set theory, the disjoint union (or discriminated union) of a family of sets is a modified union operation that indexes the elements according to which set they originated in.
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Equaliser (mathematics)
In mathematics, an equalizer is a set of arguments where two or more functions have equal values.
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Equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive.
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Fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in British English, fibre bundle) is a space that is locally a product space, but globally may have a different topological structure.
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Fibred category
Fibred categories (or fibered categories) are abstract entities in mathematics used to provide a general framework for descent theory.
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Fondements de la Géometrie Algébrique
Fondements de la Géometrie Algébrique (FGA) is a book that collected together seminar notes of Alexander Grothendieck.
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Grothendieck connection
In algebraic geometry and synthetic differential geometry, a Grothendieck connection is a way of viewing connections in terms of descent data from infinitesimal neighbourhoods of the diagonal.
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Grothendieck topology
In category theory, a branch of mathematics, a Grothendieck topology is a structure on a category C which makes the objects of C act like the open sets of a topological space.
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Jacobian matrix and determinant
In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Moduli space
In algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects.
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Monad (category theory)
In category theory, a branch of mathematics, a monad (also triple, triad, standard construction and fundamental construction) is an endofunctor (a functor mapping a category to itself), together with two natural transformations.
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Pullback (category theory)
In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms and with a common codomain.
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Quotient space (topology)
In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or "gluing together" certain points of a given topological space.
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Representable functor
In mathematics, particularly category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets.
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Springer Science+Business Media
Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
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Stack (mathematics)
In mathematics a stack or 2-sheaf is, roughly speaking, a sheaf that takes values in categories rather than sets.
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Surjective function
In mathematics, a function f from a set X to a set Y is surjective (or onto), or a surjection, if for every element y in the codomain Y of f there is at least one element x in the domain X of f such that f(x).
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Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M. As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector.
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Tensor field
In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold).
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Topological space
In topology and related branches of mathematics, a topological space may be defined as a set of points, along with a set of neighbourhoods for each point, satisfying a set of axioms relating points and neighbourhoods.
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Topology
In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.
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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).
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Up to
In mathematics, the phrase up to appears in discussions about the elements of a set (say S), and the conditions under which subsets of those elements may be considered equivalent.
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Vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every point x of the space X we associate (or "attach") a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over X.
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Descent (category theory), Descent data, Descent theory, Faithfully-flat descent.
References
[1] https://en.wikipedia.org/wiki/Descent_(mathematics)
