In mathematics, the idea of descent has come to stand for a very general idea, extending the intuitive idea of 'gluing' in topology. Since the topologists' glue is actually the use of equivalence relations on topological spaces, the theory starts with some ideas on identification. Euclid, a famous Greek mathematician known as the father of geometry, is shown here in detail from The School of Athens by Raphael. ... Topology (Greek topos, place and logos, study) is a branch of mathematics concerned with spatial properties preserved under bicontinuous deformation (stretching without tearing or gluing); these are the topological invariants. ... In mathematics, an equivalence relation on a set X is a binary relation on X that is reflexive, symmetric and transitive, i. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...

At a fundamental level, the passage to a quotient space is not very well-behaved in topology. A current theory that addresses the issue is non-commutative geometry. Alexander Grothendieck encountered related issues at the end of the 1950s, in laying the modern foundations of algebraic geometry. There the situation on passing to a quotient space is much worse, because (roughly speaking) polynomials are less 'flexible' than general continuous functions. Grothendieck's paradigm is simply stated as this: pass from a space X to the category Sh(X) of all sheaves of sets on it. This doesn't lose much (see soberification). Then try to understand continuous mappings f from Y to X in terms of 'descending' sheaves from Y to X, in other words the image of f^*. (The intuition is that Y lies above X as a sort of covering space, and the 'down' direction is from Y to X.) 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 space. ... In mathematics, there is a close relationship between spaces, which are geometric in nature, and the numerical functions on them. ... Alexander Grothendieck (Berlin, March 28, 1928) is one of the most important mathematicians of the 20th century. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... In mathematics, a sheaf F on a topological space X is something that assigns a structure F(U) (such as a set, group, or ring) to each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and... In mathematics, specifically topology, a covering map is a continuous surjective map p : C → X, with C and X being topological spaces, which has the following property: to every x in X there exists an open neighborhood U such that p -1(U) is a union of mutually disjoint open...

A sophisticated theory resulted. It was a tribute to the efforts to use category theory to get round the alleged 'brutality' of imposing equivalence relations within geometric categories. One out-turn was the eventual definition adopted in topos theory of geometric morphism, to get the correct notion of surjectivity. Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... For discussion of topoi in literary theory, see literary topos. ... A surjective function. ...

Descent of vector bundles

The case of the construction of vector bundles from data on a disjoint union of topological spaces is a straightforward place to start. In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... In set theory, a disjoint union (or discriminated union) is a union of a collection of sets whose members are pairwise disjoint. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ...

Suppose X is a topological space covered by open sets X_i. Let Y to be the disjoint union of the X_i, so that there is a natural mapping In set theory, a disjoint union (or discriminated union) is a union of a collection of sets whose members are pairwise disjoint. ...

p : Y → X.

We think of Y as 'above' X, with the X_i projection 'down' onto X. With this language, descent implies a vector bundle on Y (so, a bundle given on each X_i), and our concern is to 'glue' those bundles V_i, to make a single bundle V on X. What we mean is that V should, when restricted to X_i, give back V_i, up to a bundle isomorphism. Look up Up to on Wiktionary, the free dictionary In mathematics, the phrase up to xxxx indicates that members of an equivalence class are to be regarded as a single entity for some purpose. ...

The data needed is then this: on each overlap

X_ij,

intersection of X_i and X_j, we'll require mappings

f_ij

to use to identify V_i and V_j there, fiber by fiber. Further the f_ij must satisfy conditions based on the reflexive, symmetric and transitive properties of an equivalence relation (gluing conditions). For example the composition

f_ijof_jk = f_ik

for transitivity (and choosing apt notation). The f_ii should be identity maps and hence the symmetry becomes invertibility of f_ij (so that it is fiberwise an isomorphism).

These are indeed standard conditions in fiber bundle theory (see transition function). One important application to note is change of fiber: if the f_ij are all you need to make a bundle, then there are many ways to make an associated bundle. That is, we can take essentially same f_ij, acting on various different fibers. In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ... In mathematics, a transition function has several different meanings: In topology, a transition function is a homeomorphism from one coordinate chart to another. ... In mathematics, the theory of fiber bundles with a structure group (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from to , which are both topological spaces with a group action of . ...

Another major point is the relation with the chain rule: the discussion of the way there of constructing tensor fields can be summed up as 'once you learn to descend the tangent bundle, for which transitivity is the Jacobian chain rule, the rest is just 'naturality of tensor constructions'. In calculus, the chain rule is a formula for the derivative of the composite of two functions. ... In mathematics, physics and engineering, a tensor field is a very general concept of variable geometric quantity. ... In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x ∈ M and v ∈ Tx(M), the tangent space... In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ...

To move closer towards the abstract theory we need to interpret the disjoint union of the

X_ij

now as

Y×_XY,

the fiber product (here an equalizer) of two copies of the projection p. The bundles on the X_ij that we must control are actually V′ and V", the pullbacks to the fiber of V via the two different projection maps to X. In category theory, a branch of mathematics, the pullback (also called the fiber product) is the limit of a diagram consisting of two morphisms f : X → Z and g : Y → Z with a common codomain. ... Equalizer can mean: Equalizer, an audio processing tool. ...

Therefore by going to a more abstract level one can eliminate the combinatorial side (that is, leave out the indices) and get something that makes sense for p not of the special form of covering with which we began. This then allows a category theory approach: what remains to do is to re-express the gluing conditions.

History

The ideas here flourished in the period 1955-1965 (which was roughly the time at which the requirements of algebraic topology were met but those of algebraic geometry were not). From the point of view of abstract category theory the work of comonads of Beck was a summation of those ideas; see Beck's monadicity theorem. Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... Algebraic geometry is a branch of mathematics which, as the name suggests, combines abstract algebra, especially commutative algebra, with geometry. ... Category theory is a mathematical theory that deals in an abstract way with mathematical structures and relationships between them. ... In category theory, a monad or triple is a type of functor, together with two associated natural transformations. ... In category theory, a branch of mathematics, Becks monadicity theorem asserts that a functor is monadic if and only if U has a left adjoint; U reflects isomorphisms; and C has co-equalizers of U-split co-equalizer pairs, and U preserves those co-equalizers. ...

The difficulties of algebraic geometry with passage to the quotient are acute: it is like doing the non-commutative geometry of Connes, to mention the currently-fashionable theory in the area of 'bad quotients', but with polynomials to separate points, rather than general continuous functions. The urgency (to put it that way) of the problem for the geometers accounts for the title of the 1959 Grothendieck seminar TDTE on theorems of descent and techniques of existence connecting the descent question with the representable functor question in algebraic geometry in general, and the moduli problem in particular. In mathematics, there is a close relationship between spaces, which are geometric in nature, and the numerical functions on them. ... Alexander Grothendieck (Berlin, March 28, 1928) is one of the most important mathematicians of the 20th century. ... In mathematics, especially in category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. ... In algebraic geometry, the moduli problem is to describe the parameters on which algebraic varieties depend. ...

As with a number of the more abstract flights of the Grothendieck school, later work relied on some of this and bypassed other parts (to the extent that the papers, published only in mimeographed form, may have already become hard to find). There were six TDTE Bourbaki seminars given by Grothendieck, which were incorporate in the FGA collection. That is now posted as online PDF. The Séminaire Nicolas Bourbaki (Bourbaki Seminar) is a series of seminars (in fact public lectures with printed notes distributed) that has been held in Paris since 1948. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

The work a few years later of David Mumford in his geometric invariant theory spectacularly mixed scheme and categorical techniques with more concrete geometry, to construct moduli spaces for curves and abelian varieties (for the first time, in the required technical sense of 'moduli'). David Bryant Mumford (born 11 June 1937) is an American mathematician known for distinguished work in algebraic geometry, and then for research into vision and pattern theory. ... In mathematics, geometric invariant theory in algebraic geometry is a (technically complex) development building on nineteenth century invariant theory. ... In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. ... In algebraic geometry, a moduli space is a parameter space for families of algebraic objects (such as algebraic varieties, morphisms, vector bundles). ...