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In mathematics, the idea of descent extends the intuitive idea of 'gluing' in topology. Since the topologists' glue is the use of equivalence relations on topological spaces, the theory starts with some ideas on identification.
The case of the construction of vector bundles from data on a disjoint union of topological spaces is a straightforward place to start.
Xi. Let Y be the disjoint union of the Xi, so that there is a natural mapping
We think of Y as 'above' X, with the Xi projection 'down' onto X. With this language, descent implies a vector bundle on Y (so, a bundle given on each Xi), and our concern is to 'glue' those bundles Vi, to make a single bundle V on X. What we mean is that V should, when restricted to Xi, give back Vi, up to a bundle isomorphism.
The data needed is then this: on each overlap
intersection of Xi and Xj, we'll require mappings
to use to identify Vi and Vj there, fiber by fiber. Further the fij must satisfy conditions based on the reflexive, symmetric and transitive properties of an equivalence relation (gluing conditions). For example, the composition
for transitivity (and choosing apt notation). The fii should be identity maps and hence symmetry becomes (so that it is fiberwise an isomorphism).
These are indeed standard conditions in fiber bundle theory (see transition map). One important application to note is change of fiber: if the fij 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 fij, acting on various fibers.
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'.
To move closer towards the abstract theory we need to interpret the disjoint union of the
now as
the fiber product (here an equalizer) of two copies of the projection p. The bundles on the Xij that we must control are V′ and V", the pullbacks to the fiber of V via the two different projection maps to X.
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.
The ideas were developed 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.
The difficulties of algebraic geometry with passage to the quotient are acute. 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 (see FGA) connecting the descent question with the representable functor question in algebraic geometry in general, and the moduli problem in particular.
Let . Each sheaf F on X gives rise to a descent data:
where satisfies the cocycle condition: [1]
The fully faithful descent says: is fully faithful. The descent theory tells conditions for which there is a fully faithful descent.
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.
In mathematics, the inverse limit is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits can be defined in any category although their existence depends on the category that is considered. They are a special case of the concept of limit in category theory.
In mathematics, a direct limit is a way to construct a object from many objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any category. The way they are put together is specified by a system of homomorphisms between those smaller objects. The direct limit of the objects , where ranges over some directed set , is denoted by . This notation suppresses the system of homomorphisms; however, the limit depends on the system of homomorphisms.
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition.
In mathematics, and particularly topology, a fiber bundle is a space that is locally a product space, but globally may have a different topological structure. Specifically, the similarity between a space and a product space is defined using a continuous surjective map, that in small regions of behaves just like a projection from corresponding regions of to The map called the projection or submersion of the bundle, is regarded as part of the structure of the bundle. The space is known as the total space of the fiber bundle, as the base space, and the fiber.
A CW complex is a kind of a topological space that is particularly important in algebraic topology. It was introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial nature that allows for computation. The C stands for "closure-finite", and the W for "weak" topology.
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities and allowing "varieties" defined over any commutative ring.
This is a glossary of terms specific to differential geometry and differential topology. The following three glossaries are closely related:
In mathematics, an algebraic stack is a vast generalization of algebraic spaces, or schemes, which are foundational for studying moduli theory. Many moduli spaces are constructed using techniques specific to algebraic stacks, such as Artin's representability theorem, which is used to construct the moduli space of pointed algebraic curves and the moduli stack of elliptic curves. Originally, they were introduced by Alexander Grothendieck to keep track of automorphisms on moduli spaces, a technique which allows for treating these moduli spaces as if their underlying schemes or algebraic spaces are smooth. After Grothendieck developed the general theory of descent, and Giraud the general theory of stacks, the notion of algebraic stacks was defined by Michael Artin.
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.
In mathematics, a Grothendieck universe is a set U with the following properties:
Fibred categories are abstract entities in mathematics used to provide a general framework for descent theory. They formalise the various situations in geometry and algebra in which inverse images of objects such as vector bundles can be defined. As an example, for each topological space there is the category of vector bundles on the space, and for every continuous map from a topological space X to another topological space Y is associated the pullback functor taking bundles on Y to bundles on X. Fibred categories formalise the system consisting of these categories and inverse image functors. Similar setups appear in various guises in mathematics, in particular in algebraic geometry, which is the context in which fibred categories originally appeared. Fibered categories are used to define stacks, which are fibered categories with "descent". Fibrations also play an important role in categorical semantics of type theory, and in particular that of dependent type theories.
In mathematics, an LF-space, also written (LF)-space, is a topological vector space (TVS) X that is a locally convex inductive limit of a countable inductive system of Fréchet spaces. This means that X is a direct limit of a direct system in the category of locally convex topological vector spaces and each is a Fréchet space. The name LF stands for Limit of Fréchet spaces.
This is a glossary of properties and concepts in category theory in mathematics.
In mathematics—more specifically, in differential geometry—the musical isomorphism is an isomorphism between the tangent bundle and the cotangent bundle of a pseudo-Riemannian manifold induced by its metric tensor. There are similar isomorphisms on symplectic manifolds. The term musical refers to the use of the symbols (flat) and (sharp).
In mathematics, the Gauss–Manin connection is a connection on a certain vector bundle over a base space S of a family of algebraic varieties . The fibers of the vector bundle are the de Rham cohomology groups of the fibers of the family. It was introduced by Yuri Manin for curves S and by Alexander Grothendieck in higher dimensions.
In mathematics a stack or 2-sheaf is, roughly speaking, a sheaf that takes values in categories rather than sets. Stacks are used to formalise some of the main constructions of descent theory, and to construct fine moduli stacks when fine moduli spaces do not exist.
In algebraic geometry, the Nisnevich topology, sometimes called the completely decomposed topology, is a Grothendieck topology on the category of schemes which has been used in algebraic K-theory, A¹ homotopy theory, and the theory of motives. It was originally introduced by Yevsey Nisnevich, who was motivated by the theory of adeles.
In algebraic geometry, a prestackF over a category C equipped with some Grothendieck topology is a category together with a functor p: F → C satisfying a certain lifting condition and such that locally isomorphic objects are isomorphic. A stack is a prestack with effective descents, meaning local objects may be patched together to become a global object.
In algebraic geometry, a torsor or a principal bundle is an analogue of a principal bundle in algebraic topology. Because there are few open sets in Zariski topology, it is more common to consider torsors in étale topology or some other flat topologies. The notion also generalizes a Galois extension in abstract algebra. Though other notions of torsors are known in more general context this article will focus on torsors over schemes, the original setting where torsors have been thought for. The word torsor comes from the French torseur. They are indeed widely discussed, for instance, in Michel Demazure's and Pierre Gabriel's famous book Groupes algébriques, Tome I.
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