Descent (category theory)
Encyclopedia
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 relation
s on topological space
s, the theory starts with some ideas on identification.
A sophisticated theory resulted. It was a tribute to the efforts to use category theory
to get around the alleged 'brutality' of imposing equivalence relations within geometric categories. One outcome was the eventual definition adopted in topos theory of geometric morphism, to get the correct notion of surjectivity.
s from data on a disjoint union
of topological space
s is a straightforward place to start.
Suppose X is a topological space covered by open sets 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 the symmetry becomes invertibility of fij (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 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 field
s 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 actually 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.
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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the idea of descent has come to stand for a very general idea, extending the intuitive idea of 'gluing' in topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
. Since the topologists' glue is actually the use of equivalence relation
Equivalence relation
In mathematics, an equivalence relation is a relation that, loosely speaking, partitions a set so that every element of the set is a member of one and only one cell of the partition. Two elements of the set are considered equivalent if and only if they are elements of the same cell...
s on topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
s, the theory starts with some ideas on identification.
A sophisticated theory resulted. It was a tribute to the efforts to use category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
to get around the alleged 'brutality' of imposing equivalence relations within geometric categories. One outcome was the eventual definition adopted in topos theory of geometric morphism, to get the correct notion of surjectivity.
Descent of vector bundles
The case of the construction of vector bundleVector 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 : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
s from data on a disjoint union
Disjoint union
In mathematics, the term disjoint union may refer to one of two different concepts:* In set theory, a disjoint union is a modified union operation that indexes the elements according to which set they originated in; disjoint sets have no element in common.* In probability theory , a disjoint union...
of topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
s is a straightforward place to start.
Suppose X is a topological space covered by open sets Xi. Let Y be the disjoint union
Disjoint union
In mathematics, the term disjoint union may refer to one of two different concepts:* In set theory, a disjoint union is a modified union operation that indexes the elements according to which set they originated in; disjoint sets have no element in common.* In probability theory , a disjoint union...
of the Xi, so that there is a natural mapping
- p : Y → X.
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
Up to
In mathematics, the phrase "up to x" means "disregarding a possible difference in x".For instance, when calculating an indefinite integral, one could say that the solution is f "up to addition by a constant," meaning it differs from f, if at all, only by some constant.It indicates that...
a bundle isomorphism.
The data needed is then this: on each overlap
- Xij,
intersection of Xi and Xj, we'll require mappings
- fij
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
- fijofjk = fik
for transitivity (and choosing apt notation). The fii should be identity maps and hence the symmetry becomes invertibility of fij (so that it is fiberwise an isomorphism).
These are indeed standard conditions in fiber bundle
Fiber bundle
In mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...
theory (see transition function
Transition function
In mathematics, a transition function has several different meanings:* In topology, a transition function is a homeomorphism from one coordinate chart to another...
). 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
Associated bundle
In mathematics, the theory of fiber bundles with a structure group G 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...
. That is, we can take essentially same fij, acting on various fibers.
Another major point is the relation with the chain rule
Chain rule
In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...
: the discussion of the way there of constructing tensor field
Tensor field
In mathematics, physics and engineering, a tensor field assigns a tensor to each point of a mathematical space . Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the physical...
s can be summed up as 'once you learn to descend the tangent bundle
Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...
, for which transitivity is the Jacobian
Jacobian
In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector- or scalar-valued function with respect to another vector. Suppose F : Rn → Rm is a function from Euclidean n-space to Euclidean m-space...
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
- Xij
now as
- Y×XY,
the fiber product (here an equalizer) of two copies of the projection p. The bundles on the Xij that we must control are actually 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.
History
The ideas were developed in the period 1955-1965 (which was roughly the time at which the requirements of algebraic topologyAlgebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
were met but those of algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
were not). From the point of view of abstract category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
the work of comonads of Beck was a summation of those ideas; see Beck's monadicity theorem
Beck's monadicity theorem
In category theory, a branch of mathematics, Beck's monadicity theorem asserts that a functorU: C \to Dis monadic if and only if# U has a left adjoint;# U reflects isomorphisms; and...
.
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
Alexander Grothendieck
Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...
seminar TDTE on theorems of descent and techniques of existence (see FGA
Fondements de la Géometrie Algébrique
FGA, or Fondements de la Géometrie Algébrique, is a book that collected together seminar notes of Alexander Grothendieck. It is animportant source for his pioneering work on scheme theory, which laid foundations for algebraic geometry in its modern technical developments.The title is a translation...
) connecting the descent question with the representable functor
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. Such functors give representations of an abstract category in terms of known structures In mathematics, particularly category theory, a...
question in algebraic geometry in general, and the moduli problem in particular.