Descent (category theory)

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.

At a fundamental level, the passage to a , and the down direction is from Y to X .)

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.

=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.

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

: 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.

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 ij of 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 . That is, we can take essentially same f ij , acting on various different fibers.

Another major point is the relation with the 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

: X ij

now as

: Y × X Y ,

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 .

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.

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 Alexander 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.

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 work a few years later of David Mumford in his geometric invariant theory spectacularly mixed scheme (mathematics) 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 ).