Descent (category Theory) - Descent of Vector Bundles

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 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_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 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

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.

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.