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.
Suppose X is a topological space covered by open sets Xi.
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 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.
satisfies the cocycle condition[1] The fully faithful descent says: The functor