Descent (category theory)

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.

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

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

intersection of X_i and X_j, we'll require mappings

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

for transitivity (and choosing apt notation). The f_ii should be identity maps and hence symmetry becomes ${\displaystyle f_{ij}=f_{ji}^{-1}}$ (so that it is fiberwise an isomorphism).

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