Editing Section (fiber bundle) (section)

==== Generalizations ====

Obstructions to extending local sections may be generalized in the following manner: take a [[topological space]] and form a [[Category (mathematics)|category]] whose objects are open subsets, and morphisms are inclusions. Thus we use a category to generalize a topological space. We generalize the notion of a "local section" using sheaves of [[abelian group]]s, which assigns to each object an abelian group (analogous to local sections).

There is an important distinction here: intuitively, local sections are like "vector fields" on an open subset of a topological space. So at each point, an element of a ''fixed'' vector space is assigned. However, sheaves can "continuously change" the vector space (or more generally abelian group).

This entire process is really the [[global section functor]], which assigns to each sheaf its global section. Then [[sheaf cohomology]] enables us to consider a similar extension problem while "continuously varying" the abelian group. The theory of [[characteristic class]]es generalizes the idea of obstructions to our extensions.