Generalizations
Obstructions to extending local sections may be generalized in the following manner: take a topological space and form a 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 groups, 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 classes generalizes the idea of obstructions to our extensions.
Read more about this topic: Section (fiber Bundle), Local and Global Sections, Extending To Global Sections