Section (fiber bundle)

is a continuous right inverse of the projection function

is a fiber bundle over a base space,

: then a section of that fiber bundle is a continuous map, such that A section is an abstract characterization of what it means to be a graph.

can be identified with a function taking its values in the Cartesian product

The language of fibre bundles allows this notion of a section to be generalized to the case when

is a fibre bundle, then a section is a choice of point

simply means that the section at a point

In particular, a vector field on a smooth manifold

is a choice of tangent vector at each point of

is assumed to be a smooth fiber bundle over

In this case, one considers the space of smooth sections of

It is also useful in geometric analysis to consider spaces of sections with intermediate regularity (e.g.,

A local section of a fiber bundle is a continuous map

is the fiber), then local sections always exist over

in bijective correspondence with continuous maps from

The (local) sections form a sheaf over

The space of continuous sections of a fiber bundle

Sections are studied in homotopy theory and algebraic topology, where one of the main goals is to account for the existence or non-existence of global sections.

An obstruction denies the existence of global sections since the space is too "twisted".

More precisely, obstructions "obstruct" the possibility of extending a local section to a global section due to the space's "twistedness".

For example, a principal bundle has a global section if and only if it is trivial.

However, it only admits a nowhere vanishing section if its Euler class is zero.

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).

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.