Sheaf theory

In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data is well behaved in that it can be restricted to smaller open sets, and also the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set. (Intuitively, every piece of data is the sum of its parts.)

Quotes

• The purpose of sheaf theory is quite general: it is to obtain global information from local information, or else to define “obstructions” which characterize the fact that a local property does not hold globally any more: for example a manifold is not always orientable, or a differential equation can be locally solvable, but not globally.
  • Masaki Kashiwara; Pierre Schapira (1 May 2002). Sheaves on Manifolds: With a Short History. Springer Science & Business Media. p. 1. ISBN 978-3-540-51861-7.