What is sheaves meaning?

In mathematics, a sheaf is a mathematical object that is used to describe local data attached to a topological space. It is a flexible tool to study local structures and to glue them together into global structures. Sheaves can be thought of as a collection of functions defined on subsets of a space that obey certain rules of compatibility. They are often used in algebraic geometry, topology, representation theory and other areas of mathematics.

More precisely, a sheaf is a functor that associates with every open set of a topological space X a certain algebraic structure (a module over a ring, a vector space, etc.) and with every inclusion of open sets a certain homomorphism (restriction map) between the corresponding algebraic structures, satisfying certain axioms. Sheaves are designed to capture the idea of a manifold being locally similar to Euclidean space, but with glueing maps between the local patches.

Sheaves provide a powerful set of tools for working with geometric objects, and have led to many important developments in modern mathematics. They are also important in physics, especially in mathematical physics and quantum field theory.