Sheaf (Mathematics)

A sheaf is a tool for systematically tracking data, such as sets, abelian groups or rings, attached to the open sets of a topological space and defined locally with respect to them; for instance, the data assigned to each open set might be the ring of continuous functions defined on it. Such data are well-behaved in that they can be restricted to smaller open sets, and the data assigned to an open set is equivalent to any collection of compatible data assigned to smaller open sets that cover it. Sheaves of a given type, together with the maps or morphisms between them, form a category on a fixed topological space, and sheaves have applications across topology, algebraic geometry and differential geometry: geometric structures such as differentiable manifolds or schemes can be expressed as a sheaf of rings on a space, and sheaves provide the framework for sheaf cohomology, a general cohomology theory that encompasses classical topological cohomology theories.