Sheaf

A data assignment that associates to each open set of a topological space an algebraic object (sets, groups, rings, modules, etc.) together with restriction maps, satisfying locality (sections equal locally are equal) and gluing (compatible local sections glue uniquely).

A data assignment that associates to every open set U of a topological space X an algebraic or analytic object (set, group, ring, module, etc.) together with restriction maps, satisfying locality (sections equal on overlaps are equal) and gluing (compatible local sections on a cover uniquely glue to a global section).