Sheaf Theory

Sheaves were introduced some 20 years ago by Jean Leray and have had a profound effect on several mathematical disciplines. Their major virtue is that they unify and give a mechanism for dealing with many problems concerned with passage from local information to global information. This is very useful when dealing with, say, differentiable manifolds, since locally these look like Euclidean space, and hence localized problems can be dealt with by means of all the tools of classical analysis. Piecing together “solutions” of such local problems in a coherent manner to describe, e.g., global invariants, is most easily accomplished via sheaf theory and its associated cohomology theory. The major virtue of sheaf theory is information-theoretic in nature. Most problems could be phrased and perhaps solved without sheaf theory, but the notation would be enormously more complicated and difficult to comprehend.