Sheaf

In mathematics a sheaf means, firstly, a topological construction F on a given space X giving one a set or richer structure F(U) for each open set U of X and subject to a patching condition. It can also mean a more refined generalisation, for which see Grothendieck topology.

The sheaf axiom

The formal definition of a sheaf divides into two parts. The first, which defines the idea of presheaf, is simple category theory. The second carries the burden of saying patching works, and is easy to understand in examples such as analytic continuation, where it amounts to saying that a consistent definition of an analytic function on each of several open sets in the complex plane gives rise to an analytic function on the union.

Definition of presheaf of sets on X: it is a contravariant functor from the open subsets of X, ordered by inclusion, to the category of sets. More explicitly if U and V are open subsets of X with U contained in V, we are given the data of a function from F(V) to F(U), which we can name 'restriction from V to U. The functor condition implies some basic axioms for restrictions.

Sheaf axiom: this explains the relation of F(V) with all the F(U) when U runs over an open cover of V. Within the cartesian product of the F(U) we look at the self-consistent families - those that restrict compatibly on the intersections of two of the open sets U. That is a necessary condition for a family to come by restriction from a single element of F(V). The sheaf axiom says that it is also sufficient.

There are many examples in which the sheaf axiom can be routinely checked: for example the sheaf C(X) of real-valued continuous functions on X, for which C(U) is the set of real-valued continuous functions on U.

Stalk of a sheaf

If we fix a point x of X and consider F(N) as N runs over open neighbourhoods of x, we can take the (inverse) limit, in the categorical sense. We call that F_x, the stalk of F at x. In the theory it is shown in what sense F can be reconstructed from its stalks.

Space associated to a sheaf

In early developments of sheaf theory, it was shown that giving a sheaf F as a functor is as good as giving a certain topological space Y together with a mapping from Y to X. Here Y is supposed to be the space of stalks of X: each stalk is given the discrete topology but Y is given the topology such that F can be recovered from Y as the sheaf of sections of the mapping. Of course, this accounts for the agricultural terminology.

What that means is that for open sets U we construct the sections, i.e. mappings from U to Y such that the composition with the given mapping from Y to X is the identity on U. Given any mapping Z to X we can in fact in this way construct a sheaf of sections: that is, sections of any continuous mapping obey the axioms for a sheaf. The statement is that sheaves as functors can be considered to arise in this way, up to a natural transformation, the isomorphism concept for functors.

The correct topology for Y makes the mapping to X a local homeomorphism. This therefore gives a complete picture of sheaves on X, as worked out for example in the book of Godement. Much of the categorical language came later than that.

The space associated to the sheaf of sections of Z, with its given continuous mapping to X, is therefore endowed with a local homeomorphism to X, and in a sense deals with all the 'ramification' in the mapping, in the 'best possible way'. This may be expressed by adjoint functors; but is also important as an intuition about sheaves of sets. This collection of ideas is related to topos theory, but in a sense that more general notion of sheaf moves away from geometric intuition.