Sheaf (mathematics)

Alternate meanings: River Sheaf, King Sceaf

In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain a bigger one. A presheaf is similar to a sheaf, but it may not be possible to glue. Sheaves, it turns out, enable one to discuss in a refined way what is a local property, as applied to a function.

Introduction

Sheaves are used in topology, algebraic geometry and differential geometry whenever one wants to keep track of algebraic data that vary with every open set of the given geometrical space. They are a global tool to study objects which vary locally (i.e., depending on the open set). As such, they are a natural instrument to study the global behaviour of entities which are of local nature, such as open sets, continuous, analytic, differentiable functions, and so on.

For a typical example, consider a topological space X, and for every open set U in X, let F(U) be the set of all continuous functions U → R. If V is an open subset of U, then the functions on U can be restricted to V, and we get a map F(U) → F(V). "Gluing" describes the following process: suppose the U_i are given open sets with union U, and for each i we are given an element f_i ∈ F(U_i), i.e. a continuous function f_i : U_i → R. If these functions agree where they overlap, then we can glue them together in a unique way to form a continuous function f : U → R which agrees with all the given f_i. The collection of the sets F(U) together with the restriction maps F(U) → F(V) then form a sheaf of sets on X. Indeed, the F(U) are commutative rings and the restriction maps are ring homomorphisms, and F is therefore even a sheaf of rings on X.

For a very similar example, consider a differentiable manifold X, and for every open set U of X, let F(U) be the set of differentiable functions U → R. Here too, gluing works and we obtain a sheaf of rings on X. Another sheaf on X assigns to every open set U of X the vector space of all differentiable vector fields defined on U. Restriction and gluing of vector fields works like that of functions, and we obtain a sheaf of vector spaces on the manifold X.

Timeline of the history of sheaf theory

The first origins of sheaf theory are hard to pin down - they may be co-extensive with the idea of analytic continuation. It took about 15 years for a recognisable, free-standing theory of sheaves to emerge from the foundational work on cohomology.

1936 Eduard Čech introduces the nerve construction, for associating a simplicial complex to an open covering.
1938 Whitney gives a 'modern' definition of cohomology, summarizing the work since Alexander and Kolmogorov defined cochains.
1943 Steenrod publishes on homology with local coefficients.
1945 Jean Leray publishes work carried out as a POW, motivated by proving fixed point theorems for application to PDE theory; it is the start of sheaf theory and spectral sequences.
1947 Henri Cartan reproves the de Rham theorem by sheaf methods, in correspondence with André Weil. Leray gives a sheaf definition in his courses via closed sets (the later carapaces).
1948 The Cartan seminar writes up sheaf theory for the first time.
1950 The 'second edition' sheaf theory from the Cartan seminar: the sheaf space (éspace étalé) definition is used, with stalkwise structure. Supports are introduced, and cohomology with supports. Continuous mappings give rise to spectral sequences. At the same time Kiyoshi Oka introduces an idea (adjacent to that) of a sheaf of ideals, in several complex variables.
1951 The Cartan seminar proves the Theorems A and B based on Oka's work.
1953 The finiteness theorem for coherent sheaves in the analytic theory is proved by Cartan and Serre, as is Serre duality.
1954 Serre's paper Faisceaux algébriques cohérents (published 1955) introduces sheaves into algebraic geometry. These ideas are immediately exploited by Hirzebruch, who writes a major book on topological methods.
1955 Grothendieck in lectures in Kansas defines abelian category and presheaf, and by using injective resolutions allows direct use of sheaf cohomology on all topological spaces, as derived functors.
1957 Grothendieck's Tohoku paper rewrites homological algebra; he proves Grothendieck duality (i.e., Serre duality for singular varieties).
1958 Godement's book on sheaf theory is published. At around this time Mikio Sato proposes his hyperfunctions, which will turn out to have sheaf-theoretic nature.
1957 onwards: Grothendieck extends sheaf theory in line with the needs of algebraic geometry, introducing: schemes and general sheaves on them, local cohomology, the derived category (with Verdier), and the Grothendieck topology. There emerges also his influential schematic idea of 'six operations' in homological algebra.

At this point sheaves had become a mainstream part of mathematics, with use by no means restricted to algebraic topology. It was later discovered that the logic in categories of sheaves is intuitionistic logic (this observation is now often referred to as Kripke-Joyal semantics, but probably should be attributed to a number of authors). This shows that some of the facets of sheaf theory can also be traced back as far as Leibniz.

The formal definition

To define sheaves we will proceed in two steps. The first step is to introduce the concept of a presheaf, which captures the idea of associating local information to a topological space. The second step is to introduce an additional axiom, called the gluing axiom or the sheaf axiom, which captures the idea of gluing local information to get global information.

Definition of a presheaf

Suppose X is a topological space, and C is a category (often, this is the category of sets, the category of Abelian groups, the category of commutative rings, or the category of modules over a fixed ring). A presheaf F of objects in C on the space X is given by the following data:

for every open set U in X, an object F(U) in C
for every inclusion of open sets V ⊂ U, a morphism F(U) → F(V) in the category C. This is called the "restriction from U to V". We will write it as res_U,V.

We require two properties:

for every open set U in X, we have res_U,U = id_F(U), i.e., the restriction of F(U) to U is the identity.
given any three open sets W ⊂ V ⊂ U, we have res_V,W o res_U,V = res_U,W, i.e. the restriction of F(U) to F(V) and then to F(W) is the same as the restriction of F(U) directly to F(W).

This definition can be expressed naturally in terms of category theory. First we define the category of open sets on X to be the category Top_X whose objects are the open sets of X and whose morphisms are inclusions. Top_X is then the category corresponding to the partial order ⊂ on the open sets of X. A C-presheaf on X is then a contravariant functor from Top_X to C.

If F is a C-valued presheaf on X, and U is an open subset of X, then F(U) is called the sections of F over U. (This is by analogy with sections of fiber bundles; see below) If C is a concrete category, then each element of F(U) is called a section. F(U) is also often denoted Γ(U,F).

The gluing axiom

Sheaves are presheaves on which sections over small open sets can be glued to give sections over larger open sets. We will first state the gluing axiom in a form that requires C to be a concrete category.

Let U be the union of the collection of open sets {U_i}. For each U_i, choose a section f_i on U_i. We say that the f_i are compatible if for any i and j,

res_{U_i,U_i∩U_j}(f_i) = res_{U_j,U_i∩U_j}(f_j).

Intuitively speaking, if the f_i represent functions, this says that any two functions agree where they overlap. The sheaf axiom says that we can produce from the f_i a unique section f over U whose restriction to each U_i is f_i, i.e., res_{U,U_i}(f)=f_i. Sometimes this is split into two axioms, one guaranteeing existence, and the other guaranteeing uniqueness.

To rephrase this definition in a way that will work in any category, we note that we can write the objects and morphisms involved in the definition above in a diagram that looks like this:

{\mathcal {F}}(U)\rightarrow \prod _{i}{\mathcal {F}}(U_{i}){\rightarrow  \atop \rightarrow }\prod _{i,j}{\mathcal {F}}(U_{i}\cap U_{j})

Here the first map is the product of the restriction maps res_{U,U_i,}:F(U)→F(U_i) and each pair of arrows represents the two restrictions res_{U_i,U_i∩U_j}:U_i→U_i∩U_j and res_{U_j,U_i∩U_j}:U_j→U_i∩U_j. It is worthwhile to note that these maps exhaust all of the possible restriction maps among U, the U_i, and the U_i∩U_j.

The condition that F be a sheaf is exactly that F(U) is the limit of the rest of the diagram. This suggests that we should rephrase the notion of covering in a categorical context. When we do, we get a diagram that looks similar to the one above:

\prod _{i,j}U_{i}\cap U_{j}{\rightarrow  \atop \rightarrow }\prod _{i}U_{i}\rightarrow U

(It is worth noting here that to form the products in the diagram, we must embed the category Top_X in a complete category) The condition that U is the union of the U_i is that U is a colimit of the rest of the diagram.

The gluing axiom is now that F turns all colimits into limits.

Examples

In addition to the sheaves of continuous functions, differentiable functions and vector fields given in the introduction, sheaves of sections are very important examples. Suppose E and X are topological spaces and π : E → X is a continuous map. For every open set U in X, let F(U) be the set all continuous maps f : U → E such that π(f(x)) = x for all x in U. Such a function f is called a section of π. It is not difficult to check that F is a sheaf of sets on X. In fact, every sheaf of sets on X is essentially of this type, for very special maps π; see below.

Given a sheaf F on X, the elements of F(X) are also called the global sections, a terminology motivated by the previous example.

Further examples:

Any fiber bundle gives rise to a sheaf of sets, by taking sections.
See how sheaves are used in the article on Riemann surfaces.
Ringed spaces are sheaves of commutative rings; especially important are the locally ringed spaces where all stalks (see below) are local rings.
Schemes are special locally ringed spaces important in algebraic geometry; sheaves of modules are important in the associated theory.

Stalks of a sheaf at a point and germs of functions

Fix a point x of X. We would like to study the behavior of F near the point x. In analytical terms, we would like to somehow take the limit as we get nearer and nearer to the point x. The corresponding concept is to take the direct limit of F(N) as N runs over the open neighbourhoods of x ordered by inclusion (in categorical terminology, this is the colimit). We denote this limit by F_x and call it the stalk of F at x. If F is a C-valued sheaf on X, then the stalk F_x is an object of C.

For any open set U containing x there is a morphism from F(U) to F_x. If C is a concrete category, then applying this morphism to an element f in F(U) gives an element of F_x called the germ of f at x.

This corresponds to the notion of germ of a function used elsewhere in mathematics. Intuitively, the germ of the function f at x describes the local behavior of f at the point x; it is a kind of 'ghost' of f, looked at only very near x. See also the detailed example given at local ring.

For some sheaves, germs behave well, and can give good local information; the germ of an analytic function around a point determines the function in a small neighboorhood of the point, using its power series expansion. However, some sheaves do not have well; the germ of a smooth function at any point does not determine the function in any small neighboorhood of the point. As an example, take any bump function. Its local behavior on the interval where it is one is that of a constant function, but knowing that a bump function is the constant one near a given point does not tell you where the function begins to decay; from its local behavior, you cannot even conclude that it is a bump function!

The Étalé space of a sheaf

In early developments of sheaf theory, it was shown that giving a sheaf F on X is as good as giving a certain topological space E together with a continuous map from E to X. More precisely: to every sheaf F of sets on X there exists a local homeomorphism π: E → X such that F is isomorphic (in the sense of natural isomorphism, the isomorphism concept for functors) to the sheaf of sections of π that was described in the example section above.

Furthermore, the space E is determined up to homeomorphism by F. It is the space of stalks of F: each stalk is given the discrete topology, and we take the disjoint union of all the stalks, with π mapping all of the stalks F_x to x. The topology on this space of stalks can be chosen so that the sheaf F can be recovered as the sheaf of sections of π.

At a higher level of abstraction, we can say that the category of sheaves of sets on X is equivalent to the category of local homeomorphisms to X.

The space E was called espace étalé in Godement's influential book about algebraic geometry and sheaf theory (Topologie Algebrique et Theorie des Faisceaux, R. Godement); in that book, sheaves are in fact defined as coming from sections of local homeomorphisms; the functorial approach we gave above came later and is more common nowadays.

The above considerations remain true for sheaves of C on X: we can still form the space of stalks, each stalk is an object in C, and the sections naturally become objects in C as well.

Given an arbitrary continuous map g : Z → X, the corresponding sheaf of sections gives rise in the above manner to a space of stalks E and a local homeomorphism π : E → X. In a sense this deals with all the 'ramification' in the map g, 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.

Generalizations

It is possible to define a cohomology theory for sheaves of abelian groups that gives much useful information. The main issue is the existence of the long exact sequence coming from an exact sequence of sheaves. In applications emphasis was placed on sheaves on spaces that were less well-behaved than finite complexes. For example, in algebraic geometry spaces carrying the Zariski topology are rarely Hausdorff.

The algebraic geometry case was first tackled by Jean-Pierre Serre by developing an analogue of Cech cohomology; this worked, though in general the construction doesn't have such good properties. Then Alexander Grothendieck used derived functors of the global section functor, providing a more definitive solution.

Grothendieck was motivated to develop a cohomology theory for sheaves that would give stronger results, and that would, in particular, allow a proof of the Weil conjectures. By precisely analyzing the properties of X needed to define sheaves, he defined the notion of a Grothendieck topology on a category (this came in a somewhat roundabout fashion — see background and genesis of topos theory).

A category together with a Grothendieck topology is called a site. It is possible to define the notion of a sheaf on any site. The notion of sites later led Lawvere to develop the notion of an elementary topos.