# Sheaf mathematics

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 (that is depend on the open sets). As such, they are a natural instrument to study the global behaviour of entities which are of local nature, such as open sets, analytic functions, manifolds, 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 UR. 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 Ui are given open sets with union U, and for each i we are given an element fiF(Ui), a continuous function fi : UiR. If these functions agree where they overlap, then we can glue them together in a unique way to form a continuous function f : UR which agrees with all the given fi. The collection of the sets F(U) together with the restriction maps F(U) → F(V) then form a sheaf of sets on X. Furthermore, each F(U) is a commutative ring and the restriction maps are ring homomorphisms, making F 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 UR. 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.

## 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 VU, a morphism resU,V : F(U) → F(V) in the category C. This is called the restriction morphism. The restriction morphism is required to satisfy two properties:
• for every open set U in X, we have resU,U = idF(U), i.e., the restriction from U to U is the identity morphism on F(U).
• given any three open sets WVU, we have resV,W ○ resU,V = resU,W, i.e. the restriction from U to V and then to W is the same as the restriction from U directly to 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 TopX whose objects are the open sets of X and whose morphisms are inclusions. TopX 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 TopX 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

See main article gluing axiom for a higher-level discussion

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

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

resUi,UiUj(fi) = resUj,UiUj(fj).

Intuitively speaking, if the fi represent functions, this says that any two compatible functions agree where they overlap. The sheaf axiom says that we can produce from the fi a unique section f over U whose restriction to each Ui is fi, i.e., resU,Ui(f)=fi. Sometimes this is split into two axioms, one guaranteeing existence, and the other guaranteeing uniqueness. A presheaf satisfying only the uniqueness part of the sheaf axiom is sometimes called a monopresheaf.

## 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 : EX is a continuous map. For every open set U in X, let F(U) be the set all continuous maps f : UE 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:

## Morphisms of sheaves

Let F and G be two sheaves on X both with values in the category C. We define a morphism from G to F to be a family of morphisms φU : G(U) → F(U) in the category C for all opens U in X which commute with the restriction maps. That is, the following diagram must commute

File:SheafMorphism-01.png

for each pair of open sets UV in X. If F and G are considered as contravariant functors from TopX to C then a morphism between them is nothing more than a natural transformation. With this definition the set of all C-valued sheaves on X forms a category (a functor category). An isomorphism of sheaves on X is just an isomorphism in this category.

One can generalize this notion to morphisms between sheaves on different spaces. Let f : XY be a continuous function between two topological spaces, and let F be a sheaf on X and G a sheaf on Y both with values in C. Then a morphism from G to F relative to f is given by a family of morphisms φU : G(U) → F(f−1(U)) for each open set U in Y such that the diagram

File:SheafMorphism-02.png

commutes for each pair of open sets UV in Y. The previous definition is the special case resulting when f is the identity map on X.

The category theoretical description is slightly more complicated in the general case. Let Top be the contravariant functor from the category of topological spaces Top to the category of small categories Cat which sends each space X to the poset category of its open sets TopX. Here Top(f) is a covariant functor from TopY to TopX sending each open set to its preimage. Composing F with Top(f) we obtain a contravariant functor from TopY to C. A morphism from G to F relative to f is then a natural transformation from G to F ○ Top(f).

Note that all of the above makes sense if we are working only with presheaves instead of sheaves.

## 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 an example of a colimit). We denote this limit by Fx and call it the stalk of F at x. If F is a C-valued sheaf on X, then the stalk Fx is an object of C, for C a category such as the category of abelian groups or the category of commutative rings.

For any open set U containing x there is a morphism from F(U) to Fx. If C is a concrete category, then applying this morphism to an element f in F(U) gives an element of Fx 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 neighbourhood of the point, using its power series expansion. However, some sheaves do not behave well; the germ of a smooth function at any point does not determine the function in any small neighbourhood 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 étale 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: EX

such that F is isomorphic 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 Fx 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. (One can also consider such a space in the light of the theory of representable functors; the history shows that this theory developed also in the mid-1950s.)

The space E was called espace étalé in Godement's influential book about homological algebra 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 now more common.

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 : ZX, the corresponding sheaf of sections gives rise in the above manner to a space of stalks E and a local homeomorphism π : EX. 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 (sheaf cohomology) that can give much useful, more concrete 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 Čech 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.

## History

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.

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.