Sheaf

Alternate meaning: River Sheaf

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 patching smaller open sets to obtain a bigger one.

Table of contents

1 Introduction 2 Timeline of the history of sheaf theory 3 The formal definition 3.1 Definition of a presheaf 3.2 The sheaf axiom 4 Examples 5 Stalks of a sheaf at a point and germs of functions 6 Local homeomorphisms: an equivalent approach 7 Generalization

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 object.

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 in F(U) can be restricted to V, and we get a map F(U) → F(V). "Patching" 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 are compatible, i.e. if any two of them agree on the intersection of their domains, then we can patch 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, patching 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 patching 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 theory

• 1936 Eduard Čech introduces the nerve construction, for associating a simplicial complex to an open covering.
• 1938 Whitney gives a 'modern' definition of cohomology, as a summation of 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) refounds algebraic geometry.
• 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.

The formal definition

The formal definition of a sheaf consists of two parts. The first is the concept of presheaf, which formalizes the idea of restriction, and can be formulated in terms of elementary category theory. The second part, the "sheaf axiom", formalizes the idea that patching works and is technically more involved.

Definition of a presheaf

Suppose X is a topological space, and C is a concrete category (think of the examples we already encountered above: the category of sets, the category of commutative rings or the category of real vector spaces). A presheaf F of C on X is given by the following data:

• for every open set U in X, an object F(U) in C
• for every two open sets V ⊂ U, a morphisms F(U) → F(V) in the category C. We call this the "restriction of F(U) to V" and 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 V and then to W is the same as the restriction of F(U) directly to W.

In the language of category theory, all of this can be summarized as follows: a presheaf of C on X is a contravariant functor from the category of open subsets of X, with inclusions as morphisms, to C. ("Contravariant" because the restriction morphisms F(U) → F(V) go in the opposite direction of the inclusion V ⊂ U.)

The sheaf axiom

A sheaf is a presheaf satisfying an additional axiom which captures the idea of pasting together the structures F(U). To state this axiom we need to first define what compatible elements of F(U) and F(V) are: elements f of F(U) and g of F(V) are compatible if f and g restrict to the same element h of F(U∩V).

Now, if U is the union of a collection of open sets {U_i}, the presheaf condition implies that any element f of F(U) gets restricted to compatible elements f_i of the F(U_i). The sheaf axiom states that the converse is also true: given elements f_i in F(U_i) for each i, such that any two of those elements are compatible, then there is precisely one element f of F(U) which restricts to the given f_i.

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

If we fix a point x of X and consider F(N) as N runs over open neighbourhoods of x, we can take the (direct) limit, in the categorical sense. We denote this limit by F_x and call it the stalk of F at x. If F is a sheaf of C on X, then the stalk F_x is an object of C.

Given an open set U containing x and an element f in F(U), then by applying the natural limit homorphism to f one obtains an element in F_x, 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.

Another example is given by analytic functions, for which power series serve as germs; but note that the germ of a differentiable function contains more information that its Taylor expansion.

Local homeomorphisms: an equivalent approach

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 stalk 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.

Generalization

One can define cohomology for sheaves of abelian groups on a given topological space. The idea here is that a sheaf serves as a "measuring rod" for the space, and the cohomology groups of the sheaf serve as rough measures of the space when measured with that rod.

By precisely analyzing the properties of X needed to define sheaves, Alexander Grothendieck came up with the concept of a Grothendieck site, defined generalized sheaves on these sites and with that also very general cohomology theories.