Sheaves and Presheaves

Posted on 2025-06-19 In Math

Outline

I. Sheaves and Dierential Manifolds:

Denitions and Examples

In Geometry as well as in Physics, one has often to use the tools of dierential

and integral calculus on topological spaces which are locally like open subsets of the

Euclidean space R

, but do not admit coordinates valid everywhere. For example, the

sphere {(x, y, z) ∈ R

: x

+ y

+ z

= 1}, or more generally, the space {(x

, . . . , x

) ∈

∑

= 1} are clearly of geometric interest. On the other hand, constrained motion

has to do with dynamics on surfaces in R

. In general relativity, one studies ‘space-

time’ which combines the space on which motion takes place and the time parameter

in one abstract space and allows reformulation of problems of Physics in terms of a

4-dimensional object. All these necessitate a framework in which one can work with

the tools of analysis, like dierentiation, integration, dierential equations and the like,

on fairly abstract objects. This would enable one to study Dierential Geometry in

its appropriate setting on the one hand, and to state mathematically the equations of

Physics in the required generality, on the other. The basic objects which accomplish

this are called dierential manifolds. These are geometric objects which are locally like

domains in the Euclidean space, so that the classical machinery of calculus, available in

, can be transferred, rst, to small open sets and then patched together. The main

tool in the patching up is the notion of a sheaf. We will rst dene sheaves and discuss

basic notions related to them, before taking up dierential manifolds.

1 Sheaves and Presheaves

At the outset, it is clear that functions of interest to us belong to a class such as

continuous functions, innitely dierentiable (or C

∞

) functions, real analytic functions,

holomorphic functions of complex variables, and so on. Some properties common to all

these classes of functions are that

i) they are all continuous, and

ii) the condition for a continuous function to belong to the class is of a local nature.

Chapter I: Sheaves and Differential Manifolds:

Definitions and Examples

By this we mean that for a continuous function to be dierentiable, real analytic or

holomorphic, it is necessary and sucient for it to be so in the neighbourhood of every

point in its domain of denition. We will start with an axiomatisation of the local nature

of the classes of functions we seek to study.

Denition 1.1. Let X be a topological space. An assignment to every open subset

U of X, of a set F(U) and to every pair of open sets U, V with V ⊂ U, of a map (to be

called restriction map) res

: F(U) → F(V ) satisfying

res

V W

◦ res

= res

for every triple W ⊂ V ⊂ U of open sets, is called a presheaf of sets. If F(U) are

all abelian groups, rings, vector spaces, …and the restriction maps are homomorphisms

of the respective structures, then we say that F is a presheaf of abelian groups, rings,

vector spaces, ….

Denition 1.2. A presheaf is said to be a sheaf if it satises the following additional

conditions. Let U =

∪

i∈I

be any open covering of an open set U. Then

: Two elements s, t ∈ F(U) are equal if res

s = res

t for all i ∈ I.

: If s

∈ F(U

) satisfy res

∩U

= res

∩U

for all i, j ∈ I, then there exists

an element s ∈ F(U ) with res

s = s

for all i. We will also assume that F(∅)

consists of a single point.

Example 1.3.

1) As indicated above, the concepts of dierentiability, real analyticity, …are all local

in nature, so that it is no surprise that if X is an open subspace of R

, then the

assignment to every open subset U of X, of the set A(U ) of dierentiable functions

with the natural restriction of functions as restriction maps, gives rise to a sheaf.

This sheaf will be called the sheaf of dierentiable functions on X. Obviously this

sheaf is not merely a sheaf of sets, but a sheaf of R-algebras.

2) The assignment of the set of bounded functions to every open set U of X, and the

natural restriction maps dene a presheaf on X. However, if we are given as in

any compatible set of bounded functions s

, then while such a data does dene

a unique function s on U, there is no guarantee that it will be bounded. Thus

this presheaf satises S

but not S

and so is not a sheaf. (Can one modify this in

order to obtain a sheaf?)

3) Consider the assignment of a xed abelian group to every nonempty open set U ,

all restriction maps being identity. We will also assign the trivial group to the

empty set. This denes a presheaf, but is not a sheaf in general. (Why?)

Section 1: Sheaves and Presheaves

4) The standard n-simplex ∆

is dened to be {(x

, x

, . . . , x

) ∈ R

n+1

∑

1, x

≥ 0} with the induced topology. In algebraic topology, one way of studying

the topology of a space X is to look at continuous maps of the standard simplices

into X and studying their geometry. We will give the basic denitions here. A

singular n-simplex in a topological space X is a continuous map of the standard

n-simplex ∆

into X. If A is a xed abelian group, then an A-valued singular

cochain in X is an assignment of an element of A to every singular simplex. Now,

to every open subset U of X, associate the set S

(U) of all singular cochains in

U. If V is an open subset of U , then we have an obvious inclusion of the set of

singular simplices in V into that in U. Consequently there is also a restriction

map S

(U) → S

(V ). This makes S

a presheaf, which we may call the presheaf

of singular cochains in X. It is obvious that this does not satisfy the axiom S

for sheaves. For if X =

∪

is a nontrivial open covering, then we can dene a

nonconstant cochain which is zero on all simplices whose images are contained in

some U

. On the other hand, if a cochain is dened on simplices with images in

some U

, then one can extend this cochain to all singular simplices by dening the

cochain to be zero on simplices whose images are not contained in any of the U

Thus this presheaf satises S

5) The most characteristic example of a sheaf from our point of view is the one which

associates to any open set U the algebra of continuous functions on U . This can

be generalised as follows. Let Y be any xed topological space and consider the

assignment to any open set U, of the set of continuous maps from U to Y with

the obvious restriction

6) We will generalise this example a little further. Let E be a topological space and

π : E → X a continuous surjective map. Then associate to each open subset U of

X the set of continuous sections of π over U (namely, continuous maps σ : U → E

such that π ◦σ = Id

). This, together with the obvious restriction maps, is a sheaf

called the sheaf of sections of π. Example 5) is obtained as a particular case on

taking E = Y × X and π to be the second projection.

We will see below that every sheaf arises in this way, that is to say, to every sheaf

on X, one can associate a space E = E(F) and a map π : E → X as above such that

the sheaf of sections of π may be identied with F.

1.4. Sheaf associated to a presheaf. In the theory of holomorphic functions, one

talks of a germ of a function at a point, when one wishes to study its properties in

an (unspecied) neighbourhood of a point. This means the following. Consider pairs

(U, f) consisting of open sets U containing the given point x and holomorphic functions

f dened on U. Introduce an equivalence relation in this set by declaring two such pairs

(U, f), (V, g) to be equivalent if f and g coincide in some neighbourhood of x which

Chapter I: Sheaves and Differential Manifolds:

Definitions and Examples

is contained in U ∩ V . An equivalence class is called a germ. This procedure can be

imitated in the case of a presheaf and this leads to the concept of a stalk of a presheaf

at a point.

1.5. Denition. Let F be a presheaf on a topological space X. Then the stalk F

of F at a point x ∈ X is the quotient set of the set consisting of all pairs (U, s) where

U is an open neighbourhood of x and s is an element of F(U) under the equivalence

relation:

(U, s) is equivalent to (V, t) if and only if there exists an open neighbourhood W of

x contained in U ∩ V such that the restrictions of s and t to W are the same.

If s ∈ F(X) then, for any x ∈ X, the pair (X, s) has an image in the stalk F

, namely

the equivalence class containing it. It is called the germ of s at x. We will denote it by

Let E = E(F) be the germs of all elements at all points of X, that is to say, the

disjoint union of all the stalks F

, x ∈ X. What we intend to do now is to provide the

set E with a topology such that

a) the map π : E → X which maps all of F

to x, is continuous;

b) if s ∈ F(U) then the section ˜s of E over U, dened by setting ˜s(x) = s

, is

continuous.

We achieve this by associating to each pair (U, s), where U is an open subset of X and

s ∈ F(U ), the set ˜s(U ), and dening a topology on E whose open sets are generated

by sets of the form ˜s(U ). In order to check that with this topology, the maps ˜s are

continuous, we have only to show that ˜s

−1

(

t(V )) is open for every open subset V of X

and t ∈ F(V ). This is equivalent to the following

1.6. Lemma. If s ∈ F(U ) and t ∈ F(V ), then the set of points x ∈ U ∩ V such that

= t

is open in X.

Proof. From the denition of the equivalence relation used to dene F

, we deduce

that if s

= t

for some point a ∈ X, then the restrictions of s and t to some open

neighbourhood N of a are the same. Hence we must have s

= t

for every point x of

N as well. Thus N is contained in {x ∈ U : s

= t

}, proving the lemma.

Finally if U is an open set in X, then π

−1

(U) =

∪

x∈U

∪

˜s(V ) where the latter

union is over all open subsets V ⊂ U and all s ∈ F(V ). This shows that π

−1

(U) is open

in E and hence that π is continuous.

1.7. Denition. The étale space associated to the presheaf F is the set E(F) =

∪

x∈X

provided with the topology generated by the sets ˜s(U), where U is any open

set in X and s ∈ F(U). Moreover, if π is the natural map E(F) → X which has F

Section 1: Sheaves and Presheaves

bre over x for all x ∈ X, then the sheaf of sections of π is called the sheaf associated

to the presheaf F.

1.8. Remarks.

1) If we look at the bre of π over x ∈ X, namely F

, we see that the topology

induced on it is nothing serious, since ˜s(U) (being a section of π), intersects F

only in one point, namely s

. In other words, the induced topology on the bres

is discrete.

2) Consider the constant presheaf A dened by the abelian group A. Clearly the

stalk at any point x ∈ X is again A so that E can in this case be identied with

A × X. Any a ∈ A gives rise to an element of A(U ) for every open subset U of

X. The corresponding section ˜a over U of π : E = A ×X → X is given simply by

x 7→ (a, x) over U. Hence the image, namely {a}×U is open in E. From this one

easily concludes that the topology on E = A × X is the product of the discrete

topology on A and the given topology on X. Hence its sections over any open set

U are continuous maps of U into the discrete space A. This is the same as locally

constant maps on U with values in A.

3� Notice that in the construction of the étale space we did not make full use of the

data of a presheaf. It is enough if we are given F(U) for open sets U running

through a base of open sets.

1.9. Denition. If F

, F

are presheaves on a topological space X, then a homomor-

phism f : F

→ F

is an association to each open subset U of X of a homomorphism

(

) :

(

)

→ F

(

)

such that whenever

U, V

are open sets with

⊂

, we have a

commutative diagram

(U) F

(U)

(V ) F

(V )

f(U)

res

f(V )

When we consider presheaves of abelian groups, rings, . . ., and talk of homomorphisms of

such sheaves, we require that all the maps f (U ) be homomorphisms of these structures.

1.10. Examples.

1) The inclusion of the set of dierentiable functions (on a domain in R

) in the set

of continuous functions is clearly a homomorphism of the sheaf of dierentiable

functions into that of continuous functions.

2) Consider the sheaf of dierentiable functions in a domain of R

. The map f 7→

∂f

∂x

induces a sheaf homomorphism of the sheaf of dierentiable functions into itself,

the homomorphism being one of sheaf of abelian groups but not of rings.

Chapter I: Sheaves and Differential Manifolds:

Definitions and Examples

We will rephrase this as follows. Let V be a (nite-dimensional) vector space over

R. Then one has a natural sheaf of dierentiable functions on V . One can for

example choose a linear isomorphism of V with R

and consider functions of V as

functions of the coordinate variables x

, x

, . . . , x

. Then dierentiability makes

sense, independent of the isomorphism chosen. For let y

, y

, . . . , y

is a set of

variables obtained by some other isomorphism of V with R

, that is to say,

= a

+ a

+ ··· + a

where (a

) is an invertible matrix. Then f is dierentiable with respect to (x

)

if and only if it is so with respect to (y

). Indeed dierentiation can be dened

intrinsically as follows. For any v ∈ V , dene ∂

f(x) = lim

t→0

f(x+tv) −f (x)

. Then

f 7→ ∂

f gives rise to a sheaf homomorphism of abelian groups.

3) The inclusion of constant functions in dierentiable functions gives a homomor-

phism of the constant sheaf R in the sheaf of dierentiable functions.

If F is a presheaf and

F the sheaf of sections of E(F), then for every open set U

in X, we get a natural homomorphism of F(U) into

F(U ), which maps any s to the

section ˜s. If V ⊂ U , then for every x ∈ V , the element of F

given by (U, s) is the same

as that given by (V, res

s) by the very denition of F

. This implies that the diagram

F(U )

F(V )

res

is commutative, proving that the natural homomorphisms F(U ) →

F(U ) dene a ho-

momorphism of presheaves.

1.11. Proposition. A presheaf F satises Axiom S

if and only if the induced map

F(U ) →

F(U ) is injective, for every open set U of X.

Proof. To say that F(U) →

F(U ) is injective is equivalent to saying that if s

, s

∈ F(U)

with (s

)

= (s

)

for all x ∈ U , then s

= s

. But then the assumption assures us

that res

= res

for some open neighbourhood N

⊂ U of x. Now Axiom S

applied to the covering U =

∪

says precisely that s

= s

. Conversely, if U =

∪

and s, t are elements of F(U) satisfying res

(s) = res

(t) for all i, then the same is

true of ˜s and

t. But since

F satises S

, it follows that ˜s =

t. Now if we assume that

F(U ) →

F(U ) is injective, it follows that s = t, so that we conclude that F satises

Section 1: Sheaves and Presheaves

1.12. Proposition. A presheaf F is a sheaf if and only if the natural maps F(U ) →

F(U ) are all isomorphisms.

Proof. In fact, if all these maps are isomorphisms, then F is isomorphic to the sheaf

F and it follows that F is itself a sheaf. On the other hand, if F is a sheaf, then we

conclude from the above proposition that the maps F(U) →

F(U ) are injective and

we need only to verify that they are surjective. An element σ ∈

F(U ) is a section

over U of the étale space of F. Hence, for every x ∈ U , there is a neighbourhood N

and an element s

(x)

∈ F(N

) which represents the equivalence class σ(x) ∈ F

. The

section ˜s

(x)

over N

of the étale space given rise to by s

(x)

, and the section σ, coincide

at x. It follows that the two sections coincide in a neighbourhood N

′

of x, contained

in N

. This means that there exist an open covering U =

∪

′

and s

(x)

∈ F(N

′

)

such that (s

(x)

)

= σ(a) for all a ∈ N

′

. In particular, s

(x)

and s

(y )

give rise to the

same section of

F over N

′

∩ N

′

. But, thanks to the injectivity of the natural map

F(N

′

∩N

′

) →

F(N

′

∩N

′

) we conclude that the restrictions of s

(x)

and s

(y )

to N

′

∩N

′

are the same. Since F is actually a sheaf, this implies that there exists s ∈ F(U ) whose

restriction to N

′

is s

(x)

for all x ∈ U. Thus we have s

= σ(x) for all x ∈ U , or, what

is the same,

, proving that

(

)

→

(

)

is surjective.

Somewhat subtler is the relationship between Axiom S

and the surjectivity of

F(U ) →

F(U ). If F satises S

, then any section of

F gives rise, as above, to an

open covering {N

} and elements s

(x)

of F(N

). In order to piece all these elements

together and obtain an element of F(U ) we need to check that the restrictions of s

(x)

and s

(y )

to N

∩ N

coincide, at least after passing to a smaller covering. We have the

following set-topological lemma.

1.13. Lemma. Let {U

}

i∈I

be a locally nite open covering of a topological space U

and {V

}

i∈I

be a shrinking. Then for every x ∈ U, there exists an open neighbourhood

such that I

= {i ∈ I : M

∩ V

6= ∅} is nite, and if i ∈ I

then x belongs to V

and M

is a subset of U

. If M

and M

intersect, then there exists i ∈ I such that

∪ M

⊂ U

Proof. Since {U

} is locally nite, so is the shrinking, and the existence of M

such

that the corresponding I

is nite, is trivial. We will now cut down this neighbourhood

further in order to satisfy the other conditions. We intersect M

with U \V

for all i ∈ I

for which x /∈ V

. We thus obtain an open neighbourhood of x, and the closures of all

, i ∈ I

then contain x. It can be further intersected with

∩

i∈I

, and the resulting

neighbourhood satises the rst assertion of the lemma. Now if M

∩ M

6= ∅, then for

any z ∈ M

∩ M

, choose i ∈ I such that z ∈ V

. Then M

intersects V

and hence

⊂ U

. Similarly M

is also contained in U

proving the second assertion.

Chapter I: Sheaves and Differential Manifolds:

Definitions and Examples

This lemma can be used to deduce that under a mild topological hypothesis, the

natural maps F(U) →

F(U ) are surjective for all open sets U , if the presheaf F satises

Axiom S

1.14. Proposition. If every open subset U of X is paracompact, and the presheaf F

satises S

, then the map F(U) →

F(U ) is surjective for all U .

Proof. Firstly, given an element σ of

F(U ), there is a locally nite open covering {U

} of

U, and elements s

∈ F(U

) for all i, with the property that for all x ∈ U

, the elements

have the image σ(x) in

. Let {V

} be a shrinking of {U

}. For every x ∈ U, choose

as in 1.13. We may also assume that the restrictions to M

of any of the s

for

which i ∈ I

, is the same, say s

(x)

. It follows that the restrictions of s

(x)

and s

(y )

∩M

are the same as the direct restriction of some s

to M

∩M

, proving in view

of Axiom S

, that there exists s ∈ F(U) whose restriction to M

is s

(x)

for all x ∈ U.

This proves that F(U ) →

F(U ) is surjective.

1.15. Exercises.

1) Let X be a topological space which is the disjoint union of two proper open sets

and U

. Dene F(U) to be (0) whenever U is an open subset of either U

. For all other open sets U, dene F(U ) = A, where A is a nontrivial abelian

group. If U ⊂ V and F(U) = A, then dene the restriction map to be the identity

homomorphism. All other restriction maps are zero. Show that F is a presheaf

such that

F = (0).

2) In the above, does F satisfy Axiom S

Subsheaves.

1.16. Denition. A sheaf G is said to be a subsheaf of a sheaf F if we are given a

homomorphism G → F satisfying either of the following equivalent conditions.

1) G

→ F

is injective for all x ∈ X.

2) For any open subset U of X, G(U) → F(U) is injective.

To see that the above conditions are equivalent, note that 1) implies that E(G) →

E(F) is injective and hence the set of sections of G over any set is also mapped injectively

into the set of sections of F. Conversely, assume 2), and let a, b ∈ G

have the same

image in F

. Then there exist a neighbourhood U of x and elements s, t ∈ G(U) such

that s

= a, t

= b. Moreover, by replacing U with a smaller neighbourhood we may

also assume that the images of s and t are the same in F(U ). This implies by our

assumption that s = t as elements of G(U), as well. Hence s

= t

in G