sheaf

sheaf

Definition 1

.

Let

T

be a

category

, and let

C

be another category. Then a

presheaf

on

T

is a

contravariant functor

F

:

T

→

C

. If

U

⁢

→

𝑖

⁢

V

is a

morphism

in

T

, we call

F

⁢

(

i

)

:

F

⁢

(

V

)

→

F

⁢

(

U

)

the

restriction

map

obtained from

i

. A

morphism of presheaves

is a

natural transformation

.

To understand this definition, and what it has to do with the more familiar definition of a presheaf (

http://planetmath.org/Sheaf

), let

T

be a site. Then a presheaf maps the “

open sets

” to objects in

C

and provides restriction maps whenever the underlying “open sets” have an “

inclusion

”. Of course, on a site, there may be many non-isomorphic inclusions.

In practice, to avoid set-theoretic difficulties, one usually fixes a

universe

𝒰

and then requires

T

to be a small (

http://planetmath.org/Small

) category.

Remark 1

.

If

T

is a

small category

and

C

is a

U

-category, then presheaves on

T

form a category.

If

T

is not small, that is if the isomorphism classes of objects do not form a

U

-set, then (normally) there are as many

functors

as subsets of

T

, which leads to the resulting category failing to be a

U

-category. If one removes the various qualifications in terms of universes, this means that there are too many functors: Russell’s

paradox

arises. If one uses universes, and qualifies everything in sight appropriately, a problem remains: there are too many functors from sets to sets to be allowed. While your answer is described in terms of sets, they are too big to be inside the universe of discourse. This problem underlies the set-theoretic problems with

direct limits

.

Let

T

be a site, and suppose either

C

is an

abelian category

or

C

is the

category of sets

.

Definition 2

.

Let

F

:

T

→

C

be a presheaf. Then

F

is a

sheaf

if for every

covering

U

i

→

U

in

T

, the

products

∏

U

i

F

⁢

(

U

i

)

and

∏

U

i

×

U

U

j

F

⁢

(

U

i

×

U

U

j

)

exist, and the

sequence

F

⁢

(

U

)

→

∏

U

i

F

⁢

(

U

i

)

⇉

∏

U

i

×

U

U

j

F

⁢

(

U

i

×

U

U

j

)

is exact.

This “exactness” really needs explaining. First of all, for every

i

, we have a morphism

F

⁢

(

U

)

→

F

⁢

(

U

i

)

obtained from the covering. The map on the left (call it

f

) is the product of all these maps. Second, for each pair

i

and

j

, we have a morphism

g

1

:

F

⁢

(

U

i

)

→

F

⁢

(

U

i

×

U

U

j

)

and a morphism

g

2

:

F

⁢

(

U

j

)

→

F

⁢

(

U

i

×

U

U

j

)

obtained from the two

projection maps

of the fibred product. The two maps on the right-hand side are obtained as products of all the

g

1

and

g

2

respectively.

If

C

is the category of sets, then the exactness condition requires that the left-hand map is

injective

and that the two right-hand maps agree exactly on its image.

If

C

is an abelian category, then the kernel of the left-hand map should be zero, and the kernel of the

difference

of the two right-hand maps should be its image. Perhaps this would be clearer if the

diagram

added a “

0

→

” on the left, but this usage is very standard.

This definition of a sheaf is the motivation for the definition of a site: a site captures precisely those features of a

topological space

that are required to form sheaves. The category of sheaves on a topological space is an example of a topos; the original definition of a topos was “a category

equivalent

to the category of sheaves on a site”, although the

concept

was generalized in later work. The motivation was that while a site may be badly behaved and pose set-theoretic difficulties, the topos associated to it is generally better behaved.

The category of sheaves on a site has a long list of nice properties; among them is “has enough injectives”, so that

derived functors

can be extracted and

cohomology

calculated. This is how, for example, étale cohomology is obtained.

References

1

Grothendieck et al.,

Séminaires en Gèometrie Algèbrique 4

, tomes 1, 2, and 3, available on the web at

http://www.math.mcgill.ca/ archibal/SGA/SGA.html

http://www.math.mcgill.ca/ archibal/SGA/SGA.html

Title

sheaf

Canonical name

Sheaf1

Date of creation

2013-03-22 14:12:47

Last modified on

2013-03-22 14:12:47

Owner

archibal (4430)

Last modified by

archibal (4430)

Numerical id

9

Author

archibal (4430)

Entry type

Definition

Classification

msc 14F20

Classification

msc 18F10

Classification

msc 18F20

Synonym

sheaves

Related topic

Sheaf

Related topic

Site

Related topic

Sheafification2

Related topic

NaturalTransformation

Related topic

FundamentalGroupoid2

Defines

presheaf

Defines

presheaves

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