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Home , Category (mathematics), Cokernel, Epimorphism, Image (category theory), Kernel (category theory), Morphism

Basics of Sheaves

Eric Auld

Revised October 8, 2016

Contents

1 , , and in the of Presheaves 1

2 and Kernel, Cokernel, and Image 2

3 Sheaves 4

4 Kernel, Cokernel, and Image in the Category of Sheaves 6

5 Sheaves and the Definition of a Manifold 10

6 The Pullback 10

7 With Covering Spaces and Lifting 12

1 Kernel, Cokernel, and Image in the Category of Presheaves

Let F ÝÑΦ G be a of presheaves of abelian groups. For each U Ă X, associate the abelian kerpΦU q ď FpUq (resp ImpΦU q ď GpUq, resp cokpΦU q).

Claim. U ÞÑ kerpΦU q (resp U ÞÑ ImpΦU q, resp U ÞÑ cokpΦU q) naturally has the structure of a presheaf. These are the categorical kernel, cokernel, image, in the category of presheaves of abelian groups.

Proof. Look at the following diagrams, and convince yourself that it works. In each case note that the on sections should exist uniquely, and then just confirm the maps comprise a map of presheaves.

1 ΨU

iU ΦU HpUq kerpΦU q FpUq GpUq

r r r r

iV ΦV HpV q kerpΦV q FpV q GpV q

ΨV

Above, the restriction map on ker is just that of F. So it’s not surprising that the map to the kernel is a map of presheaves.

ΨU

ΦU πU FpUq GpUq cokpΦU q HpUq

r r r r

ΦV πV FpV q GpV q cokpΦV q HpV q

ΨV

In the above case, the restriction map should work if the restriction map respects representatives, i.e., if 1 1 g ´ g P ImpΦU q ùñ g|V ´ g |V P ImpΦV q. This is true, and in fact it’s why the restriction map works below.

ΨU

iU πU HpUq ImpΦU q GpUq pcok ΦqpUq

r r r r

iV πV HpV q ImpΦV q GpV q pcok ΦqpV q

ΨV

2 Stalk and Kernel, Cokernel, and Image

Let F ÝÑΦ G be a map of presheaves of abelian groups.

Claim. The stalk of the presheaf kernel (resp. cokernel, resp. image) is isomorphic to the kernel (resp. cokernel, resp. image) of the stalk map, i.e.

• pKer Φqx – KerpΦxq

• pIm Φqx – ImpΦxq

• pCok Φqx – CokpΦxq

2 Remark. Clearly a map of presheaves is injective (surjective) on all sections iff its presheaf kernel (cokernel) is zero. Since the stalk of the kernel is the kernel of the stalk, we might think that this implies that a map of presheaves is injective on all sections iff it is injective on all stalks. But when we try to make it work, we see

Φ is monic (epic) in presheaves

Φ injective (surjec- kerpre Φ “ 0 pcokpre Φ “ 0q tive) on all sections

Φ injective (surjec- kerpΦ q “ 0 pcokpΦ q “ 0q, all x tive) on all stalks x x

Stalk of kerpre Φ (cokpre Φ) is zero everywhere and the dashed does not work, because it is very possible for a (unseparated) presheaf to have zero stalks but not be the zero presheaf. It just has to be “locally” zero, like the presheaf cokernel of the exponential map. On the other hand, it holds for the kernel if F is separated. If we work in the category of sheaves (once we find out what the kernel and cokernel are), we will find that the diagram has a different failure:

Φ is monic (epic) in sheaves

Φ injective (surjective) on all sections ker Φ “ 0 pcok Φ “ 0q

Φ injective (surjective) on all stalks kerpΦxq “ 0 pcokpΦxq “ 0q, all x and we find that the dashed arrow holds only in the injective case, when the presheaf kernel is the sheaf kernel. Claim. Presheaves of abelian groups form an .

pre Let Γp‚,Xq be the global AbX Ñ Ab. Claim.Γ p‚,Xq is exact. Proof. This is clear, since Γ commutes with the taking of kernels and images. (Note that this works in presheaves though it fails in sheaves because it is harder for a sequence to be exact in the category of presheaves. Some wily exact sequences sneak in, in the category of sheaves, whose global sections are not exact.)

3 3 Sheaves

Example. Let ϕ : X Ñ Y be continuous. Then for each V Ă Y , let ΓpV q be the continuous sections of ϕ. This is a sheaf. (Note glueability might fail if we just talked about general subsets (incoherent cover).)

Example. In fact, we need not even take continuous sections. Just any old sections as functions form a sheaf (even sections of a discontinuous ). Claim that the ”any old sections” sheaf is flabby if ϕ is surjective, and so any sheaf is a subsheaf of a flabby sheaf (the discontinuous sections of the map from the etale ).

Given a presheaf F on X, we may form its sheafification in the ”compatible germs” way. Another method is the “etale space” way. Here we form the etale space EF of the presheaf, which as a set is Fx, and has the strongest (?) such that f ÞÑ EF is a for all f. xPX

š Let U Ă X, and let µ P Fx. xPU Claim. ś

(a) µ represents a compatible choice of germs ðñ µ is a continuous section U Ñ EF .

´1 (b) Stalks of ΓpEF Ñ Xq are isomorphic to Fx “ p rxs.

1 Proof. (a) is omitted. For (b), for any sections σ, σ : U Ñ EF , take pf, V q compatible around x, and

1 1 σpxq “ σ pxq σx “ σx

D nbd where their compatible nbds are the same D nbd where σ “ σ1

´1 1´1 with the LHS vertical arrow holding because take σ pV q X σ pV q where V is a nbd of x in EF .

Claim. A presheaf F is separated ðñ Every section is determined by its germs.

Proof. ùñ If two distinct sections f, f 1 give rise to the same germs, then for each point x in their domain we may choose a neighborhood Ux where they agree. Therefore F can’t be separated.

ðù If distinct f, f 1 agree on an open cover, then they surely give rise to the same germs.

Remark. Note that this implies that if F Ñ G is a map of presheaves, and G is separated, then Φ is determined by its actions on the stalks. (Though note that germs don’t determine a section in F, so Φ could for example be injective on stalks but not on sections, if F has some sections which are locally zero.)

Claim. A presheaf F is separated ðñ pF ÝÑshf F shf q (the map sending each section to its collection of germs) is injective on each section, i.e., iff it is a of presheaves.

Proof. This follows from the previous claim.

4 In general, the glueability does not play as nicely with the ”compatible collection of germs” notion. We’d like to say

Every collection pf ,U q agreeing Glueable i i on sections glues in the presheaf

Every collection

Sheafification pfi,Uiq agreeing surjective on stalks glues in the presheaf

Unfortunately, the dashed arrow doesn’t necessarily hold: if F is not separated, then germs do not determine sections, so there may be more collections agreeing on germs than there are collections agreeing on sections. However, we can see from this that

Claim. A presheaf F is glueable ðù pF ÝÑshf F shf q is surjective on each section.

One way to get the dashed arrow to work (even in the absence of F being separated) is to require that we’re working on a paracompact hausdorff space (that is, to assume paracompactness of the open set you’re concerned with. Or if you want it to work for all sections, assume hereditary paracompactness.) If we’re given an element i in the sheafification (a compatible collection of germs represented by a cover f ,Ui agreeing on germs), we seek to find another cover with the same germs, which actually agrees as sections, and then use that F is assumed glueable.

Our strategy for doing this will be to take our cover Ui, and to pick for each x a neighborhood Mx, and a subset Ix of the index set I, such that

(a) Ix is finite for all x

(b) Mx meets My ùñ Ix meets Iy.

(c) i P Ix ùñ Mx Ă Ui.

x i Then, if we can do this, we can just choose f to be a representative of tf : i P Ixu. (We may have to shrink x Mx in order to say that f is defined on Mx, but note that shrinking the Mx does not affect any of the above x y conditions.) Condition (b) will then guarantee that if Mx meets My, then f and f are both restrictions of x y some common Ui for i P Ix X Iy, and so f agrees with f as sections.

Note also that if we have conditions (a) and (b), we can get condition (c) easily, since we can just shrink

Mx to be inside the open set XiPIx Ui, remembering that shrinking Mx does not affect our conditions. So condi- tions (a) and (b) suffice. Given an open cover Ui, we might call such a choice x ÞÑ pIx,Mxq an indexed refinement of Ui.

Lemma. Any locally finite open cover on a normal has an indexed refinement.

5 df Proof. Take a shrinking of the cover Ui to a locally finite closed cover Zi, and take Ix “ ti P I : x P Ziu.

Then YIzIx Zi is closed since Zi is locally finite, and we may choose Mx to meet only tZi : i P Ixu. Then conditions (a) and (b) are satisfied, since if z P Mx X My, then take an index i such that z P Zi, then i P Ix X Iy.

The beauty of the shrinking Zi is that we can kind of play the Ui and the Zi off of one another: we can say that all the ”tags” i P Ix we’ve chosen are such that Mx Ă Ui, but at the same time, we’ve managed to tag every i where Vx meets Zi. (The push/pull is that requirements (a) and (c) above want us to tag fewer, where requirement (b) wants us to tag more.)

Cf. Swan p. 79, where he talks about presheaves with the “(approximate) collation property with respect to ” various kinds of coverings.

Claim. If F is a sheaf, then F ÝÑshf F shf is an of sheaves. (Note that the converse is obvious, if we mean an isomorphism of presheaves.)

Proof. We need only show that in a sheaf,

Sections f P FpUq ÝÑ Continuous sections U Ñ EF is a . Since F is separated, we have injectivity as above. Also since F is separated, the dashed arrow above works, and we can conclude surjectivity.

Claim. For any presheaf, sheafification induces an isomorphism on stalks.

Proof. See proof that he stalks of ΓpEF Ñ Xq are – Fx.

4 Kernel, Cokernel, and Image in the Category of Sheaves

Claim. Let F ÝÑΦ G be a map of sheaves. Then kerpΦq is in fact a sheaf.

Proof. The fact that K is separated is trivial, since it is a sub-presheaf of a sheaf. For gluability, let fi be compatible sections, all mapped to zero. They glue in F uniquely. If they did not glue to a section in K, then the image of this section would have a nonzero somewhere, contradiction. (Note the necessity of G separated to conclude that K is a sheaf.)

Remark. This establishes that kerpΦq is in fact the categorical kernel in the category of presheaves (i.e. the equalizer with the zero map), because given a sheaf map H ÝÑ F ÝÑΦ G with composition zero, there is a unique presheaf map H ÝÑ kerpΦq factoring through, and that of course will be a sheaf map. (That is, it’s easier to be the kernel in the category of sheaves than in the category of presheaves, because the condition is phrased in terms of ”for all maps, there exists a unique map...” and there are more presheaf maps than there are sheaf maps. You might argue existence is harder in sheaves, but remember they are a full .)

The sheafification of the presheaf cokernel is easily seen to be the categorical sheaf cokernel, by a quick argument:

6 0

F Φ G Ψ H Ψ1 π Ψ2 cokprepΦq shf cokpΦq

Now consider the image sheaf. Let F ÝÑΦ G be a map of sheaves, and consider the image presheaf ImprepΦq, and how it may fail to be a sheaf. It is a subpresheaf of the sheaf G, so it is separated. The problem comes with glueability. Given pgi,Viq, agreeing on sections, which are all in the image of Φ, they glue in G, but they may not glue to something in the image. (Just because a collection pΦpfiq,Uiq agree on sections does not mean pfi,Uiq agree on sections. Think of the image of the exponential map.) We might say the image is downward closed, but not upward closed. Now we think about what the categorical image should be in the category of sheaves.

pre How is kerpG ÝÝÑπ cokprepΦqq different from kerpG ÝÑπ cokpΦqq?

(Note kernel is the same in both categories)

The answer is: kerpG ÝÑπ cokpΦqq is bigger, because it includes things whose images are ”locally zero”. This is exactly the failure of glueability of ImprepΦq, coming up again, leading to cokpre not being separated. This shows that

Claim. The sheaf image is the sheafification of the presheaf image. (Sheafifying will allow us to glue the things we need to glue.)

We could also give another proof, by focusing on the stalks. The kernel of the map π is those g P G pre who up as the zero collection of germs. We know that the map on stalks G ÝÝÑπ ImprepΦq is just πpre pre shf modding out by ImpΦxq, and G ÝÝÑ Im pΦq ÝÑ ImpΦq is just adding an isomorphism of stalks at the end. So the kernel of π is just those sections whose images in ImprepΦq have zero germs everywhere. (They are ”locally zero”.) This is just compatible collections of germs which are taken to zero by πpre, which is exactly compatible collections of germs in impΦq.

There is a little detail to take care of here. Let P be the image presheaf. We need to show that a collection of

germs µ, each in the image of some Φx, are P-compatible whenever they are G-compatible.

Lemma. µ is P-compatible. (Note of course any P-compatible collection is G-compatible.)

Proof. Let pg, Uq, g P GpUq be a compatible neighborhood. We know µy “ gy for all y. Let x P U. Since µx 1 1 1 is in the image of Φx, there is a representative f ÞÑ g . where gx “ µx. This means that g and g agree after restriction, and of course this restriction remains in P, because P is closed downward. Thereby we have found an P-compatible structure.

We know that the presheaf image is not always the same as the sheaf image. But there is a limited case where this is so:

Claim. Let F ÝÑΦ G be a monomorphism of sheaves. Then Im Φ is in fact a sheaf.

7 Proof. Presheaves are an abelian category, so any monomorphism is isomorphic to its image. F – ImpΦq. But F is a sheaf.

Let F ÑΦ G be a morphism of sheaves, and let G ÝÑπ C be the sheaf kernel (cokernel, image).

Claim. The stalk of the kernel (cokernel, image) is the same as the kernel (cokernel, image) of the stalk map.

Proof. We know this is true for the presheaves, and sheafification induces an isomorphism on stalks.

Remark. As before, if F ÝÑΦ G is a map between two sheaves, it should be harder for it to be a monomorphism () of presheaves than it is to be a monomorphism (epimorphism) of sheaves, because the definitions are in terms of the existence of maps H Ñ F, or G Ñ J and there are more such presheaf maps than there are sheaf maps. However, since we know that the presheaf kernel of Φ is its sheaf kernel, and in any are characterized by their kernels being zero, we can say that Φ is monic in presheaves iff it is monic in sheaves. This establishes that

Claim. If F ÝÑΦ G is a sheaf monomorphism, then it is injective on all sections. (Note that the converse is clear.)

If we wanted this result outside an abelian category, we still have some play: (use the OX i!OU , or whatever object represents the U-section functor).

It is probably worth recalling both pictures:

Φ is monic (epic) in presheaves

Φ injective (surjec- kerpre Φ “ 0 tive) on all sections (cokpre Φ “ 0)

Φ injective (surjec- kerpΦxq “ 0 tive) on all stalks (cokpΦxq “ 0) all x

Stalk of kerpre Φ (cokpre Φ) is zero everywhere

(Dashed line does not hold, since an unseparated presheaf may be have zero stalks but not be zero (it is ”locally zero”).

8 Φ is monic (epic) in sheaves

Φ injective (surjec- ker Φ “ 0 resp, tive) on all sections cok Φ “ 0

Φ injective (surjec- kerpΦxq “ 0 resp, tive) on all stalks cokpΦxq “ 0, all x

(Dashed arrow holds only in the injective case, since the presheaf kernel is the sheaf kernel.) Claim. Sheaves of abelian groups are an abelian category. Claim. If Φ is an isomorphism on all sections, then it is an isomorphism of sheaves. (Note the converse is clear.)

Proof 1: It is monic and epic in an abelian category.

´1 Proof 2: We only need to show that the maps ΦU form a morphism of sheaves, i.e. we need to show that they commute with the restriction map rUV .

Φ Ψ FpUq U GpUq U FpUq

r r r Φ Ψ FpV q V GpV q V FpV q

We already know that Φ do, so ΦU ; r;ΨV “ r;ΦV ;ΨV “ r, and ΦU ;ΨU ; r “ r, so r;ΨV “ ΨU ; r when precomposed with ΦU , but ΦU is an isomorphism. Φ Claim. If F Ñ G be a morphism of sheaves such that Φx : Fx Ñ Gx is an isomorphism for all x, then Φ is an isomorphism of sheaves. (Note the converse is clear.) Proof. It is monic and epic in an abelian category.

Claim. The global section functor Γp‚,Xq : AbX Ñ Ab is left exact. Proof. The slick way is to say that ”the from sheaves to presheaves is left exact, and the section functor is exact on presheaves, so ΓpX, ´q “ For; ΓprepX, ´q is left exact.” To go through the details explicitly, 0 Ñ A ÝÑΦ B ÝÑΨ C being exact says that Φ is a monomorphism and ker Ψ “ im Φ. Then Φ is a monomorphism in the category of presheaves as well, and ker Ψ is the same in the category of presheaves, and im Φ is the same in the category of presheaves, because Φ is a monomorphism. So the sequence is in fact exact in the category of presheaves. (In other words, the forgetful functor from sheaves to presheaves is left exact.) And the global section functor is exact on presheaves.

9 5 Sheaves and the Definition of a Manifold

Suppose we take as models (open set of Rn, continuous functions; open set of Rn, smooth functions; open set of Cn, analytic functions). Define a (topological, smooth, complex) manifold to be a sheaf on a Hausdorff space X which is locally isomorphic to the above.

An atlas is an open covering Uα of X where for each α we have a homeomorphism ϕα so that the compositions of the ϕα are (continuous, smooth, analytic) wherever they are both defined.

Claim. The data of an atlas is equivalent to the data of manifold as a sheaf.

Proof. We do the case of smooth manifolds. Given such a sheaf, for each x take a homeo fx : Vx Ñ n 8 Ux Ă R . The definition of an isomorphism of sheaves is that ϕ P ΓpUxq “ C pUxq iff fx; ϕ P ΓpVxq. ´1 (Or, ψ P ΓpVxq iff fx ; ψ P ΓpUxq.) ´1 r r 8 We want to show that these are smooth wherever they agree, i.e. that fx ; fy : Ux Ñ Uy is C . (We know it is a composition of and so a homeomorphism.) For this it suffices to show ´1 8 r 8 r that fx ; fy; ϕ P C pUxq for any ϕ P C pUy q (in particular for ϕ being the coordinate functions). But 8 r ´1 8 r ϕ P C pUy q ùñ fy; ϕ P ΓpVx,yq ùñ fx ; fy; ϕ P C pUx q. Now suppose that we start with the data of an atlas, and we want to define a sheaf. We have a ´1 presheaf on a base (i.e. maps on sets inside a Uα which are s.t. fα ; ϕ are smooth, consistent because of the compatibility requirement) which we can sheafify and get the right sheaf.

6 The Pullback Sheaf

´1 pre pre Let ϕ : X Ñ Y be continuous. Define the inverse image presheaf functor ϕpre : AbY Ñ AbX by

ϕ´1 F U : lim F V . p pre qp q “ ÝÑ p q V ĄϕpUq

´1 Claim. ϕpreF is indeed a presheaf.

Proof. Let U Ą W Ą T .

lim F V ÝÑ p q V ĄϕpUq

lim F V lim F V ÝÑ p q ÝÑ p q V ĄϕpW q V ĄϕpT q

Since A Ă B ùñ ϕpAq Ă ϕpBq, we have a well-defined commutative map of diagrams, that gives a map of the direct limits.

Note that this need not be a sheaf, even if F is a sheaf. For instance, a sheaf of abelian groups F on a one-point space is nothing but an A. If I map a two-point X into my

10 one-point space Y , I get that the section over each nonempty open set in X is A...in other words, the constant presheaf. This is not a sheaf, since there are disconnected open subsets.

´1 Claim. pϕ Gqx – Gy.

There is another way to describe this construction, as just the pullback of the Etale space. Let G be a sheaf on Y , and ϕ : X Ñ Y be a continuous map. Let Eϕ´1G be defined as the subspace of X ˆ EG such that the germ lies over x.

´1 Claim. A collection of germs in xPX GϕpXq is compatible in the sense of ϕ ðñ it represents a continuous section of E ´1 . ϕ G ś ´1 Claim. ϕ˚ $ ϕ , i.e. given F P |AbX | and G P |AbY |, we have a bijection

´1 homAbX pπ G, Fq Ø homAbY pG, π˚Fq.

´1 Example. In the particular case that ϕ : ˚ Ñ X is the inclusion of a point, then ϕpreFp˚q is just Fp. It is a sheaf already (the over Fp). Since sheaves of abelian groups over a point can be identified with the abelian groups themselves, in this case we may consider ϕ´1 as a functor from sheaves on X to abelian groups. The right adjoint is the skyscraper sheaf on a group A at p.

´1 Claim. The functor ϕ : AbY Ñ AbX is exact. Proof. Consider 0 Ñ F ÝÑΦ G ÝÑΨ H Ñ 0 exact in AbY , and we want to show that

0 Ñ ϕ´1F ÝÑ ϕ´1G ÝÑ ϕ´1H Ñ 0 is exact in AbY . We know the stalk maps of 0 Ñ F Ñ G Ñ H are exact, and how do the stalk maps of 0 Ñ ϕ´1F Ñ ϕ´1G Ñ ϕ´1H behave?

´1 ´1 pϕ Fqx Ñ pϕ Gqx

Fϕpxq Ñ Gϕpxq

Let U Ă X be an inclusion of ringed spaces. Let F be an OU -module and G be an OX -module.

´1 Claim. There is a natural bijection HomOY pi!F, Gq Ø HomOU pF, i Gq

Proof. For each V I need a map pi!FqpV q Ñ GpV q, i.e. a map from each collection pfi,Wiq agreeing on sections, and such that fi “ 0 if Wi Ć U, to GpV q.(finish)

´1 Remark. Since we know i G is just G|U , and since HomOU pOU , G|Uq are just GpUq, this means that we can identify Hompi!OU , Gq with GpUq as well, i.e. i!OU represents the U-section functor.

11 7 Relation With Covering Spaces and Lifting

We know that a covering map has the unique . For this, we needed local homeomorphism for openness, and Hausdorff total space for closedness. So we may get the unique lifting property also for a local homeomorphism from a Hausdorff space to a Hausdorff space. (Unique lifting not just of paths, but from any connected space.)

Claim. A local homeomorphism from a Hausdorff space to a Hausdorff space has the unique lifting property.

p Claim. Let Y ÝÑ X be a local homeomorphism from a Hausdorff space to a manifold. Then p is a covering map.

A presheaf F is said to satisfy the axiom if for any open connected subset U of X,

f, g P FpUq have the same germs at a point ùñ f “ g.

Let X be a locally connected Hausdorff space, and F be a presheaf on X satisfying the identity axiom.

Claim. The etale space EpFq is Hausdorff.

12