Given a topological space X
and a continuous map π:X→Y
,
there is an adjunction between the categories of sheaves on X
and sheaves on Y
.
The adjunction is somewhat mysterious to me as I begin to write this,
so the hope is that by writing about it, I’ll begin to understand.
The functors in question are the pushforward and inverse image functors,
which arise naturally in algebraic geometry (so I am led to believe).
What makes the adjunction difficult to understand is that the definitions
of the functors in question don’t seem to play nicely with each other.
The proof (with a little setting the stage) appears below.
The method of proof, like most category-theoretic proofs,
is a story of careful bookkeeping with functors and natural transformations,
along with simple calculations to check.
The idea essentially seems to be that the
unit ηF:F→π∗π−1F
“shreds a section f
into pieces,” while the counit
ϵG:π−1π∗G→G
recognizes that these “shredded up” sections assemble into sections of G
.
Adjunctions
Remember, given categories C
and D
, a pair of functors
F:C→D
and G:D→C
form an adjoint pair
when there exists an isomorphism
D(Fc,d)≅C(c,Gd)
which is natural in c
and d
.
We say F
is left adjoint to G
,
or equivalently that G
is right adjoint to F
.
In particular, setting d=Fc
, we have
D(Fc,Fc)≅C(c,GFc).
which gives us a natural transformation η:1C⟹GF
.
Similarly, we have ϵ:FG⟹1D
.
The natural transformation η
is called the unit of the adjunction,
and ϵ
is the counit.
In fact, the existence of these natural transformations
could be taken to be the definition of an adjunction,
provided the natural transformations satisfy the identities
ϵF∘Fη=1FandGϵ∘ηG=1G,
which says that for all c∈C
and d∈D
, the compositions
FcFηcFGFcϵFcFc
and
GdηGdGFGdGϵdGd
are the identity.
My goal in this blog post is to show that the functors I am interested in
satisfy this latter definition of an adjunction.
Sheaves
Given a pair of spaces X
and Y
, and a continuous map π:X→Y
,
we have functors π∗:ShX→ShY
and π−1:ShY→ShX
from the category
of sheaves on X
to the category of sheaves on Y
and vice versa.
The former is simple to describe: if G
is a sheaf on X
,
the pushforward sheafπ∗G
on Y
is the assignment
U↦F(π−1(U))
for all open subsets U
of Y
.
If ψ:G→G′
is a morphism of sheaves on X
,
the morphism π∗ψ
is defined by the rule
(π∗ψ)(U)=ψ(π−1(U)).
The latter is slightly more complicated.
Recall that if F
is a sheaf on Y
,
there is a space F
equipped with a local homeomorphism ρ:F→Y
.
The sheaf F
is the sheaf of local sections of ρ
;
F
is called the espace étalé for the sheaf F
.
The set of points of F
above y∈Y
is the stalkFy
.
As usual, a section s:U→F
of the projection ρ:F→X
is an element (sy)y∈U
of ∏y∈UFy
.
In our case, continuity of s
is the following condition
$$\begin{equation}\label{star}\tag{$\ast$}
\text{for all } y \in U, \text{ there exists an open neighborhood }
U_y \text{ of } y\\ \text{ and } f\in\mathscr{F}(U_y) \text{ such that }
f_x = s_x \text{ for all } x \in U_y.
\end{equation}$$
Given that F
and X
admit maps to Y
, we can form their pullback,
π−1F↓⏐XπF↓⏐ρY
which as a set is {(x,fy)∈X×F:π(x)=y}
.
The topology on π−1F
is induced from the product topology on X×F
.
The inverse image sheafπ−1F
is the sheaf of sections of π−1F→X
.
The universal property of the pullback ensures that this really defines a functor
from the category of sheaves on Y
to the category of sheaves on X
.
Condition $\eqref{star}$
for continuity of a section s:V→π−1F
—that is,
a tuple (sπ(x))x∈V
in ∏x∈VFπ(x)
—says
that for all x∈V
there exists an open neighborhood Vx
of x
and an open neighborhood Ux
in Y
such that π(Vx)⊂Ux
.
Furthermore there exists a section f∈F(Ux)
such that fπ(v)=sπ(v)
for all v∈Vx
.
If ϕ:F→F′
is a morphism of sheaves on Y
,
the resulting morphism π−1ϕ:π−1F→π−1F′
is defined by
(π−1ϕ)(V)[(sπ(x))x∈V]=(ϕπ(x)(sπ(x)))x∈V.
One needs to demonstrate that the right-hand side satisfies condition $\eqref{star}$
;
the check is simple, so I’ll leave it to the reader.
The claim is that π−1
is left adjoint to π∗
.
Therefore we should expect natural transformations
η:1ShY⟹π∗π−1
and ϵ:π−1π∗⟹1ShX
.
The Unit of the Adjunction
So let F
be a sheaf on Y
,
and let U
be an open set in Y
.
Since π∗π−1F(U)=π−1F(π−1(U))
,
we can think of s∈π∗π−1F(U)
as
a section s:π−1(U)→π−1F
,
i.e. a tuple (sπ(x))x∈π−1(U)
satisfying the compatibility condition above.
Note that in particular if U⊂Y
is open
and f∈F(U)
, the continuity condition $\eqref{star}$
for a section π−1(U)→π−1F
is satisfied by the section
(fπ(x))x∈π−1(U)
,
and thus defines an element of π∗π−1F(U)
.
Let us thus define a map ηF(U):F(U)→π∗π−1F(U)
as
ηF(U)(f)=(fπ(x))x∈π−1(U).
(Let’s parse the notation: a natural transformation η
yields
a map of sheaves ηF
for each sheaf F
.
This is the definition of that map on the open subset U⊂Y
.)
To prove the claim that η
is a natural transformation,
we need to show that given a map ϕ:F→F′
of sheaves on Y
, the following diagram commutes
Fϕ↓⏐F′ηFηF′π∗π−1F↓⏐π∗π−1ϕπ∗π−1F′.
Given f∈F(U)
, commutativity of the square is the claim that
the following equality holds
This is in fact true, since taking the stalk at x
is functorial.
The Counit of the Adjunction
Now suppose that G
is a sheaf on X
.
Let π∗G
denote the espace étalé over Y
for the sheaf π∗G
.
A section s:V→π−1π∗G
is a tuple (sπ(x))x∈V
taking values in π∗Gπ(x)
satisfying the continuity condition $\eqref{star}$
,
which in our particular case asserts the existence,
for each x∈V
, of an open neighborhood Vx
of x
and an open neighborhood
Ux
in Y
satisfying π(Vx)⊂Ux
.
Furthermore there exists g∈π∗G(Ux)=G(π−1(Ux))
such that the image of sπ(x)
in Gx
under the natural map
π∗Gπ(x)→Gx
is equal to gx
.
Since the sets π−1(Ux)
cover V
by assumption,
sheafiness of G
implies the existence of a unique g∈G(V)
such that gx
is equal to the image of sπ(x)
under the natural map π∗Gπ(x)→Gx
for allx∈V
.
Let us thus define a map ϵG(V):π−1π∗G(V)→G(V)
sending (sπ(x))x∈V
to g
.
We claim that ϵ
defines a natural transformation from π−1π∗G
to G
.
To prove the claim, we need to show that given a map ψ:G→G′
of sheaves on X
, the following diagram commutes
π−1π∗Gπ−1π∗ψ↓⏐π−1π∗G′ϵGϵG′G↓⏐ψG′.
Given (sπ(x))x∈V∈π−1π∗G(V)
,
the upper right path of the square sends us first to some g∈G(V)
such that gx
is the image of sπ(x)
in Gx
for all x∈V
,
and then sends g
to ψ(V)(g)∈G′(V)
.
On the other hand, the lower left path of the square sends
(sπ(x))x∈V
first to ((π∗ψ)π(x)(sπ(x)))x∈V
in π−1π∗G′(V)
, and then to
some g′∈G′(V)
such that gx′
is the image of
(π∗ψ)π(x)(sπ(x))
in Gx′
for all x∈V
.
If U⊂Y
is an open set containing π(x)
such that
s∈π∗G(U)=G(π−1(U))
represents sπ(x)
for π(x)∈U
,
we have (π∗ψ)π(x)(sπ(x))=ψ(π−1(U))(s)π(x)
.
Furthermore we know that g∈G(V)
is such that the image of sπ(x)
in Gx
is gx
for all x∈V
.
Therefore we have that
the image of ψ(π−1(U))(s)π(x)
in Gx′
is equal to ψ(V)(g)x
,
as desired.
The inverse image–pushforward adjunction
To conclude, we just need to show that two double compositions
of natural transformations are the identity.
The first, translated into our particular case,
says that if F
is a sheaf on Y
,
we have that
π−1Fπ−1ηFπ−1π∗π−1Fϵπ−1Fπ−1F
is equal to the identity.
The difficulty here seems to be largely notational.
Let V⊂X
be an open set.
Suppose (sπ(x))x∈V
is a section in π−1F(V)
.
We have
First observe that the continuity condition $\eqref{star}$
tells us that
there is, for each x∈V
, an open neighborhood Vx
of x
and
an open set Ux⊂Y
containing π(Vx)
.
On this open set we have s∈F(Ux)
such that
((ηF)(Ux)(s))π(x)=(ηF)π(x)(sπ(x))
.
We saw above that (ηF)(Ux)(s)
is the section
(sπ(x))x∈π−1(Ux)∈π∗π−1F(Ux)=π−1F(π−1(U))
.
The map ϵπ−1F
recognizes that the sections
(sπ(x))x∈π−1(Ux)
satisfy the compatibility conditions
to glue up to form a section (sπ(x))x∈V
in π−1F(V)
,
demonstrating that the double composition is indeed the identity. So far so good.
The second double composition we need to show is equal to the identity is defined
for a sheaf G
on X
as follows
π∗Gηπ∗Gπ∗π−1π∗Gπ∗ϵGπ∗G.
If U⊂Y
is an open set and g∈π∗G(U)
is a section,
the first map sends g
to the section
(gπ(x))x∈π−1(U)∈π∗π−1π∗G(U)
.
The second map recognizes that the above section and g∈G(π−1(U))=π∗G
agree on each stalk, and so sends our section back to g
.
This completes the proof that inverse image and pushforward form an adjoint pair of functors.