Given a topological space XX and a continuous map π ⁣:XY\pi\colon X \to Y , there is an adjunction between the categories of sheaves on XX and sheaves on YY . 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 ⁣:Fππ1F\eta_\mathscr{F} \colon \mathscr{F} \to \pi_\ast\pi^{-1}\mathscr{F} “shreds a section ff into pieces,” while the counit ϵG ⁣:π1πGG\epsilon_{\mathscr{G}} \colon \pi^{-1}\pi_\ast\mathscr{G} \to \mathscr{G} recognizes that these “shredded up” sections assemble into sections of G\mathscr{G} .

Adjunctions

Remember, given categories CC and DD , a pair of functors F ⁣:CDF \colon C \to D and G ⁣:DCG \colon D \to C form an adjoint pair when there exists an isomorphism

D(Fc,d)C(c,Gd)D(Fc,d) \cong C(c,Gd)

which is natural in cc and dd . We say FF is left adjoint to GG , or equivalently that GG is right adjoint to FF . In particular, setting d=Fcd = Fc , we have

D(Fc,Fc)C(c,GFc).D(Fc,Fc) \cong C(c,GFc).

which gives us a natural transformation η ⁣:1CGF\eta \colon 1_C \Longrightarrow GF . Similarly, we have ϵ ⁣:FG1D\epsilon\colon FG \Longrightarrow 1_D . The natural transformation η\eta is called the unit of the adjunction, and ϵ\epsilon 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

ϵFFη=1FandGϵηG=1G,\epsilon F \circ F\eta = 1_F \quad\text{and}\quad G\epsilon \circ \eta G = 1_G,

which says that for all cCc \in C and dDd \in D , the compositions

FcFηcFGFcϵFcFc \begin{CD} Fc @>{F\eta_c}>> FGFc @>{\epsilon_{Fc}}>> Fc \end{CD}

and

GdηGdGFGdGϵdGd\begin{CD} Gd @>{\eta_{Gd}}>> GFGd @>{G\epsilon_d}>> Gd \end{CD}

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 XX and YY , and a continuous map π ⁣:XY\pi \colon X \to Y , we have functors π ⁣:ShXShY\pi_\ast \colon \operatorname{Sh}_X \to \operatorname{Sh}_Y and π1 ⁣:ShYShX\pi^{-1}\colon \operatorname {Sh}_Y \to \operatorname{Sh}_X from the category of sheaves on XX to the category of sheaves on YY and vice versa. The former is simple to describe: if G\mathscr{G} is a sheaf on XX , the pushforward sheaf πG\pi_\ast\mathscr{G} on YY is the assignment

UF(π1(U))U \mapsto \mathscr{F}(\pi^{-1}(U))

for all open subsets UU of YY . If ψ ⁣:GG\psi \colon \mathscr{G} \to \mathscr{G}' is a morphism of sheaves on XX , the morphism πψ\pi_\ast\psi is defined by the rule

(πψ)(U)=ψ(π1(U)).(\pi_\ast\psi)(U) = \psi(\pi^{-1}(U)).

The latter is slightly more complicated. Recall that if F\mathscr{F} is a sheaf on YY , there is a space FF equipped with a local homeomorphism ρ ⁣:FY\rho \colon F \to Y . The sheaf F\mathscr{F} is the sheaf of local sections of ρ\rho ; FF is called the espace étalé for the sheaf F\mathscr{F} . The set of points of FF above yYy \in Y is the stalk Fy\mathscr{F}_y .

As usual, a section s ⁣:UFs\colon U \to F of the projection ρ ⁣:FX\rho \colon F \to X is an element (sy)yU(s_y)_{y \in U} of yUFy\prod_{y \in U} \mathscr{F}_y . In our case, continuity of ss 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 FF and XX admit maps to YY , we can form their pullback,

π1FFρXπY\begin{CD} \pi^{-1}F @>>> F \\ @VVV @VV{\rho}V \\ X @>{\pi}>> Y \end{CD}

which as a set is {(x,fy)X×F:π(x)=y}\{ (x,f_y) \in X \times F : \pi(x) = y\} . The topology on π1F\pi^{-1}F is induced from the product topology on X×FX \times F . The inverse image sheaf π1F\pi^{-1}\mathscr{F} is the sheaf of sections of π1FX\pi^{-1}F \to X . The universal property of the pullback ensures that this really defines a functor from the category of sheaves on YY to the category of sheaves on XX .

Condition $\eqref{star}$ for continuity of a section s ⁣:Vπ1Fs\colon V \to \pi^{-1}F —that is, a tuple (sπ(x))xV(s_{\pi(x)})_{x \in V} in xVFπ(x)\prod_{x \in V}\mathscr{F}_{\pi(x)} —says that for all xVx \in V there exists an open neighborhood VxV_x of xx and an open neighborhood UxU_x in YY such that π(Vx)Ux\pi(V_x) \subset U_x . Furthermore there exists a section fF(Ux)f \in \mathscr{F}(U_x) such that fπ(v)=sπ(v)f_{\pi(v)} = s_{\pi(v)} for all vVxv \in V_x .

If ϕ ⁣:FF\phi\colon \mathscr{F} \to \mathscr{F}' is a morphism of sheaves on YY , the resulting morphism π1ϕ ⁣:π1Fπ1F\pi^{-1}\phi \colon \pi^{-1}\mathscr{F} \to \pi^{-1}\mathscr{F}' is defined by

(π1ϕ)(V)[(sπ(x))xV]=(ϕπ(x)(sπ(x)))xV.(\pi^{-1}\phi)(V)\left[(s_{\pi(x)})_{x \in V}\right] = \left(\phi_{\pi(x)}(s_{\pi(x)})\right)_{x \in 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 π1\pi^{-1} is left adjoint to π\pi_\ast . Therefore we should expect natural transformations η ⁣:1ShYππ1\eta \colon 1_{\operatorname{Sh}_Y} \Longrightarrow \pi_\ast\pi^{-1} and ϵ ⁣:π1π1ShX\epsilon \colon \pi^{-1}\pi_\ast \Longrightarrow 1_{\operatorname{Sh}_X} .

The Unit of the Adjunction

So let F\mathscr{F} be a sheaf on YY , and let UU be an open set in YY . Since ππ1F(U)=π1F(π1(U))\pi_\ast\pi^{-1}\mathscr{F}(U) = \pi^{-1}\mathscr{F}(\pi^{-1}(U)) , we can think of sππ1F(U)s \in \pi_\ast\pi^{-1}\mathscr{F}(U) as a section s ⁣:π1(U)π1Fs \colon \pi^{-1}(U) \to \pi^{-1}F , i.e. a tuple (sπ(x))xπ1(U)(s_{\pi(x)})_{x \in \pi^{-1}(U)} satisfying the compatibility condition above.

Note that in particular if UYU \subset Y is open and fF(U)f \in \mathscr{F}(U) , the continuity condition $\eqref{star}$ for a section π1(U)π1F\pi^{-1}(U)\to \pi^{-1}F is satisfied by the section (fπ(x))xπ1(U)(f_{\pi(x)})_{x \in \pi^{-1}(U)} , and thus defines an element of ππ1F(U)\pi_\ast\pi^{-1}\mathscr{F}(U) .

Let us thus define a map ηF(U) ⁣:F(U)ππ1F(U)\eta_{\mathscr{F}}(U)\colon \mathscr{F}(U) \to \pi_\ast\pi^{-1}\mathscr{F}(U) as

ηF(U)(f)=(fπ(x))xπ1(U).\eta_{\mathscr{F}}(U)(f) = (f_{\pi(x)})_{x\in\pi^{-1}(U)}.

(Let’s parse the notation: a natural transformation η\eta yields a map of sheaves ηF\eta_{\mathscr{F}} for each sheaf F\mathscr{F} . This is the definition of that map on the open subset UYU \subset Y .)

To prove the claim that η\eta is a natural transformation, we need to show that given a map ϕ ⁣:FF\phi \colon \mathscr{F} \to \mathscr{F}' of sheaves on YY , the following diagram commutes

FηFππ1Fϕππ1ϕFηFππ1F.\begin{CD} \mathscr{F} @>{\eta_{\mathscr{F}}}>> \pi_\ast\pi^{-1}\mathscr{F} \\ @V{\phi}VV @VV{\pi_\ast\pi^{-1}\phi}V \\ \mathscr{F}' @>{\eta_{\mathscr{F}'}}>> \pi_\ast\pi^{-1}\mathscr{F}'. \end{CD}

Given fF(U)f \in \mathscr{F}(U) , commutativity of the square is the claim that the following equality holds

((ϕ(U)(f))π(x))xπ1(U)=(ϕπ(x)(fπ(x)))xπ1(U). \left((\phi(U)(f))_{\pi(x)}\right)_{x \in \pi^{-1}(U)} = \left(\phi_{\pi(x)}(f_{\pi(x)})\right)_{x \in \pi^{-1}(U)}.

This is in fact true, since taking the stalk at xx is functorial.

The Counit of the Adjunction

Now suppose that G\mathscr{G} is a sheaf on XX . Let πG\pi_\ast G denote the espace étalé over YY for the sheaf πG\pi_\ast\mathscr{G} .

A section s ⁣:Vπ1πGs \colon V \to \pi^{-1}\pi_\ast G is a tuple (sπ(x))xV(s_{\pi(x)})_{x \in V} taking values in πGπ(x)\pi_\ast\mathscr{G}_{\pi(x)} satisfying the continuity condition $\eqref{star}$ , which in our particular case asserts the existence, for each xVx \in V , of an open neighborhood VxV_x of xx and an open neighborhood UxU_x in YY satisfying π(Vx)Ux\pi(V_x) \subset U_x . Furthermore there exists gπG(Ux)=G(π1(Ux))g \in \pi_\ast\mathscr{G}(U_x) = \mathscr{G}(\pi^{-1}(U_x)) such that the image of sπ(x)s_{\pi(x)} in Gx\mathscr{G}_x under the natural map πGπ(x)Gx\pi_\ast\mathscr{G}_{\pi(x)} \to \mathscr{G}_x is equal to gxg_x .

Since the sets π1(Ux)\pi^{-1}(U_x) cover VV by assumption, sheafiness of G\mathscr{G} implies the existence of a unique gG(V)g \in \mathscr{G}(V) such that gxg_x is equal to the image of sπ(x)s_{\pi(x)} under the natural map πGπ(x)Gx\pi_\ast\mathscr{G}_{\pi(x)} \to \mathscr{G}_x for all xVx \in V . Let us thus define a map ϵG(V) ⁣:π1πG(V)G(V)\epsilon_{\mathscr{G}}(V) \colon \pi^{-1}\pi_\ast\mathscr{G}(V) \to \mathscr{G}(V) sending (sπ(x))xV(s_{\pi(x)})_{x\in V} to gg .

We claim that ϵ\epsilon defines a natural transformation from π1πG\pi^{-1}\pi_\ast\mathscr{G} to G\mathscr{G} . To prove the claim, we need to show that given a map ψ ⁣:GG\psi\colon \mathscr{G} \to \mathscr{G}' of sheaves on XX , the following diagram commutes

π1πGϵGGπ1πψψπ1πGϵGG.\begin{CD} \pi^{-1}\pi_\ast\mathscr{G} @>{\epsilon_{\mathscr{G}}}>> \mathscr{G} \\ @V{\pi^{-1}\pi_\ast\psi}VV @VV{\psi}V \\ \pi^{-1}\pi_\ast \mathscr{G}' @>{\epsilon_{\mathscr{G}'}}>> \mathscr{G}'. \end{CD}

Given (sπ(x))xVπ1πG(V)(s_{\pi(x)})_{x \in V} \in \pi^{-1}\pi_\ast\mathscr{G}(V) , the upper right path of the square sends us first to some gG(V)g \in \mathscr{G}(V) such that gxg_x is the image of sπ(x)s_{\pi(x)} in Gx\mathscr{G}_x for all xVx \in V , and then sends gg to ψ(V)(g)G(V)\psi(V)(g) \in \mathscr{G}'(V) . On the other hand, the lower left path of the square sends (sπ(x))xV(s_{\pi(x)})_{x \in V} first to ((πψ)π(x)(sπ(x)))xV\left((\pi_\ast\psi)_{\pi(x)}(s_{\pi(x)})\right)_{x \in V} in π1πG(V)\pi^{-1}\pi_\ast\mathscr{G}'(V) , and then to some gG(V)g' \in \mathscr{G}'(V) such that gxg'_x is the image of (πψ)π(x)(sπ(x))(\pi_\ast\psi)_{\pi(x)}(s_{\pi(x)}) in Gx\mathscr{G}'_x for all xVx \in V .

If UYU \subset Y is an open set containing π(x)\pi(x) such that sπG(U)=G(π1(U))s \in \pi_\ast\mathscr{G}(U) = \mathscr{G}(\pi^{-1}(U)) represents sπ(x)s_{\pi(x)} for π(x)U\pi(x) \in U , we have (πψ)π(x)(sπ(x))=ψ(π1(U))(s)π(x)(\pi_\ast\psi)_{\pi(x)}(s_{\pi(x)}) = \psi(\pi^{-1}(U))(s)_{\pi(x)} . Furthermore we know that gG(V)g \in \mathscr{G}(V) is such that the image of sπ(x)s_{\pi(x)} in Gx\mathscr{G}_x is gxg_x for all xVx \in V . Therefore we have that the image of ψ(π1(U))(s)π(x)\psi(\pi^{-1}(U))(s)_{\pi(x)} in Gx\mathscr{G}'_x is equal to ψ(V)(g)x\psi(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\mathscr{F} is a sheaf on YY , we have that

π1Fπ1ηFπ1ππ1Fϵπ1Fπ1F\begin{CD} \pi^{-1}\mathscr{F} @>{\pi^{-1}\eta_{\mathscr{F}}}>> \pi^{-1}\pi_\ast\pi^{-1}\mathscr{F} @>{\epsilon_{\pi^{-1}\mathscr{F}}}>> \pi^{-1}\mathscr{F} \end{CD}

is equal to the identity. The difficulty here seems to be largely notational. Let VXV \subset X be an open set. Suppose (sπ(x))xV(s_{\pi(x)})_{x \in V} is a section in π1F(V)\pi^{-1}\mathscr{F}(V) . We have

(π1ηF)(V)[(sπ(x))xV]=((ηF)π(x)(sπ(x)))xVπ1ππ1F(V).(\pi^{-1}\eta_{\mathscr{F}})(V)\left[(s_{\pi(x)})_{x\in V}\right] = \left((\eta_{\mathscr{F}})_{\pi(x)}(s_{\pi(x)})\right)_{x \in V} \in \pi^{-1}\pi_\ast\pi^{-1}\mathscr{F}(V).

First observe that the continuity condition $\eqref{star}$ tells us that there is, for each xVx \in V , an open neighborhood VxV_x of xx and an open set UxYU_x \subset Y containing π(Vx)\pi(V_x) . On this open set we have sF(Ux)s \in \mathscr{F}(U_x) such that ((ηF)(Ux)(s))π(x)=(ηF)π(x)(sπ(x))\left((\eta_{\mathscr{F}})(U_x)(s)\right)_{\pi(x)} = (\eta_{\mathscr{F}})_{\pi(x)}(s_{\pi(x)}) .

We saw above that (ηF)(Ux)(s)(\eta_{\mathscr{F}})(U_x)(s) is the section (sπ(x))xπ1(Ux)ππ1F(Ux)=π1F(π1(U))(s_{\pi(x)})_{x \in \pi^{-1}(U_x)} \in \pi_\ast\pi^{-1}\mathscr{F}(U_x) = \pi^{-1}\mathscr{F}(\pi^{-1}(U)) . The map ϵπ1F\epsilon_{\pi^{-1}\mathscr F} recognizes that the sections (sπ(x))xπ1(Ux)(s_{\pi(x)})_{x \in \pi^{-1}(U_x)} satisfy the compatibility conditions to glue up to form a section (sπ(x))xV(s_{\pi(x)})_{x \in V} in π1F(V)\pi^{-1}\mathscr{F}(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\mathscr{G} on XX as follows

πGηπGππ1πGπϵGπG.\begin{CD} \pi_\ast\mathscr{G} @>{\eta_{\pi_\ast\mathscr{G}}}>> \pi_\ast\pi^{-1}\pi_\ast\mathscr{G} @>{\pi_\ast\epsilon_{\mathscr{G}}}>> \pi_\ast\mathscr{G}. \end{CD}

If UYU \subset Y is an open set and gπG(U)g \in \pi_\ast\mathscr{G}(U) is a section, the first map sends gg to the section (gπ(x))xπ1(U)ππ1πG(U)(g_{\pi(x)})_{x \in \pi^{-1}(U)} \in \pi_\ast\pi^{-1}\pi_\ast\mathscr{G}(U) .

The second map recognizes that the above section and gG(π1(U))=πGg \in \mathscr{G}(\pi^{-1}(U)) = \pi_\ast\mathscr{G} agree on each stalk, and so sends our section back to gg .

This completes the proof that inverse image and pushforward form an adjoint pair of functors.