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Stream: learning: questions

Topic: Sheafification, adjunction between bundles and presheaves


view this post on Zulip Peva Blanchard (May 17 2024 at 22:44):

Another side quest from @David Egolf 's thread on reading Baez topos series.

Let XX be an arbitrary topological space.

First, we defined a functor Λ\Lambda from the category Psh(X)\text{Psh}(X) of presheaves over XX to the category Top/X\text{Top}/X of bundles over XX. Λ\Lambda maps a presheaf FF over XX to an étale space Λ(F)\Lambda(F) over XX.

Second, we defined a functor Γ:Top/XPsh(X)\Gamma : \text{Top}/X \rightarrow \text{Psh}(X). Γ\Gamma maps a bundle EXE \rightarrow X to its sheaf of sections over XX.

I was wondering if there were an adjunction ΛΓ\Lambda \dashv \Gamma.

I managed to get a proof that, indeed, Λ\Lambda is left adjoint to Γ\Gamma. But my proof is relatively long, and I am not sure I checked everything right. An external confirmation/refutation is welcome.

ps: According to John in the other thread, the composite ΓΛ\Gamma \circ \Lambda is called "sheafification", and appears as the left adjoint to the forgetful functor from sheaves to presheaves. This led me to wonder if sheafification were a monad.

view this post on Zulip Eric M Downes (May 17 2024 at 22:56):

Peva Blanchard said:

I managed to get a proof that, indeed, Λ\Lambda is left adjoint to Γ\Gamma. But my proof is relatively long, and I am not sure I checked everything right. An external confirmation/refutation is welcome.

Was your proof too long to fit in the margins of this stream? :P

(Seriously, where do I find it? I'd like to read it.)

view this post on Zulip John Baez (May 17 2024 at 23:13):

Peva Blanchard said:

I was wondering if there were an adjunction ΛΓ\Lambda \dashv \Gamma.

Try the course notes: the end of Part 2 and also the section "The adjunction between presheaves and bundles" in Part 3.

view this post on Zulip John Baez (May 17 2024 at 23:16):

This led me to wonder if sheafification were a monad.

Since it's a left adjoint followed by a right adjoint, it's a monad. But even better, I believe it's an [[idempotent monad]]. That is, if you sheafify a presheaf that's already a sheaf, you don't get anything new, just something isomorphic to what you had.

view this post on Zulip Peva Blanchard (May 18 2024 at 05:29):

John Baez said:

Peva Blanchard said:

I was wondering if there were an adjunction ΛΓ\Lambda \dashv \Gamma.

Try the course notes: the end of Part 2 and also the section "The adjunction between presheaves and bundles" in Part 3.

Oops ! I should have known the answer was already there. I read the series last year: perhaps I was just remembering the adjunction without realizing that I read about it there.

Anyway, thank you for the link.

view this post on Zulip Peva Blanchard (May 18 2024 at 05:35):

Eric M Downes said:

Peva Blanchard said:

I managed to get a proof that, indeed, Λ\Lambda is left adjoint to Γ\Gamma. But my proof is relatively long, and I am not sure I checked everything right. An external confirmation/refutation is welcome.

Was your proof too long to fit in the margins of this stream? :P

(Seriously, where do I find it? I'd like to read it.)

Yes sure :)

I chose to prove the adjunction by building a natural isomorphism

(Top/X)(Λ(F),E)Psh(F,Γ(E)) (Top/X)(\Lambda(F), E) \cong Psh(F, \Gamma(E))

where FF is a presheaf over XX, and π:EX\pi : E \rightarrow X a bundle over XX.

Some notations. Given a section sFUs \in FU over an open subset UXU \subseteq X, [s]x[s]_x is the germ of ss at xx, and [s]:UΛ(F)[s] : U \rightarrow \Lambda(F) is the continuous function x[s]xx \mapsto [s]_x. We also denote by [U,s]={[s]x  xU}[U, s] = \{[s]_x ~|~ x \in U \} the set of germs of ss over UU, which is a member of a base of open sets of Λ(F)\Lambda(F).

Forward direction

Consider a bundle morphism f:Λ(F)Ef : \Lambda(F) \rightarrow E, i.e., a continuous function that preserves the base point. We want to define a natural transformation η\eta from the presheaf FF to the sheaf Γ(E)\Gamma(E) of sections of EE.

Components

Let UU be an open subset of XX, and sFUs \in FU a section of FF over UU.
We want to define a section zz of EE over UU, i.e., a continuous function z:UEz : U \rightarrow E such that πz=idU\pi\circ z = id_U. For every xUx \in U, let's define

z(x)=f([s]x)E z(x) = f([s]_x) \in E

Because ff is a morphism of bundles over XX, we have πz=idU\pi \circ z = id_U.
It remains to show that zz is continuous. By construction of Λ\Lambda, we know that the map

[s]:UΛ(F)x[s]x\begin{align*} [s] : U &\rightarrow \Lambda(F) \\ x &\mapsto [s]_x \\ \end{align*}

is continuous. By the continuity of ff and the fact that the composition of two continuous functions is continuous, we conclude that zz is also continuous.

To sum up, we defined a function

ηU:FUΓ(E)Usf[s]\begin{align*} \eta_U : FU &\rightarrow \Gamma(E)U \\ s &\mapsto f \circ [s] \end{align*}

Naturality

Consider an inclusion UVU \subseteq V. We want to show that the following diagram commutes.
image.png

This amounts to show that

f[sU]=f[s]U=(f[s])U\begin{align*} f \circ [s|_U] &= f \circ [s]|_U \\ &= \left(f \circ [s]\right)|_U \end{align*}

Therefore, we have a natural transformation η:FΓ(E)\eta : F \rightarrow \Gamma(E).

view this post on Zulip Peva Blanchard (May 18 2024 at 05:51):

Backward direction

Let η:FΓ(E)\eta : F \rightarrow \Gamma(E) be a natural transformation. We want to define a morphism f:Λ(F)Ef : \Lambda(F) \rightarrow E of bundles, i.e. a continuous function that preserves the base point.

Definition

Fix a germ [s]xΛ(F)[s]_x \in \Lambda(F), for some section sFUs \in FU over an open neighborhood UXU \subseteq X of xx. We define

f([s]x)=[ηUs]x f([s]_x) = [\eta_U s]_x

We need to show that ff is well-defined. Assume [s]x=[t]x[s]_x = [t]_x for another section tFVt \in FV over another open neighborhood VXV \subseteq X of xx. Without loss of generality, we may assume that VUV \subseteq U. In that case t=sVt = s|_V. Then

[ηVt]x=[ηVsV]x=[(ηUs)V]x (naturality)=[ηUs]x\begin{align*} [\eta_V t]_x &= [\eta_V s|_V]_x \\ &= [(\eta_U s)|_V]_x \text{ (naturality)}\\ &= [\eta_U s]_x \\ \end{align*}

Therefore, ff is well-defined. Note also that, by definition, ff preserves the base point.

Continuity

Let [W,r][W, r] be an open neighborhood of f([s]x)f([s]_x) in Γ(E)\Gamma(E). Without loss of generality, we may, again, assume that WUW \subseteq U. Then,

r=(ηUs)W=ηWsW r = \left(\eta_U s\right)|_W = \eta_W s|_W

is the restriction of ηUs\eta_U s on WW.

Consider t=sWFWt = s|_W \in FW, and the open neighborhood [W,t][W, t] of [t]x=[s]x[t]_x = [s]_x. For every yWy \in W,

f([t]y)=[ηWt]y=[ηWsW]y=[(ηUs)W]y=[r]y[W,r]\begin{align*} f([t]_y) &= [\eta_W t]_y \\ &= [\eta_W s|_W]_y \\ &= [\left(\eta_U \circ s\right)|_W]_y \\ &= [r]_y \\ &\in [W, r] \end{align*}

In other words,

[W,t]f1[W,r] [W, t] \subseteq f^{-1}[W, r]

I.e., ff is continuous.

view this post on Zulip Peva Blanchard (May 18 2024 at 05:51):

Natural bijection

We have shown that

(Top/X)(Λ(F),E)APsh(F,Γ(E))fη=(sηUf[s])UOX\begin{align*} (Top/X)(\Lambda(F), E) &\xrightarrow{A} Psh(F, \Gamma(E)) \\ f &\mapsto \eta = \left( s \xmapsto{\eta_U} f \circ [s] \right)_{U \in \mathcal{O}X} \end{align*}

and

Psh(F,Γ(E))B(Top/X)(Λ(F),E)η([s]x with sFUf[ηUs]x)\begin{align*} Psh(F, \Gamma(E)) &\xrightarrow{B} (Top/X)(\Lambda(F), E) \\ \eta &\mapsto \left([s]_x \text{ with } s \in FU \xmapsto{f} [\eta_U s]_x \right) \end{align*}

are both well-defined functions.

Prove that BA=idB \circ A = id

Let f:Λ(F)Ef : \Lambda(F) \rightarrow E be a morphism of bundles over XX, and g=B(A(f))g = B(A(f)).
By definition, for every sFUs \in FU for some open neighborhood UU of xx,

g([s]x)=[A(f)Us]x=[f[s]]x=f([s]x)\begin{align*} g([s]_x) &= [A(f)_U s]_x \\ &= [f \circ [s]]_x \\ &= f([s]_x) \end{align*}

I.e., g=fg = f, and thus BA=idB \circ A = id.

Prove that AB=idA \circ B = id

Let η:FΓ(E)\eta : F \rightarrow \Gamma(E) be a natural transformation of presheaves, and ν=A(B(η))\nu = A(B(\eta)).
By definition, for every open subset UU of XX,

νU:FU Γ(E)Us B(η)[s]=(xB(η)([s]x))=(x[ηUs]x)=ηUs\begin{align*} \nu_U : FU \rightarrow &~ \Gamma(E)U \\ s \mapsto &~ B(\eta) \circ [s] \\ &= \left(x \mapsto B(\eta)([s]_x) \right)\\ &= \left(x \mapsto [\eta_U s]_x \right)\\ &= \eta_U s \end{align*}

I.e., νU=ηU\nu_U = \eta_U, and thus AB=idA \circ B = id.

view this post on Zulip Peva Blanchard (May 18 2024 at 08:00):

Naturality of AA

With respect to EE

We show that A=AF,EA = A_{F, E} transforms naturally with respect to EE.

Consider a morphism of bundles g:EEg : E \rightarrow E'.
The naturality condition is expressed in the following diagram

image.png

I.e., we want to show that, for every open subset UXU \subseteq X,

Γ(g)UηU=ηU \Gamma(g)_U \circ \eta_U = \eta'_U

I.e., for every sFUs \in FU,

Γ(g)U(ηU(s))=ηU(s)    Γ(g)U(f[s])=(gf)[s]    g(f[s])=(gf)[s]\begin{align*} &\Gamma(g)_U(\eta_U(s)) = \eta'_U(s) \\ \iff &\Gamma(g)_U(f \circ [s]) = (g \circ f) \circ [s] \\ \iff &g \circ (f \circ [s]) = (g \circ f) \circ [s] \end{align*}

which holds by the associativity.

With respect to FF

We show that A=AF,EA = A_{F, E} transforms naturally with respect to FF.

Consider a natural transformation λ:FF\lambda : F \rightarrow F' of presheaves over XX. The naturality condition is expressed in the following diagram
image.png

I.e., we want to show that, for every open subset UXU \subseteq X,

ηU=ηUλ \eta'_U = \eta_U \circ \lambda

I.e., for every section sFUs \in F'U,

ηU(s)=ηU(λU(s))    (fΛ(λ))(s)=f[λU(s)]    f(Λ(λ)(s))=f[λU(s)]    f[λU(s)]=f[λU(s)]\begin{align*} &\eta'_U(s) = \eta_U(\lambda_U(s)) \\ \iff &(f \circ \Lambda(\lambda))(s) = f \circ [\lambda_U(s)] \\ \iff &f \circ (\Lambda(\lambda)(s)) = f \circ [\lambda_U(s)] \\ \iff &f \circ [\lambda_U(s)] = f \circ [\lambda_U(s)] \\ \end{align*}

view this post on Zulip Peva Blanchard (May 18 2024 at 08:47):

Naturality of BB

We have shown that every component BF,EB_{F,E} is the inverse of the component AF,EA_{F,E} (as functions in SetSet). Because AA is natural in EE and FF, I think we can conclude that BB is also natural in EE and FF.

view this post on Zulip Peva Blanchard (May 18 2024 at 09:10):

This concludes the proof that Λ\Lambda is left adjoint to Γ\Gamma.

view this post on Zulip John Baez (May 18 2024 at 09:13):

Wow, great! When @David Egolf reaches this material in the course, he can either try to figure out himself why Λ\Lambda is left adjoint to Γ\Gamma and check his work here, or just read this!