The Associated Sheaf Functor

40 第3章 Grothendieck位相・サイト上の層・層化関手に関するノート By uniqueness, z= (P f)(x). On the other hand, since {x_f}f∈S is a matching family for S, (P u)(x_f) = x_{f u} = ((P f u)(x)) = y. Therefore,z =x_f and we obtain (P f)(x) =x_f(f ∈S).

The proof is complete.

Definition 3.2.6 Let (C, J) be a site. We shall denote by Sh(C, J) the category whose objects are sheaves on (C, J) and morphism are natural transformations.

Therefore, Sh(C, J) is a full subcategory of the category of presheavesSets^Cop and there exists the inclusion functori: Sh(C, J),→Sets^Cop. 3 Definition 3.2.7 (Grothendieck topoi) We shall call a categoryGwhich is equiv-alent to the category of sheaves on a site (C, J), i.e. G ∼= Sh(C, J), a Grothendieck

topos. 3

Lemma 3.2.1 Let(C, J)be a site andI∋i7→P_i∈Sets^Cop a diagram of presheaves P_i(i ∈ I). If for all i ∈I, P_i ∈ Sh(C, J), then the limit lim←−ⁱ∈IP_i in Sets^Cop is a

sheaf on Sh(C, J). 2

Proof Note that forPi ∈Sets^Cop(i∈I), the limit lim←−^i∈IPi inSets^Cop is given by pointwise:

(lim←−_i∈IPi)(C)∼= (lim←−_i∈I)(PiC) (C∈C)

and sheaves are obtained by equalizers as (3.2.3), in particular, limits. By the com-mutativity of limits [2, p. 227], the statements follows. The proof is complete.

3.3 The Associated Sheaf Functor 41 Definition 3.3.2 Letx={x_f}f∈S ∈Match(S, P) and y={y_g}g∈T ∈Match(T, P)

for some S, T∈J(C) (C∈C). We definex∼Cy as follows:

x∼Cy⇐⇒ ∃^def R∈J(C), R⊆S∩T and ∀r∈R, xr=yr. (3.3.3) Fact 3.3.1 ∼C is an equivalence relation on⨿

S∈J(C)Match(S, P)for eachC∈C.2 Proof The reflexivity and symmetry may be obvious. We shall show only the tran-sitivity here. Letx={xf}f∈S ∈Match(S, P), y={yg}g∈T ∈Match(T, P) andz= {zh}h∈U ∈Match(U, P) be three matching families for someS, T, U ∈J(C) (C ∈C) such that x∼C y, y ∼C z. Then there exists R, R^′ ∈ J(C) such that R ⊆S∩T, R^′⊆T∩U and

∀r∈R, x_r=y_R, ∀r^′∈R^′, y_r′ =z_r′.

ConsiderW :=R∩R^′. ThenW ∈J(C) and for allw∈W, xw=yw=zw. Therefore, x∼Cz. The proof is complete.

We shall denote the equivalence class with a representative matching family xby [x].

Definition 3.3.3 LetP be a presheaf onC. Define a mappingP⁺:C^op→Setsby P⁺C:= ⨿

S∈J(C)

Match(S, P)/∼C . (3.3.4)

Fact 3.3.2 P⁺C ∼= lim−→^S∈J(C)Match(S, P), where the colimit is taken over all S ∈ J(C) ordered by reverse inclusion, and then there exists a morphism iRS : Match(S, P)∋ {xf}f∈S 7→ {xf}f∈R∈Match(R, P)if R⊆S. 2 Proof To proveP⁺Chas the universal mapping property, suppose that there exists a cocone{τS : Match(S, P)→L}S∈J(S)under Lsuch thatτT =τSiST for allS, T ∈ J(C) withS⊆T. We must show that there exists a unique mapt:P⁺C →Lsuch that τS =tπ for all τS(S ∈J(C)), where π : Match(S, P) →P⁺C is the quotient map for the equivalence relation∼C, as in the following diagram:

P⁺C_ _ _ _ _^∃_^!t_ _ _ _ _ _//

⟳

L

Match(S, P).

π

ffMMMMMMMMMMM ^τS

99s

ss ss ss ss ss

Define t:P⁺C→Lforx={xf}f∈S ∈Match(S, P)(S ∈J(C)) by t([x]) :=τS(x).

We must verify that the definition oftdoes not depend on the choice of a particular representative x∈ Match(S, P). To this end, lety ={yg}g∈T ∈ Match(T, P) (T ∈ J(C)) such that x ∼C y. Then there exists R ∈ J(C) such that R ⊆ S ∩T and xr=yrfor allr∈R. Hence,iRS(x) ={xr}r∈R={yr}r∈R=iRT(y). Therefore, t([y]) =τT(y) = (τRiRT)(y) =τR(iRT(y)) =τR(iRS(x)) = (τRiRS)(x) =τS(x) =t([x]).

42 第3章 Grothendieck位相・サイト上の層・層化関手に関するノート Since t([x]) = (tπ)(x), t satisfies the commutativity τ_S = tπ for all τ_S. To prove the uniqueness oft, suppose that there is another mappingt^′ :P⁺C →Lsuch that τ_S =t^′π for allτ_S. Thent^′([x]) =t^′π(x) = τ_S(x) =t([x]) for allx∈Match(S, P).

Therefore,t^′ =t. The proof is complete.

Forh:C^′ →C, P⁺h:P⁺C→P⁺C^′ is defined by

(P⁺h)([{x_f}f∈S]) := [{x_hf′}f^′∈h^∗(S)] ({x_f}f∈S ∈Match(S, P) (S ∈J(C))).

We claim that{xhf^′}f^′∈h^∗(S) ∈Match(h^∗(S), P). Indeed, for allf^′ : D^′ →C^′ ∈ h^∗(S), hf^′ ∈ S and for all g^′ : E^′ → D^′, (P g^′)(xhf^′) = xhf^′g^′, since {xf}f∈S is a matching family.

We claim that the definition ofP⁺his well-defined. Leth:C^′ →C,x={xf}f∈S

and y = {y_g}g∈T be two matching families such that x ∼C y, i.e., there exists R ∈ J(C) such that R ⊆ S∩T and x_r = y_r for all r ∈ R. We must show that {x_hf}f^′∈h^∗(S)∼C^′ {y_hg′}g^′∈h^∗(T). Considerh^∗(R)⊆h^∗(S)∩h^∗(T), by the stability axiom ofJ,h^∗(R)∈J(C^′). It is sufficient to show that

∀r^′∈h^∗(R), xhr^′ =yhr^′.

Note thath^∗(R) ={r^′|hr^′∈R}. Since for allr∈R, xr=yr and for allr^′ ∈h^∗(R), we havexhr^′ =yhr^′. Therefore,P⁺his well-defined.

We claim thatP⁺:C^op→Setsis a contravariant functor.

(i) LetS∈J(C). Note that for idC, id^∗_C(S) =S. Then (P⁺idC)([{xf}f∈S]) = [{xf}f∈S].

(ii) Leth:C^′→C andk:C^′′→C^′. Since (hk)^∗(S) ={f^′|hkf^′ ∈S}, (P⁺(hk))([{xf}f∈S]) = [{xhkf^′|f^′ ∈(hk)^∗(S)}].

On the other hand, since k^∗(h^∗(S)) ={g|kg ∈ h^∗(S)} = {f^′|hkf^′ ∈ S} = (hk)^∗(S),

(P⁺(k))((P⁺h)([{x_f}f∈S)]) = (P⁺k)([{x_hf′}f^′∈h^∗(S)])

= [{xhkf^′}f^′∈k^∗(h^∗(S))].

LetP, Qbe two presheaves and ϕ:P →Qa natural transformation. We define a mappingϕ⁺:P⁺→Q⁺ forC∈Cby

ϕ⁺_C:P⁺C →Q⁺C

ϕ⁺_C([{xf}f∈S]) := [{ϕ_dom(f)(xf)}f∈S] (S∈J(C),[{xf}f∈S]∈P⁺C).(3.3.5) First, we must verify the well-definedness ofϕ⁺. We claim that{ϕ_dom(f)(xf)}f∈S is a matching family ofQforS. Letf :D→C∈S andg:E→D. Then, sinceϕis a natural transformation andxis a matching family, we obtain

(Qg)(ϕ_D(x_f)) =ϕ_E((P g)(x_f)) =ϕ_E(x_{f g}) =ϕ_{dom(f g)}(x_{f g}).

3.3 The Associated Sheaf Functor 43 Hence, {ϕ_dom(f)(x_f)}f∈S is a matching family of Q for S. Let x = {x_f}f∈S and

y={y_g}g∈T be two matching families such thatx∼Cy, i.e., there exists R∈J(C) such thatR⊆S∩T andx_r=y_rfor allr∈R. Thenϕ_dom(r)(x_r) =ϕ_dom(r)(y_r). This implies that [{ϕ_dom(f)(x_f)}f∈S] = [{ϕ_dom(g)(y_g)}g∈T]. Therefore,ϕ⁺ : P⁺ →Q⁺ is well-defined.

Next, we shall prove that ϕ⁺ is a natural transformation. Let {xf}f∈S ∈ Match(S, P) (S∈J(C)). Then for anyh:D→C,

(Q⁺h)(ϕ⁺_C([{xf}f∈S])) = (Q⁺h)([{ϕ_dom(f)(xf)}f∈S]) = [{ϕ_dom(hf′)(xhf^′)}f^′∈h^∗(S)] and

ϕD(P⁺h([{xf}f∈S])) =ϕ⁺_D([{xhf^′}f^′∈h^∗(S)]) = [{ϕ_dom(hf′)(xhf^′)}f^′∈h^∗(S)] : P⁺C

P⁺h

ϕ⁺_C

//

⟳

Q⁺C

Q⁺h

P⁺D

ϕ_D //Q⁺D.

Hence, (Q⁺h)(ϕ⁺_C([{xf}f∈S])) = ϕ⁺_D(P⁺h([{xf}f∈S])). Therefore, ϕ⁺ is a natural transformation.

The functorial property of·⁺, i.e., id⁺_P = id_P+and (ψ◦ϕ)⁺=ψ⁺◦ϕ⁺for composable two natural transformations ϕand ψmay be clear by definition.

From the above, the mapping

+ :Sets^Cop∋P 7→P⁺∈Sets^Cop is a functor and is called theplus construction.

Definition 3.3.4 (canonical maps of presheaves) LetPbe a presheaf onC. We define a family of mappingsηP = ((ηP)C)C∈C:P →P⁺ for eachC∈C by

(ηP)C(x) := [{(P f)(x)}f∈t_C] (x∈P C). (3.3.6) We claim thatηP is a natural transformation fromP toP⁺, i.e., for anyh:C^′→C, the following diagram is commutative:

P C

P h

(η_P)_C

//

⟳

P⁺C

P⁺h

P C^′

(ηP)C′

//P⁺C^′.

(3.3.7)

Sincet_C′ =h^∗(t_C), we have

(P⁺h)((η_P)_C(x)) = (P⁺h)([{(P f)(x)}f∈tC])

= [{(P(hg))(x)}g∈h∗(t_C)]

= [{(P g)((P h)(x))}g∈tC′].

On the other hand,

(ηP)C^′((P h)(x)) = [{(P g)((P h)(x))}g∈t_C′].

44 第3章 Grothendieck位相・サイト上の層・層化関手に関するノート Hence, (P⁺h)((η_P)_C(x)) = (η_P)_C′((P h)(x)). Therefore,η_P = ((η_P)_C)_C∈C is a natu-ral transformation. We shall callη_P thecanonical mapofP.

Lemma 3.3.1 Let P be a presheaf onC andη_P the canonical map ofP. Then (i) P is a separated presheaf on (C, J)⇔ηP is a monomorphism.

(ii) P is a sheaf on (C, J)⇔ηP is an isomorphism. 2 Proof (i) (⇒) Suppose thatP is separated. Letx, y∈P C(C∈C) such that

(η_P)_C(x) = (η_P)_C(y)⇔[{(P f)(x)}f∈tC] = [{(P f)(y)}f∈tC]. (3.3.8) Then ∃R ∈J(C) such that∀r∈R, (P r)(x) = (P r)(y). Since P is separated x=y. Hence, (η_P)_C is injective for allC∈C. Therefore, η_P is a monomor-phism.

(⇐) Conversely, suppose thatη_P is a monomorphism, i.e., for allC∈C,(η_P)_C is injective. Let x, y ∈ P C (C ∈ C) and S ∈ J(C) such that for all f ∈ S, (P f)(x) = (P f)(y). Since [{(P f)(x)}f∈S] = [{(P f)(x)}f∈t_C] = (ηP)C(x) and [{(P f)(y)}f∈S] = [{(P f)(y)}f∈t_C] = (ηP)C(y), (ηP)C(x) = (ηP)C. SinceηC is injective,x=y. Therefore,P is separated.

(ii) (⇒) Suppose thatP is a sheaf on (C, J). ThenP is separated. Hence, by (i), ηP is a monomorphism. Hence, it is sufficient to show thatηP is an epimor-phism. Let C∈C, S ∈J(C) and{xf}f∈S ∈Match(S, P). Then, sinceP is a sheaf, there exists a unique x∈P C such that{(P f)(x)}f∈S ={xf}f∈S. On the other hand, (η_P)_C(x) = [{(P f)(x)}f∈tC] = [{(P f)(x)}f∈S] = [{x_f}f∈S].

Hence, (η_P)_C is surjective for all C ∈ C. Therefore, η_P is an epimorphism.

Consequently,η_P is an isomorphism.

(⇐) Conversely, suppose that η_P is an isomorphism. In particular, η_P is a monomorphism. By (i), P is separated. Thus, it is sufficient to show that P has an amalgamation for all matching families. Let {xf}f∈S ∈ Match(S, P) (C ∈ C, S ∈ J(C)). Since (ηP)C is surjective, there exists x ∈ P C such that (ηP)C(x) = [{(P f)(x)}f∈t_C] = [{xf}f∈S].

Hence, there exists R ∈ J(C) such that R ⊆ S and xr = (P r)(x) for all r ∈ R. Let f : D → C ∈ S. Then f^∗(R) = {h|f h ∈ R} ∈ J(D), by the stability axiom of J. Thus, for all h ∈ f^∗(R), (P(f h))(x) = xf h. This implies that {(P f h)(x))}h∈tD ∼D {x_{f h}}h∈tD = {(P h)(x_f)}h∈tD. Hence, (η_P)_D((P f)(x)) = [{(P h)((P f)(x))}h∈tD] = [{(P h)(x_f)}h∈tD] = (η_P)_D(x_f).

Since η_P is a monomorphism, (P f)(x) = x_f (f ∈ S). Therefore, x is an amalgamation of {x_f}f∈S.

The proof is complete.

Lemma 3.3.2 Let F be a sheaf on (C, J) and P a presheaf on C. Then for any natural transformation ϕ : P → F, there exists a unique natural transformation

3.3 The Associated Sheaf Functor 45 ϕ˜:P⁺→F such thatϕ= ˜ϕ◦η_P, i.e., η_P has the universal mapping property:

P ^ηP //

∀ϕ

A

AA AA AA AA AA AA AA

A P⁺

∃! ˜ϕ

⟳

F.

(3.3.9)

Proof LetC∈C, S∈J(C) and [{xf}f∈S]∈P⁺C. For anyh:D→C inS, by the definition of P⁺,

(P⁺h)([{xf}f∈S]) = [{xhf^′}f^′∈h^∗(S)] = [{xhf^′}f^′∈t_D], sinceh∈S impliesh^∗(S) =t_D. On the other hand, by the definition ofη_P,

(ηP)D(xh) = [{(P f^′)(xh)}f^′∈tD] = [{xhf^′}f^′∈tD].

Hence,

∀h:D→C∈S, (P⁺h)([{xf}f∈S]) = (ηP)D(xh). (3.3.10) If the natural transformation ˜ϕ:P⁺ →F such thatϕ= ˜ϕ◦η_P were to exist, for anyh:D→Cin S,

(F h)( ˜ϕC)([{xf}f∈S])

= ˜ϕD((P⁺h)([{xf}f∈S])) (by the naturality of ˜ϕ)

= ˜ϕ_D((η_P)_D)(x_h) (by (3.3.10))

=ϕD(xh).

On the other hand,{ϕ_dom(h)(xh)}h∈S is a matching family ofF forS and sinceF is a sheaf,

∃!y∈P C,∀h∈S, (F h)(y) =ϕ_dom(h)(xh).

Accordingly, we define ˜ϕ:P⁺ →F for eachC∈Cand [{x_f}f∈S]∈P⁺C by ϕ˜_C([{x_f}f∈S]) =y.

To verify that ˜ϕ is a natural transformation from P⁺ to F, we must show that for anyg:D→Cin C, the following diagram is commutative:

P⁺C

P⁺g

ϕ˜C //

⟳

F C

F g

P⁺D

ϕ˜_D

//F D.

Let [{xh}h∈S] ∈ P⁺C. Then ˜ϕD((P⁺g)([{xh}h∈S])) = ˜ϕD([{xgh^′}h′∈g∗(S)]) is a uniquez∈F Dsuch that

∀h^′∈g^∗(S), (F h^′)(z) =ϕ_dom(gh′)(xgh^′).

46 第3章 Grothendieck位相・サイト上の層・層化関手に関するノート On the other hand, by the definition of ˜ϕ, for anyh^′∈g^∗(S),

(F h^′)((F g)( ˜ϕ_C)([{x_h}h∈S])) = (F(gh^′))( ˜ϕ_C)([{x_h}h∈S])) =ϕ_dom(gh′)(x_gh′).

Hence,

∀h^′∈g^∗(S), (F h^′)( ˜ϕD((P⁺g)([{xh}h∈S]))) = (F h^′)((F g)( ˜ϕC)([{xh}h∈S])).

Sinceg^∗(S)∈J(C) andF is separated, this implies that

ϕ˜D((P⁺g)([{xh}h∈S]))) = ((F g)( ˜ϕC)([{xh}h∈S])), i.e., ˜ϕis a natural transformation fromP⁺ toF.

Next, we verify ϕ = ˜ϕ◦ηP. Let C ∈ C and x ∈ P C. Then ( ˜ϕC◦(ηP)C)(x) = ϕ˜C({(P h)(x)}h∈t_C) is a uniquey∈F C such that

∀h∈t_C, (F h)(y) =ϕ_dom(h)(P h)(x). (3.3.11) On the other hand, sinceϕis a natural transformation fromP to F,

∀h∈t_C, (F h)((ϕ_C)(x)) =ϕ_dom(h)((P h)(x)) = (F h)(y) (by (3.3.11)).

Since tC ∈ J(C) andF is separated, this implies that y = ϕC(x). Therefore, ϕ = ϕ˜◦ηP. The proof is complete.

The following two lemmas are central results to prove Theorem 3.3.1.

Lemma 3.3.3 Let P be a presheaf on C. Then P⁺ is a separated presheaf on

(C, J). 2

Proof LetC ∈C, Q ∈J(C) and x={x_f}f∈S,y ={y_g}g∈T matching families for some S, T ∈ J(C) such that for allh∈ Q, (P⁺h)([x]) = (P⁺h)([y]). Namely, by the definition ofP⁺,

∀h∈Q, [{xhf^′}f′∈h∗(S)] = [{yhg^′}g′∈h∗(T)].

For eachh:D→C∈Q, by the definition of ∼C, there exists Rh∈J(D) such that Rh⊆h^∗(S)∩h^∗(T) andxhr=yhrfor allr∈Rh. To show that P⁺ is separated, we must show that there existsR∈J(C) such thatR⊆S∩T andxr=yr∈P C for all r∈R. To this end, let R:={hr|h∈Q, r∈Rh}. ThenR is a sieve on C and for allh∈Q, h^∗(R) ={s|hs∈R} ⊇Rh ∈J(D), so h^∗(R)∈J(D). By the transitivity axiom ofJ, R∈J(C). On the other hand, let hr∈R for someh∈Qandr ∈Rh. Thenr∈Rh⊆h^∗(S)∩h^∗(T), i.e.,hr∈S∩T. Therefore,J(C)∋R⊆S∩T and for allr∈R, x_r=y_r, i.e., [x] = [y].The proof is complete.

Lemma 3.3.4 Let P be a separated presheaf on (C, J). Then P⁺ is a sheaf on

(C, J). 2

Proof LetC∈Cand {[xf]}f∈S a matching family of P⁺ forS ∈J(C). Note that for allf :D→C∈S, xf is a matching family ofP for someSf ∈J(D) and denote it byxf ={xf,h}h∈S_f. The condition that{[xf]}f∈S∈Match(S, P⁺) implies that

∀f :D→C∈S, ∀g:D^′→D, (P⁺g)([xf]) = [xf g],

3.3 The Associated Sheaf Functor 47 that is,

∀f :D→C∈S, ∀g:D^′ →D, [{xf,gh^′}h^′∈g^∗(Sf)] = [{xf g,h}h∈S_{f g}].

By the definition of the equivalence relation, there exists R_f,g ∈ J(D^′) such that Rf,g⊆g^∗(Sf)∩Sf g and

∀r∈R_f,g, x_f,gr=x_{f g,r}. (3.3.12) Let T :={f g|f ∈S, g ∈Sf}. Then T is a sieve on C. For each f :D → C ∈S, f^∗(T) ={t|f t∈T} ⊇Sf ∈J(D). By the transitivity axiom ofJ,T ∈J(C). Now, we define a matching family y={yt}t∈T ofP forT by

yt:=xf,g (t=f g, f∈S, g∈Sf).

Firstly, we must show that the definition does not depend on the choice of f andg such that t =f g ∈T. Let t =f g =f^′g^′ for some f, f^′ ∈ S, g ∈ S_f and g^′ ∈ S_f′. Then there existsR_f,g, R_f′,g^′ ∈J(D^′) andR_f,g∩R_f′,g^′ ∈J(D^′). If r∈R_f,g∩R_f′,g^′, then

(P r)(x_f,g) =x_f,gr

=xf g,r (by (3.3.12))

=x_f′g^′,r

=xf^′,g^′r (by (3.3.12))

= (P r)(x_f′,g^′).

SinceP is separated,xf,g=xf^′,g^′. Hence,y is well-defined.

Next, we verify that y is a matching family. To this end, let f g ∈ T for some f :D→C∈S andg:D^′→D∈S_f andh:E→D^′. Then

(P h)(yf g) = (P h)(xf,g) =xf,gh=yf gh. Hence,y∈Match(T, P). Therefore, [y]∈P⁺C.

We shall show that [y] is an amalgamation of {xf}f∈S, i.e., for allf :D → C ∈ S, (P⁺f)([y]) = [xf], i.e.,

[{yf g}g∈f^∗(T)] = [{xf,g}g∈S_f]∈P⁺D.

Since for all f : D → C ∈ S, Sf ⊆ f^∗(T) = {t|f t ∈ T} and for all g ∈ Sf ∈ J(D), yf g =xf,g. This implies that{yf g}g∈f^∗(T) ∼D {xf,g}g∈S_f. Hence, [y] is an amalgamation of [{xf}f∈S]. By Lemma 3.3.3,P⁺is separated. Therefore, this amal-gamation is unique. Consequently,P⁺ is a sheaf on (C, J). The proof is complete.

Definition 3.3.5 (the associated sheaf functor) We define a mapping a:Sets^Cop→Sh(C, J)

as follows:

(i) a(P) := (P⁺)⁺ forP ∈Sets^Cop;

48 第3章 Grothendieck位相・サイト上の層・層化関手に関するノート (ii) a(ϕ) := (ϕ⁺)⁺ for a natural transformationϕ:P→Qin Sets^Cop. 3 Since plus construction + is a functor, so isaand is calledthe associated sheaf functor.

By applying Lemma 3.3.2 twice, for any presheafP onC, ˜η_P :=η_P+◦η_P also has the universal mapping property.

We claim that ˜η:= (˜ηP)_P∈SetsCop : id_SetsCop →iais a natural transformation. To show this, it is sufficient to show the following commutativity inSets^Cop:

P

ϕ

η_P //

⟳

P⁺

ϕ⁺

Q _η

Q

//Q⁺

for all P, Q ∈Sets^Cop and all natural transformations ϕ: P → Q. To this end, let C∈Candx∈P C. Then

((ϕ⁺)C◦(ηP)C)(x) = (ϕ⁺)C((ηP)C(x)))

= (ϕ⁺)C([{(P f)(x)}f∈t_C])

= [{ϕ_dom(f)((P f))(x))}f∈(t_C)].

On the other hand,

((ηQ)C◦(ϕC))(x) = (ηQ)C(ϕC(x))

= [{(Qf)(ϕ_C(x))}f∈tC]

= [{ϕ_dom(f)((P f)(x))}f∈t_C] (by the naturality ofϕ).

Thus, ((ϕ⁺)_C◦(η_P)_C)(x) = ((η_Q)_C◦(ϕ_C))(x) for allC∈Cand allx∈P C. Hence, ϕ⁺◦η_P =η_Q◦ϕ. By replacingP, Q andϕbyP⁺, Q⁺ andϕ⁺, respectively, we have (ϕ⁺)⁺◦η_P+=η_Q+◦ϕ⁺. Therefore, we obtain the following commutative diagram:

P

⟳ ϕ

η_P //P⁺

ϕ⁺

η_P+//

⟳

(P⁺)⁺

(ϕ⁺)⁺

Q _η

Q

//Q⁺

η_Q+

//(Q⁺)⁺.

Consequently, ˜η is a natural transformation from id_SetsCop toia.

In conclusion, for the inclusion functor i : Sh(C, J) ,→ Sets^Cop, there exists the associated sheaf functor a : Sets^Cop → Sh(C, J) and a natural transformation

˜

η : id_SetsCop → ia such that for each P ∈ Sets^Cop, ˜ηP : P → ia(P) has the uni-versal mapping property. Consequently, a is a left adjoint of the inclusion functor i: Sh(C, J),→Sets^Cop, i.e., a⊣i and ˜η is the unit ofa⊣i.

By Lemma 3.3.1 (ii), for any sheafF on (C, J),ηF is an isomorphism. Hence, we can construct a natural isomorphism ˜ε= (˜ε_F)_F∈Sh(C,J) : ai →id_Sh(C,J) by setting

˜

ε_F := (˜η_F)⁻¹. Then ˜εis the counit ofa⊣i. We have the following corollary:

Corollary 3.3.1 Let (C, J) be a site, i: Sh(C, J),→Sets^Cop the inclusion functor anda:Sets^Cop →Sh(C, J)the associated sheaf functor. Then

a◦i: Sh(C, J)→Sh(C, J) (3.3.13)

49

is naturally isomorphic to id_Sh(C,J). 2

To complete the proof of Theorem 3.3.1, we shall show the following lemma:

Lemma 3.3.5 The associated sheaf functora preserves finite limits. 2 Proof It is sufficient to show that the plus construction + preserves finite limits.

To this end, let C ∈ C and S ∈ J(C). Then Match(S,−) : Sets^Cop ∋ P 7→

Match(S, P) ∈ Sets is a functor. Since every matching family can be identified with a natural transformation, i.e., there is a natural isomorphism

Match(S, P)∼= Hom_SetsCop(S, P),

where we identify S with the subfunctor S ⊆ y(C), and the Hom-functor Hom_SetsCop(S,−) preserves limits, we have

Match(S,lim←−^i∈IPi)∼= lim←−^i∈IMatch(S, Pi)

for any finite diagram {P_i}i∈I (I is finite) in Sets^Cop. Recall that P⁺C ∼= lim−→^S∈J(C)Match(S, P) and J(C) is a filtered category equipped with reverse inclu-sions, since for all S, T ∈J(C),S∩T ∈J(C). Since filtered colimits commute with finite limits inSets[2, p. 211], we obtain

(lim←−ⁱ∈IP_i)⁺C∼= lim−→^S∈J(C)Match(S,lim←−ⁱ∈IP_i)

∼= lim−→^S∈J(C)lim←−^i∈IMatch(S, Pi)

∼= lim←−^i∈Ilim−→^S∈J(C)Match(S, Pi)

∼= lim←−ⁱ∈I(P_i⁺C)

∼= (lim←−ⁱ∈IP_i⁺)C (∵limits in Sets^Cop are computed pointwise).

This implies that (lim←−^i∈IPi)⁺ ∼= lim←−^i∈IP_i⁺, i.e., + preserves finite limits. The proof is complete.

With the above lemmas, the proof of Theorem 3.3.1 is complete.

参考文献

[1] S. Mac Lane and Iake Moerdijk,Sheaves in Geometry and Logic, corrected 2nd.

ed., Springer, 1994.

[2] S. Mac Lane,Categories for the Working Mathematician, Springer-Verlag, 1971.

51

There is a large cardinal in St. Louis,

and there is no word about how mad his family is.

They are inaccessible, indescribable, ineffable, shrewd, ethereal, subtle, and remarkable.

It is what forced him to disclose his vice.

淡中 圏 本名：田中健策 冬コミは忙しくて大したものが書けないが、この口惜しさが次の夏コミ の原動力になるのである、否、原動力にしなくてはいけないのである。と言うわけで次回予 告。次回教育における「論証」の扱いへの疑問を述べるのと、いくつかの唾棄すべき数学関 連書籍を一刀両断しようと思う。もう少し高等な数学の話としては「層化」の話をしたい。

本当は去年の冬コミに書く予定だったのにね。まあとにかくこうご期待である。

 よく分からないブログ http://blog.livedoor.jp/kensaku_gokuraku/

鈴木 佑京  東大大学院総合文化研究科修士一年鈴木佑京（@otb btb)^です. ^{今回も前回同様マニ} アックな内容になりましたが、マニアックなだけならまだしも、数学的実質に乏しくなって しまい反省してます。次はもっと読んだ人を幸せにできる原稿を書きます。アンジュ様格好 いいです。

才川 隆文 Redmine^ちゃんのslave bot^{であり号令係でした。}commit log^{が飛び交いチケットが} 増減するダイナミズムに身を任せるのは快感です。

古賀 実 今回初めて表紙絵を担当しました．構図の元ネタはSTAR WARS episode VII^のキービ ジュアルです．同人誌作成で使用し始めたgitが便利で今では常用してます．

発行者 ： The dark side of Forcing

連絡先 ： http://forcing.nagoya/ , http://proofcafe.org/forcing/

発行日 ： 2014年12月30日

ドキュメント内 i There is a small cat from Topoi, And her name is Joy. She looks for the path to the darkside of math. It s going to be her favorite toy. (ページ 46-61)