GENERALIZED RINGED SPACES AND SCHEMES
DAVID I. SPIVAK
November 29, 2011
Contents
1. Introduction
2. Categorical Preliminaries
2.1. Basic category theory
2.2. Natural transformation diagrams
2.3. Correspondences
2.4. Downarrow categories
3. Sieves and Covering Sieves
3.1. First definitions
3.2. Covering Sieves
3.3. (C, CovC )-spaces
3.4. Sheaves
4. R-ringed Categories and R-ringed Spaces
4.1. R-ringed Categories
4.2. Examples
4.3. R-ringed spaces
5. The Structure theorem
6. Simplifying assumptions
6.1. Op(X)
6.2. C ∞ -rings
7. Deleted stuff
7.1. 1
7.2. 2
7.3. 3
7.4. 4
7.5. 5
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1. Introduction
2. Categorical Preliminaries
2.1. Basic category theory. For n ∈ N, let [n] denote the “subdivided interval”
category with n + 1 objects {0, 1, . . . , n} and exactly one morphism i → j for every
i ≤ j ≤ n.
1
2
DAVID I. SPIVAK
Let C be a category. A diagram in C is a functor X : I → C, where I is a
small category (such a functor is also called an I-shaped diagram in C). The Ishaped diagrams in C form the objects of a category, whose morphisms are natural
transformations. If i ∈ I is an object, we sometimes denote X(i) by Xi .
There is an isomorphism of categories C ∼
= C [0] ; that is we may identify objects
in C with functors [0] → C, and morphisms with natural transformations between
them.
Any functor F : X → Y naturally induces a functor F I : X I → Y I , which we
may denote simply by F : X I → Y I .
Let F : C → D be a functor. By an object in F we mean a functor [0] → D which
factors through F . More generally, we refer to a diagram I → D which factors
through F as an I-shaped diagram in F . If C = D and F is the identity, then
diagrams in F are simply diagrams in C.
Let Pre(C) denote the category of contravariant functors from C to Sets. It is
called the category of presheaves on C. There is a natural functor r : C → Pre(C)
given by
r(C) = Hom(−, C)
called the representation functor, or the Yoneda imbedding.
Let C and D be categories. An adjunction from C to D is a triple (L, R, φ), where
L : C → D and R : D → C are functors, and φ is a natural isomorphism of functors
φ : HomD (L(−), −) → HomC (−, R(−)).
That is, for any C ∈ C and D ∈ D, there is a natural isomorphism Hom(LC, D) ∼
=
Hom(C, RD). The functor L is called a left adjoint and the functor R is called a
right adjoint. If φ(β [ ) = β ] , then we refer to a pair of morphisms
(β [ : LC → D, β ] : C → RD)
as a φ-partnership or (L, R)-partnership, and we refer to β [ and β ] as the left and
right partners, respectively . We denote such a φ-partnership by β : C → D. We
say that φ is the adjunction isomorphism of the adjoint pair
Co
L
/D
R.
We sometimes suppress mention of φ when it is inconvenient.
If F : A → B and X : A → C are functors, depicted in the diagram
A
F

B
X
/C
?

 λF X
then the left Kan extension of X along F (if it exists) is denoted
λF X : B → C.
A natural transformation ηF (X) : X → λF X ◦ F is part of the data of a left Kan
extension, but is typically suppressed.
By the term Basic Category Theory or BCT we will mean anything that can be
proven using the following facts (all of which can be found in [?]).
Lemma 2.1. Let C and D be categories, and let I be a small category.
GENERALIZED RINGED SPACES AND SCHEMES
3
(1) Suppose that Y : I → C is a functor which has a limit Y C . Then for any
object X ∈ C, one has a natural isomorphism
HomC (X, Y C ) ∼
= lim(HomSets (X, Y )).
(2) Suppose that X : I → C is a functor which has a colimit X B . Then for any
object Y ∈ C, one has a natural isomorphism
HomC (X B , Y ) ∼
= lim(HomSets (X, Y )).
(3) Suppose that
Co
/D
L
R
is a pair of adjoint functors. Then L commutes with colimits and R commutes with limits. Explicitly, if a diagram X : I → C has a colimit X B ,
then the diagram LX : I → D has a colimit (LX)B , and there is a natural
isomorphism
∼
=
(LX)B −
→ L(X B ),
(and similarly for R and limits).
L /
(4) If C o
D is a pair of adjoint functors and I is a small category, then
R
CI o
L
/
DI are also adjoint.
R
(5) The Yoneda imbedding r : C → Pre(C) is fully faithful. Moreover, for any
presheaf F on C and object C ∈ C, we have an isomorphism of sets
HomPre(C) (rC, F ) ∼
= F (C).
w
2.2. Natural transformation diagrams. Let A, B, C, and D be categories, A −
→
y
x
z
B−
→ D and A −
→C−
→ D pairs of composable functors, and β : zx → yw a natural
transformation of functors. We can express these, data in a natural transformation
square:
/B
w
A
;C
β

x
C
y
/ D.
z
This is an abbreviation of the diagram
β
x
C
/B
w
A
zx

z
;C
yw
y
+ / D.
4
DAVID I. SPIVAK
A natural transformation square in which yw = zx and β is the identity transformation is called a commutative square (of categories), and is written
w
A
/B
y
x
C
z
/ D.
Similarly, there are natural transformation diagrams of any given shape; for example, see Lemma 2.8.
There is one other special case worth mentioning. Suppose that y : B → D has a
right adjoint y 0 : D → B. Then if A = C and x : A → C is the identity, then there
is a natural isomorphism
HomDA (yw, z) ∼
= HomB A (w, y 0 z).
Any natural transformation β [ : yw → z has a right partner β ] : w → y 0 z, and
these two transformations represent the same data. We represent this data by the
diagram
w
/B
A
O
id
A
β
z
y
y0
/D
2.3. Correspondences.
Definition 2.2. Let C and D be categories. A correspondence F from C to D,
written F : C < D is a functor F ] : D → Pre(C). The correspondence F is called
full (resp. faithful) if F ] is full (resp. faithful). The correspondence F can also be
regarded as a functor F [ : C op × D → Sets via the Cartesian adjunction.
Remark 2.3. There are several reasons for the notation F : C < D to denote a
correspondence F : C op × D → Sets. First, it is good to write C before D, since it is
the contravariant variable. Second, correspondences generalize functors by saying
that the correspondence underlying a functor F : D → C is
[−, F (−)]C : C < D.
Thus the direction on the tip on the arrow is preserved by the less-than symbol.
We will sometimes denote a correspondence F : C < D by some variant of the
notation [−, −]. There is often a way to do so which makes F clear. See Example
2.4.
Example 2.4. Let G : D → C be a functor. It naturally induces two correspondences,
[−, G(−)]C : C op × D → Sets and [G(−), −]C : Dop × C → Sets,
defined for C ∈ C and D ∈ D by [C, G(D)]C : = HomC (C, GD) and [G(D), C]C : =
HomC (GD, C), respectively. In particular, the identity functor idC : C → C gives
the correspondence [−, −]C = HomC : C op × C → Sets.
More generally, functors G : D → C and H : E → C induce correspondences
[G(−), H(−)]C and [H(−), G(−)]C in the obvious way.
GENERALIZED RINGED SPACES AND SCHEMES
5
As seen in Example 2.4, a functor G : D → C induces two correspondences. In
this work, we emphasize the correspondence [−, G(−)]C : C op × D → Sets.
Definition 2.5. Let G : D → C be a functor. The correspondence associated to G
is the composition of G with the Yoneda imbedding,
G
r
D−
→C−
→ Pre(C),
and is denoted rG.
Note that correspondences can be composed. If F : D < E and G : C < D are
correspondences, we can define a correspondence F ◦ G : C < E as follows. Consider
F as a functor E → Pre(D) and consider G as a functor D → Pre(C). The left Kan
extension of G along the Yoneda imbedding is a functor λr G : Pre(D) → Pre(C),
and we define
F ◦ G := F ◦ λr G.
Let C and D be categories, and let F, G : C < D be two correspondences between
them. A natural transformation of correspondences a : F → G is simply a natural
transformation of left partners
(F [ ⇒ G [ ) : C op × D → Sets
or equivalently of right partners
(F ] ⇒ G ] ) : D → Pre(C).
Note that both of these ares equivalent to a correspondence a : (C × [1]) < D.
Lemma 2.6. Let F : C < D be a correspondence. It determines an adjunction
Pre(D) o
λr F
/ Pre(C)
ρr F
of presheaf categories in which λr F(rD) = F(−, D) for any D ∈ D.
Proof. Let F = F ] : D → Pre(C), L = λr F and R = ρr F. For a presheaf
P ∈ Pre(D), one defines L(P ) = colimrX→P F (X). For a presheaf Q ∈ Pre(C),
one defines R(Q) = HomPre(C) (F (−), Q). The result follows from the chain of
isomorphisms
HomPre(C) (L(P ), Q) = HomPre(C) (colim F (X), Q)
rX→P
∼
= lim R(Q)(X)
rX→P
∼
= lim HomPre(D) (rX, RQ)
rX→P
∼
= HomPre(D) (colim rX, RQ)
rX→P
∼
= HomPre(D) (P, RQ).
2.4. Downarrow categories.
Definition 2.7. Let G : C → D and H : E → D be functors. The downarrow
category (G ↓ H) is the category whose objects consist of triples (C, f, E), where
C ∈ C and E ∈ E are objects and f : GC → HD is a morphism in D. The morphism
set Hom(G↓H) ((C, f, E), (C 0 , f 0 , E 0 )) is the subset of HomC (C, C 0 ) × HomE (E, E 0 )
6
DAVID I. SPIVAK
consisting of those pairs (c, e) for which the induced square in D commutes, i.e. for
which f 0 c = ef .
If C = D and G = idD , we may denote (G ↓ H) simply by (D ↓ H).
There are canonical functors π1 : (F ↓ G) → C and π2 : (F ↓ G) → E, called the
first and second projection functors.
Lemma 2.8. A natural transformation diagram
/Do
F
C

α
c
C0
F
;C
0
G
????β
#
d
/ D0 o
0
G
E
e
E0
induces a functor
(α ⇓d β) : (F ↓ G) → (F 0 ↓ G0 ).
Proof. Let C ∈ C and E ∈ E be objects, and let h : F C → GE be an object in
(F ↓ G). Define (α ⇓d β)(h) to be the composition
d(h)
α
β
F 0 c(C) −
→ dF C −−−→ dGE −
→ G0 eE.
This is clearly functorial.
In case dF = F 0 c and α = id, we denote (α ⇓d β) by (F ⇓d β); similarly, if
eG = G0 d and β = id, we denote (α ⇓d β) by (α ⇓d G). If C = D and F = idD ,
then we denote (F ⇓d β) by (⇓d β), and similarly for G.
Lemma 2.9. Let F : C → D and G : E → D be functors.
(1) If C and E are cocomplete and F is cocontinuous then (F ↓ G) is cocomplete
and the projections π1 and π2 are also cocontinuous.
(2) If C and E are complete and G is continuous then (F ↓ G) is complete and
the projections π1 and π2 are also continuous.
Proof. Since (Gop ↓ F op ) is the opposite category of (F ↓ G), it suffices to prove
item 1.
X : I → (F ↓ G), L = colim(π1 X) ∈ C, M = colim(π2 X) ∈ E.
F π1 X
FL
F L0
/ Gπ2 X
induced
/ GM
/ GM 0
Let F : C → D and F 0 : C 0 → D0 be functors, and let L : (D ↓ F ) → (D0 ↓ F 0 ) be
a functor. Consider the correspondence
H : (D ↓ F ) < (D0 ↓ F 0 )
GENERALIZED RINGED SPACES AND SCHEMES
7
induced by L. There is a natural functor from H to the constant correspondence
HomD (LF, F 0 ), which sends a commutative square
a
LD
/ LF
f
d
D0
a0
/ F0
to f : LF → F 0 .
3. Sieves and Covering Sieves
3.1. First definitions. Monomorphisms will play a large role in this paper. If
G : C → D and H : E → D are functors, let [G(−), H(−)]m
D denote the subfunctor
of [G(−), H(−)]D consisting only of monomorphisms. Likewise, let (G ↓m H)
denote the the full subcategory of (G ↓ H) whose objects are triples (C, f, E) in
which f : GC → HE is a monomorphism in D.
Definition 3.1. Let C be a category and X : A → Pre(C) a functor to the presheaf
category on C. The category of sieves on X in C, denoted SieveC (X), is the
category (Pre(C) ↓m X). If C : [0] → C is an object, we write SieveC (C) to
denote SieveC (rC), where r : C → Pre(C) is the Yoneda imbedding. Thus an
object in SieveC (C) is simply a subfunctor of HomC (−, C). If A = Pre(C) and
X : A → Pre(C) is the identity, then we denote SieveC (X) by SieveC .
Let X : A → Pre(C). The forgetful functor
SieveC (X) = (Pre(C) ↓m X) → (Pre(C) ↓ X)
has a left adjoint, called the image-sieve functor, which we denote by
(−)m : (Pre(C) ↓ X) → SieveC (X).
Let us make this more explicit.
An object f : P → Xa in (Pre(C) ↓ X) provides, for each c ∈ C, a map of sets
f (c) : P (c) → Xa (c). Let im(f )(c) denote the set-theoretic image of f (c), and let
f m (c) be the monomorphism im(f )(c) ,→ Xa (c). This defines (−)m on objects, and
it is clear how to define it on morphisms. We refer to f m as the image-sieve of f .
We use the same notation in a similar situation. Given a functor F : (Pre(C) ↓
X) → (Pre(D) ↓ Y ), let F m denote the induced functor
F m : SieveC (X) → SieveD (Y )
which is obtained by the composition
F
(−)m
SieveC (X) → (Pre(C) ↓ X) −
→ (Pre(D) ↓ Y ) −−−→ SieveD (Y ).
Example 3.2. Suppose that f : C → C 0 is a morphism in C. It induces a morphism
of presheaves f : rC → rC 0 , which gives rise to a sieve f m : im(f ) ,→ rC 0 . Explicitly,
for any object c ∈ C, one sees that im(f )(c) is the set of morphisms c → C 0 which
factor through f . If f is a monomorphism, then im(f ) = rC and f m = f .
8
DAVID I. SPIVAK
Let F : Pre(D) → Pre(C) be a functor, A a category, and Y : A → Pre(D) a
functor.
Pre(D) o
Y
A
idA
F
Pre(C) o
FY
A.
One defines
F∗ = (⇓F Y )m : SieveD (Y ) → SieveC (F Y ).
Lemma 3.3. Let F : Pre(D) → Pre(C) be a functor, A a category, and Y : A →
Pre(D) a functor. The functor F∗ has a right adjoint F −1 : SieveC (F Y ) → SieveD (Y ).
Proof. For a sieve j : τ ,→ F Y , define F −1 (j) : F −1 (τ ) ,→ Y to be the sieve whose
value on T ∈ D is the set
F −1 (τ )(T ) = {h : T → Y |F (h) ∈ τ (F T ).}
Let i : σ ,→ Y be a sieve on Y . By the nature of sieve categories, the sets
HomSieveD (Y ) (σ, F −1 (τ )) and HomSieveC (F Y ) (F∗ σ, τ ) are either empty or singleton.
The former set is empty if and only if there exists an object T ∈ D and a morphism
h : T → X such that h ∈ σ(T ) and h 6∈ F −1 (τ )(T ). This is the case if and
only if F (h) ∈ F∗ σ(F T ) and F (h) 6∈ τ (F T ), which is the case if and only if
HomSieveC (F Y ) (F∗ σ, τ ) is empty.
Definition 3.4. Let F : Pre(D) → Pre(C) be a functor, A a category, and Y : A →
Pre(D) a functor. We refer to F∗ : SieveD (Y ) → SieveC (F Y ) as the pushforward
along F functor, and to F −1 : SieveC (F Y ) → SieveD (Y ) as the pullback along F
functor.
Let X : A → Pre(D) and Y : A → Pre(D) be functors, and let β : X → Y be a
natural transformation between them
Pre(D) o
X
β
Pre(D) o
A
Y
A.
Then β induces (⇓ β) : (Pre(D) ↓ X) → (Pre(D) ↓ Y ), and we denote (⇓ β)m by
β∗ : SieveD (X) → SieveD (Y ).
With a proof similar to that of Lemma 3.3, one can show that β∗ has a right adjoint
β −1 : SieveD (Y ) → SieveD (X).
We refer to β∗ as the pushforward along β functor and to β −1 as the pullback along
β functor.
GENERALIZED RINGED SPACES AND SCHEMES
9
3.2. Covering Sieves.
Definition 3.5. Let C be a category. Choose a subset CovC ⊂ Ob(SieveC ), and
for each presheaf X ∈ Pre(C), let CovC (X) denote the subset of CovC that are
sieves on X. Elements in CovC are called covering sieves, and elements in CovC (X)
are called covering sieves on X.
A functor F : Pre(D) → Pre(C) is said to push forward coverings if, for every
commutative diagram
Pre(D) o
Y
A
idA
F
Pre(C) o
FY
A
the dotted arrow in the diagram of sets
CovD (Y ) _ _ _ _ _ _/ CovC (F Y )
Ob(SieveD (Y ))
/ Ob(SieveC (F Y ))
F∗
exists, and makes the diagram commute (note that because the vertical arrows are
monomorphisms, if the dotted arrow exists then it is unique). The functor F is
said to pull back coverings if, for every commutative diagram as above, the dotted
arrow in the diagram
CovC (F Y ) _ _ _ _ _ _/ CovD (Y )
Ob(SieveC (F Y ))
/ Ob(SieveD (Y ))
F −1
exists, and makes the diagram commute (note that because the vertical arrows are
monomorphisms, if the dotted arrow exists then it is unique).
A correspondence F : C < D is said to push forward coverings (resp. pull back
coverings) if its left Kan extension λr F : Pre(D) → Pre(C) does so. Note that to
say that F pushes forward coverings means that it takes coverings on D to coverings
on C. (One should think of the < as pointing from D to C. See remark 2.3.)
Suppose X, Y : A → Pre(D) are functors and β : X → Y is a natural transformation between them
X
Pre(D) o
A
β
Pre(D) o
Y
A.
Then β is said to push forward (resp. pull back) coverings if β∗ (resp. β −1 ) preserves
coverings as above.
A localizing system on C is a subset CovC ⊂ Ob(SieveC ) satisfying the following
condition:
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DAVID I. SPIVAK
• If P ∈ Pre(C) is a presheaf and α : σ ,→ P is a sieve on P , then α is
a covering sieve if and only if for every object X ∈ C and morphism of
presheaves f : rX → P , the pullback sieve
f −1 α : f −1 σ ,→ rX
is a covering sieve.
Let X ∈ Pre(C) be a presheaf, let i : σ ,→ X and j : τ ,→ X be sieves on X,
and let g : τ → σ be a morphism of sieves on X (note that g is automatically a
monomorphism of presheaves). Define g : τ → σ to be a covering morphism if, when
considered as a sieve on σ, it is a covering sieve (i.e. g ∈ CovC (σ)).
Fix an object X ∈ Pre(C) and a localizing system CovC . One would like to
think of CovC (X) as a category, whose objects are covering sieves on X and whose
morphisms are covering morphisms of sieves on X. However, suppose that i : σ ,→
X is a covering sieve and that g : τ ,→ σ is a covering sieve. It is not guaranteed that
the composition gi : τ ,→ X is a covering sieve. In fact, it is not even guaranteed
that idX : X → X is a covering sieve. To remedy these facts, and the fact that
covering sieves should be akin to epimorphisms, we have the following definition.
Definition 3.6. Let C be a category and B ⊂ C a subcategory. Suppose that for
f
g
every pair of composable morphisms A −
→B−
→ C in C, if the composition gf is in
B, then g is also in B. Then we say that B is an epitype subcategory of C.
Definition 3.7. Let C be a category and let CovC be a localizing system such that,
for each C ∈ C, the identity morphism idC is a covering sieve on C. If, for each
X ∈ Pre(C), the covering sieves and covering morphisms on X form an epitype
subcategory of SieveC (X), then we say that CovC is a Grothendieck Topology on
C. In this case, we let CovC (X) denote the category of covering sieves and covering
morphisms on X, for each X, and we refer to the pair (C, CovC ) as a site.
The above definition is equivalent to the usual one, for example as found in [?,
3.2.4], which we show in the following Lemma.
Lemma 3.8. Let C be a category and let CovC be a localizing system on C. Then
CovC is a Grothendieck topology if and only if the following conditions hold.
(1) For each C ∈ C, the identity sieve C → C is in CovC (C).
(2) If σ is in CovC (C) and f : D → C is a morphism in C, then f −1 σ is in
CovC (D).
(3) Let C ∈ C be an object, τ ∈ SieveC (C) a sieve on C, and σ ∈ CovC (C) a
covering sieve. If for each composition f : D → σ → C, the pullback f −1 τ
is in CovC (D), then τ ∈ CovC (C).
Proof. Axiom 3 is equivalent to axiom
3’: Let C ∈ C be an object, τ ,→ C a sieve on C and σ ,→ C a covering sieve.
If the pullback τ ×C σ ,→ σ is a covering sieve of σ, then τ is a covering
sieve of C.
First, suppose that CovC is a Grothendieck topology in our sense. We prove each
of the three axioms.
(1) By definition.
(2) This follows from the definition of CovC being a localizing system.
GENERALIZED RINGED SPACES AND SCHEMES
11
(3) Consider the diagram
τ ×C D
p
/D
τ ×C σ
p
/σ
τ
/ C.
By assumption, the top map is a covering sieve for any map D → σ.
Hence, by definition, the middle map τ ×C σ → σ is a covering sieve. Since
covering sieves form a category, the composition τ ×C σ → σ → C is a
covering sieve. Finally, since CovC is an epitype subcategory, τ → C must
also be a covering sieve.
Now suppose that the three axioms hold. It suffices to show that for any presheaf
X ∈ Pre(C), the covering sieves on X form an epitype subcategory of SieveC (X).
First we need to show that it forms a subcategory. If f : A → B and g : B → C are
covering sieves, then form the pullback square
f
A
/B
p
A
id
p
f
/B
/B
g
g
/C
Since the top composition is a covering sieve, the bottom one is too by axiom 3.
Finally, to show that it is an epitype subcategory, suppose that the composition
A → B → C is a covering sieve. Form the pullback diagram
/A
A
p
B
/B
p
B
/ C.
Since the vertical composition is a covering sieve, axiom 3’ implies that B → C is
a covering sieve if A → A is.
Definition 3.9. Suppose that (C, CovC ) is a Grothendieck site and F : Pre(C) →
Pre(D) is a functor. Then the F -weak topology on Pre(D) is the weakest Grothendieck
topology under which F∗ preserves covering sieves. Given a functor G : Pre(D) →
Pre(C), the G-strong topology on Pre(D) is the strongest topology under which
G−1 preserves covering sieves.
Definition 3.10. Let (C, CovC ) be a site. A morphism f : P → Q of presheaves
on C is called a covering presheaf on Q if the image sieve f m : im(f ) ,→ Q is a
covering sieve on Q.
12
DAVID I. SPIVAK
Definition 3.11. Let C be a category. Given a morphism f : P → Q of presheaves
on C, let f −1 : (Pre(C) ↓ Q) → (Pre(C) ↓ P ) denote the functor which sends each
object g : R → Q to its pullback
f −1 (g) : P ×Q R → P
along f . That is, define f −1 to be the functor which sends right arrows to left
arrows in the diagram
/R
P ×Q R
p
−1
g
f
(g)
P
/ Q.
f
f
g
Lemma 3.12. Let C be a category, and P −
→ Q ←
− R a diagram of presheaves
on C. The image-sieve functor commutes with pullback in the sense that there is a
natural isomorphism of sieves on P ,
f −1 (g m )) ∼
= f −1 (g)m .
Proof. Consider the diagram
f −1 (g)
P ×Q R
q
q
p
qq
qqq
q
q
xq i
im(π1 ) _ _ _/ P ×Q im(g)
MMM
MMM
p
MM
f −1 (g m )
−1
f (g)m MMM
M& P
f
/R
/ im(g)
gm
/ Q,
in which π1 : P ×Q R → P is the first projection. Since pullbacks preserve monomorphisms, the map f −1 (g m ) : P ×Q im(g) → P is a monomorphism. The dotted arrow
i exists because im(π1 ) → P is the initial monomorphism through which π1 factors.
We need to show that i is an isomorphism. It must be a monomorphism, so it
suffices to show that i is an epimorphism. For C ∈ C, an element of (P ×Q im(g))(C)
consists of a morphism a : C → P for which a dotted arrow exists in the diagram
C _ _ _ _/ R
g
a
P
f
/Q
making it commute. Given such a morphism a, choose a morphism C → R making
the diagram commute, and one gets an element of (P ×Q R)(C), which then factors
through im(π1 )(C), providing the necessary lift.
Lemma 3.13. Let (C, CovC ) be a site, and let g : R → Q be a morphism of
presheaves on C. Then g is a covering presheaf if and only if, for all objects C ∈ C
and maps f : rC → Q, the pullback f −1 (g) : f −1 R → rC is a covering presheaf.
GENERALIZED RINGED SPACES AND SCHEMES
13
Proof. Follows from Lemma 3.12.
Proposition 3.14. Let (C, CovC ) be a site, and let g : R → Q be a morphism of
presheaves on C. Then g is a covering presheaf if and only if it is a generalized
cover in the sense of [?]
Proof. Recall that g : R → Q is a generalized cover in the sense of [?] if it has
the following property: given any C ∈ C and map rC → Q, there is a covering
sieve i : σ → rC such that for every element U → C in σ(U ), the composite
rU → rC → Q lifts through g. No compatibility between the various compositions
rU → R is required.
If g is a covering presheaf, then for any map f : rC → Q, the pullback
f −1 (g) : rC ×Q R → rC
is a covering presheaf by Lemma 3.13, and its image f −1 (g)m : im(f −1 (g)) → rC
is a covering sieve with the above property; i.e. g is a generalized cover.
On the other hand, suppose that g is a generalized cover. Suppose that i : σ → rC
is a covering sieve with the required property. Then for any rU → σ, there is a lift
rU → R, hence a lift rU → rC ×Q R, hence a lift rU → im(π1 ); see the diagram
rC ×Q R
8/ R
p p
{=
p
{
p
{
{
p p
p
{
g
{
p mp6 im(π1 )
{
{ p mp m
{ pm
π1m
{mp mp m
{mp
/σ
/ Q.
/ rC
rU
i
f
The map rU → im(π1 ) is unique since π1m is a monomorphism. Thus there is
an induced morphism σ → im(π1 ) through which i factors. Since (C, CovC ) is a
site, the covering morphisms form an epitype subcategory. Therefore, since i is a
covering sieve, so is π1m ; hence g is a covering presheaf by Lemma 3.13.
3.3. (C, CovC )-spaces. In this section, one should think of (C, CovC ) as the category
of topological spaces with the usual topology. Note that the category of subsheaves
of a representable sheaf rX (where X ∈ Top) is isomorphic to the category Op(X)
of open subsets of X.
Definition 3.15. Let C = (C, CovC ) be a site. Define a C-space to be a site
(Sub(X), CovSub(X) ), where X ∈ C is an object, Sub(X) is the category of subsheaves of rX, and CovSub(X) is the induced topology. Write CovX instead of
CovSub(X) . A morphism f : (Sub(X), CovX ) → (Sub(Y ), CovY ) is a functor Sub(Y ) →
Sub(X) which pushes forward covering sieves. We denote the category of C-spaces
by C − spc.
There is always a faithful functor C → C − spc. If it is an isomorphism of
categories, then we say that C is sober, and we denote C-spaces (Sub(X), CovSub(X) )
simply by X.
14
DAVID I. SPIVAK
Example 3.16. LetC = Sob be the category of sober topological spaces with the
standard (open covers) topology. This is a sober site; that is, the category Sob−spc
is simply the category of sober topological spaces.
3.4. Sheaves. perhaps put this after definition 4.3
Definition 3.17. Let I be a Reedy category, let I B denote the right-coning of I,
and let (C, CovC ) be a site. Suppose given a cofibrantly generated model structure (W, C, F ) on the functor category SetsI . In this case, we will say that
(I, (C, CovC ), (W, C, F )), or just I if the rest is understood, is a ready indexing
category.
In this case, Pre(C)I has an induced model structure in which the weak equivalences are determined objectwise on C. A functor F : I B → Pre(C) is called an
I-hypercover if, for all i ∈ I, the natural map
F (i) → lim
F
←
−
∂(i↓I B )
is a covering presheaf.
Given an I-hypercover F , let ∂F : I → Pre(C) be the restriction of F to I, let
F 0 ∈ Pre(C) be the restriction of F to the cone point, and let dF : hocolim(∂F ) →
F 0 be the induced map. Let
0
dm
F : im(dF ) ,→ F
denote the image-sieve of dF . We call the localization of Pre(C)I at the set
{dm
F |F is an I − hypercover} the Jardine category of I-presheaves on C and denote
it IPre(C).
Example 3.18. The one morphism category I = [0] is a Reedy category and the
trivial model structure on Sets = SetsI is cofibrantly generated. Note that a covering [0]-hypercover is the same as a covering presheaf. Thus, the Jardine category
of [0]-presheaves is simply the ordinary category of sheaves on C.
The simplicial indexing category ∆op is also a Reedy category, and SetsI has a
model structure, the usual model structure on the category of simplicial sets; thus
∆op is a ready indexing category. A covering ∆op -hypercover is similar to a hypercover in the usual sense. The only difference is that, according to [?], a hypercover
must be the coproduct of representables in every degree whereas a ∆op -hypercover
need not be. However, it is easy to see that localizing Pre(C) at the class of hypercovers is the same as localizing it at the class of ∆op -hypercovers. Indeed, every
hypercover is a ∆op -hypercover, and yet all ∆op -hypercovers are acyclic fibrations
(hence weak equivalences) after localizing at the class of hypercovers citation? .
Let C be a category and C ∈ C an object. The Yoneda imbedding r : C → Pre(C)
induces a functor r0 : Pre(C) → Pre(Pre(C)), sending rX to rrX. More explicitly,
if F ∈ Pre(C), then evaluating r0 (F ) on a presheaf G gives
r0 (F )(G) = colim F (X),
rX→G
where the colimit is taken over (r ↓ G). It is sometimes convenient to not mention
r0 explicitly, just as one often does not mention the Yoneda imbedding explicitly.
Definition 3.19. Let (C, CovC ) be a site and I a ready indexing category. An
I-sheaf on C is a cofibrant-fibrant object in IPre(C).
GENERALIZED RINGED SPACES AND SCHEMES
15
Thus if I = [0], then an I-sheaf is just a sheaf, and if I = ∆op then an I-sheaf is
a cofibrant-fibrant simplicial presheaf in the Jardine model structure.
4. R-ringed Categories and R-ringed Spaces
4.1. R-ringed Categories.
Definition 4.1. A category of affines is a triple R = (R, CovR , L), where (R, CovR )
is a Grothendieck site and L is a class of limit cones on R.
Definition 4.2. Let R be a category and let L be a class of limit cones in R. Let
C be a category and let F : C < R be a correspondence. Then F is said to preserve
limit cones in L if, when considered as a functor F : R → Pre(C), each limit cone
in R is sent to a limit cone in Pre(C).
Definition 4.3. Let R = (R, CovR , L) be a category of affines. An R-ringed
category is a triple (C, CovC , F), in which (C, CovC ) is a Grothendieck site, and
F : C < R is a correspondence which preserves limit cones in L. A morphism
(C, CovC , F) → (D, CovD , G)
of R-ringed categories is a pair (H, a), where H : C → Pre(D) is a correspondence
which pushes forward coverings, and a : (H ◦ F) → G is a natural transformation
of correspondences R → Pre(D).
A local R-ringed category is an R-ringed category (C, CovC , F) such that F∗ : SieveR →
SieveC preserves coverings. A morphism of local R-ringed categories is a morphism
(H, a) of R-ringed categories, such that a∗ preserves coverings.
4.2. Examples. We now give several short examples and a few long examples to
help motivate the preceding definitions.
Example 4.4. Fix a category of affines R. The initial object in the category of Rringed categories (and the initial object in the category of local R-ringed categories)
is (R, CovR , idR ).
Example 4.5. The one-morphism category R = [0] has only one possible Grothendieck
topology, and every functor R → Sets is automatically limit-preserving. Thus an
[0]-ringed category can be regarded as just a Grothendieck site (C, CovC ) together
with a presheaf of sets F.
Definition 4.6. Let Aff = Ringsop denote the category of affine schemes, and let
CovAff denote the Zariski topology on Aff . Let LAff denote the class of all limit
cones that exist on Aff . Let Aff = (Aff , CovAff , LAff ), and call it the algebraic
category of affines. An Aff -ringed category is called an algebraic-ringed category
and a local Aff -ringed category is called a local algebraic-ringed category.
Lemma 4.7. Let R denote the category of correspondences F : [0] < Aff for which
the triple ([0], Cov[0] , F) is an algebraic-ringed site. Then R is equivalent to the
category of rings.
Proof. A correspondence F : [0] < Aff is simply a functor F : Aff → Sets. Let
RF = F (Spec(Z[x])). If F preserves all limits in Aff , it is easy to show that the set
RF has the structure of a ring; see [?, ]. On the other hand, every ring A gives rise
to a functor HomAff (Spec(A), −) : Aff → Sets which preserves all limits in Aff .
One can show that these two operations are mutually inverse.
16
DAVID I. SPIVAK
Proposition 4.8. Let X be a topological space and Op(X) its category of open sets;
let CovOp(X) denote the Zariski topology on Op(X). Let RS(X) denote the category
of correspondences F : Op(X) < Aff for which the triple (Op(X), CovOp(X) , F) is
an algebraic-ringed site. Then RS(X) is equivalent to the category of ringed-space
structures on X.
Proof. A correspondence F : Op(X) < Aff is the same thing as a presheaf on
Op(X) with values in the category SetsAff of functors Aff → Sets. The condition
that F preserve all limits in Aff is equivalent to the condition that, for all open
sets U ∈ Op(X), the functor F(U, −) : Aff → Sets preserves limits in Aff ; in other
words, F(U, −) is a ring by Lemma 4.7. This completes the proof.
Proposition 4.9. Let X, Op(X), and CovOp(X) be as in Proposition 4.8. Let
LRS(X) denote the category of correspondences F : Op(X) < Aff for which the
triple (Op(X), CovOp(X) , F) is a local algebraic-ringed site. Then LRS(X) is equivalent to the category of local ringed-space structure on X.
Proof.
Example 4.10. Let E denote the category whose objects are Euclidean spaces Rn and
whose morphisms are smooth maps. Let CovE denote the Grothendieck topology
inherited from the inclusion i : E → Top. Finally, let L denote the set of finiteproduct diagrams in E (e.g. Rn ← Rn+m → Rm is in L). Then E = (E, CovE , L)
is called the C ∞ -category of affines and an E-ring is called a C ∞ -ring. These have
been studied by Lawvere [?], Moerdijk and Reyes [?], and many others.
More generally, for any algebraic theory T , we get a notion of T -ringed space and
local T -ringed space. Recall that a theory T is a category with objects T0 , T1 , . . .
and isomorphisms Tn ∼
= (T1 )n identifying the nth object with the nth power of the
first object. A T -algebra is a product preserving functor from T to Sets. In other
words, if CovT is the indiscrete topology on T and L is the set of product cones,
then the category of T -algebras is equivalent to the category of (T, CovT , L)-ringed
category structures on the terminal category ∗. A presheaf of T -algebras on a site
is a (T, CovT , L)-ringed category structure on that site.
4.3. R-ringed spaces. In the following, one should think of (C, CovC ) as the usual
category of topological spaces.
Definition 4.11. Let R = (R, CovR , L) be a category of affines, let C = (C, CovC )
be a site, and let F : R → C be a functor such that R has the F -strong topology
and such that the functor
F ∗ : Sub(F (R)) → Sub(R)
is an isomorphism of categories for any R ∈ R. These conditions can probably be simplified.
An R-ringed C-space is a 4-tuple (Sub(X), CovX , OX , ν), such that
(1) X ∈ C is an object, X = Sub(X) is the category of subsheaves of the
representable presheaf, iX : Pre(X) → Pre(C) is the (left Kan extension)
functor taking a presheaf of X to its underlying sheaf on C, and CovX :=
CovX is the induced topology on X,
(2) OX : R → Pre(X) is a correspondence under which (X, CovX , OX ) is an
R-ringed category, and
GENERALIZED RINGED SPACES AND SCHEMES
17
(3) ν : iX ◦ OX → F is a natural transformation of functors R → Pre(C).
In other words, an R-ringed C-space is a C-space which is simultaneously an Rringed category over C.
A morphism of R-ringed C-spaces is a morphism of C-spaces which is simultaneously a morphism of R-ringed categories over C.
A local R-ringed C-space is a C-space which is simultaneously a local R-ringed
category over C. A morphism of local R-ringed C-spaces is a morphism of R-ringed
C-spaces which is local as a morphism of R-ringed categories.
Example 4.12. Let (Top, CovTop ) be the Zariski topology on Top and let Aff
be the algebraic category of affines. Let F : Aff → Top be the functor which
takes an affine scheme to its underlying topological space. Note that Aff has
theF -strong topology, so Top underlies Aff . An Aff -ringed (Top, CovTop , F )space is just a ringed (topological) space in the usual sense, and a local Aff -ringed
(Top, CovTop , F )-space is a local ringed space in the usual sense.
In general, if R is a category of affines and a correspondence F : Top < R is
understood, then we refer to a (local) R-ringed (Top, CovTop , F )-space simply as
a (local) R-ringed space.
5. The Structure theorem
Fix a category of affines R = (R, CovR , L), a site (C, CovC ) and a functor
F : R → C such that (C, CovC , F ) underlies R. Let LRS denote the category
of local R-ringed C-spaces; here we just call them local ringed spaces. If X ∈ C is
an object in C, let X = Sub(X) denote the category of subsheaves of X.
Let H : Rop ×R → Sets denote the functor HomR (−, −). For any object R ∈ R,
let HR = Hom(R, −) : R → Sets.
The most fundamental local ringed spaces are those coming from R. For any
R ∈ R, there is an underlying space F (R). Let R = Sub(F (R)), and let HR : Rop ×
R → Sets be the correspondence given by
HR (U, S) = HomPre(R) (U, S).
Theorem 5.1. Let R ∈ R and C ∈ C. Then there is a natural homotopy equivalence
MapLRS (????)
???
F : R → C. what is the natural presheaf on F (R) so that the local-global theorem
will hold?
6. Simplifying assumptions
6.1. Op(X). Replacing Sieve(X) with Op(X) in some cases.
6.2. C ∞ -rings. Show, e.g., that a lot of what can be done with smooth rings can
be done with C ∞ rings.
18
DAVID I. SPIVAK
7. Deleted stuff
7.1. 1.
Proposition 7.1. Let Pre(D) o
L
/ Pre(C) be an adjunction, let A = [0], let
R
X : A → Pre(C) and Y : A → Pre(D) be objects, and let β [ : LX → Y be a natural
transformation. The functors
β∗
SieveD (Y ) o
β −1
/ Sieve (X)
C
are adjoint.
Proof. Let f : σ ,→ Y and g : τ ,→ X be sieves. We need to show that there is a
natural isomorphism
HomSieve (X) (β∗ (f ), g) ∼
= HomSieve (Y ) (f, β −1 (g)).
C
D
We begin by writing out commutative diagrams representing each side.
Consider the diagram
(1)
Lσ
Lf
im(βLf ) 
i τ 
/ LY
β[
/X
g
/ X.
The set I of dotted arrows i making the diagram commute is equal to HomSieveC (X) (β∗ (f ), g).
Consider also the diagram
(2)
σ 
j
−1 
β τ
p

Rτ f
/Y
/Y
β]
Rg
/ RX.
The set J of dotted arrows j making the diagram commute is equal to HomSieveD (Y ) (f, β −1 (g)).
We must show that there is a natural isomorphism I ∼
= J.
The set J of maps j : σ → β −1 τ which make Diagram 4 commute is in one-to-one
correspondence with the set of maps
{j 0 : σ → Rτ |(Rg)j 0 = β ] f }
which in turn is in one-to-one correspondence with the set
S = {i0 : Lσ → τ |gi0 = β [ Lf }.
Since g is a monomorphism, the set S is either empty or consists of a single element.
If it is empty then one easily sees that I is empty; similarly if S consists of a single
element then so does I. Thus I ∼
= J.
GENERALIZED RINGED SPACES AND SCHEMES
19
/ Pre(C) be an adjunction, let X : A → Pre(C)
L
Proposition 7.2. Let Pre(D) o
R
and Y : A → Pre(D) be functors, and let β [ : LX → Y be a natural transformation.
The functors
β∗
SieveD (Y ) o
β −1
/ Sieve (X)
C
are adjoint.
Proof. Let a, b ∈ A be objects and let f : σ ,→ Ya and g : τ ,→ Xb be sieves. We
need to show that there is a natural isomorphism
HomSieveC (X) (β∗ (f ), g) ∼
= HomSieveD (Y ) (f, β −1 (g)).
We begin by writing out commutative diagrams representing each side.
Consider the diagram
(3)
Lf
Lσ
/ LYa
β[
im(β [ Lf )
i 
τ

g
/ Xa
X
k
/ Xb
in SieveC (X). The set I of pairs (i, k) of morphisms making Diagram 3 commute
is equal to HomSieveC (X) (β∗ (f ), g). Consider also the diagram
(4)

σ j
β −1 τ
f

p

Rτ
/ Ya
Y
k
/ Yb
β]
Rg
/ RXb
in SieveD (Y ). The set J of pairs (j, k) of morphisms making Diagram 4 commute is equal to HomSieveD (Y ) (f, β −1 (g)). We must show that there is a natural
isomorphism I ∼
= J.
Fix a morphism k : a → b in A. Since β ] is a natural transformation, the diagram
Ya
β]
Yk
Yb
/ RXa
RXk
β]
/ RXb
20
DAVID I. SPIVAK
commutes. The set Jk of maps j : σ → β −1 τ which make Diagram 4 commute is in
one-to-one correspondence with the set of maps
{j 0 : σ → Rτ |(Rg)j 0 = β ] Yk f } = {j 0 : σ → Rτ |(Rg)j 0 = (RXk )β ] f }
which in turn is in one-to-one correspondence with the set
Sk = {i0 : Lσ → τ |gi0 = Xk β [ (Lf )}.
Since g is a monomorphism, the set Sk is either empty or consists of a single element.
By the universal property of the image functor im, one sees that if it is empty then
one easily sees that Ik is empty; similarly if Sk consists of a single element then so
does Ik . Thus Ik ∼
= Jk and I ∼
= J.
Example 7.3. We will give a concrete example of β∗ and β −1 .
Recall that any correspondence F : C < D gives rise to an adjunction
L
Pre(D) o
/ Pre(C) .
R
For concreteness, let F : D → C be a functor and F = [−, F (−)]C . Also suppose
that A = [0] and that X ∈ C and Y ∈ D are objects. Let β [ : LY → X be a
morphism and let β ] be its right partner.
Let g : σ ,→ Y be a sieve. Then β∗ g is the bottom arrow in the diagram
Lσ
Lg
/ LY
β[
/ X,
I
where I = im(β [ ◦ (Lg)). It is easy to see that for any c ∈ C, one has
I(c) = {h : c → X|∃h0 : c → Lσ such that β [ (Lg)h0 = h}.
That is, β∗ gives the maps to X which factor through Lσ.
Now let k : τ ,→ X be a sieve. The sieve β −1 k is the top arrow in the pullback
diagram
β −1 k
β −1 τ
p
Rτ
/Y
β]
Rk
/ RX.
For any d ∈ D, one has
β −1 τ (d) = {h : d → Y |β ] h ∈ Rτ (d)}.
That is, β −1 gives the maps to Y whose composition with β ] factor through Rτ .
Unfortunately, the variance in Proposition 7.2 is not quite fitting. For example,
recall that a functor F : D → C induces a correspondence F = [−, F (−)]C . However,
given (γ : C → F (D)) ∈ F(C, D) one cannot apply Proposition 7.2 because requires
a map F (D) → C, rather than a map C → F (D). Again, the variance is backwards.
There is a remedy, but it requires mixing left and right adjoints.
Perhaps the following can be generalized to accomidate A not necessarily [0].
GENERALIZED RINGED SPACES AND SCHEMES
21
Construction 7.4. Let C and D be categories. Suppose given a correspondence
F : C < D, objects C ∈ C and D ∈ D, and an element γ ∈ F(C, D). Recall that we
have an adjunction
L:=λr F
/ Pre(C)
Pre(D) o
ρr F =:R
and that L(rD) = F(−, D). We have a natural transformation diagram
o
Pre(D)
O
L
rD
R
Pre(C) o
id
F (−,D)
KS
Pre(C) o
[0]
[0]
γ
rC
[0]
This gives rise to two functors:
(5)
(6)
L
γ −1
γ∗
L−1
∗
SieveD (D) −−→
SieveC (F(−, D)) −−→ SieveC (C)
SieveC (C) −→ SieveC (F(−, D)) −−−→ SieveD (D).
7.2. 2.
Definition 7.5. Let (C, CovC ) and (D, CovD ) be sites, let F : C < D be a correspondence. We say that F is a morphism of sites if it pulls back covering sieves.
Warning 7.6. This definition can be a bit misleading. Suppose X and Y are
topological spaces and F : Op(Y ) → Op(X) is a functor; say F = f −1 is induces
by a continuous map f : X → Y . The functor F is a morphism of sites if covering
sieves on Y are taken to covering sieves on X. While it may seem that F is “pushing
forward” covering sieves, one must remember to phrase everything in the language
of correspondences. The functor F induces a correspondence F : Op(X) < Op(Y )
by
F(U, V ) = HomOp(X) (U, f −1 (V )).
Again, F (and hence F ) is a morphism of sites if it pulls back coverings on Y to
covering on X.
7.3. 3. Grothendieck sites are often given in a certain coherent form.
Definition 7.7. Let C be a category. A subcategory i : D ⊂ C is called a universal
monotype subcategory (or simply unimono) if
(1) B is a monotype subcategory, and
(2) if f : A → B is a map in D and g : B 0 → B is a map in C, then the fiber
product A0 in the diagram
(7)
g0
A0
f0
B0
p
/A
f
g
/B
22
DAVID I. SPIVAK
exists in C, and the map f 0 is in D.
A limit diagram as in 7 is called a base change diagram (of f by g).
Example 7.8. The subcategory SieveC (X) ,→ (Pre(C) ↓ X) is a unimono subcategory.
Let Op ⊂ Top be the subcategory consisting of open inclusions in Top. Then
Op is unimono, because the pullback of an open inclusion under a continuous map
is an open inclusion.
Lemma 7.9. Let i : D ⊂ C be a unimono subcategory. Then D has fiber products
(i.e. each Λ22 diagram A → B ← C in D has a limit in D), and i preserves fiber
products.
f
g
Proof. Let A −
→B ←
− C be a Λ22 diagram in D. Then the fiber product A ×B C
exists in C and, with reference to the diagram
f0
A ×B C
p
0
g
A
/C
g
f
/ B,
both f 0 and g 0 are in D. It suffices to show that A ×B C (which is an object in D)
is a limit in D. Suppose one has an object D and maps G : D → A and F : D → B
in D such that f G = gF . Then one obtains a map ` : D → A ×B C in C such that
g 0 ` = G and f 0 ` = F . Since D ⊂ C is a monotype subcategory, the morphism `
must in fact be in D, which renders A ×B C a limit in D.
Let i : D ,→ C be a unimono subcategory. For every object X ∈ C, one can
consider X as an object in D, and has the category D/X := (D ↓ X) (we use
this notation to make clear that we are not referring to the very different category
(i ↓ X)). For any X ∈ C, let iX : D/X → C be the composition of i with the
evaluation functor D/X → D given by (U → X) 7→ U .
Endow D with the i-strong topology, and endow D/X with the iX -strong topology.
A morphism f : X → Y in C induces a functor in the opposite direction denoted
f −1 : D/Y → D/X by pullback.
Lemma 7.10. Let i : D ,→ C be a unimono subcategory, and let X ∈ C be an object.
A sieve j : σ ,→ U in D/X is a covering sieve if and only if it is a covering sieve in
C.
Proof. Recall that D/X has the strong topology relative to the evaluation D/X → C.
In other words, every covering sieve in C (with a map to X) is a covering sieve in
D/X . To show the converse, we need only show that composition and pullback of
sieves in D/X evaluate to composition and pullback of sieves in C. This is clear.
Definition 7.11. Let C be a category, let CovC be a Grothendieck topology on
C, and let i : D ⊂ C be a unimono subcategory. Given an object X ∈ C and
sieve g : σ ,→ rX, let g 0 : σ 0 → rX be the preimage g 0 = i∗ (g). We say that D
GENERALIZED RINGED SPACES AND SCHEMES
23
canonically generates C if the following condition is satisfied. For every X ∈ C and
sieve g : σ ,→ rX, the preimage sieve g 0 : σ 0 → rX satisfies the property that
• for all objects Y ∈ C, the map of sets
Hom(g 0 , rY ) : Hom(rX, rY ) → Hom(σ 0 , rY )
is an isomorphism.
In this case, we call the functor i : D → C a comprehensible site.
A morphism (F, f ) : i → i0 of comprehensible sites is a square
D
i
C
f
/ D0
i0
F
/ C0
such that
(1) F is a morphism of sites, and
(2) F preserves base change diagrams of morphisms in D.
There are many examples of this idea. I know of no useful sites which are not of
this form. there may be none; look at Giraud’s little theorem. If there are none, reformulate the “topos” item b
Note that any canonically generated topology is subcanonical.
Example 7.12.
(1) Let C be a category. The canonical topology on C is canonically generated by i = idC .
(2) The discrete topology (resp. the indiscrete topology) on C is generated by
the pair (idC , Ob(Pre(C))) (resp. the pair (idC , ∅). The discrete topology
is also canonically generated by taking D to be the discrete category on
Ob(C).
(3) The Zariski topology on Top (or on the category of schemes) is canonically
generated by taking D ⊂ C to be the subcategory of open inclusions.
(4) Any topos T can be represented as sheaves on a site (C, CovC ). Let D = C
and take E = Ob(T ). Then T is isomorphic to the topos of sheaves on
the site generated by (idC , E). Note, however, that T may have other such
representations wherein i : D → C comes in to play.
In this paper, we shall only be interested in subcanonical topologies. The results
should hold more generally, but we restrict to the subcanonical case as a gift to our
weary readers.
Definition 7.13. Let C be a category, i : D ,→ C a unimono subcategory, and
CovC = Covi the topology generated by i. Then an (C, i)-space is a site consisting
of the induced topology CovD/X on D/X for some object X ∈ C. A morphism of
(C, i)-spaces is simply a functor which pushes forward covering sieves. We denote
the category of (C, i)-spaces by (C, i) − spc.
7.4. 4.
Lemma 7.14. Let i : D → C be a unimono subcategory and E ⊂ Ob(Pre(C) a set
of presheaves. The assignment D/− : C op → (C, i) − spc is functorial.
24
DAVID I. SPIVAK
Proof. Let f : Y → X be a morphism in C, let p : U → X be a morphism in D,
and let j : σ ,→ U be a covering sieve in D/X . Since, D/X has the strong topology
relative to the evaluation map D/X → C, the morphism i(j) is a covering sieve in
C.
The morphism f induces a functor F : D/X → D/Y by pullback; it suffices to
show that F∗ preserves covering sieves. The image of j under F∗ is the pullback
F∗ (σ) ,→ V in the all-Cartesian diagram
/V
F∗ (σ)
p
σ
/Y
p
/U
j
f
p
/ X.
Since σ ,→ U is a covering sieve on U , the pullback F∗ (σ) ,→ V is a covering sieve
on V , in C. Finally, since D/Y has the strong topology relative to the evaluation
map D/Y → C, the sieve F∗ (σ) ,→ V is a covering sieve in D/Y as well.
Definition 7.15. Let i : D ⊂ C be a unimono subcategory. We say that C is i-sober
if the functor D/− : C op → (C, i) − spc is fully faithful.
7.5. 5.
Lemma 7.16. Suppose F : A → B is a full functor. Then the induced functor
L = λr (rF ) : Pre(A) → Pre(B) is also full.
Proof. Let P, Q ∈ Pre(A) be presheaves. Recall that for any object A ∈ A, one
has an isomorphism L(A) ∼
= rF A. Therefore we have
HomPre(B) (LP, LQ) = HomPre(B) (colim rF X, colim rF Y )
rX→P
rY →Q
∼
= lim HomPre(B) (rF X, colim rF Y )
rY →Q
rX→P
∼
= lim colim HomPre(B) (rF X, rF Y ).
Because F is full, HomPre(B) (rF X, rF Y ) ∼
= HomPre(A) (rX, rY ). A chain of isomorphisms similar to the one above proves that
lim
colim HomPre(A) (rX, rY ) ∼
= HomPre(A) (P, Q),
rX→P
rY →Q
which in turn completes the proof.
Proposition 7.17. Suppose F : D → C is a full functor inducing an adjunction
Pre(C) o
L
/ Pre(D)
R.
Let D ∈ D be an object and let β = id : LrD → rF D. Then the composition
β −1 β∗ : SieveD (D) → SieveD (D)
is isomorphic to the identity.
GENERALIZED RINGED SPACES AND SCHEMES
25
Proof. Let f : σ ,→ D be an object in SieveD (D). The pushforward of f under β
is f m : im(Lf ) → LD. Let η : D → RLD be the unit morphism. Then the pullback
of f m under β is the top morphism in the diagram
M
β−1β∗ f
p
Rim(Lf )
/D
η
/ RLD.
R(f m )
Since both M and σ are subfunctors of D, we will compare them to each other
instead of comparing the morphisms f and β −1 β∗ f.
For any A ∈ D, we have
M (A) = {g : A → D|ηg ∈ Rim(Lf )(A)}
= {g|Lg ∈ im(Lf )(LA)}
= {g|Lg factors through Lf }.
Since F is full, so is L, by Lemma 7.16. Therefore the last set is {g : A →
D|g factors through f } which is precisely σ(A).