Category Theory - abstractnonsensedotcom.files.wordpress.com · of f. Every map in every category has a deﬁnite domain and codomain. Deﬁnition 1.0.5. A map f: A!Bin a category

Category Theory

2

Category Theory

Travis Dirle

December 12, 2017

2

Contents

1 Categories 1

2 Construction on Categories 7

3 Universals and Limits 11

4 Adjoints 23

5 Limits 31

6 Generators and Projectives 37

7 Categories of Fractions 43

8 Flat Functors and Cauchy Completeness 47

9 Bicategories and Distributors 51

10 Internal Category Theory 57

11 Abelian Categories 61

12 Regular Categoriies 71

13 Algebraic Theories 75

14 Monads and Algebras 81

15 Enriched Category Theory 87

16 Fibred Categories 93

17 Locales 95

18 Sheaves 111

i

CONTENTS

19 Grothendieck Toposes 125

20 The Classifying Topos 131

21 Elementary Toposes 133

22 Internal Logic of a Topos 139

ii

Chapter 1

Categories

Definition 1.0.1. A universe is a set U with the following properties:i) x ∈ y and y ∈ U ⇒ x ∈ U ,ii) I ∈ U and ∀i ∈ I xi ∈ U ⇒ ∪i∈Ixi ∈ U ,iii) x ∈ U ⇒ P(x) ∈ U ,iv) x ∈ U and f : x −→ y surjective function⇒ y ∈ U ,v) N ∈ U ,where N denotes the set of finite ordinals.

Proposition 1.0.2. i) x ∈ U and y ⊂ x⇒ y ∈ U ,ii) x ∈ U and y ∈ U ⇒ x, y ∈ U ,iii) x ∈ U and y ∈ U ⇒ x× y ∈ U ,iv) x ∈ U and y ∈ U ⇒ xy ∈ U .

Definition 1.0.3. A category A consists of:• a collection ob(A ) of objects;• for eachA,B ∈ ob(A ), a collection A (A,B) of maps/arrows/morphisms

from A to B• for each A,B,C ∈ ob(A ), a function

A (B,C)×A (A,B)→ A (A,C)

(g, f) 7→ g f,called composition;• for each A ∈ ob(A ), an element 1A of A (A,A), called the identity on A,

satisfying the following axioms:• associativity: for each f ∈ A (A,B), g ∈ A (B,C) and h ∈ A (C,D),

we have (h g) f = h (g f);• identity laws: for each f ∈ A (A,B), we have f 1A = f = 1B f .

1

CHAPTER 1. CATEGORIES

Definition 1.0.4. If f ∈ A (A,B), we call A the domain and B the codomainof f . Every map in every category has a definite domain and codomain.

Definition 1.0.5. A map f : A→ B in a category A is an isomorphism if thereexists a map g : B → A in A such that gf = 1A and fg = 1B. We call g theinverse of f and write g = f−1 which is unique.

The objects of a category need not be remotely like sets and also, the maps in acategory need not be remotely like functions.

Definition 1.0.6. A category that has no maps, except for the identities, is calleda discrete category.

Definition 1.0.7. Let A and B be categories. A functor F : A → B consistsof:• a function

ob(A )→ ob(B),

written as A 7→ F (A);• for each A,A′ ∈ A , a function

A (A,A′)→ B(F (A), F (A′)),

written as f 7→ F (f),satisfying the following axioms:

• F (f ′ f) = F (f ′) F (f) whenever Af−→ A′

f ′−→ A′′;• F (1A) = 1F (A) whenever A ∈ A .

Definition 1.0.8. A functor F : A → B is faithful (respectively, full) if foreach A,A′ ∈ A , the function

A (A,A′)→ B(F (A), F (A′))

f → F (f)

is injective (respectively, surjective).

2


Note the roles of A and A′ in the definition. Faithfulness does not say that if f1

and f2 are distinct maps in A then F (f1) 6= F (f2). F is faithful if for eachA,A′

and g : F (A) → F (A′), there is at most one map from A to A′ that F sends tog. It is full if for each such A,A′ and g, there is at least one map from A to A′

that F sends to g.

Definition 1.0.9. Let A be a category. A subcategory I of A consists of asubclass ob(I ) of ob(A ) together with, for each S, S ′ ∈ ob(I ), a subclassI (S, S ′) of A (S, S ′), such that I is closed under composition and identities.It is a full subcategory if I (S, S ′) = A (S, S ′) for all S, S ′ ∈ ob(I ).

A full category therefore consists of a selection of the objects, with all of themaps between them. So a full subcategory can be specified simply by sayingwhat its objects are. Whenever I is a subcategory of A , there is an inclusionfunctor. It is automatically faithful, and it is full iff I is a full subcategory. Notethat the image of a functor need not be a subcategory.

Definition 1.0.10. Let A and B be categories and let F,G : A → B be func-tors. A natural transformation α : F → G is a family

(F (A)

αA−→ G(A))A∈A

of maps in B such that for every map Af−→ A′ in A , the square

F (A)F (f)−−−→ F (A′)

αA

y yαA′

G(A) −−−→G(f)

G(A′)

commutes. The maps αA are called the components of α.

Definition 1.0.11. For any two categories A and B, there is a category whoseobjects are the functors from A to B and whose maps are the natural transfor-mations between them. This is called the functor category from A to B, andwritten as [A ,B] or BA .

Definition 1.0.12. A natural isomorphism between functors from A to B is anisomorphism in [A ,B].

3


Lemma 1.0.13. Let α : F → G be a natural transformation between functorsF,G : A → B. Then α is a natural isomorphism if and only if αA : F (A) →G(A) is an isomorphism for all A ∈ A .

Definition 1.0.14. Given functors F,G : A → B, we say that they are natu-rally isomorphic if there exists a natural isomorphism from F to G. Also,

F (A) ∼= G(A) naturally in A

if F and G are naturally isomorphic.

This terminology can be understood as follows. If F (A) ∼= G(A) naturally inA then certainly F (A) ∼= G(A) for each individual A, but more is true: we canchoose isomorphisms αA : F (A) → G(A) in such a way that the naturality ax-iom is satisfied, i.e. the above commutative diagram. There are many examplesof categories and functors such that F (A) ∼= G(A) for all A ∈ A , but not natu-rally in A.

Definition 1.0.15. An equivalence between categories A and B consists of apair F : A → B and G : B → A of functors together with a pair of naturalisomorphisms

η : 1A → G F ε : F G→ 1B.

If there exists an equivalence between A and B, we say that A and B areequivalent, and write A ' B. We also say that the functors F and G areequivalences.

Definition 1.0.16. A functor F : A → B is essentially surjective on objects iffor all B ∈ B, there exists A ∈ A such that F (A) ∼= B.

Proposition 1.0.17. A functor is an equivalence if and only if it is full, faithfuland essentially surjective on objects.

4


Corollary 1.0.18. Let F : C → D be a full and faithful functor. Then C isequivalent to the full subcategory C ′ of D whose objects are those of the formF (C) for some C ∈ C .

Definition 1.0.19. A category A is small if the collection of all maps in A issmall, i.e. if it is a set, otherwise it is called large. We call A locally small iffor each A,B ∈ A , the collection A (A,B) is small. So small implies locallysmall. A category is small if and only if it is locally small and its class of objectsis small. It is essentially small if it is equivalent to some small category.

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6

Chapter 2

Construction on Categories

Definition 2.0.1. Every category A has an opposite/dual category A op, de-fined by reversing the arrows. Formally, ob(A op) = ob(A ) and A op(B,A) =A (A,B) for all objects A and B. Identities in A op are the same as in A .Composition in A op is the same as in A , but with the arguments reversed.

Definition 2.0.2. Given categories A and B, there is a product category A ×B, in which

ob(A ×B) = ob(A )× ob(B),

(A ×B)((A,B), (A′, B′)) = A (A,A′)×B(B,B′).

Put another way, an object is a pair (A,B) and a map (A,B) → (A′, B′) is apair (f, g) where f : A→ A′ in A and g : B → B′ in B.

Definition 2.0.3. Let A and B be categories. A contravariant functor from Ato B is a functor A op → B. An ordinary functor A → B is sometimes calleda covariant functor from A to B, for emphasis.

Functors C → D correspond one-to-one with functors C op → Dop, and (A op)op =A , so a contravariant functor from A to B can also be described as a functorA → Bop.

Definition 2.0.4. An equivalence of the form A op ' B is sometimes called aduality between A and B. One says that they are dual.

7

CHAPTER 2. CONSTRUCTION ON CATEGORIES

Definition 2.0.5. Given categories and functors

B

Q

AP// C

the comma category written as (P ⇒ Q) or (P ↓ Q), is the category definedas follows:• objects are triples (A, h,B) with A ∈ A , B ∈ B, and h : P (A)→ Q(B)

in C ;• maps (A, h,B)→ (A′, h′, B′) are pairs (f : A→ A′, g : B → B′) of maps

such that the square

P (A)P (f)

//

h

P (A′)

h′

Q(B)Q(g)

// Q(B′)

commutes.

Definition 2.0.6. The slice category of A over A, denoted by A /A, is the cate-gory whose objects are maps into A and whose maps are commutative triangles.More precisely, an object is a pair (X, h) with X ∈ A and h : X → A in A ,and a map (X, h) → (X ′, h′) in A /A is a map f : X → X ′ in A making thetriangle

Xf//

h

X ′

h′

A

commute. Slice categories are a special case of comma categories. Dually, thereis a coslice category A/A ∼= (A⇒ 1A ), whose objects are the maps out of A.

Definition 2.0.7. A congruence on a category A is an equivalence relation onarrows such that

i) f ∼ g implies dom(f) = dom(g) and cod(f) = cod(g)ii) f ∼ g implies bfa ∼ bga for all arrows a : A → X and b : Y → B,

where dom(f) = X = dom(g) and cod(f) = Y = cod(g).

8


Definition 2.0.8. Let ∼ be a congruence on the category A , and define thecongruence category A ∼ by: ob(A ∼) = ob(A ) and arrows is the set <f, g >: f ∼ g with 1A =< 1A, 1A > and < f ′, g′ > < f, g >=< f ′f, g′g >.

Definition 2.0.9. We define the quotient category A / ∼ as follows: ob(A / ∼) = ob(A ) and arrows are (HomA )/ ∼.

Definition 2.0.10. Suppose we have a functor F : A → B, then F determinesa congruence ∼F on A by setting

f ∼F g iff dom(f) = dom(g), cod(f) = cod(g), and F (f) = F (g).

We write ker(F ) = A ∼F for this congruence category and call it the kernelcategory of F .

Definition 2.0.11. A group in a category C consists of objects and arrows thatsatisfy the same commutative diagrams that typical groups satisfy. That is, forall (generalized) elements, x, y, z : Z → G, the following equations hold:

m(m(x, y), z) = m(x,m(y, z))

m(x, u) = x = m(u, x)

m(x, ix) = u = m(ix, x)

Definition 2.0.12. A homomorphism h : G → H of groups in C consists ofan arrow in C that preserves structure and whose properties can be shown incommutative diagrams.

With these identities and composites, we thus have a category of groups in C ,denoted by Group(C ). For example, a group in the usual sense is a group in thecategory Sets. Also, the groups in Group are exactly the abelian groups.

Definition 2.0.13. Enlarging Group to include also categories with more thatone object, but still having inverses for all arrows, gives us a category calledgroupoids.

Definition 2.0.14. A strict monoidal category is a category C equipped with abinary operation ⊗ : C × C → C which is funtorial and associative,

A⊗ (B ⊗ C) = (A⊗B)⊗ C,

together with a distinguished object I that acts as a unit,

I ⊗ C = C = C ⊗ I.

A strict monoidal category is exactly the same thing as a monoid in Cat.

9


Definition 2.0.15. A monoidal category consists of a categor C equipped witha functor ⊗ : C × C → C and a distinguished object I , together with naturalisomorphisms

αABC : A⊗ (B ⊗ C)→ (A⊗B)× C,λA : I ⊗ A→ A, ρA : A⊗ I → A.

These satisfy commutative diagrams as well. A monoidal category is thus acategory that is strict monoidal “up to natural isomorphism”.

10

Chapter 3

Universals and Limits

Definition 3.0.1. Let A be a category. A presheaf on A is a functor A op →Set.

Definition 3.0.2. Let A be a category. An object I ∈ A is initial if for everyA ∈ A , there is exactly one map I → A. An object T ∈ A is terminal if forevery A ∈ A , there is exactly one map A→ T .

A category need not have an initial object, but if it does, it is unique up to iso-morphism.

Lemma 3.0.3. Let I and I ′ be initial objects in a category. Then there is aunique isomorphism I → I ′. In particular, I ∼= I ′.

Definition 3.0.4. Let A be a locally small category and A ∈ A . we define afunctor

HA = A (A,−) : A → Set

as follows:• for objects B ∈ A , put HA(B) = A (A,B);• for maps B

g−→ B′ in A , define

HA(g) = A (A, g) : A (A,B)→ A (A,B′)

byp 7→ g p

for all p : A→ B. Sometimes HA(g) is written as g − or g∗.

11

CHAPTER 3. UNIVERSALS AND LIMITS

Definition 3.0.5. Let A be a locally small category. A functorX : A → Set isrepresentable if X ∼= HA for some A ∈ A . A representation of X is a choiceof an object A and an isomorphism between HA and X .

Definition 3.0.6. Let A be a locally small category. The functor

H• : A op → [A ,Set]

is defined on objects A by H•(A) = HA and on maps f by H•(f) = Hf .

More precisely, a map A′f−→ A induces a natural transformation Hf : HA →

HA′ whose B-component is the function

HA(B) = A (A,B)→ HA′(B) = A (A′, B)

p 7→ p f

Definition 3.0.7. Let A be a locally small category and A ∈ A . We define afunctor

HA = A (−, A) : A op → Set

as follows:• for objects B ∈ A , put HA(B) = A (B,A);• for maps B′

g−→ B in A , define

HA(g) = A (g, A) = g∗ = − g : A (B,A)→ A (B′, A)

byp 7→ p g

for all p : B → A.

Definition 3.0.8. Let A be a locally small category. A functor X : A op → Setis representable if X ∼= HA for some A ∈ A . A representation of X is achoice of object A ∈ A and an isomorphism between HA and X .

12


Definition 3.0.9. Let A be a locally small category. The Yoneda embedding ofA is the functor

H• : A → [A op,Set ]

defined on objects A by H•(A) = HA and on maps f by H•(f) = Hf .

More precisely, a map Af−→ A′ induces a natural transformation Hf : HA →

HA′ , whose B-component is

HA(B) = A (B,A)→ HA′(B) = A (B,A′)

p 7→ f p

As a summary:

For each A ∈ A , we have a functor AHA

−−→ Set.Putting them all together gives a functor A op H•−→ [A ,Set ].

For each A ∈ A , we have a functor A op HA−−→ Set.Putting them all together gives a functor A

H•−→ [A op,Set ].

The second pair of functors is the dual of the first.

Definition 3.0.10. Let A be a locally small category. The functor

HomA : A op ×A → Set

is defined as follows: HomA (A,B) = A (A,B) and (HomA (f, g))(p) = g p f , whenever A′

f−→ Ap−→ B

g−→ B′.

We see that HomA carries the same information as H• (or H•), presentedslightly differently.

Definition 3.0.11. Let A be an object of a category. A generalized element ofA is a map with codomain A. A map S → A is a generalized element of A ofshape S.

Theorem 3.0.12. (Yoneda) Let A be a locally small category. Then

[A op,Set ](HA, X) ∼= X(A)

naturally in A ∈ A and X ∈ [A op,Set ].

13


Informally, then, the Yoneda lemma says that for any A ∈ A and presheaf X onA :

A natural transformation HA → X is an element of X(A).

Corollary 3.0.13. Let A be a locally small category and X : A op → Set. Thena representation of X consists of an object A ∈ A together with an elementu ∈ X(A) such that:

for each B ∈ A and x ∈ X(B), there is a unique map x : B → A such that(Xx)(u) = x.

Recall that by definition, a representation of X is an object A ∈ A togetherwith a natural isomorphism α : HA → X . The above corollary states that suchpairs (A,α) are in natural bijection with pairs (A, u) satisfying the last condi-tion. Pairs (B, x) are sometimes called elements of the presheaf X . The Yonedalemma tells us that x amounts to a generalized element of X of shape HB. Anelement u satisfying the above condition, is sometimes called a universal ele-ment of X . So the corollary says that a representation of a presheaf X amountsto a universal element of X .

Corollary 3.0.14. Let A be a locally small category and X : A → Set. Thena representation of X consists of an object A ∈ A together with an elementu ∈ X(A) such that:

for each B ∈ A and x ∈ X(B), there is a unique map x : A→ B such that(Xx)(u) = x.

Corollary 3.0.15. For any locally small category A , the Yoneda embedding

H• : A → [A op,Set ]

is full and faithful.

Informally, this says that for A,A′ ∈ A , a map HA → HA′ of presheaves is thesame thing as a map A→ A′ in A .

Lemma 3.0.16. Let J : A → B be a full and faithful functor and A,A′ ∈ A .Then

i) a map f in A is an isomorphism if and only if the map J(f) in B is anisomorphism;

14


ii) for any isomorphism g : J(A) → J(A′) in B, there is a unique isomor-phism f : A→ A′ in A such that J(f) = g;

iii) the objects A and A′ of A are isomorphic if and only if the objects J(A)and J(A′) of B are isomorphic.

Corollary 3.0.17. Let A be a locally small category and A,A′ ∈ A . Then

HA∼= HA′ ⇐⇒ A ∼= A′ ⇐⇒ HA ∼= HA′ .

Definition 3.0.18. Let A be a category and X, Y ∈ A . A product of X and Yconsists of an object P and maps

Xp1←− P

p2−→ Y

with the property that for all objects and maps

Xf1←− A

f2−→ Y

in A , there exists a unique map f : A→ P such that

Af1

~~

f

f2

X Pp1oo

p2// Y

commutes. The maps p1 and p2 are called the projections.

Definition 3.0.19. Let A be a category, I a set, and (Xi)i∈I a family of objectsof A . A product of (Xi)i∈I consists of an object P and a family of maps(

Ppi−→ Xi

)i∈I

with the property that for all objects A and families of maps(A

fi−→ Xi

)i∈I

there exists a unique map f : A→ P such that pi f = fi for all i ∈ I . We callthe maps fi the components of the map (fi)i∈I .

15


Some examples of products (each in a different category) are minx, y forx, y ∈ (R,≤), X ∩ Y for X, Y ∈P(S), or gcd(x, y) in the poset (N, |).

A fork in a category consists of objects and maps

f : A→ s, t : X → Y

such that sf = tf

Definition 3.0.20. Let A be a category and let s, t : X → Y be objects andmaps in A . An equalizer of s and t is an object E together with a map E i−→ Xsuch that

i : E → s, t : X → Y

is a fork, and with the property that for any fork, there exists a unique mapf : A→ E such that

A

f

f

Ei// X

commutes.

An equalizer describes the set of solutions of a single equation, but by combin-ing equalizers with products, we can also describe the solution set of any systemof simultaneous equations.

Definition 3.0.21. Let A be a category, and take objects and maps

Y

t

X s// Z

in A . A pullback of this diagram is an object P ∈ A together with mapsp1 : P → X and p2 : P → Y such that

Pp2//

p1

Y

t

X s// Z

commutes, and with the property that for any commutative square

Af2//

f1

Y

t

X s// Z

16


in A , there is a unique map f : A → P such that the resulting diagram com-mutes. Namely, p1f = f1 and p2f = f2. The first square above is called thepullback square

Definition 3.0.22. Let A be a category and I a small category. A functor I→ Ais called a diagram in A of shape I.

Definition 3.0.23. Let A be a category, I a small category, and D : I→ A adiagram in A .

i) A cone on D is an object A ∈ A (the vertex of the cone) together with afamily (

AfI−→ D(I)

)I∈I

of maps in A such that for all maps I u−→ J in I, the triangle

D(I)

Du

A

fI==

fJ// D(J)

commutes.

ii) A limit of D is a cone(L

PI−→ D(I))I∈I

with the property that for any

cone on D, there exists a unique map f : A → L such that pI f = fI for allI ∈ I. The maps pI are called the projections of the limit.

In general, the limit of a diagram D is the terminal object in the category ofcones on D, and is therefore an extremal example of a cone on D. The word’limit’ can be understood as meaning ’on the boundary’.

Definition 3.0.24. i) Let I be a small category. A category A has limits ofshape I if for every diagram D of shape I in A , a limit of D exists.

ii) A category has all limits (or properly, has small limits) if it has limits ofshape I for all small categories I.

Definition 3.0.25. A finite limit is a limit of shape I for some finite category I.For instance, binary products, terminal objects, equalizers and pullbacks are allfinite limits.

17


Proposition 3.0.26. Let A be a category.i) If A has all products and equalizers then A has all limits.ii) If A has binary products, a terminal object and equalizers then A has

finite limits.

Definition 3.0.27. Let A be a category. A map Xf−→ Y in A is monic (or a

monomorphism if for all objects A and maps x, x′ : A→ X ,

f x = f x′ ⇒ x = x′

In Set, a map is monic if and only if it is injective. In categories of algebrassuch as Grp, Vectk, Ring, etc, it is also true that the monic maps are exactly theinjections.

Lemma 3.0.28. A map f : X → Y is monic if and only if the square

X 1 //

1

X

f

Xf// Y

is a pullback.

Definition 3.0.29. Let A be a category and I a small category. Let D : I → Abe a diagram in A , and write Dop for the corresponding functor I op → A op. Acocone on D is a cone on Dop, and a colimit of D is a limit of Dop.

Explicitly, a cocone onD is an objectA ∈ A (the vertex of the cocone) togetherwith a family (

D(I)fI−→ A

)I∈I

of maps in A such that for all maps I u−→ J in I, the diagram

D(I)fI //

Du

A

D(J)

fJ

==

18


commutes. A colimit of D is a cocone(D(I)

pI−→ C)I∈I

with the property that for any cocone on D, there is a unique map f : C → Asuch that f pI = fI for all I ∈ I. We write (the vertex of) the colimit aslim→I D, and call the maps pI coprojections.

Definition 3.0.30. A sum/coproduct is a colimit over a discrete category. (Thatis, it is a colimit of shape I for some discrete category I.)

Definition 3.0.31. A coequalizer is a colimit of shape E (shape of a fork/equalizer).

A coequalizer is a generalization of a quotient by an equivalence relation.

Definition 3.0.32. A pushout of a diagram

Xs //

t

Y

Z

is (if it exists) a commutative square

X

t

s // Y

Z // •

that is universal as such. In other words still, a pushout in a category A is apullback in A op. It is a colimit.

Definition 3.0.33. Let A be a category. A mapXf−→ Y in A is epic/epimorphism

if for all objects Z and maps g, g′ : Y → Z,

g f = g′ f ⇒ g = g′.

An epic in A is a monic in A op.

In categories of algebras, any surjective map is certainly epic. In some cate-gories, the coverse holds as well. However, there are examples where this fails,

19


like in Ring, the inclusion Z → Q is epic but not surjective. This is also anexample of a map that is monic and epic but not an isomorphism.

We have that any isomorphism in any category is both monic and epic.

Definition 3.0.34. A split mono (epi) is an arrow with a left (right) inverse.Given arrows e : X → A and s : A → X such that es = 1A, the arrow s iscalled a section/splitting of e, and the arrow e is called a retraction of s. Theobject A is called a retract of X .

The condition that ’every epimorphism splits’ or ’every surjection has a section’is the categorical version of the axiom of choice.

Definition 3.0.35. An object P is said to be projective if for any epi e : E → Xand arrow f : P → X there is some (not necessarily unique) arrow f : P → Esuch that e f = f . One says that f lifts across e i.e. the diagram commutes:

E

e

P

f>>

f// X

Definition 3.0.36.

i) Let I be a small category. A functor F : A → B preserves limits ofshape I if for all diagrams D : I → A and all cones

(A

pI−→ D(I))I∈I

on D,(A

pI−→ D(I))I∈I

is a limit cone on D in A

⇒(F (A)

FpI−−→ FD(I))I∈I

is a limit cone on F D in B.

ii) A functor F : A → B preserves limits if it preserves limits of shape I forall small categories I.

iii) Reflection of limits is defined as in i), but with⇐ in place of⇒.

Definition 3.0.37. A functor F : A → B creates limits (of shape I) if wheneverD : I → A is a diagram in A ,

i) for any limit cone(B

qI−→ FD(I))I∈I

on the diagram F D, there is a

unique cone(A

pI−→ D(I))I∈I

on D such that F (A) = B and F (pI) = qI for

all I ∈ I;

20


ii) this cone(A

pI−→ D(I))I∈I

is a limit cone on D.

Lemma 3.0.38. Let F : A → B be a functor and I a small category. Supposethat B has, and F creates, limits of shape I. Then A has, and F preserves,limits of shape I.

Since Set has all limits, it follows that all our categories of algebras have alllimits, and that the forgetful functors preserve them.

Definition 3.0.39. Consider a functor G : D → A with colimit

(L, (pD)D∈D).

That colimit is absolute when for every functor F : A → B,

(FL, (FpD)D∈D)

is the colimit of F G.

Definition 3.0.40. A functor G : C → D is final when the following conditionsare satisfied for every category A and every functor F : D → A :

i) if the limit (L, (pD)D∈D) of F exists, then (L, (pGC)C∈C ) is the limit ofF G;

ii) if the limit (L, (qC)C∈C ) of F G exists, then the limit of F exists as well.

21


22

Chapter 4

Adjoints

Definition 4.0.1. Let F : A → B be a functor and B an object of B. Areflection of B along F is a pair (RB, ηB) where

i) RB is an object of A and ηB : B → F (RB) is a morphism of B,ii) if A ∈ |A | and b : B → F (A) is a morphism of B, there exists a unique

morphism a : RB → A in A such that F (a) ηB = b.

Proposition 4.0.2. Let F : A → B be a functor and B an object of B. Whenthe reflection of B along F exists, it is unique up to isomorphism.

Proposition 4.0.3. Consider a functor F : A → B and assume that, for everyB ∈ B, ’the’ reflection of B along F exists and such a reflection (RB, ηB) hasbeen choosen. In that case, there exists a unique functor R : B → A satisfyingthe two properties

i) ∀B ∈ B R(B) = RB,ii) (ηB : B → FRB)B∈B is a natural transformation.

Definition 4.0.4. A functor R : B → A is left adjoint to the functor F : A →B when there exists a natural transformation η : 1B ⇒ F R such that for everyB ∈ B, (RB, ηB) is a reflection of B along F . In an analogous way a functorR : B → A is right adjoint to F when there exists a natural transformationε : F R⇒ 1B such that for eachB ∈ B, (RB, εB) is a coreflection ofB alongF , where coreflection is the dual notion of reflection.

23

CHAPTER 4. ADJOINTS

Definition 4.0.5. Let F : A → B and G : B → A be functors. We say that Fis left adjoint to G, and G is right adjoint to F , and write F a G, if

B(F (A), B) ∼= A (A,G(B))

naturally in A ∈ A and B ∈ B. An adjunction between F and G is a choiceof natural isomorphism.

’Natually in A and B’ means that there is a specified bijection for each A and B,and that it satisifies a naturality axiom. To state it, we need some notation. Givenobjects A ∈ A and B ∈ B, the correspondence between maps F (A)→ B andA→ G(B) is denoted by a horizontal bar, in both directions:(

F (A)g−→ B

)7→(A

g−→ G(B)),(

F (A)f−→ B

)←[(A

f−→ G(B)).

So ¯f = f and ¯g = g. We call f the transpose of f , and similarly for g. Thenaturality axiom has two parts:(

F (A)g−→ B

q−→ B′)

=(A

g−→ G(B)G(q)−−→ G(B′)

)(that is, q g = G(q) g) for all g and q, and(

A′p−→ A

f−→ G(B))

=

(F (A′)

F (p)−−→ F (A)f−→ B

)for all p and f .

The concept of left and right adjoint are dual to each other. Adjunctions canbe composed as well.

Definition 4.0.6. For each A ∈ A , we have a map(A

ηA−→ GF (A))

=(F (A)

1−→ F (A)).

Dually, for each B ∈ B, we have a map(FG(B)

εB−→ B)

=(G(B)

1−→ G(B)).

These define natural transformations

η : 1A → G F, ε : F G→ 1B.

called the unit and counit of the adjunction, respectively.

24

CHAPTER 4. ADJOINTS

Definition 4.0.7. Given an adjunction F a G with unit η and counit ε, thetriangles

FFη//

1F""

FGF

εF

F

GηG//

1G""

GFG

Gε

G

commute. These are called the triangle identities. They are commutative dia-grams in the functor categories [A ,B] and [B,A ], respectively.

The unit and counit determines the whole adjunction, even though they appearto know only the transpose of identities.

Lemma 4.0.8. Let F a G be an adjunction, with unit η and counit ε. Then

g = G(g) ηA

for any g : F (A)→ B, and

f = εB F (f)

for any f : A→ G(B).

Theorem 4.0.9. Take functors F : A → B and G : B → A . There is aone-to-one correspondence between:

i) adjunctions between F and G (with F on the left and G on the right);ii) pairs

(1A

η−→ GF, FGε−→ 1B

)of natural transformations satisfying the

triangle identities.

Recall that by definition, an adjunction between F and G is a choice of iso-morphism for each A and B, satisfying the naturality equations.

Corollary 4.0.10. We have that F a G if and only if there exist natural trans-formations 1

η−→ GF and FG ε−→ 1 satisfying the triangle identities.

Lemma 4.0.11. Take and adjunction F a G and an object A ∈ A . Then theunit map ηA : A→ GF (A) is an initial object of (A⇒ G).

25

CHAPTER 4. ADJOINTS

Theorem 4.0.12. Take categories and functors F : A → B and G : B → A .There is a one-to-one correspondence between:

i) adjunctions between F and G (with F on the left and G on the right);ii) natural transformations η : 1A → GF such that ηA : A → GF (A) is

initial in (A⇒ G) for every A ∈ A .

Corollary 4.0.13. Let G : B → A be a functor. Then G has a left adjoint ifand only if for each A ∈ A , the category (A⇒ G) has an initial object.

Lemma 4.0.14. Adjunctions give rise to representable functors in the followingway. Let F : A → B and G : B → A with F a G between locally smallcategories. Then the functor

A (A,G(−)) : B → Set

(that is, the composite BG−→ A

HA

−−→Set) is representable.

Proposition 4.0.15. Any set-valued functor with a left adjoint is representable.

Definition 4.0.16. Consider two functors F : A → B and G : A → C . Theleft Kan extension of G along F , if it exists, is a pair (K,α) where• K : B → C is a functor,• α : G⇒ K F is a natural transformation,

satisfying the following universal property: if (H, β) is another pair with• H : B → C a functor,• β : G⇒ H F a natural transformation,

there exists a unique natural transformation γ : K ⇒ H satisfying the equaltiy(γ ∗ F ) α = β.

We shall use the notation LanFG to denote the left Kan extension of G alongF . The notation RanFG is used for the dual notion of right Kan extension. Wewrite (γ ∗ F ) instead of γ ∗ 1F .

26

CHAPTER 4. ADJOINTS

Theorem 4.0.17. Consider two functors F : A → B and G : A → C , withA small and C cocomplete. Under these conditions, the left Kan extension of Galong F exists.

Proposition 4.0.18. Consider a full and faithful functor F : A → B with A asmall category. Let C be a cocomplete category. Given a functor G : A → C ,the canonical natural transformation G⇒ (LanFG) F is an isomorphism.

Proposition 4.0.19. Consider a functor G : A → C , with A a small category.Write 1 for the category with a single object and a single arrow, and F : A → 1for the corresponding functor. The functor G has a colimit if and only if the leftKan extension LanFG of G along F exists.

Proposition 4.0.20. Consider a functor F : A → B between small categories.The following are equivalent:

i) F has a right adjoint G;ii) LanF1A exists and, for every functor L : A → C , the isomorphism

L LanF1A∼= LanFL holds;

iii) LanF1A exists and the isomorphism F LanF1A∼= LanFF holds.

Definition 4.0.21. A functor F : A → B satisfies the solution set conditionwith respect to an object B ∈ B when there exists a set SB ⊂ |A | of objectssuch that ∀A ∈ A ∀b : B → FA ∃A′ ∈ SB ∃a : A′ → A ∃b′ : B → FA′

F (a) b′ = b

Theorem 4.0.22. (Adjoint functor theorem) Consider a complete category Aand a functor F : A → B. The following are equivalent:

i) F has a left adjoint functor.ii) The following conditions hold:a) F preserves small limits;b) F satisfies the solution set condition for every object B ∈ B.

27

CHAPTER 4. ADJOINTS

Proposition 4.0.23. (Adjoint functor theorem for ordered sets) Let A be an or-dered set, B a complete ordered set, and G : B → A an order-preserving map.Then

G has a left adjoint ⇔ G preserves meets

Theorem 4.0.24. (Special adjoint functor theorem) Consider a functor F :A → B and suppose the following conditions are satisfied:

i) A is complete;ii) F preserves small limits;iii) A is well-powered;iv) A has a cogenerating family.Under these conditions, F has a left adjoint functor.

Definition 4.0.25. Let C be a category. A weakly initial set in C is a set S ofobjects with the property that for each C ∈ C , there exist an element S ∈ S anda map S → C.

Theorem 4.0.26. (General Adjoint Functor Theorem) Let A be a category, B acomplete category, and G : B → A a functor. Suppose that B is locally smalland that for each A ∈ A , the category (A⇒ G) has a weakly initial set. Then

G has a left adjoint ⇔ G preserves limits.

Definition 4.0.27. A full subcategory A of a category B is replete when, withevery A ∈ A , A also contains every object B ∈ B isomorphic to A .

Definition 4.0.28. A reflective subcategory of a category B is a full repletesubcategory A of B whose inclusion i : A → B in B admits a left adjointr : B → A , called the reflection.

Definition 4.0.29. A localization of a category B with finite limits is a reflectivesubcategory A of B whose reflection preserves finite limits.

28

CHAPTER 4. ADJOINTS

Definition 4.0.30. An essential localization of a category B is a reflective sub-category A of B whose reflection itself admits a left adjoint.

29

CHAPTER 4. ADJOINTS

30

Chapter 5

Limits

Definition 5.0.1. A category C is filtered wheni) ∃C ∈ Cii) ∀C1, C2 ∈ C ∃C3 ∈ C ∃f : C1 → C3 ∃g : C2 → C3,iii) ∀C1, C2 ∈ C ∀f, g : C1 → C2 ∃C3 ∈ C ∃h : C2 → C3 h f = h g.

By a filtered colimit we mean the colimit of a functor defined on a filtered cat-egory. We say that a category A has filtered colimits when for every smallfiltered category C and every functor F : C → A , the colimit of F exists.

Theorem 5.0.2. Consider a small filtered category C and a finite category D .Given a functor F : C × D → Set to the category of sets and mappings, thefollowing mixed interchange property holds:

colimC∈D( limD∈D

F (C,D)) ∼= limD∈D

(colimC∈DF (C,D)).

Definition 5.0.3. Given categories I and A and an object A ∈ A , there is afunctor ∆A : I→ A with constant value A on objects and 1A on maps. Thisdefines, for each I and A , the diagonal functor

∆ : A → [I ,A ]

Now, given a diagram D : I→ A and an object A ∈ A , a cone on D with vertexA is simply a natural transformation ∆A → D, and we write Cone(A,D) forthe set of cones on D with vertex A, we therefore have

Cone(A,D) = [I ,A ](∆A,D).

31

CHAPTER 5. LIMITS

Proposition 5.0.4. Let I be a small category, and D : I→ A a diagram. Thenthere is a one-to-one correspondence between limit cones on D and representa-tions of the functor

Cone(−, D) : A op → Setwith the representing objects of Cone(−, D) being the limit objects (that is, thevertices of the limit cones) of D.

Briefly put: a limit of D is a representation of [I ,A ](∆−, D). The propositionformalizes the thought that cones on a diagram D correspond one-to-one withmaps into lim←I D. It implies that if D has a limit then

Cone(A,D) ∼= A(A, lim←I

D)

naturally in A.

Corollary 5.0.5. Limits are unique up to isomorphism.

Proposition 5.0.6. Let I be a small category and A a category with all limitsof shape I. Then lim←I defines a functor [I ,A ] → A , and this functor is rightadjoint to the diagonal functor.

This functor is defined as such: choose for each D ∈ [I,A ] a limit cone D, andcall its vertex lim←I D. For each map α : D → D′, we have a canonical maplim←I α : limD → limD′. Thus

[I,A ](∆A,D) = Cone(A,D) ∼= A (A, limD)

naturally in A and D.

Lemma 5.0.7. Let I be a small category, A a locally small category, D : I →A a diagram, and A ∈ A . Then

Cone(A,D) ∼= lim←I

A (A,D)

naturally in A and D.

Here, A (A,D) is the functor

A (A,D) : I→ Set

32

CHAPTER 5. LIMITS

I 7→ A (A,D(I))

Proposition 5.0.8. (Representables preserve limits) Let A be a locally smallcategory and A ∈ A . Then A (A,−) : A → Set preserves limits.

This proposition tells us that

A(A, lim←I

D)∼= lim←I

A (A,D).

To dualize this, a limit in A op is a limit in A , so A (−, A) transforms colimitsin A into limits in Set:

A(

lim→I

D,A)∼= lim←I

A (D,A).

Note the right hand side is a limit, not a colimit.

Definition 5.0.9. Let A and I be categories. For each A ∈ A, there is a functor

evA : [A ,I ]→ I

X 7→ X(A),

called evaluation at A.

Theorem 5.0.10. (Limits in functor categories) Let A and I be small categoriesand I a locally small category. Let D :I→ [ A ,I ] be a diagram, and supposethat for each A ∈ A, the diagram D(−)(A) : I → I has a limit. Then there isa cone on D whose image under evA is a limit cone on D(−)(A) for each A ∈A. Moreover, any such cone on D is a limit cone.

Limits in a functor category are computed pointwise (meaning the objects ofA). For example, given two functors X, Y ∈ [A,I ], their product can be com-puted by first taking the product X(A) × Y (A) in I for each ’point’ A, thenassembling them to form a functor X × Y .

The pointwise character of this construction is precisely expressed by theformula

( limD∈D

F (D))(C) = limD∈D

(F (D)(C)).

In other words, the value of the limit limD∈D F (D) at an object C is the limit ofthe values of F (D) at C.

33

CHAPTER 5. LIMITS

Theorem 5.0.11. Consider a complete category A and a small category C .Under these conditions, the category Fun(C ,A ) is complete and limits in it arecomputed pointwise.

Theorem 5.0.12. Consider a small category C and a functor F from C to Set.In the category Fun(C , Set), F can be presented as the colimit of a diagram justconstituted of representable functors and representable natural transformations.

Corollary 5.0.13. Let I and A be small categories, and I a locally small cate-gory. If I has all limits (respectively, colimits) of shape I then so does [ A ,I ],and for A ∈ A, the evaluation functor evA : [A ,I ]→ I preserves them.

Take categories I,J and I . There are isomorphisms of categories

[I, [J,I ]] ∼= [I× J,I ] ∼= [J, [I,I ]].

Under these isomorphisms, a functor D :I×J→ I corresponds to the functorsD• mapping I 7→ D(I,−) and D• mapping J 7→ D(−, J).

Proposition 5.0.14. (Limits commute with limits) Let I and J be small cate-gories. Let I be a locally small category with limits of shape I and shape J.Then for all D : I × J → I , we have

lim←J

lim←I

D• ∼= lim←I×J

D ∼= lim←I

lim←J

D•,

and all these limits exist. In particular, I has limits of shape I × J.

Corollary 5.0.15. Let A be a small category. Then [A op,Set ] has all limits andcolimits, and for each A ∈ A, the evaluation functor evA : [A op,Set ] → Setpreserves them.

Corollary 5.0.16. The Yoneda embedding preserves limits, for any small cate-gory.

34

CHAPTER 5. LIMITS

Definition 5.0.17. Let A be a category and X a presheaf on A. The category ofelements E(X) of X is the category in which:

i) objects are pairs (A, x) with A ∈ A and x ∈ X(A);ii) maps (A′, x′) → (A, x) are maps f : A′ → A in A such that (Xf)(x) =

x′.

There is a projection functor P : E(X) → A defined by P (A, x) = A andP (f) = f .

Theorem 5.0.18. (Density) Let A be a small category and X a presheaf on A.Then X is the colimit of the diagram

E(X)P−→ A H•−→ [Aop,Set]

that is, X ∼= lim→I(H• P ).

Theorem 5.0.19. Let F a G be an adjunction. Then F preserves colimits andG preserves limits.

The previous theorem is often used to prove that a functor does not have an ad-joint.

Definition 5.0.20. A category is complete (or properly, small complete) if ithas all limits.

Theorem 5.0.21. A category C is complete precisely when each family of objectshas a projuct and each pair of parallel arrows has an equalizer.

Proposition 5.0.22. For a category C , the following conditions are equivalent:i) C is finitely complete;ii) C has a terminal object, binary products and equalizers;iii) C has a terminal object and pullbacks.

35

CHAPTER 5. LIMITS

Definition 5.0.23. A category D is finitely generated wheni) D has finitely many objects,ii) there are finitely many arrows f1, . . . , fn such that each arrow of D is the

composite of finitely many of these fi.

Proposition 5.0.24. Let F : D → A be a functor, with A finitely complete andD finitely generated. Then the limit of F exists.

Definition 5.0.25. A category A is cartesian closed if it has finite products andfor each B ∈ A , the functor −×B : A → A has a right adjoint. We write theright adjoint as (−)B, and, for C ∈ A , call CB an exponential.

Theorem 5.0.26. For any small category A, the presheaf category is cartesianclosed.

36

Chapter 6

Generators and Projectives

Definition 6.0.1. Consider a category A and an object A ∈ A . Two monomor-phisms f : R A and g : S A are equivalent when there exists anisomorphism r : R→ S such that g r = f . An equivalence class of monomor-phisms with codomain A is called a subobject of A. The dual notion is that of a“quotient of A”.

Definition 6.0.2. A category A is well-powered when the subobjects of everyobject constitute a set.

In Set, the subobjects of a set X are in bijection with the subsets of X . In Gr,they are in bijection with subgroups.

Given an object A of a category C , let us consider the class Mono(A) ofall monomorphisms with codomain A. A monomorphism r : R A issmaller than a monomorphism s : S A when there exists a (mono)morphismt : R S such that s t = r. Performing the quotient on Mono(A) which iden-tifies isomorphic monomorphisms, we obtain a partial order on the class Sub(A)of subobjects of A. We recall that C is well-powered when, for each A ∈ C ,Sub(A) is a set.

Definition 6.0.3. Consider an object A of a category C . By the intersection ofa family of subobjects of A, we mean their infimum in Sub(A). By the union ofa family of subobjects of A, we mean their supremum in Sub(A).

37

CHAPTER 6. GENERATORS AND PROJECTIVES

Proposition 6.0.4. Consider an object A ∈ C and suppose Sub(A) is a set. Thefollowing are equivalent:

i) the intersection of every family of subobjects of A exists;ii) the union of every family of subobjects of A exists.

Definition 6.0.5. In a category, an epimorphism is called regular when it is thecoequalizer of a pair of arrows.

Definition 6.0.6. An epimorphism f : A → B in a category is called extremalwhen it does not factor through any proper subobject of B; i.e., given f = i pwith i a monomorphism, i is necessarily an isomorphism.

Definition 6.0.7. In a category A , an epimorphism f : A → B is called astrong epimorphism when, for every commutative square z u = v f , withz : X → Y a monomorphism, there exists a (unique) arrow w : B → X suchthat w f = u, z w = v.

Proposition 6.0.8. In a category A ,i) the composite of two strong epimorphisms is a strong epimorphism,ii) if a composite f g is a strong epi, f is a strong epi,iii) a morphism which is both a mono and a strong epi, is an isomorphism,iv) every regular epi is strong,v) every strong epi is extremal.

Proposition 6.0.9. Let F : A → B be a functor admitting a left adjoint functorG : B → A . The functor F preserves strong monomorphisms, and the functorG preserves strong epimorphisms and regular epimorphisms.

Definition 6.0.10. A category C is finitely well-complete wheni) C is finitely complete,ii) given an object C ∈ C , the intersection of an arbitrary class of subobjects

of C always exists.

38


Proposition 6.0.11. In a finitely well-complete category, every morphism f fac-tors as f = i p, where i is a monomorphism and p is a strong epimorphism.

Definition 6.0.12. A category C has strong-epi-mono-factorizations when ev-ery morphism f of C factors as f = i p, with p a strong epimorphism and i amonomorphism. The monomorphism i is also called the image of f .

Definition 6.0.13. Let C be a category. A family (Gi)i∈I of objects of C is calleda family of generators when, given any two parallel morphisms u, v : A → Bin C ,

∀i ∈ I∀g : Gi −→ A u g = v g ⇒ u = v.

Generators are important because of the following property: every object can berecaptured as a “quotient of a coproduct of generators”.

Proposition 6.0.14. Let C be a category with coproducts and (Gi)i∈I a familyof objects of C . The following are equivalent:

i) (Gi)i∈I is a family of generators;ii) for every object C ∈ C , the unique morphism

γC :∐

i∈I,f∈C (Gi,C)

(domain of f) −→ C

such that γC sf = f is an epimorphism.

Definition 6.0.15. Let C be a category with coproducts and (Gi)i∈I a family ofobjects of C . The family (Gi)i∈I is a strong family of generators when, forevery object C ∈ C , the morphism γC is a strong epimorphism. The family is aregular family of generators when, for every object C ∈ C , the morphism γCis a regular epimorphism. When the family is reduced to a single element G,we say that G is a strong or a regular generator, according to the case.

39


Definition 6.0.16. Let C be a category and (Gi)i∈I a family of objects of C . Letus write G for the full subcategory of C generated by the Gi’s and G /C for thefull subcategory of C /C generated by the objects of the form f : Gi → C. Thefamily is a dense family of generators when for every object C ∈ C , the colimitof the functor

ΓC : G /C −→ G , (f : Gi −→ C) 7→ Gi,

is precisely (C, (f)f∈G /C). When the family is reduced to a single element G,G is called a dense generator.

Proposition 6.0.17. In a category with coproducts, every dense family of gener-ators is regular and every regular family of generators is strong.

Definition 6.0.18. i) A family of functors (Fi : A → Bi)i∈I is collectivelyfaithful when given morphisms f, g : A→ A′ in A

(∀i ∈ I Fi(f) = Fi(g))⇒ (f = g).

ii) A family of functors (Fi : A → Bi)i∈I collectively reflects isomorphismswhen, given a morphism f : A→ A′ in A ,

(∀i ∈ I Fi(f) is an isomorphism )⇒ (f is an isomorphism ).

Definition 6.0.19. Let C be a category (with finite limits). A family (Gi)i∈Iof objects of C is a strong family of generators when the family of functorsC (Gi,−) : C → Set collectively reflects isomorphisms. When the family isreduced to a single object G, G is called a strong generator.

Definition 6.0.20. An object P of a category C is projective when, given astrong epimorphism p : X → Y and a morphism f : P → Y , there exists afactorization g : P → X such that p g = f .

Proposition 6.0.21. For an object P of a category C , the following conditionare equivalent:

i) P is projective;ii) the functor C (P,−) : C → Set preserves epimorphisms.

40


Definition 6.0.22. A category C has enough projectives when every object is astrong quotient of a projective object.

41


42

Chapter 7

Categories of Fractions

A graph is, roughly speaking, a “category without a composition law”.

Definition 7.0.1. A graph G consists ofi) a class |G | whose elements are called the objects (or vertices) of the graph.ii) for each pair (A,B) ∈ |G |×|G |, a set G (A,B) whose elements are called

the morphisms (or arrows) from A to B.The graph G is small when |G | itself is a set.

Definition 7.0.2. A morphism of graphs F : F → G between two graphsconsists of

i) a mapping F : |F | → |G |,ii) for each pair (A,B) ∈ F ×F of objects, a mapping

F (A,B) −→ G (FA, FB).

Obviously, every category is a graph (just forget composition).

Definition 7.0.3. Let G be a graph. A path in G is a nonempty finite sequence(A1, f1, A2, f2, . . . , An) alternating objects and arrows in G ; each arrow fi hasdomain Ai and codomain Ai+1.

Definition 7.0.4. Let G be a graph. A commutativity condition on G is a pairof paths both defined from some given object A to some given object B.

We now formally add some inverse arrows of a given category.

43

CHAPTER 7. CATEGORIES OF FRACTIONS

Definition 7.0.5. Consider a category C and a class Σ of arrows of C . Thecategory of fractions C [Σ−1] is said to exist when a category C [Σ−1] and afunctor φ : C → C [Σ−1] can be found, with the following properties:

i) ∀f ∈ Σ φ(f) is an isomorphism;ii) if D is a category and F : C → D is a functor such that for all morphisms

f ∈ Σ, F (f) is an isomorphism, there exists a unique functor G : C [Σ−1]→ Dsuch that G φ = F .

Proposition 7.0.6. Consider a category C and a set Σ of arrows of C . Thecategory of fractions C [Σ−1] exists. Moreover when C is small, C [Σ−1] is smallas well.

Definition 7.0.7. Consider a category C and a class Σ of morphisms of C . Theclass Σ admits a right calculus of fractions when the following holds:

i) ∀C ∈ C 1C ∈ Σ;ii) given s : A→ B and t : B → C, (s ∈ Σ and t ∈ Σ)⇒ (t s ∈ Σ);iii) if f : A→ B is in C and s : C → B is in Σ, there exist g : D → C in C

and t : D → A in Σ such that f t = s g;iv) if f, g : A → B are in C and s : B → C is in Σ with the property

s f = s g, there exists t : D → A in Σ with the property f t = g t.

Definition 7.0.8. Let C be a category and Σ ⊂ C a class of morphisms suchthat the category of fractions φ : C → C [Σ−1] exists. The class Σ is saturatedwhen for every morphism f ∈ C

φ(f) is an isomorphism iff f ∈ Σ.

Definition 7.0.9. Consider two arrows f : A → B, g : C → D in a categoryC . We say that f is orthogonal to g and write f⊥g when, given arbitrarymorphisms u, v such that v f = g u there exists a unique morphism w suchthat w f = u, g w = v.

An epimorphism f is strong when, for every monomorphism g, f⊥g.

44


Definition 7.0.10. Given an arrow f : A → B and objects X, Y of a categoryC :

i) we say that f is orthogonal toX and write f⊥X when for every morphisma : A→ X , there exists a unique morphism b : B → X such that b f = a;

ii) we say that Y is orthogonal to f and write Y⊥f when for every morphismc : Y → B there exists a unique morphism d : Y → A such that f d = c.

Definition 7.0.11. Let C be a category and Σ a class of morphisms of C . By theorthogonal subcategory of C determined by Σ, we mean the full subcategoryCΣ of C whose objects are those X ∈ C such that f⊥X for every f ∈ Σ.

Theorem 7.0.12. Let C be a cocomplete category in which every object is pre-sentable. Given a set Σ of morphisms of C , the corresponding orthogonal sub-category CΣ is reflective in C .

Definition 7.0.13. Let C be a cocomplete category and E a class of morphismsof C . The class E is closed under colimits when given a small category D ,two functors F,G : D → C and a natural transformation α : F ⇒ G, if allthe morphisms αD : FD → GD are in E , then the corresponding factorizationcolimαD : colimFD → colimGD is in E as well.

Definition 7.0.14. By a factorization system on a category B we mean a pair(E ,M) where both E andM are classes of morphisms of B and

i) every isomorphism belongs to both E andM,ii) both E andM are closed under composition,iii) ∀e ∈ E ∀m ∈M e⊥m,iv) every morphism f ∈ B can be factored as f = m e, with e ∈ E and

m ∈M.

Definition 7.0.15. Consider a finitely complete category B. A univeral closureoperation on B consists in giving, for every subobject S B in B, anothersubobject S B called the closure of S in B; these assignments have tosatisfy the following properties, where S, T are subobjects of B and f : A→ Bis a morphism of B;

45


i) S ⊂ S;ii) S ⊂ T ⇒ S ⊂ T ;iii) S = S;iv) f−1(S) = f−1(S).

Definition 7.0.16. Consider a finitely complete category B provided with a uni-versal closure operation.

i) A subobject S B is dense when S = B;ii) a subobject S B is closed when S = S.

Definition 7.0.17. Consider a finitely complete category B with strong-epi-mono factorizations. Given a universal closure operation on B, a morphismf : A → B is bidense when its image is dense and the equalizer of its kernelpair is dense.

46

Chapter 8

Flat Functors and CauchyCompleteness

Definition 8.0.1. Consider two finitely complete categories A ,B. A functorF : A → B is left exact when it preserves finite limits.

Theorem 8.0.2. Let A be a small category. The category Lex(A , Set) of leftexact functors is reflective in the category Fun(A , Set) of all functors.

Definition 8.0.3. For an arbitrary category A , a functor F : A → Set is flatwhen the category Elts(F ) of elements of F is cofiltered. Given an arbitraryfunctor F : A → B, F is flat when for each object B ∈ B, the functorB(B,F−) : A → Set is flat.

Proposition 8.0.4. Given a category A , every representable functor

A (A,−) : A → Set

is flat.

Proposition 8.0.5. Let F : A → B be a functor with a left adjoint. Then F isflat.

47

CHAPTER 8. FLAT FUNCTORS AND CAUCHY COMPLETENESS

Definition 8.0.6. An infinite cardinal α is regular when it satisfies

(#I < α and ∀i ∈ I #Xi < α)⇒ # (∪i∈IXi) < α

where I,Xi are arbitrary sets.

Definition 8.0.7. Let α be a regular cardinal. A category C is α-filtered wheni) there exists at least one object in C ,ii) given a set I with #I < α and a family (Ci ∈ C )i∈I of objects of C , there

exist an object C ∈ C and morphisms fi : Ci → C in C ,iii) given a set I with #I < α and a family (fi : C → C ′)i∈I in C , there exist

an object C ′′ ∈ C and a morphism f : C ′ → C ′′ such that f fi = f fj , forall indices i, j.

Definition 8.0.8. Let α be a regular cardinal.i) By an α-filtered colimit in a category C , we mean the colimit of a functor

F : D → C where the category D is α-filtered.ii) By an α-limit in a category C , we mean the limit of a functor F : D → C

where D is a small category and #Ar(D) < α, where Ar(D) indicates the setof arrows of D .

In general, we shall write #D < α to indicate that the small category D has aset Ar(D) of arrows of card less than α.

Definition 8.0.9. Let α be a regular cardinal. Consider two α-complete cate-gories A ,B. A functor F : A → B is α-left-exact when it preserves α-limits.

Definition 8.0.10. Let α be a regular cardinal.i) A functor F : A → Set is α-flat when its category of elements is α-

cofiltered.ii) A functor F : A → B is α-flat when, for every B ∈ B, the functor

B(B,F−) : A → Set is α-flat.

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Definition 8.0.11. A functor F : A → B is absolutely flat when it is α-flat forevery regular cardinal α.

Definition 8.0.12. In a category C , a morphism e : C → C is idempotent whene e = e.

Definition 8.0.13. In a category C an idempotent e : C → C splits when thereexists a retract r, i : R→ C of C such that i r = e.

Definition 8.0.14. A category C is Cauchy complete when all idempotents ofC split.

49


50

Chapter 9

Bicategories and Distributors

In the case of the category A =Cat of small categories, the set A (A,B) offunctors can be given the structure of a category, with natural transformations asarrows. With that example in mind, we make the following definition.

Definition 9.0.1. A 2-category A consists ofi) a class |A |,ii) for each pair A,B of elements of |A |, a small category A (A,B),iii) for each triple A,B,C of elements of |A |, a bifunctor

cABC : A (A,B)×A (B,C) −→ A (A,C),

iv) for each element A ∈ |A |, a functor

uA : 1 −→ A (A,A),

where 1 is the terminal object of the category of small categories.

These data are required to satisfy the following axioms.i) Associativity axiom: given four elements A,B,C,D ∈ A , the following

equality holds:

cACD (cABC × 1) = cABD (1× cBCD)

ii) Unit axiom: given two elementsA,B ∈ |A |, the following equalities hold:

cAAB (uA × 1) ∼= 1 ∼= cABB (1× uB)

We will now clarify some terminology. Given a 2-category A :The elements of the set |A | are called 0-cells or objects;The objects of the category A (A,B) are called 1-cells or arrows;The arrows of the category A (A,B) are called 2-cells;

51

CHAPTER 9. BICATEGORIES AND DISTRIBUTORS

Now, βα denotes composition in the category A (A,B), or just βα. Whilel f denotes the image of the pair (f, l) of arrows under the composition functorcABC , or just lf . Also, φ∗α denotes the image of the pair (α, φ) of 2-cells underthe composition functor cABC . 1A : A → A denotes the image of the uniqueobject of 1 under the unit functor uA; we write iA instead of i1A to denote theunit on the arrow 1A in the category A (A,A).

Definition 9.0.2. In a 2-category A , consider two arrows f : A → C g : A →B. The Kan extension of f along g, when it exists, is a pair (h, α) where

i) h : B → C is an arrow and α : f ⇒ h g is a 2-cell,ii) given any pair (k, β) with k : B → C an arrow and β : f ⇒ k g a 2-cel,

there exists a unique 2-cell γ : h⇒ k such that

(γ ∗ ig) α = β.

While in an ordinary category most diagrams in which we are interested arecommutative, very often in a 2-category one considers non-commutative dia-grams of arrows “filled in” with 2-cells.

Definition 9.0.3. Given two 2-categories A ,B, a 2-functor F : A → Bconsists in giving

i) for each object A ∈ A , an object FA ∈ B,ii) for each pair of objects A,A′ ∈ A , a functor

FA,A′ : A (A,A′) −→ B(FA, FA′).

We often write F instead of FA,A′ . These data are required to satisfy the follow-ing axioms.

i) Compatibility with composition: given three objects A,A′, A′′ ∈ A , thefollowing equality holds:

FA,A′′ cAA′A′′ = cFA,FA′,FA′′ (FAA′ × FA′A′′)

ii) Unit: for every object A ∈ A , the following equality holds:

FAA uA = uFA

Definition 9.0.4. Consider two 2-categories A ,B and two 2-functors betweenthem F,G : A → B. A 2-natural transformation θ : F ⇒ G consists ingiving, for each object A ∈ A , an arrow θA : FA→ GA such that the equality

B(1FA, θA′) FAA′ = B(θA, 1GA′) GAA′

holds for each pair of objects A,A′ ∈ A .

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Proposition 9.0.5. Small 2-categories, 2-functors and 2-natural transformationsthemselves constitute a 2-category.

Definition 9.0.6. Consider 2-categories A ,B, 2-functors F,G : A → B and2-natural transformations α, β : F ⇒ G. A modification

Ξ : α β

consists in giving, for every object A ∈ A , a 2-cell

ΞA : αA ⇒ βA,

in such a way that the following axiom is satisfied: for every pair of morphismsf, g : A→ A′ and every 2-cell α : f ⇒ g in A , the equality

ΞA′ ∗ Fα = Gα ∗ ΞA

holds in B.

Definition 9.0.7. A 3-category consists of the following data:i) a class |A |;ii) for each pair A,B of elements of |A |, a small 2-category A (A,B);iii) for each triple A,B,C of elements of |A |, a 2-functor

cABC : A (A,B)×A (B,C) −→ A (A,C);

iv) for each element A of |A | a 2-functor

uA : 1 −→ A (A,A)

where 1 is the terminal 2-category.These data following the usual associativity and unit axioms.

Proposition 9.0.8. There exists a 3-category structure on the following data;• Objects: the 2-categories.• Arrows: the 2-functors• 2-cells: the 2-natural transformations• 3-cells: the modifications

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Clearly one could now define 3-functors, 3-natural trans and 3-mods. And thefact of having 3-cells in the 3-categories will allow the definition of “morphismsof 3-modifications”. The 3-cats, 3-functors, 3-nat trans, 3-mods and morphismsof 3-mods will now organize themselves in what is called a 4-category, whosedefinition can be easily adopted from previous definitions.

The process can be iterated, yielding the notion of an n-category, for n ∈ N,n > 0; those n-categories organize themselves in an (n+1)-category. One couldeven define a 0-category as being a set and 0-functor as being a mapping; apply-ing the previous process yields the notions of 1-category and 1-functor, whichare just the usual notions of category and functor.

Given a 2-functor F : A → B and an object B ∈ B, we shall write 2-Cone(B,F ) to denote the category whose objects are the 2-natural transforma-tions ∆B ⇒ F (the “2-cones on F with vertex B”) and whose morphisms arethe modifications between them.

Definition 9.0.9. The 2-limit of F , if it exists, is a pair (L, π) where L ∈ B isan object and π : ∆L ⇒ F is a 2-natural transformation such that the functor

B(B,L) −→ 2− Cone(B,F )

of composition with π is an isomorphism of categories, for each object B ∈ B.

Definition 9.0.10. The bilimit of a 2-functor F : A → B, if it exists, is a pair(L, π) where L ∈ B is an object and π : ∆L ⇒ F is a 2-natural transformationsuch that the functor

B(B,L) −→ 2− cone(B,F )

of composition with π is an equivalence of categories, for each object B ∈ B.

Definition 9.0.11. A bicategory A is specified by the following data:i) a class |A | of ’objects’ (also called 0-cells);ii) for each pair A,B of objects, a small category A (A,B) where objects

are called ’arrows’ (or morphisms or 1-cells) and whose morphisms are called’2-cells’; we write α β for the composite of the 2-cells α, β.

iii) for each triple A,B,C of objects, a composition law given by a functor

cABC : A (A,B)×A (B,C) −→ A (A,C);

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given arrows f : A → B, g : B → C of the bicategory A , we write g f fortheir composite cABC(f, g); given other arrows f ′ : A → B, g′ : B → C and2-cells γ : f ⇒ g′, we write δ ∗ γ for their composite cABC(γ, δ);

iv) for each object A ∈ A , an ’identity arrow’ 1A : A → A; we write iA forthe identity 2-cell on 1A.

The associativity and identity axioms are now replaced by the existence of someisomorphisms, which is part of the data:

i) Associativity isomorphisms: for each quadruple of objects, a natural iso-morphism

αABCD : cACD (cABC × 1)⇒ cABD (1× cBCD)

ii) Unit isomorphisms: for each pair of objects, two natural isomorphisms

λAB : 1⇒ cAAB (iA × 1), ρAB : 1⇒ cABB (1× iB)

These various data are required to satisfy coherence conditions expressed by thefollowing axioms.

i) Associativity coherence: coming from a diagram.ii) Identity coherence: coming from a diagram.

Roughly speaking, a distributer is to a functor what a relation is to a mapping.

Definition 9.0.12. By a distributor/profunctor/bimodule from a category Ato a category B, we mean a bifunctor

φ : B∗ ×A −→ Set.

55


56

Chapter 10

Internal Category Theory

Definition 10.0.1. Let C be a category with pullbacks. By an internal categoryA in C we mean

i) an object A0 ∈ |C |, called the “object of objects”,ii) an object A1 ∈ |C |, called the “object of arrows,iii) two morphisms d0, d1 : A1 → A0 in C , called respectively “source” and

“target”,iv) an arrow i : A0 → A1 in C called “identity”,v) an arrow c : A1 ×A0 A1 → A1 in C , called “composition”, where the

pullback (A1 ×A0 A1, π1, π0) is that of d0, d1

These data must satisfy the following axioms:i) d0 i = 1A0 = d1 i;ii) d1 π1 = d1 c, d0 π0 = d0 c;

iii) c (

1A1

i d0

)= 1A1 = c

(i d1

1A1

);

iv) c (1A1 ×A0 c) = c (c×A0 1A1).

Definition 10.0.2. Let C be a category with pullbacks. Given two internal cate-gories A ,B, and internal functor F : A → B is a pair of morphisms

F0 : A0 −→ B0, F1 : A1 −→ B1

which satisfies the following conditions:i) d0 F1 = F0 d0, d1 F1 = F0 d1;ii) F1 i = i F0;iii) F1 c = c (F1 ×F0 F1).

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CHAPTER 10. INTERNAL CATEGORY THEORY

Definition 10.0.3. Let C be a category with pullbacks. Given two internal cat-egories A ,B and two internal functors F,G : A → B, an internal naturaltransformation α : F ⇒ G is a morphism α : A0 → B1 which satisfies thefollowing conditions:

i) d0 α = F0, d1 α = G0;ii) c (α d1, F ) = c (G,α d0).

Proposition 10.0.4. Let C be a category with pullbacks. For every object C ∈C , the representable functor

C (C,−) : C −→ Set

maps internal categories, internal functors and internal natural transformationsrespectively.

Definition 10.0.5. Let C be a category with pullbacks and let A be an internalcategory. By an internal A -valued functor P : A → C we mean

Definition 10.0.6. A functor F : A → B satisfies the solution set conditionwith respect to an object B ∈ B when there exists a set SB ⊂ |A | of objectssuch that ∀A ∈ A ∀b : B → FA ∃A′ ∈ SB ∃a : A′ → A ∃b′ : B → FA′

F (a) b′ = b

i) an object P ∈ |C | together with a morphism p0 : P → A0 of C ,ii) an arrow p1 : A1×A0 P −→ P where (A1×A0 P, πA1 , πP ) is the pullback

of d0, p0.

These data are requred to satisfy the following:i) p0 p1 = d1 πA1;ii) p1 (i p0, 1P ) = 1P ;iii) p1 (1A1 ×A0 p1) = p1 (c×A0 1P ).

Definition 10.0.7. Let C be a category with pullbacks, A an internal categoryand P,Q : A → C two internal C -valued functors, written explicitly as P =

58


(P, p0, p1),Q = (Q, q0, q1). By an internal natural transformation α : P ⇒ Qwe mean an arrow α : P → Q such that

i) q0 α = p0;ii) α p1 = q1 (1A1 ×A0 α).

Definition 10.0.8. Let C be a finitely complete category, A an internal categoryin C and P : A → C a C -valued internal functor. With the previous notation,

by an internal limit of P , we mean a coreflection of P along the functor ∆A .by an internal colimit of P , we mean a reflection of P along the functor ∆A .

C is said to be internally (co)complete when the internal (co)limit exists forevery A and every P .

Definition 10.0.9. Let C be a finitely complete category.-By an internal product in C we mean the internal limit of a C -valued in-

ternal functor P : A → C , where A is a discrete internal category.-Let X,A be objects of C . By the internal power XA ∈ C we mean, if it

exists, the internal limit of the constant C -valued functor on X defined on thediscrete internal category on A.

59


60

Chapter 11

Abelian Categories

Definition 11.0.1. By a zero object in a category C , we mean an object 0 whichis both an initial and a terminal object.

Definition 11.0.2. Consider a category C with a zero object 0. A morphismf : A → B is called a zero morphism when it factors through the zero object0.

Proposition 11.0.3. In a category C with a zero object 0, there is exactly onezero morphism from each object A to each object B.

Proposition 11.0.4. In a category C with a zero object 0, the composite of azero morphism with an arbitrary morphism is again a zero morphism.

Definition 11.0.5. In a category C with a zero object 0, the kernel of an arrowf : A→ B is -when it exists- the equalizer of f and the zero morphism 0: A→B. The cokernel of f is defined dually.

Proposition 11.0.6. Let f be a monomorphism in a category with a zero object.If f g = 0 for some morphism g, then g = 0.

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CHAPTER 11. ABELIAN CATEGORIES

Proposition 11.0.7. In a category with a zero object, the kernel of a monomor-phism f : A→ B is just the zero arrow 0→ A.

Proposition 11.0.8. In a category with a zero object, the kernel of a zero mor-phism 0 : A→ B is just, up to isomorphism, the identity on A.

Definition 11.0.9. By a preadditive category we mean a category C togetherwith an abelian group structure on each set C (A,B) of morphisms, in such away that the composition mappings

cABC : C (A,B)× C (B,C) −→ C (A,C), (f, g) 7→ g f

are group homomorphisms in each variable. We shall write the group structureadditively.

Proposition 11.0.10. In a preadditive category C , the following are equivalent:i) C has an initial object;ii) C has a terminal object;iii) C has a zero object.In that case, the morphisms factoring through the zero object are exactly the

identities for the group structure.

Proposition 11.0.11. Given two objects A,B in a preadditve category C , thefollowing are equivalent:

i) the product (P, pA, pB) of A,B exists;ii) the coproduct (P, sA, sB) of A,B exists;iii) there exists an object P and morphisms

pA : P → A, pB : P → B, sA : A→ P, sB : B → P

with the properties

pA sA = 1A, pB sB = 1B, pA sB = 0, pB sA = 0,

sA pA + sB pB = 1P .

Moreover, under these conditions

sA = Ker pB, sB = Ker pA, pA = Coker sB, pB = Coker sA.

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Definition 11.0.12. Given two objects A,B in a preadditive category, a quintu-ple (P, pA, pB, sA, sB) as in the previous proposition, is called a biproduct of Aand B. The object P will generally be written A⊕B.

Definition 11.0.13. By an additive category we mean a preadditive categorywith a zero object and binary biproducts.

Proposition 11.0.14. On a category C , any two additive structures are neces-sarily isomorphic.

Proposition 11.0.15. Let f, g : A → B be two morphisms in a preadditivecategory. The following conditions are equivalent:

i) the equalizer Ker(f, g) exists;ii) the kernel Ker(f − g) exists;iii) the kernel Ker(g − f) exists.When this is the case, those three objects are isomorphic.

Definition 11.0.16. Given two preadditive categories A and B, a functor F :A → B is additive when, for all objects A,A′ in A , the mapping

FAA′ : A (A,A′) −→ B(F (A), F (A′)), f 7→ F (f)

is a group homomorphism.

Proposition 11.0.17. For a functor F : A → B between additive categories,the following are equivalent:

i) F is additive;ii) F preserves biproducts;iii) F preserves finite products;iv) F preserves finite coproducts.

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Proposition 11.0.18. Given a ring R with unit, the category ModR of left R-modules is isomorphic to the category Add(R,AB) of additive functors where Ris just the ring R viewed as a preadditive category.

Proposition 11.0.19. If A is a preadditive category and A ∈ A , the repre-sentable functor HA : A → Ab is additive.

Proposition 11.0.20. (Additive Yoneda Lemma) If A is a preadditive category,A ∈ A and F : A → Ab is an additive functor, there exist isomorphisms ofabelian groups

Nat(A (A,−), F ) ∼=θF,AF (A)

where A (A,−) stands for the additive representable functor. These isomor-phisms are natural both in A and in F .

Definition 11.0.21. A category C is abelian when it satisfies the following:i) C has a zero object;ii) every pair of objects of C has a product and a coproduct;iii) every arrow of C has a kernel and a cokernel;iv) every monomorphism of C is a kernel; every epimorphism of C is a cok-

ernel.

Proposition 11.0.22. (Abelian duality principle) The dual notion of “abeliancategory” is again “abelian category”.

Proposition 11.0.23. Let C be an abelian category and A a small category. Inthat case the category of all functors and natural transformations Fun(A ,C ) isagain abelian.

Proposition 11.0.24. Let C be an abelian category and A a small additive cat-egory. In that case the category of additive functors and natural transformationsAdd(A ,C ) is again abelian.

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Proposition 11.0.25. In an abelian category, the following are equivalent:i) f is an isomorphism;ii) f is both a monomorphism and an epimorphism.

Proposition 11.0.26. In an abelian category C , the intersection of two subob-jects always exists.

Proposition 11.0.27. An abelian category is finitely complete and finitely co-complete.

Proposition 11.0.28. For a morphism f : A→ B in an abelian category C , thefollowing are equivalent:

i) f is a monomorphism;ii) Kerf = 0;iii) ∀C ∈ C ∀g : C → A f g = 0⇒ g = 0.

Theorem 11.0.29. Every morphism f in an abelian category can be factoreduniquely (up to isomorphism) as f = i p, where i is a monomorphism and p isan epimorphism. Moreover, i = Ker (Coker f) and p = Coker (Ker f).

Lemma 11.0.30. Consider an object A of an abelian category, the diagonalmorphism ∆ : A → A × A and its cokernel q : A × A → Q. The object Q isisomorphic to A.

Definition 11.0.31. Consider two arrows f, g : B → A in an abelian categoryC . With the previous notation, we write σA for the composite

A× A q−→ Qr−1

−−→ A

65


and call it subtraction on A. We write f − g for the composite

B

fg

−−−−→ A× A σA−→ A

and define f + g = f − (0− g).

Theorem 11.0.32. Every abelian category is additive.

Proposition 11.0.33. In an abelian category, the intersection and union of afinite family of subobjects always exists.

Definition 11.0.34. Consider a category C with binary intersection and binaryunions of subobjects. The union of two subobjects r : R A and s : S A iseffective when R∪S is the pushout of R, S over their common subobject R∩S.

Proposition 11.0.35. In an abelian category, binary unions are effective.

Definition 11.0.36. In an abelian category C , a composable pair of morphisms

Af−→ B

g−→ C

is called an exact sequence when the image of f coincides with the kernel of g.

Definition 11.0.37. A finite or infinite sequence of morphisms

· · · → Anfn−→ An+1

fn+1−−→ An+2 → · · ·

in an abelian category is called exact when each pair of consecutive morphismsis an exact sequence.

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Proposition 11.0.38. In an abelian category, the following equivalences hold:

i) 0→ Af−→ B is an exact sequence iff f is a monomorphism;

ii) Bf−→ A→0 is an exact sequence iff f is an epimorphism;

iii) 0→ Af−→ B

g−→ C is an exact sequence iff f = Ker g;

iv) Cg−→ B

f−→ A→0 is an exact sequence iff f = Coker g.

Definition 11.0.39. By a short exact sequence in an abelian category is meantan exact sequence of the form

0→ Af−→ B

g−→ C → 0

Proposition 11.0.40. Consider a short exact sequence

0→ Af−→ B

g−→ C → 0

in an abelian category. The following conditions are equivalent.i) there exists a morphism s : C → B such that g s = 1C;ii) there exists a morphism r : B → A such that r f = 1A;iii) there exists morphisms s : C → B, r : B → A such that the quintuple

(B, r, g, f, s) is the biproduct of A and C.

Definition 11.0.41. In an abelian category, a split exact sequence is a shortexact sequence which satisfies the previous conditions.

Definition 11.0.42. In an abelian category C , consider an object A and a mor-phism f : A→ B.

i) A pseudo-element of A is an arrow • a−→ A with codomain A; we shallwrite simply a ∈∗ A;

ii) two pseudo-elements X a−→ A and X ′ a′−→ A are pseudo-equal when there

exist epimorphisms Yp−→ X, Y

p′−→ X ′ with the property a p = a′ p′; we shallwrite simply a =∗ a′;

iii) the pseudo-image under f : A → B of a pseudo-element • a−→ A of A isthe composite f a; we shall write simply f(a).

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Some famous lemmas that require some diagram chasing are the following: Thekernels’ lemma, The five lemma, The short five lemma, The nine lemma, Therestricted snake lemma, The snake lemma.

Lemma 11.0.43. (First Noether isomorphism theorem) In an abelian category,consider subobjects A B C. In this case B/A is a subobject of C/A and(C/A)/(B/A) is isomorphic to C/B.

Lemma 11.0.44. (Second Noether isomorphism theorem) Consider two subob-jects R A and S A in an abelian category. The following isomorphismholds:

S/(R ∩ S) ∼= (R ∪ S)/R

Definition 11.0.45. Consider an additive functor F : A → B between twoabelian categories. We say:

i) F is left exact when it preserves exact sequences of the form

0→ A→ B → C;

ii) F is right exact when it preserves exact sequences of the form

A→ B → C → 0;

iii) F is exact when it preserves exact sequences of the form

0→ A→ B → C → 0.

Proposition 11.0.46. Consider an additive functor F : A → B between twoabelian categories. The following equivalences hold:

i) F is left exact iff F preserves finite limits;ii) F is right exact iff F preserves finite colimits;iii) F is exact iff F preserves finite limits and finite colimits.

Proposition 11.0.47. Consider an additive functor F : A → B between twoabelian categories. The following are equivalent:

i) F is exact;ii) F preserves all exact sequences.

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Proposition 11.0.48. Consider a left exact functor F : A → B between abeliancategories. The following are equivalent:

i) F is exact;ii) F preserves epimorphisms.

Theorem 11.0.49. (The faithful embedding theorem) Every small abelian cate-gory admits a faithful and exact embedding in the category of abelian groups.

Theorem 11.0.50. (The full and faithful embedding theorem) Every small abeliancategory A has a full, faithful, and exact embedding in a category ModR of mod-ules over a ring R.

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70

Chapter 12

Regular Categories

Definition 12.0.1. A category C is regular when it satisfies the following con-ditions:

i) every arrow has a kernel pair;ii) every kernel pair has a coequalizer;iii) the pullback of a regular epimorphism along any morphism exists and is

again a regular epimorphism.

Theorem 12.0.2. In a regular category, every morphism factors as a regularepimorphism followed by a monomorphism and this factorization is unique upto isomorphism.

Proposition 12.0.3. In a regular category, the following conditions are equiva-lent:

i) f is a regular epimorphism;ii) f is a strong epimorphism.

Corollary 12.0.4. In a regular category:i) the composite of two regular epimorphisms is a regular epimorphism;ii) if a composite f g is a regular epimorphism, f is a regular epimorphism;iii) a morphism which is both a regular epimorphism and a monomorphism

is an isomorphism.

71

CHAPTER 12. REGULAR CATEGORIES

Proposition 12.0.5. A category C is regular when it satisfies the following con-ditions:

i) every arrow has a kernel pair;ii) every arrow f can be factored as f = i p with i a monomorphism a p a

regular epimorphism;iii) the pullback of a regular epimorphism along any morphism exists and is

a regular epimorphism.

Proposition 12.0.6. Let C be a category with finite limits. The category is reg-ular if and only if it satisfies the following:

i) every arrow f can be factored as f = i p with i a monomorphism and pa strong epimorphism;

ii) the pullback of a strong epimorphism along any morphism is again astrong epimorphism.

Definition 12.0.7. By an exact sequence in a regular category we mean a dia-gram

u, v : P → Af−→ B

where (u, v) is the kernel pair of f and f is the coequalizer of (u, v).

Definition 12.0.8. Let F : C → D be a functor between regular categories. Thefunctor F is exact when it preserves:

i) all finite limits which happen to exist in C ;ii) exact sequences.

Proposition 12.0.9. Let F : C → D be an exact functor between regular cate-gories. The functor F preserves:

i) regular epimorphisms;ii) kernel pairs;iii) coequalizers of kernel pairs;iv) mono-regular-epi factorizations.

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Proposition 12.0.10. Let F : C → D be a functor between regular categories.The following are equivalent:

i) F is exact;onii) F preserves finite limits and regular epimorphisms.

Definition 12.0.11. By a relation on an object A of a category C , we mean anobject R ∈ C together with a monomorphism pair of arrows

r1, r2 : R→ A

(i.e., given arrows x, y : X → R, x = y iff r1 x = r1 y and r2 x = r2 y).For every object X ∈ C we write

RX = (r1 x, r2 x) : x ∈ C (X,R)

for the corresponding relation (in the usual sense) generated by R on the setC (X,A).

Definition 12.0.12. By an equivalence relation on an object A of a categoryC , we mean a relation (R, r1, r2) on A such that, for every object X ∈ C ,the corresponding relation RX on the set C (X,A) is an equivalence relation.More generally, the relation R is reflexive (respectively transitive, symmetric,antisymmetric,..) when each relation RX is.

Definition 12.0.13. An equivalence relation (R, r1, r2) on an object A of a cat-egory C is effective when the coequalizer q of (r1, r2) exists and (r1, r2) is thekernel pair of q.

Definition 12.0.14. An exact category is a regular category in which equiva-lence relations are effective.

Theorem 12.0.15. The following conditions are equivalent:i) C is an abelian category;ii) C is an additive exact category;iii) C is a non-empty, preadditive exact category.

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Metatheorem: To prove that a property involving just finite limits and exact se-quences holds in every regular category, it suffices to prove it holds in everyGrothendieck topos.

Definition 12.0.16. Let C be a finitely complete category. By a relation R :A→ B, we mean a subobject R ⊂ A×B. We write

pR1 : R ⊂ A×B pA−→ A,

pR2 : R ⊂ A×B pB−→ B

for the corresponding projections.

Definition 12.0.17. Let C be a finitely complete category. Given a relation R :A → B in C , we define the opposite relation R0 : B → A as the followingcomposite:

R ⊂ B × A ∼= A×B,where the isomorphism is the canonical one. In other words, pR0

1 = pR2 andpR0

2 = pR1 .

Observe that every category can be viewed as a ’discrete’ 2-category with justidentity 2-cells. With that in mind, we get the following theorem.

Theorem 12.0.18. Let C be a regular, well-powered and finitely complete cate-gory. We get a 2-category Rel(C ) by choosing as:

-objects = those of C ;-arrows = the relations of C , with the composition defined as earlier;-2-cells = the inclusions of relations, viewed as subobjects in C .

The following are the so-called “modularity laws”:

Proposition 12.0.19. Let C be a regular, well-powered, finitely complete cate-gory. Consider three relations

R : A→ B, S : B → C, T : A→ C.

The following identities hold:

(S R) ∩ T ⊂ S (R ∩ (S0 T )),

(S R) ∩ T ⊂ (S ∩ (T R0)) R,where ∩ denotes the intersection as subobjects in C .

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Chapter 13

Algebraic Theories

Definition 13.0.1. A presentation of an algebraic theory T is a theory withequality, specified by choosing, besides a denumerable set of variables, a setOn of “n-ary operations” for each integer n ∈ N, together with a set of ax-ioms, subject to the following requirements. The terms of the theory are definedinductively

- each variable is a term;- if α ∈ On and t1, . . . , tn are terms, then α(t1, . . . , tn) is a term.

An axiom is an equality between two terms. If α ∈ O0, then α is a term. The0-ary operations are called constants.

Definition 13.0.2. Let T be a presentation of an algebraic theory in the previoussense. By a model of T we mean the choice of

- a set M ,- for all n ∈ N, for all α ∈ On, a mapping |α| : Mn → M , in such a way

that the axioms of T are realized by this interpretation. More precisely- a variable is interpreted as any element of M ,- if α ∈ On and the terms t1, . . . , tn are already interpreted as elements

|t1|, . . . , |tn| ∈M then α(t1, . . . , tn) is interpreted as the

|α|(|t1|, . . . , |tn|) ∈M,

and an axiom is satisfied in M when, for every possible interpretation of thevariables, both sides of the equality have the same interpretation in M .

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CHAPTER 13. ALGEBRAIC THEORIES

Definition 13.0.3. Let T be a presentation of an algebraic theory in the previ-ous sense. If L,M are models of T , a T -homomorphism f : L → M is amapping f : L → M such that for every operation α ∈ On and every elementsx1, . . . , xn ∈ L

f(|α|(x1, . . . , xn)) = |α|(f(x1), . . . , f(xn)).

Proposition 13.0.4. Let T be a presentation of an algebraic theory in the previ-ous sense. The models of T and their homomorphisms, together with the usualcomposition of mappings, constitute a category.

Lemma 13.0.5. Let T be a presentation of an algebraic theory. There exists asmallest equivalence relation R on the set of terms such that

i) if the axiom s = t holds, then the pair (s, t) is in R;ii) if the terms s, t are written using the variables x1, . . . , xn, the pair (s, t)

is in R and t1, . . . , tn are terms, then the pair (s′, t′) is in R, where s′, t′ areobtained from s, t by replacing xi by ti, i = 1, . . . , n;

iii) if α ∈ On and the pairs (si, ti) are in R, (i = 1, . . . , n), then the pair(α(s1, . . . , sn), α(t1, . . . , tn)) is in R.

Lemma 13.0.6. Let T be a presentation of an algebraic theory with set x1, . . . , xn, . . .of variables. Let Tn be the set of terms involving only the variable x1, . . . , xn.Let Fn be the quotient of Tn by the (restriction of the) equivalence relation Rof previous lemma. The set Fn is naturally provided with the structure of a T -model.

Lemma 13.0.7. Let T be a presentation of an algebraic theory. In the categoryModT of T -models, Fn is the n-th copower of F1.

Lemma 13.0.8. Let T be a presentation of an algebraic theory. The model Fnis the free model on n generators.

Here now is the key to a categorical approach to universal algebra.

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Proposition 13.0.9. Let T be a presentation of an algebraic theory. Using no-tation of a previous lemma, write F for the full subcategory of ModT generatedby the free models Fn on finitely many generators. The dual category F ∗ has fi-nite products and ModT is equivalent to the category of finite product-preservingfunctors from F ∗ to the category of sets, and natural transformations betweenthem.

Definition 13.0.10. By an algebraic theory T we mean a category T with adenumerable set T 0, T 1, . . . , T n, . . . of distinct objects, each object T n beingthe n-th power of the object T 1. A model of T is a functor F : T → Setwhich preserves finite products. A homomorphism of T -models is a naturaltransformation.

We shall write ModT for the category of T -models.

Proposition 13.0.11. Let T be an algebraic theory. Consider the functor

U : ModT → Set

of evaluation at T 1. Then:i) U is representable by T (T 1,−);ii) U is faithful;iii) U reflects isomorphisms;iv) each finite set with n elements (n ∈ N) admits T (T n,−) as a reflection

along U ;v) T (T 1,−) is a strong generator for ModT .

Proposition 13.0.12. Let T be an algebraic theory and consider the correspond-ing presentation T1 of this algebraic theory. The categories of models for T andT1 are equivalent and, via this equivalence, the functor

U : ModT → Set

maps a T1-model to its underlying set.

Definition 13.0.13. Let L be a complete lattice. An element k ∈ L is compactwhen k ≤

∨i∈I xi implies the existence of a finite subset J ⊂ I such that k ≤∨

i∈J xi. An algebraic lattice is a complete lattice in which every element is ajoin of compact elements.

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Definition 13.0.14. LetR and T be algebraic theories, with objects respectivelywritten R0, R1, . . . , Rn, . . . and T 0, T 1, . . . , T n, . . .. A morphism of algebraictheories is a functor F : R → T which preserves finite products and maps Rn

to T n (n ∈ N).

Proposition 13.0.15. The theory of sets is an initial object in the category ofalgebraic theories and their morphisms.

Definition 13.0.16. Let F : R → T be a morphism of algebraic theories. Thefunctor of composition with F

ModT →ModR, G 7→ G F,

is called an algebraic functor.

Theorem 13.0.17. Every algebraic functor has a left adjoint.

Definition 13.0.18. Let T be an algebraic theory, U : ModT → Set the corre-sponding forgetful functor and F : Set →ModT its left adjoint.

i) By a free T -model we mean, up to isomorphism, a model of the form F (X)for some set X;

ii) by a finitely generated T -model, we mean a modelM which is a quotientof a free model F (n) on a finite set n;

iii) by a finitely presentable T -model, we mean a model M which can beobtained via a coequalizer diagram

: F (m)→ F (n)→M

where m,n are finite sets.

Theorem 13.0.19. Let C be a category and U : C → Set a functor. Thefollowing are equivalent:

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i) C is equivalent to the category of models of some algebraic theory T , withU the corresponding forgetful functor:

ii) the following conditions are satisfied- C has coequalizers and kernel pairs;- U has a left adjoint F ;- U reflects isomorphisms;- U preserves regular epimorphisms- UF preserves filtered colimits.

Under these conditions, T ∗ is equivalent to the full subcategory of C gener-ated by the objects F (n), for n running through the finite sets. A category C asin the statement is called an algebraic category.

Definition 13.0.20. Let C be a category with finite products. If T is an algebraictheory, a model of T in C is a functor F : T → C preserving finite products.A morphism of T -models in C is just a natural transformation.

Definition 13.0.21. Let T ,R,V be algebraic theories and F : V → R, G :V → T morphisms of theories. By the tensor product R ⊗V T we mean thetheory obtained as the coequalizer of a diagram. In other words, it is the theoryobtained fromR⊗ T by adding the axiom

(F (α))(x1, . . . , xn) = (G(α))(x1, . . . , xn)

for every operation α : V n → V 1 of V .

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80

Chapter 14

Monads and Algebras

An important application of the AFT is that any equational theory T givesrise to a free a forgetful adjunction between Sets and the category of models ofthe theory, or T-algebras. In more detail, let T be a (finitary) equational theory,consisting of finitely many operation symbols, each of some finite arity, and aset of equations between terms built from these operations and variables. Forinstance, the theory of groups has a constant u, a unary operation g−1, and abinary operation g · h, and a handful of equations such as g · u = g. The theoryof rings is similar. The theory of fields is not equational, because the conditionx 6= 0 is required for an element x to have a multiplicative inverse. A T-algebrais a set equipped with operations (of the proper arities) corresponding to the op-eration symbols in T , and satisfying the equations of T . A homomorphism ofT -algebras h : A → B is a function on the underlying sets that preserve all theoperations.

Consider an adjunction F a U and the composite functor U F : C → D →C . Given any category C and endofunctor T : C → C , we can ask: When isT = U F for some adjoint functors F a U to and from another category D?Let η : 1 → T be the unit and the counit ε at FC to be εFC : FUFC → FCwith UεFC : UFUFC → UFC, which we call µ : T 2 → T .

Definition 14.0.1. A monad on a category C consists of an endofunctor T :C → C and natural transformations η : 1C → T , and µ : T 2 → T satisfyingtwo commutative diagrams, that is

µ µT = µ Tµµ ηT = 1 = µ Tη.

In fact, a monad is exactly the same thing as a monoidal monoid in the monoidalcategory [C ,C ] with composition as the monoidal product G ⊗ F = G F .For this reason, the above equations are called the associativity and unit lawsrespectively.

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CHAPTER 14. MONADS AND ALGEBRAS

Proposition 14.0.2. Every adjoint pair F a U with U : D → C , unit η : UF →1C and counit ε : 1D → FU gives rise to a monad (T, η, µ) on C with

T = U F : C → C

η : 1→ T the unit

µ = UεF : T 2 → T.

Back to our question “when does an endofunctor T arise from an adjunction”,the simple answer: just if it is the functor part of a monad.

Proposition 14.0.3. Every monad arises from an adjunction. More precisely,given a monad (T, η, µ) on the category C , there exists a category D and anadjunction F a U, η : 1→ UF, ε : FU → 1 with U : D → C such that

T = U F

η = η (the unit)

µ = UεF .

Definition 14.0.4. The Eilenberg-Moore category of T , C T is the category withobjects being the “T -algebras” which are pairs (A,α) of the form α : TA→ Ain C , such that

1A = α ηA and α µA = α Tα.A morphism of T -algebras, h : (A,α)→ (B, β) is simply an arrow h : A→ Bin C such that

h α = β T (h).

Definition 14.0.5. A comonad on a category C is a monad on C op.

Definition 14.0.6. Given an endofunctor P : S → S on any category S ,a P-algebra consists of an object A of S and an arrow α : PA → A. Ahomomorphism h : (A,α) → (B, β) of P -algebras is an arrow h : A → B inS such that h α = β P (h), namely the diagram commutes:

P (A)

α

P (h)// P (B)

β

Ah

// B

Lemma 14.0.7. Given any endofunctor P : S → S on an arbitrary categoryS , if i : P (I)→ I is an initial P -algebra, then i is an isomorphism, P (I) ∼= I .

Proposition 14.0.8. If the category S has an initial object 0 and colimits ofdiagrams of type ω = (N,≤) (call them ’ω-colimits’), and the functor P : S →S preserves ω-colimits, then P has an initial algebra.

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Definition 14.0.9. A functor of the form P (X) = C0 +C1×X+C2×X2 + · · ·+Cn ×Xn with natural number coefficients Ck, is called a (finitary) polynomialfunctor. These functors present exactly the finitary structures.

Corollary 14.0.10. Every polynomial functor P : Sets → Sets has an initialalgebra.

Proposition 14.0.11. Let the category S have finite coproducts. Given an end-ofunctor P : S → S , the following are equivalent:

i) The P -algebras are the algebras for a monad. Precisely, there is a monad(T : S → S , η, µ), and an equivalence P −Alg(S ) ∼= S T between the cate-gory of P -algebras and the category S T of algebras for the monad. Moreover,this equivalence preserves the respective forgetful functors to S .

ii) The forgetful functor U : P − Alg(S )→ S has a left adjoint F ` U .iii) For each object A of S , the endofunctor PA(X) = A+P (X) : S → S

has an initial algebra.

As a good example, looking at monoids, they can be seen as a set where “ev-ery finite sequence has been given a composite”. Let us make this a precisedefinition.

Given a set M , we write T (M) for the set of finite sequences of elementsof M . Given a morhism f : M → N, T (f) : T (M) → T (N) is the map-ping sending the sequence (x1, . . . , xn) to the sequence of corresponding images(f(x1), . . . , f(xn)).

We want to provide the set M with the structure of a monoid via a mappingξ : T (M) → M which associates with every finite sequence (x1, . . . , xn) ofelements of M a new element ξ(x1, . . . , xn) ∈M which we call the “compositeof the sequence”. First of all the “normalization condition” ξ(x) = x mustcertainly be satisfied. We shall express it by considering the mapping

εM : M → T (M), x 7→ (x)

and requiring as an axiom the commutativity of a certain triangle. Next a generalassociativity condition of the type

ξ(ξ(a11, . . . , a

1n1

), . . . , ξ(am1 , . . . , amnm

)) = ξ(a11, . . . , a

1n1, a2

1, . . . , am−1nm−1

, am1 , . . . , amnm

)

must be satisfied as well. Observe that choosing a finite sequence of finite se-quences of elements of M is the same as choosing an element of TT (M).

The previous axiom thus involves the concatenation mapping

µM : TT (M)→ T (M),

which constructs a “composite sequence” from a sequence of sequences. It isnow easy to verify that the previous definition is equivalent to the classical defi-nition of a monoid. It suffices to define a multiplication on M via

M ×M →M, (x, y) 7→ ξ(x, y).

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The associativity rule is just

(xy)z = ξ(ξ(x, y), z) = ξ(ξ(x, y), ξ(z)) = ξ(x, y, z),

x(yz) = ξ(x, ξ(y, z)) = ξ(ξ(x), ξ(y, z)) = ξ(x, y, z).

Now put 1 = ξ( ), the composite of the empty sequence. Thus M has beenprovided with the structure of a monoid and one proves inductively that ξ is justthe composition for this monoid structure:

-ξ( ) is indeed the element 1;- if ξ(x1, . . . , xn−1) = x1 . . . xn−1, then

ξ(x1, . . . , xn) = ξ(ξ(x1, . . . , xn−1), ξ(xn))

= ξ(x1 . . . xn−1, xn) = (x1 . . . xn−1)xn

= x1 . . . xn−1xn.

Definition 14.0.12. A monad on a category C is a triple (T, ε, µ) where T :C → C is a functor and ε : 1C ⇒ T, µ : TT ⇒ T are natural transformationssatisfying the commutative conditions

µ (ε ∗ 1T ) = 1T = µ (1T ∗ ε), µ (µ ∗ 1T ) = µ (1T ∗ µ).

Definition 14.0.13. Let T(T, ε, µ) be a monad on a category C . By an algebraon this monad is meant a pair (C, ξ) where C ∈ C , ξ : T (C) → C and ξ εC = 1C , ξ T (ξ) = ξ µC . If (D, ζ) is another T-algerbra, a morphismf : (C, ξ) → (D, ζ) of T-algebras is a morphism f : C → D of C such thatf ξ = ζ T (f).

Proposition 14.0.14. Let T = (T, ε, µ) be a monad on a category C . The T-algebras and their morphisms constitute a category, written C T, also called theEilenberg-Moore category of the monad.

Definition 14.0.15. Let T = (T, ε, µ) be a monad on a category C . Considerthe forgetful functor U = C T → C and its left adjoint F : C → C T, mappingC to (T (C), µC). A T-algebra is free when it is isomorphic to one of the formF (C) = (T (C), µC), for some object C ∈ C .

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Proposition 14.0.16. Let T = (T, ε, µ) be a monad on a category C . The fullsubcategory of C T generated by the free T-algebras is equivalent to the follow-ing category CT:

- the objects of CT are those of C ;- a morphism f : C → D in CT is a morphism f : C → T (D) in C ;- the composite of two morphisms f : A → B, g : B → C in CT is given in

C by the composite

Af−→ T (B)

T (g)−−→ TT (C)µC−→ T (C);

- the identity on an object C of CT is just εC : C → T (C) in C .

The category CT is also called the Kleisli category of the monad T.

Definition 14.0.17. A functor R : X → C is monadic when there exist amonad T ∼= (T, ε, µ) on C and an equivalence of categories J : X → C T suchthat U J is isomorphic to R, where U : C T → C is the forgetful functor.

Definition 14.0.18. Let T = (T, ε, µ) and S = (S, ζ, η) be two monads on asame category C . By a morphism of monads S → T, we mean a naturaltransformation λ : S ⇒ T such that λ ζ = ε, µ (λ λ) = λ η;

Definition 14.0.19. A monad T(T, ε, µ) on a category C has rank α, for someregular cardinal α, when the functor T : C → C preserves α-filtered colimits.When α = ℵ0, thus when T preserves filtered colimits, one also says that T hasfinite rank.

Definition 14.0.20. Let M be a module over a commutative ring R with unit. Mis flat when the functor

−⊗RM : ModR −→ModR

preserves monomorphisms.

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86

Chapter 15

Enriched Category Theory

Definition 15.0.1. A monoidal category V consists in giving:i) a category V ;ii) a bifunctor⊗ : V ×V → V , called the tensor product - we write A⊗B

for the image under ⊗ of the pair (A,B);iii) an object I ∈ V , called the unit;iv) for every triple A,B,C of objects, an “associativity” isomorphism

aABC : (A⊗B)⊗ C → A⊗ (B ⊗ C)

v) for every object A, a “left unit” isomorphism

lA : I ⊗ A→ A;

vi) for every object A, a “right unit” isomorphism

rA : A⊗ I → A.

These data must satisfy the following:i) the morphisms aABC are natural in A,B,C;ii) the morphisms lA are natural in A;iii) the morphisms rA are natural in A;iv) a commutative diagram for every quadruple of objects A,B,C,D (asso-

ciativity coherence);v) a commutative diagram for every pair A,B of objects (unit coherence).

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CHAPTER 15. ENRICHED CATEGORY THEORY

Definition 15.0.2. A monoidal category is symmetric when, an isomorphism

sAB : A⊗B → B ⊗ A

is given for every pair A,B of objects. These isomorphisms must be such that:i) the morphisms sAB are natural in A,B;ii) for every triple A,B,C we have associativity coherence;iii) for every object A, we have unit coherence;iv) for every pair A,B of objects we have another commutative diagram

which is our symmetry axiom.

Definition 15.0.3. A monoidal category V is biclosed when, for each objectB ∈ V , both functors

−⊗B : V → V , B ⊗− : V → V

have a right adjoint. A biclosed symmetric monoidal category is called a sym-metric monoidal closed category.

Definition 15.0.4. A category V is cartesian closed when it admits all finiteproducts and, for every object B ∈ V , the functor −× B : V → V has a rightadjoint, generally written (−)B : V → V .

Proposition 15.0.5. Every cartesian closed category is symmetric monoidal closed,with the cartesian product as a tensor product.

When V is a symmetric monoidal closed category, we write

[B,−] : V → V , C 7→ [B,C]

for the right adjoint to the functor −⊗B : V → V .

Definition 15.0.6. Let V be a monoidal category. A V -category consists in thefollowing data:

i) a class |C | of “objects”;ii) for every pair A,B ∈ |C | of objects, an object C (A,B) of V ;iii) for every triple A,B,C ∈ |C | of objects, a “composition” morphism in

V ,cABC : C (A,B)⊗ C (B,C) −→ C (A,C);

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iv) for every object A ∈ |C |, a “unit” morphism in V ,

uA : I → C (A,A).

These data must satisfy the following:i) given objects A,B,C,D,E ∈ C we get a commutative diagram, which is

the associativity axiom;ii) given objects A,B ∈ C , we get a commutative diagram, the unit axiom.

When |C | is a set, the V -category C is called a small V -category.

Definition 15.0.7. Let V be a symmetric monoidal closed category. By a V -distributor φ : A → B between small V -categories, we mean a V -functorB∗ ⊗A → V . By a morphism of V -distributors, we mean a V -natural trans-formation between the corresponding V -functors.

Proposition 15.0.8. Let V be a complete symmetric monoidal closed category.Given two V -categories A ,B, with A small, the category of V -functors A →B and V -natural transformations can be provided with the structure of a V -category, written V [A ,B].

Definition 15.0.9. Let V be a symmetric monoidal closed category. Given aV -category A and two V -functors F,G : A → V , an object N of V is calledthe object of V -natural transformations from F to G, and one writes N ∼=V − Nat(F,G), if for all V ∈ V there exist bijections, natural in the variableV ∈ V , between

i) the set of morphism V → N ,ii) the class of V -natural transformations F ⇒ [V,G−].

Theorem 15.0.10. (Enriched Yoneda lemma) Let V be a symmetric monoidalclosed category and A a small V -category. For every object A ∈ A and everyV -functor F : A → V , the object of V -natural transformations from A (A,−)to F exists and there is an isomorphism in V ,

V −Nat(A (A,−), F ) ∼= F (A),

which is V -natural both in F and in A.

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Definition 15.0.11. A morphism F : V → W of monoidal categories consistsin the following data:

i) a functor F : V → W ;ii) for each pair (A,B) of objects of V , a morphism of W ,

τAB : F (A)⊗ F (B) −→ F (A⊗B);

iii) a morphism ε : J → F (I) of W , where I stands for the unit of V and Jfor the unit of W .

These data must satisfy the following:i) the morphisms τAB are natural in A,B;ii) we have a commmutative diagram for all objects A,B,C of V , the asso-

ciativity condition;iii) we have a commutative diagram for each object A of V , the unit condi-

tions;

When V ,W are symmetric monoidal categories, F is a morphism of symmet-ric monoidal categories when moreover,

iv) we have a commutative diagram for every pair A,B of objects of V , thesymmetry condition.

Definition 15.0.12. Let V be a symmetric monoidal closed category, A a V -category and A ∈ A , V ∈ V two objects.

- The cotensor of V and A exists if there is an object [V,A] ∈ A togetherwith isomorphisms

A (B, [V,A]) ∼= [V,A (B,A)]

in V which are V -natural in B ∈ A . We say A is cotensored when [V,A]exists for all objects V ∈ V , A ∈ A .

- The tensor of V and A exists if there is an object V ⊗A ∈ A together withisomorphisms

A (V ⊗ A,B) ∼= [V,A (A,B)]

in V which are V -natural in B ∈ A . We say A is tensored when V ⊗A existsfor all objects V ∈ V , A ∈ A .

Definition 15.0.13. Let V be a symmetric monoidal closed category. Given V -functors F : A → B, G : A → V , the V -limit of F weighted by G existswhen:

90


i) for every B ∈ B, the object V -Nat(G,B(B,F−)) of V -natural trans-formations exists;

ii) there exists an object L ∈ B and isomorphisms in V ,

λB : V −Nat(G,B(B,F−)) ∼= B(B,L),

which are V -natural in B.We write in general limG F for this weighted limit. When limG F exists for

all choices of F,G with A small, B is said to be V -complete.

Definition 15.0.14. Let V be a symmetric monoidal closed category and A ,Btwo V -categories.

- By the end∫A∈A

F (A,A) of a V -functor F : A ∗ ⊗A → B, we mean theV -limit of F weighted by A : A ∗ ×A → V , when this exists.

- By the coend∫ A∈A

F (A,A) of a V -functor F : A ⊗A ∗ → B, we meanthe V -colimit of F weighted by A : A ∗ ⊗A → V , when this exists.

Definition 15.0.15. Let V be a symmetric monoidal category and A ,B two V -categories. Two V -functors F : A → B, G : G → A are V -adjoint, G leftadjoint to F and F right adjoint to G, when there exist isomorphisms in V

A (G(B), A) ∼= B(B,F (A)),

which are V -natural in A ∈ A , B ∈ B.

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Fibred Categories

Very often we have considered families of objects or morphisms in a categoryC . This suggests the study of the category Set(C ) of families of objects andmorphisms in C .

- the objects are the families (Ci)i∈I where I is a set and Ci is an object of C ;- the morphisms f : (Ci)i∈I → (Dj)j∈J are the pairs (α, (fi)i∈I) with α :

I → J a mapping in Set and each fi : Ci → Dα(i) a morphism in C .

Of special interest are the families (Ci)i∈I and (1f , (fi : Ci → Di)i∈I) for agiven set f , which constitute what will be called the fibre at f .

Definition 16.0.1. Let F : F → E be a functor. Given an object I ∈ E , thefibre of F at I is the subcategory FI of F defined in the following way:

- an object X ∈ F is in FI when F (X) = I;- if X, Y are objects in FI , a morphism f : X → Y of F is in FI when

F (f) = 1f .

Definition 16.0.2. Let F : F → E be a functor and α : J → I a morphism ofE . An arrow f : Y → X of F is cartesian over α if:

i) F (f) = α;ii) given g : Z → X is a morphism of F such that F (g) factors as αβ there

exists a unique morphism h : Z → Y in F such that F (h) = β and g = f h.

Definition 16.0.3. A functor F : F → E is a fibration when for every arrowα : J → I in E and every object X in the fibre over I , there exists in F acartesian morphism f : Y → X over α. We also call F a category fibred overE .

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CHAPTER 16. FIBRED CATEGORIES

Definition 16.0.4. Let F : F → E be a functor and α : J → I a morphism ofE . An arrow f : Y → X of F is precartesian over α if:

i) F (f) = α;ii) if g : Z → X is a morphism of F such that F (g) = α, there exists a

unique morphism h : Z → Y in the fibre FJ such that g = f h.

Definition 16.0.5. A functor F : F → E is a cofibration when the dual functorF ∗ : F ∗ → E ∗ is a fibration.

Definition 16.0.6. Let F : F → E and G : G → E be two fibrations over thesame base category E . A cartesian functor H : (F , F ) → (G , G) is a functorH : F → G such that:

i) G H = F ;ii) G maps a cartesian morphism for F to a cartesian morphism for G.

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Locales

The propositional calculus is, roughly, the theory which studies the various con-sequences one can infer from the validity of some formulas, by combining themusing the logical connectors ∧ (and), ∨ (or),⇒ (implies), ¬ (not). The predicatecalculus takes additionally into account that two quantifiers ∃ (there exists) and ∀(for all). First of all there are axioms which declare that some types of formulasare necessarily ”true”; we write ` φ to indicate the truth of φ.

Definition 17.0.1. The intuitionistic propositional calculus is the one having foraxioms:

(PC1) ` φ⇒ (ψ ⇒ φ);(PC2) ` (φ⇒ (ψ ⇒ θ))⇒ ((φ⇒ ψ)⇒ (φ⇒ θ));(PC3) ` φ⇒ (ψ ⇒ (φ ∧ ψ));(PC4) ` (φ ∧ ψ)⇒ φ;(PC5) ` (φ ∧ ψ)⇒ ψ;(PC6) ` φ⇒ (φ ∨ ψ);(PC7) ` ψ ⇒ (φ ∨ ψ);(PC8) ` (φ⇒ θ)⇒ ((ψ ⇒ θ)⇒ ((φ ∨ ψ)⇒ θ));(PC9) ` (φ⇒ ψ)⇒ ((φ⇒ ¬ψ)⇒ ¬φ);(PC10) ` ¬φ⇒ (φ⇒ ψ);and for rule of deduction, the modus ponens:if ` φ and ` φ ⇒ ψ, then ` ψ. In this definition, φ, ψ, θ are arbitrary

formulas.

Definition 17.0.2. The classical propositional calculus is the one obtained fromthe intuitionistic propositional calculus by adding the axiom

` φ ∨ ¬φ

the so-called law of the excluded middle.

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Lemma 17.0.3. In intuitionistic propositional calculus, putting

φ ≤ ψ iff ` φ⇒ ψ

provides the set F of formulas with the structure of a preordered set. This pre-ordered set is finitely complete and cocomplete and for each formula φ, the func-tor

− ∧ φ : F −→ F ; θ 7→ θ ∧ φadmits the functor

φ⇒ − : F −→ F ; ψ 7→ φ⇒ ψ

is right adjoint.

Lemma 17.0.4. In the intuitionistic propositional calculus, with the previousnotation, for every formula φ the mapping

− ⇒ ψ : F −→ F ; φ 7→ φ⇒ ψ

is a contravariant functor. When ψ =⊥ is the initial object of F , this functor isisomorphic to

¬ : F −→ F , φ 7→ ¬φ.

It is common practice to consider the quotient of the preordered set F of for-mulas by the equivalence relation identifying two formulas φ, ψ when they areisomorphic, i.e. when

` φ⇒ ψ and ` ψ ⇒ φ.

Writing [φ] for the equivalence class of the formula φ, the quotient is now anactual lattice in which each functor − ∧ [φ] has a right adjoint. This is what weshall call a Heyting algebra.

Definition 17.0.5. A Heyting algebra H is a lattice, with top and bottom ele-ments, in which for every element b ∈ H, the functor

− ∧ b : H −→ H, a 7→ a ∧ b,

has a right adjoint, which we shall denote

b⇒ − : H ⇒ H, c 7→ b⇒ c.

We shall write 1 for the top element and 0 for the bottom element.

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The adjunction indicated thus reduces to the relation

a ∧ b ≤ c iff a ≤ b⇒ c

for arbitrary elements a, b, c.

Proposition 17.0.6. In a Heyting algebra, the following relations hold:

1. a ≤ b iff (a⇒ b) = 1;

2. a = (1⇒ a);

3. a⇒ (b ∧ c) = (a⇒ b) ∧ (a⇒ c);

for arbitrary elements a, b, c.

Proposition 17.0.7. In a Heyting algebra, b ⇒ c is the greatest element suchthat b ∧ (b⇒ c) ≤ c, i.e.

1. b⇒ c =∨a : a ∧ b ≤ c;

2. (b⇒ c) ∧ b ≤ c.

Proposition 17.0.8. In a Heyting algebraH, for every element c the mapping

− ⇒ c : H −→ H, b 7→ b⇒ c,

is a contravariant functor.

Proposition 17.0.9. In a Heyting algebra, putting ¬b = (b ⇒ 0) yields thegreatest element such that b ∧ ¬b = 0, i.e.

1. ¬b =∨a : a ∧ b = 0;

2. ¬b ∧ b = 0.

The element ¬b is called the pseudo-complement of b.

Proposition 17.0.10. In a Heyting algebra, the following conditions hold:

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1. ¬0 = 1,¬1 = 0;

2. a ≤ b implies ¬b ≤ ¬a;

3. ¬a = ¬¬¬a;

4. ¬(a ∨ b) = ¬a ∧ ¬b;

5. ¬a ∨ b ≤ a⇒ b;

for all elements a, b.

Definition 17.0.11. In a Heyting algebraH, the two relations

¬(a ∨ b) = ¬a ∧ ¬b, ¬(a ∧ b) = ¬a ∨ ¬b

are referred to as the two De Morgan laws. The first one holds in every Heytingalgebra, but the second on does not in general.

Proposition 17.0.12. For a Heyting algebra H, the following conditions areequivalent:

1. H satisfies the two De Morgan laws;

2. ∀a, b ∈ H ¬(a ∧ b) = ¬a ∨ ¬b;

3. ∀a ∈ H ¬a ∨ ¬¬a = 1;

4. ∀a, b ∈ H ¬¬(a ∨ b) = ¬¬a ∨ ¬¬b.

Proposition 17.0.13. Every boolean algebra is a Heyting algebra satisfying thetwo De Morgan laws.

Proposition 17.0.14. For a Heyting algebra H, the following conditions areequivalent:

1. H is a boolean algebra;

2. ∀a ∈ H a ∨ ¬a = 1;

3. ∀a ∈ H ¬¬a = a.

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Definition 17.0.15. An element a of a Heyting algebra is regular when ¬¬a = a.

Proposition 17.0.16. The regular elements of a Heyting algebra H constitute aboolean algebra.

The next proposition can be interpreted as the fact that any Heyting algebra sat-isfies the axioms of the intuitionistic propositional calculus.

Proposition 17.0.17. In a Heyting algebra, the following relations hold for allelements a, b, c.:

1. a ≤ (b⇒ a);

2. (a⇒ (b⇒ c)) ≤ ((a⇒ b)⇒ (a⇒ c));

3. a ≤ (b⇒ (a ∧ b));

4. (a ∧ b) ≤ a;

5. (a ∧ b) ≤ b;

6. a ≤ (a ∨ b);

7. b ≤ (a ∨ b);

8. (a⇒ c) ≤ ((b⇒ c)⇒ ((a ∨ b)⇒ c));

9. (a⇒ b) ≤ ((a⇒ ¬b)⇒ ¬a);

10. ¬a ≤ (a⇒ b).

Proposition 17.0.18. In a Heyting algebra, the following relation holds:

(a⇒ (b⇒ c)) = ((a ∧ b)⇒ c).

Proposition 17.0.19. The theory of Heyting algebras is algebraic.

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CHAPTER 17. LOCALES

The union and the intersection of subsets of a set are two operations which dis-tribute over each other. When the set is provided with a topology, the lattice ofopen subsets is closed, in the lattice of all subsets, under arbitrary unions andfinite intersections. In particular, the lattice of open subsets is complete andarbitrary joins distribute over finite meets. These properties will characterize ar-bitrary locales.

Definition 17.0.20. A locale L is a complete lattice in which arbitrary joinsdistribute over finite meets, i.e. the distributivity law

a ∧

(∨i∈I

bi

)=∨i∈I

(a ∧ bi)

holds, where I is an arbitrary indexing set and a, bi are elements of L.

Proposition 17.0.21. For a lattice L, the following conditions are equivalent:

1. L is a locale;

2. L is a complete Heyting algebra.

Corollary 17.0.22. The regular elements of a locale constitute a complete booleanalgebra.

Proposition 17.0.23. Let a ∈ L be an element of a locale. The upper and lowersegments of a,

↑ a = b ∈ L : a ≤ b, ↓ a = b ∈ L : b ≤ a,

are locales as well, for the induced partial order.

Proposition 17.0.24. Let (X, T ) be a topological space and U ∈ T and opensubset.

1. The locale ↓ U is isomorphic to the locale of open subsets of the open setU ;

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CHAPTER 17. LOCALES

2. the locale ↑ U is isomorphic to the locale of open subsets of the closed setCU (the set theoretical complement of U ).

Let us recall that a continuous mapping f : (X, T )→ (Y,S) between topologi-cal spaces induces a mapping

f−1 : S −→ T , U 7→ f−1(U).

The inverse image along a mapping preserves arbitrary joins and meets of sub-sets; so between the locales of open subsets, arbitrary joins and finite meets arepreserved. But if f−1 preserves arbitrary joins, it has a right adjoint.

Definition 17.0.25. A morphism f : L →M from a locale L to a localeM isa pair of functors

f∗ : L →M, f ∗ :M→ Lsatisfying the conditions:

1. f ∗ is left adjoint to f∗;

2. f ∗ preserves finite meets.

Proposition 17.0.26. Let L,M be two locales. There is a bijection between

1. the morphisms of locales f : L →M,

2. the mappings f ∗M→ L such that

• f ∗(∨i∈I ai) =

∨i∈I f

∗(ai),• f ∗(a ∧ b) = f ∗(a) ∧ f ∗(b),• f ∗(1) = 1,

for all indexing sets I and elements ai, a, b ofM.

Definition 17.0.27. Let L be a locale.

• By an open sublocale of L we mean a monomorphismM→ L of localeswhich is isomorphic to the monomorphism i :↓ a→ L for some element ofa ∈ L.

• By a closed sublocale of L we mean a monomorphismM→ L of localeswhich is isomorphic to the monomorphism j :↑ a → L for some elementa ∈ L.

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Proposition 17.0.28. The category Loc of locales is cocomplete.

Proposition 17.0.29. The category of locales admits 0, 1 as a terminal object.

Theorem 17.0.30. The category of locales has pullbacks and is finitely complete.

Corollary 17.0.31. The product A ⊗ X of two locales is given by the poset ofthose subsets I of the set theoretical product A×X which satisfy

1. (a, x) ∈ I and (b, y) ≤ (a, x) imply (b, y) ∈ I ,

2. ∀t ∈ T (at, x) ∈ I implies (∨t∈T at, x) ∈ I ,

3. ∀t ∈ T (a, xt) ∈ I imiples (a,∨t∈T xt) ∈ I , where T is an arbitrary

indexing set.

Proposition 17.0.32. Let (X, T ) and (Y,S) be topological spaces; write (X ×Y,R) for the topological product (X, T )× (Y,S). The mapping

T ⊗ S −→ R,∨t∈T

Vt ⊗Wt 7→⋃t∈T

Vt ×Wt,

is the left adjoint of a morphism of locales. It is always surjective; it is anisomorphism as long as one of the spaces is locally compact.

Definition 17.0.33. Let L be a locale and ∆ : L −→ L⊗ L the diagonal of theproduct.

• A locale L is Hausdorff when the diagonal is a closed sublocale of L⊗L;

• a locale L is discrete when the diagonal is an open sublocale of L ⊗ L.

A morphism of locales is in particular an adjoint pair of functors, thus it inducesa monad.

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Definition 17.0.34. A nucleus on a locale L is a monad on L induced by amorphism of locales.

Proposition 17.0.35. The nuclei on a locale L are exactly the functors j : L →L which satisfy the following conditions:

(N1) a ≤ j(a);(N2) jj(a) ≤ j(a);(N3) j(a ∧ b) = j(a) ∧ j(b);for all elements a, b ∈ L.

Corollary 17.0.36. Every nucleus on a locale is idempotent.

Proposition 17.0.37. Let f :M−→ L be a morphism of locales. The followingare equivalent:

1. f is a regular monomorphism of locales;

2. f∗ is injective;

3. f ∗ is surjective;

4. f ∗ f∗ = 1M;

5. f is isomorphic to the morphism Lj −→ L induced by a nucleus j on L.

Theorem 17.0.38. The nuclei on a locale, for the pointwise partial order, con-stitute a locale. The dual of this locale is the poset of regular subobjects of thelocale L, with its usual inclusion relation.

Proposition 17.0.39. Let L be a locale and a ∈ L. The open nucleus generatedby a and the closed nucleus generated by a are the complement of each other inthe locale Nuc(L) of nuclei.

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Proposition 17.0.40. Let L be a locale and Nuc(L) its locale of nuclei. Themapping

L −→ Nuc(L), a 7→ a ∨ −,sending an element to the corresponding closed nucleus is the left adjoint partof a morphism of locales. This mapping is injective and preserves and reflectsthe partial order.

Definition 17.0.41. A morphism of locales f :M→ L is open if for every opensublocale i : N →M ofM, the regular image of f i is an open sublocale ofL.

Proposition 17.0.42. Let f :M→ L be a morphism of locales. The followingare equivalent:

1. f is an open morphism of locales;

2. f ∗ admits a left adjoint f! and the ”Frobenius identity”

f!(a ∧ f ∗(x)) = f!(a) ∧ x

is satisfied for all a ∈M, x ∈ L;

3. f ∗ preserves arbitrary meets and the identity

f ∗(x⇒ y) = f ∗(x)⇒ f ∗(y)

holds for all elements x, y ∈ L.

Corollary 17.0.43. The composite of two open morphisms of locales is still anopen morphism of locales.

Proposition 17.0.44. Let (Y,S) be a topological space. The following are equiv-alent:

1. for each element y ∈ Y , there exists an open neighborhood V of y suchthat V ∩ y = y;

2. a continuous mapping f : (X, T ) −→ (Y,S) with (X, T ) an arbitrarytopological space, is topologically open precisely when the correspondingmorphism of locales T −→ S is open.

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An etale morphism of locales is one which is locally an isomorphism.

Definition 17.0.45. A morphism of locales f : M −→ L is etale when thereexist families ai ∈M, xi ∈ L (i ∈ I) with the properties:

1.∨i∈I ai = 1;

2. for every index i ∈ I , f restricts to an isomorphism ↓ ai −→↓ xi betweenthe open sublocales generated by ai, xi.

Proposition 17.0.46. Every etale morphism of locales is open

Corollary 17.0.47. A morphism f :M→ L of locales is etale iff

1. f is open, and

2. there exist elements (ai ∈ M)i∈I with∨i∈I ai = 1, such that for each

index i ∈ I the restricted mapping f! :↓ ai −→↓ f!(ai) is an isomorphismof posets.

Proposition 17.0.48. Let f! : M→ L be a mapping between two locales. Thefollowing are equivalent:

1. f! is the direct image part of an etale morphism of locales;

2. • f! preserves arbitrary suprema, and• there exists a covering 1 =

∨i∈I ai in M such that for each index

i, f! :↓ ai −→↓ f!(ai) is an isomorphism of posets.

Proposition 17.0.49. Let f :M→ L be a morphism of locales. The followingare equivalent:

1. f is etale;

2. f is open and the diagonal ∆ :M→M⊗LM of the kernel pair of f isopen as well .

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Proposition 17.0.50. The composite of two etale morphisms of locales is againetale.

Nf

h //M

g

LTheorem 17.0.51. In the category of locales, consider the commutative dia-gram. If f and g are etale, h is etale as well.

Giving a point (= an element ) of a topological space X is just giving a contin-uous mapping from the singleton to X . So we define

Definition 17.0.52. A point of a locale L is a morphism p : 1 → L of locales,where 1 = 0, 1 is the terminal locale.

Definition 17.0.53. An element u ∈ L of a locale is prime when

1. u 6= 1 (1 here is the top element of L), and

2. ∀a, b ∈ L a ∧ b ≤ u⇒ a ≤ u or b ≤ u.

Definition 17.0.54. A subset F ⊂ L of a locale is a filter when(F1) 1 ∈ F ,(F2) 0 6∈ F ,(F3) ∀a, b ∈ L a ∈ F and a ≤ b⇒ b ∈ F ,(F4) ∀a, b ∈ L a ∈ F and b ∈ F ⇒ a ∧ b ∈ FA filter is prime when it satisfies the condition(P) ∀a, b ∈ L a ∨ b ∈ F ⇒ a ∈ F or b ∈ F .A filter is completely prime when it satisfies the stronger condition(CP) ∀ai ∈ L (i ∈ I)

∨i∈I ai ∈ F ⇒ ∃i ∈ I ai ∈ F.

Proposition 17.0.55. For a locale L there are bijections between

1. the points of L,

2. the prime elements of L,

3. the completely prime filters of L.

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Lemma 17.0.56. Let L be a locale; write Sp(L) for its set of points. For everyelement a ∈ L define

Oa = p ∈ Sp(L) : p∗(a) = 1.

The subsets Oa ⊂ Sp(L) constitute a topology on Sp(L). More precisely, thefollowing relations hold:

O0 = ∅, O1 = Sp(L), O∨i∈I ai

=⋃i∈I

Oi, Oa∧b = Oa ∩ Ob

for elements a, b, ai in L. This space Sp(L) is called the spectrum of L.

Proposition 17.0.57. Consider the functor

O : Top −→ Loc, (X, T ) 7→ T

mapping a topological space to its lattice of open subsets and a continuous map-ping to the corresponding morphism of locales. This functor has a right adjointfunctor, namely

Sp : Loc −→ Top, L 7→ Sp(L),

the functor mapping a locale L to its spectrum Sp(L).

Definition 17.0.58. A locale L has enought points when the morphism

εL : O(Sp(L)) −→ L,

counit of the adjunction, is an isomorphism.

Proposition 17.0.59. For a locale L, the following are equivalent:

1. L has enough points;

2. the mapping ε∗L : L −→ O(Sp(L)) is injective;

3. if a 6≤ b in L, there exists a point p of L such that p∗(a) = 1, p∗(b) = 0;

4. if a 6≤ b in L, there exists a prime element u of L such that a 6≤ u, b ≤ u;

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5. if a b in L there exists a completely prime filter F ⊂ L such that a ∈F, b 6∈ F .

For a topological space (X, T ), we have two notions of ”point”: the elements ofthe set X and the points of the locale T . We have a continuous mapping

η(X,T ) : X −→ Sp(T )

sending an element x ∈ X to a point px of the locale T , but this mapping, ingeneral, is neither injective nor surjective.

Definition 17.0.60. A topological space (X, T ) is sober when the mapping

η(X,T ) : (X, T ) −→ Sp(T )

is a homeomorphism.

The notion of ”point of a locale” is equivalent to that of ”prime element” of thelocale. Applying this to the case of a topological space, we get a further equiva-lence with the notion of ”coprime element” (which we call ’irreducible’) in the”colocale” of closed subsets.

Definition 17.0.61. Let (X, T ) be a topological space. A closed subset C ⊂ Xis irreducible when

1. C is not empty, and

2. for all closed subsets C1, C2 of X

C ⊂ C1 ∪ C2 ⇒ C ⊂ C1 or C ⊂ C2.

In other words, a closed subset is irreducible when its complement is aprime element in the locale of open subsets.

Proposition 17.0.62. The spectrum of a locale is a sober topological space.

Corollary 17.0.63. The category of sober topological spaces is reflective in thecategory of all topological spaces.

Theorem 17.0.64. The category of locales with enough points is equivalent tothe category of sober topological spaces.

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Corollary 17.0.65. For every topological space (X, T ), the locale T is isomor-phic to the locale of open subsets of a sober space.

Definition 17.0.66. Let L be a complete lattice and a, b elements of L. Theelement a is way below b, which we write a b, when for every family (ci)i∈Iof elements of L such that b ≤

∨i∈I ci there exist finitely many indices i1, . . . , in

such that a ≤ c1 ∨ . . . ∨ cn.

Definition 17.0.67. A continuous lattice is a complete lattice L such that, forevery element b ∈ L, the relation b =

∨a ∈ L : a b holds.

Theorem 17.0.68. Let (X, T ) be a sober topological space. The following areequivalent:

1. the space (X, T ) is locally compact;

2. the locale T is a continuous lattice.

Definition 17.0.69. A locale L is locally compact when L is a continuous lat-tice.

Theorem 17.0.70. A locally compact locale has enough points.

Definition 17.0.71. A topological space is regular when every neighborhood ofa point contains a closed neighborhood of this point.

Lemma 17.0.72. In a topological space X , the following are equivalent for twoopen subsets U, V :

1. there exists a closed subset C such that U ⊂ C ⊂ V ;

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CHAPTER 17. LOCALES

2. there exists an open subset W such that U ∩W = 0, V ∪W = X .

Theorem 17.0.73. A compact regular locale has enough points.

110

Chapter 18

Sheaves

Definition 18.0.1. A presheaf on a locale L is a contravariant functor F :L −→ Set. Given v ≤ u in L, we write

ρuv : F (u) −→ F (v), x 7→ ρuv(x) = x|v

for the action of F on v ≤ u, when no ambiguity is possible.

The functoriality of F reduces to

1. ∀u ∈ L ∀x ∈ F (u) x|u = x,

2. ∀w ≤ v ≤ u ∈ L ∀x ∈ F (u) x|w = (x|v)|w.

The mappings ρuv are called the restriction mappings.

Definition 18.0.2. Let F be a presheaf on a locale L and (ui)i∈I a family ofelements of L. A family of elements (xi ∈ F (ui))i∈I of the presheaf F is com-patible when

∀i, j ∈ I xi|ui∧uj = xj|ui∧uj .

Definition 18.0.3. A presheaf F on a locale L is separated when, given u =∨i∈I ui in L and x, y ∈ F (u) in F ,

(∀i ∈ I x|ui = y|ui)⇒ (x = y).

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Definition 18.0.4. A presheaf F on a locale L is a sheaf when, given u =∨i∈I ui in L and (xi ∈ F (ui))i∈I a compatible family in F , there exists a unique

element x ∈ F (u) such that for each index i ∈ I, x|ui = xi.

The element x ∈ F (u) is often called the gluing of the family (xi)i∈I .

Lemma 18.0.5. Let F be a presheaf on a localeL. The following are equivalent:

1. F is a sheaf;

2. F is a separated presheaf and given u =∨i∈I ui in L, every compatible

family (xi ∈ F (ui))i∈I can be ”glued” into an element x ∈ F (u) such thatfor every i ∈ I, x|ui = xi.

Lemma 18.0.6. Let F be a presheaf on a locale L.

1. If F is separated, F (0) has at most one element.

2. If F is a sheaf, F (0) has exactly one element.

Lemma 18.0.7. Let F be a presheaf on a locale L. If u =∨i∈I ui in L and

x ∈ F (u), the family (x|ui)i∈I is compatible.

Lemma 18.0.8. Let F be a sheaf on a locale L. If u =∨i∈I ui is a partition in

L (i.e. for all i, j ∈ I, i 6= j, ui ∧ uj = 0), then F (u) ∼=∏

i∈I F (ui).

Definition 18.0.9. Let L be a locale. The morphisms of presheaves, separatedpresheaves or sheaves on L are just the natural transformations between them.

So a morphism α : F −→ G between presheaves or sheaves consists in giving afamily (αu : F (u) −→ G(u))u∈L of mappings such that

∀u, v ∈ L ∀a ∈ F (u) v ≤ u⇒ αu(a)|v = αv(a|v).

We shall write Pr(L) and Sh(L) for the categories of presheaves and sheaveson L.

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Definition 18.0.10. Let F be a presheaf on a locale L. The support σ(F ) of Fis the element of L defined by

σ(F ) =∨u ∈ L : F (u) 6= ∅.

Proposition 18.0.11. LetL be a locale. The categories of sheaves and separatedpresheaves on L are closed under arbitrary limits in the category of presheaveson L.

Corollary 18.0.12. Let L be a locale. A morphism α : G ⇒ F of sheaves orseparated presheaves on L is a monomorphism iff each αu (u ∈ L) is injective.

We shall often consider the subobjectsG of F as being such that eachG(u) (u ∈L) is a subset of F (u).

Proposition 18.0.13. Let L be a locale. Every subpresheaf of a separatedpresheaf on L is itself separated.

Definition 18.0.14. Let F be a presheaf on a locale L. A subpresheaf G ⊂ F isclosed when for every covering u =

∨i∈I ui in L and element x ∈ F (u)

(∀i ∈ I x|ui ∈ G(ui))⇒ (x ∈ G(u)).

Proposition 18.0.15. Let F be a sheaf on a locale L and G ⊂ F a subpresheafof F . The following are equivalent:

1. G is a subsheaf of F ;

2. G is a closed subobject of F .

Proposition 18.0.16. Let F be a presheaf on a locale L. The closed sub-presheaves of F , ordered by inclusion, constitute a locale. Also, if F is a sheafon the locale L, then the subsheaves of F ordered by inclusion, constitute alocale.

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Lemma 18.0.17. In the category of presheaves on a localeL, for every subobjectS ⊂ F there exists a smallest closed subobject S ⊂ F containing S, namely

S(u) = x ∈ F (u) : ∃I ∃ui ∈ L u =∨i∈I

ui ∀i ∈ I x|ui ∈ S(ui)

for each u ∈ L. The subpresheaf S is called the closure of S in F .

Lemma 18.0.18. Let F be a presheaf on a locale L. The closure operation onthe subobjects of L satisfy the properties:

1. S ⊂ T ⇒ S ⊂ T ,

2. S ⊂ S,

3. S = S,

4. S ∩ T = S ∩ T ,

5. f−1(S) = f−1(S),

where S, T are subobjects of F and f : G→ F is a morphism of presheaves.

Proposition 18.0.19. Let f : G → F be a morphism of presheaves on a localeL. WritingCl(F ), Cl(G) for the locales of closed subobjects, pulling back alongf is the left adjoint part of an etale morphism of locales

φ : Cl(G) −→ Cl(F ),

which we shall write Cl(f).

Proposition 18.0.20. Let L be a locale. Then L is isomorphic to the locale ofclosed subpresheaves of the terminal presheaf. And L is isomorphic to the localeof presheaves of the terminal sheaf.

The category of sheaves on a locale constitute a basic example of what will becalled a ’topos’. We will now describe the ’topos structure’ of the category ofsheaves.

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Lemma 18.0.21. Let L be a locale, 1 the terminal sheaf on L and Ω be sheafdefined as

Ω(u) =↓ u for u ∈ L.The data

tu : 1(u) −→ Ω(u), ∗ 7→ u,

define a monomorphism t : 1→ Ω of sheaves.

S

s

γ// 1

t

Fφ// Ω

Theorem 18.0.22. Let F be a sheaf on a locale L. For every subobject S of Fthere exists a unique morphism φ : F → Ω such that the square of the diagramis a pullback.

Theorem 18.0.23. The category of sheaves on a locale is cartesian closed.

Proposition 18.0.24. Let L be a locale. The representable functors on L consti-tute a dense family of generators in the category of sheaves on L.

Definition 18.0.25. An etale mapping f : Y → X between two topologicalspaces Y and X is a mapping f : Y → X such that, for every point y ∈ Y ,there exist open neighborhoods A of y in Y and U of f(y) in X such that frestricts to a homeomorphism f : A → U . One says that the space Y is etaleover X .

Proposition 18.0.26. An etale mapping f : Y → X between topological spacesis continuous and open.

Proposition 18.0.27. Topological spaces and etale mappings constitute a cate-gory, for the usual composition of functions.

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Lemma 18.0.28. Let f : Y → X be an etale mapping between topologicalspaces and σ, ρ : U ⇒ Y two continuous sections of f on the open subsetU ⊂ X . If σ(x) = ρ(x) for some x ∈ U , then σ, ρ coincide on an openneighborhood of x.

Proposition 18.0.29. Consider three continuous functions f : Y → X, g : Z →X and h : Y → Z. If f = g h with f, g etale mappings, h is an etale mappingas well.

Proposition 18.0.30. Given a continuous mapping f : Y → X between topo-logical spaces, the following are equivalent:

1. f is an etale mapping;

2. f is open and the diagonal of the kernel pair of f is open.

Definition 18.0.31. Let F be a presheaf on a topological space (X, T ) andx ∈ X a point of the space. The stalk of F at x is the quotient set

Fx =

∐F (U) : U ∈ T , x ∈ U

≈where ≈ is the equivalence relation identifying a ∈ F (U) and b ∈ F (V ) whenthere exists an open neighborhood W ⊂ U ∩ V of x such that a|w = b|w.

Observe that Fx is just the filtered colimit of the functor F restricted to theposet of those open subsets containing x.

Definition 18.0.32. Let F be a presheaf on a topological space (X, T ). Thetotal space of F is the disjoint union

∐x∈X Fx provided with the final topology

for all the mappings

σUa : U −→∐x∈X

Fx, x 7→ [a] ∈ Fx,

where U ∈ T , a ∈ F (U) and [a] stands for the equivalence class of a in thequotient defining Fx.

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Proposition 18.0.33. Let F be a presheaf on a topological space (X, T ) and∐x∈X Fx it total space. The mapping

p :∐x∈X

Fx −→ X, [a] ∈ Fx 7→ x,

is etale.

Proposition 18.0.34. Let F be a presheaf on a topological space (X, T ). Thelocale of open subsets of the total space

∐x∈X Fx is isomorphic to the locale of

closed subpresheaves of F .

Theorem 18.0.35. For a topological space (X, T ), the category Sh(X, T ) ofsheaves over (X, T ) is equivalent to the category Et/(X, T ) of etale spacesover (X, T ).

Definition 18.0.36. Let F be a sheaf on a locale Ω. A family A of elements of∐u∈Ω F (u) is a family of generators for F when the smallest subsheaf G ⊂ F

containing all the elements of A is F itself.

Proposition 18.0.37. Let F be a sheaf on a locale Ω and A a family of elementsof∐

u∈Ω F (u). The following are equivalent:

1. A is a family of generators of F ;

2. for every u ∈ Ω and a ∈ F (u), there exist a covering u =∨i∈I ui in Ω and

elements ai ∈ A such that, for every index i ∈ I , a|ui = ai|ui

Lemma 18.0.38. Let F be a sheaf on a locale Ω. Given elements u, v ∈ Ω,a ∈ F (u), b ∈ F (v), there exists a greatest element w ∈ Ω such that a|w = b|w.This element w will be denoted by [a ≈ b].

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Proposition 18.0.39. Let F be a sheaf on a locale Ω and A a family of genera-tors of F . The following properties hold, for arbitrary generators a, b, c ∈ A:

1. [a ≈ b] = [b ≈ a];

2. [a ≈ b] ∧ [b ≈ c] ≤ [a ≈ c].

Proposition 18.0.40. Let Ω be a locale. Consider a sheaf F on Ω with family Aof generators. There is a bijection between

1. the elements of the set∐

u∈Ω F (u),

2. the mappings σ : A→ Ω which satisfy

• σ(a) ∧ σ(b) ≤ [a ≈ b],

• [a ≈ b] ∧ σ(b) ≤ σ(a),

for all elements a, b ∈ A.

Definition 18.0.41. Let Ω be a locale. An Ω-set is a pair (A, [· ≈ ·]) where A isa set and [· ≈ ·] is a mapping

[· ≈ ·] : A× A :−→ Ω, (a, b) 7→ [a ≈ b],

satisfying the following properties:(S1) [a ≈ b] = [b ≈ a];(S2) [a ≈ b] ∧ [b ≈ c] ≤ [a ≈ c];for all elements a, b, c of A.

Definition 18.0.42. Let Ω be a locale and A,B two Ω-sets. A morphism α :A→ B of Ω-sets is a mapping

B × A→ Ω, (b, a) 7→ [b ∼ α(a)],

which satifies the following:(M1) [b ≈ b′] ∧ [b′ ∼ α(a)] ≤ [b ∼ α(a)];(M2) [b ∼ α(a)] ∧ [a ≈ a′] ≤ [b ∼ α(a′)];(M3) [b ∼ α(a)] ∧ [b′ ∼ α(a)] ≤ [b ≈ b′];(M4) [a ≈ a] =

∨b∈B[b ∼ α(a)];

where a, a′ ∈ A and b, b′ ∈ B.

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The Ω-sets and their morphisms constitute a category.

Lemma 18.0.43. Let Ω be a locale and α, β : A⇒ B two morphisms of Ω-sets.The following are equivalent:

1. α = β;

2. ∀a ∈ A ∀b ∈ B [b ∼ α(a)] ≤ [b ∼ β(a)].

Definition 18.0.44. Consider a locale Ω and an Ω-set A. An Ω-subset S of A isa mapping

A −→ Ω, a 7→ [a ∈ S],

which satisfies, for all elements a, b ∈ A, the following properties:

1. [a ∈ S] ≤ [a = a];

2. [a ≈ b] ∧ [b ∈ S] ≤ [a ∈ S].

The set of Ω-subsets of A, is made a poset via the pointwise ordering i.e.,given two Ω-subsets S, T of A,

S ≤ T when ∀a ∈ A [a ∈ S] ≤ [a ∈ T ].

Proposition 18.0.45. Consider a locale Ω and an Ω-set A. There is an isomor-phism between the following posets:

1. the poset of subobjects of A in the category of Ω-sets;

2. the poset of Ω-subsets of A.

Proposition 18.0.46. Let Ω be a locale andA an Ω-set. The poset of Ω-subsets ofA is a locale in which arbitrary joins and binary meets are computed pointwise.

Theorem 18.0.47. Let Ω be a locale. The category of sheaves on Ω is equivalentto the category of Ω-sets.

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CHAPTER 18. SHEAVES

Definition 18.0.48. An ideal I of a commutative ring R with unit is a subgroupI of R satisfying the condition

∀i ∈ I ∀r ∈ R ir ∈ I.

The ideal I is prime when I 6= R and moreover

∀a, b ∈ R ab ∈ I ⇒ a ∈ I or b ∈ I.

Proposition 18.0.49. Let R be a commutative ring with unit. Given an ideal Iof R and an element r 6∈ I , there exists a prime ideal J containing I but not r.

Definition 18.0.50. Let I be an ideal of a commutative ring R with unit. Theradical of I , written

√I , is the set√I = r ∈ R : ∃n ∈ N rn ∈ I

We say I is a radical ideal when I =√I .

Lemma 18.0.51. Let R be a commutative ring with unit. The radical of everyideal is itself an ideal. Moreover, the following properties hold, where I, J, Ikare arbitrary ideals:

1. I ⊂√I;

2. I ⊂ J ⇒√I ⊂√J;

3.√I ∩ J =

√I ∩√J =√IJ;

4.√∑

k∈K Ik =√∑

k∈K√Ik;

5.√I is a radical ideal, i.e.

√√I =√I

Lemma 18.0.52. Let R be a commutative ring with unit. Every prime ideal ofR is radical.

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Proposition 18.0.53. Let R be a commutative ring with unit. For every ideal I√I =

⋂J : J prime ideal , I ⊂ J.

Proposition 18.0.54. LetR be a commutative ring with unit and S ⊂ R a subsetof R containing 1 and closed under the multiplication (m,n ∈ S implies mn ∈S). The following data define a ring R[S−1]:

R[S−1] =(a, s) : a ∈ R, s ∈ S

≈where ≈ is the equivalence relation defined by

(a, s) ≈ (b, t) iff ∃r ∈ S atr = bsr.

We write as

for the equivalence class of (a, s).

Lemma 18.0.55. LetR be a commutative ring with unit and a, b ∈ R. If√aR =√

bR, then R[a−1] is isomorphic to R[b−1].

Given a commutative ring R with unit and a prime ideal J , the set theoreticalcomplement R\J of J in R contains 1, since 1 6∈ J , and is closed under themultiplication, by primeness of J . The corresponding ring R[(R\J)−1] is gen-erally written RJ : this is ”the ring R localized at the prime ideal J”.

Proposition 18.0.56. For a commutative ring R with unit, the following areequivalent, provided 0 6= 1:

1. the set of non-invertible elements of R is an ideal;

2. R has a unique proper maximal ideal;

3. for every element r ∈ R, r is invertible or 1− r is invertible.

A ring satisfying these equivalent conditions is called a local ring.

Proposition 18.0.57. LetR be a commutative ring with unit and J a prime idealof R. The ring RJ = R[(R\J)−1], i.e. the ring R localized at J , is a local ring.

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Definition 18.0.58. The spectrum Sp(R) of a commutative ring R with unitis the set of its prime ideals provided with the topology generated by the opensubsets

Oa = J : J prime ideal of R, a 6∈ Jwhere a ∈ R.

Given any subset S ⊂ R, let us write

OS = J ∈ Sp(R) : S 6⊂ R= J ∈ Sp(R) : ∃a ∈ S a 6∈ R

=⋃a∈S

J ∈ Sp(R) : a 6∈ R =⋃a∈S

Oa.

Thus OS is an open subset of the spectrum as a union of basic open subsets.

Proposition 18.0.59. Let R be a commutative ring with unit and Rad(R) theposet of its radical ideals. The mapping

Rad(R) −→ O(Sp(R)), I 7→ OI ,

is an isomorphism of posets, withO(Sp(R)) denoting the locale of open subsetsof the spectrum.

Corollary 18.0.60. The poset of radical ideals of a commutative ring with unitis a locale.

Lemma 18.0.61. LetR be a commutative ring with unit and Sp(R) its spectrum.Given an element J ∈ Sp(R), its closure J is the complement in Sp(R) of theopen subset OJ .

Corollary 18.0.62. The spectrum of a commutative ring with unit is sober.

Proposition 18.0.63. The spectrum of a commutative ring with unit is compactand has a basis of compact open subsets.

We shall now represent the commutative ring R with unit as the ring of globalsections of a sheaf on its spectrum Sp(R).

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Definition 18.0.64. The structural space of a commutative ring R with unit isthe mapping

π :∐

J∈Sp(R)

RJ −→ Sp(R), x ∈ RJ 7→ J,

where the set∐RJ is provided with the final topology for all the mappings.

Proposition 18.0.65. Let R be a commutative ring with unit. The structuralspace π of R is an etale space over Sp(R).

Proposition 18.0.66. Let R be a commutative ring with unit. The sheaf Γ ofcontinuous sections of the structural space of R is a sheaf of rings, i.e.

• for each open subset U ⊂ Sp(R), Γ(U) is a ring.

• for each pair V ⊂ U of open subsets of Sp(R), the corresponding restric-tion mapping Γ(U)→ Γ(V ) is a ring homomorphism.

Lemma 18.0.67. Let R be a commutative ring with unit. Every section

σxy : Ob −→∐

J∈Sp(R)

RJ

of the structural space of R, for b, x, y ∈ R, has the form σabn for a ∈ R, n ∈ N.

Proposition 18.0.68. Let b ∈ R, whereR is a commutative ring with unit. Everycontinuous section σ : Ob →

∐J∈Sp(R) RJ of the structural space of R has the

form σabl , for some a ∈ R and l ∈ N.

Corollary 18.0.69. Let R be a commutative ring with unit and Γ the sheaf ofcontinuous sections of the corresponding structural space. For each elementb ∈ R, Γ(Ob) is isomorphic to the ring R[b−1].

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Theorem 18.0.70. Let R be a commutative ring with unit. The ring of globalsections of the corresponding structural space is isomorphic to the ring R.

Proposition 18.0.71. Let R be a commutative ring with unit and Γ the sheaf ofcontinuous sections of the corresponding structural space. Then Γ is ”locally alocal ring” in the sense that given U open in Sp(R) and σ ∈ Γ(U), there existsan open covering U = ∪i∈IUi such that for every i ∈ I

σ|Uiis invertible in γ(Ui) or (1− σ)|Ui

is invertible in Γ(Ui);

Definition 18.0.72. Let A,B be two categories with finite limits. A geometricmorphism φ : A → B is a pair of functors

φ∗ : A → B, φ∗ : B → A

such that

1. φ∗ is left adjoint to φ∗,

2. φ∗ preserves finite limits.

Theorem 18.0.73. Let L,M be two locales. There is a bijection between

1. the morphisms of locales f : L →M,

2. the isomorphism classes of geometric morphisms

φ : Sh(L)→ Sh(M).

Corollary 18.0.74. Let L be a locale. There is a bijection between

1. the points of the locale L,

2. the isomorphism classes of geometric morphisms Set −→ Sh(L).

Proposition 18.0.75. Let f : L → M be a morphism of locales and φ :Sh(L) → Sh(M) the corresponding geometric morphism. When viewed asgeometric morphism φ : Et/L → Et/M between the corresponding categoriesof etale morphisms, the left adjoint part φ∗ is just pulling back along f in thecategory of locales.

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Chapter 19

Grothendieck Toposes

A sheaf on a locale L is just a contravariant functor L → Set satisfying goodgluing properties with respect to the coverings in L. This definition can be car-ried over to the case where L is an arbitrary small category provided with a goodsystem of ”coverings”. The categories of sheaves obtained in this way are called”Grothendieck toposes”.

Definition 19.0.1. Let L be a poset. A subset S ⊂ L is hereditary when

∀u ∈ S ∀v ∈ L v ≤ u⇒ v ∈ S.

By a hereditary covering of an element u ∈ L we mean a hereditary subsetS ⊂ L such that u =

∨S.

For an element u ∈ L of a poset, the down segment

↓ u = v ∈ L : v ≤ u

is obviously a hereditary covering of u.

Rf

rf

fR // R

r

C (−, D)C (−,f)

// C (−, C)

Definition 19.0.2. A localizing system L on a small category C consists ingiving, for each object C of C , a family L(C) of subfunctors of the representablefunctor C (−, C); these data must satisfy the following axiom: given a pullbackas in the diagram, where f : D → C is an arrow of C and R ∈ L(C), we haveRf ∈ L(D).

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CHAPTER 19. GROTHENDIECK TOPOSES

Definition 19.0.3. Let C be a small category and L a localizing system on C .

• A presheaf on (C ,L) is just a contravariant functor F : C → Set.

• A presheaf F on (C ,L) is a sheaf when, given C ∈ C and R ∈ L(C),every natural transformation α : R⇒ F extends uniquely to C (−, C).

• A morphism of sheaves or presheaves is just a natural transformation.

Definition 19.0.4. Let C be a small category and L a localizing system on C .A presheaf F on (C ,L) is separated when, for every object C ∈ C and everypair α, β : C (−, C)⇒ F of natural transformations, α and β are equal as longas they coincide on some subobject R ∈ L(C).

Definition 19.0.5. A Grothendieck topology on a small category C consists ingiving, for each object C of C , a family L(C) of subfunctors of the representablefunctor C (−, C); these data must satisfy the following axioms:

1. for every object C ∈ C , C (−, C) ∈ L(C);

2. in the pullback of the previous diagram, with f : D → C an arrow of C , ifR ∈ L(C) then Rf ∈ L(D);

3. consider an object C ∈ C , an arbitrary subfunctor R C (−, C) andS ∈ L(C); suppose that for every object D ∈ C and every morphismf ∈ S(D), one has Rf ∈ L(D); then R ∈ L(C).

When L is a Grothendieck topology on a small category C , the pair (C ,L) iscalled a site.

Proposition 19.0.6. Let (C ,L) be a site. For every object C ∈ C ,L(C) is afilter of subobjects of C (−, C).

Proposition 19.0.7. Let L be a localizing system on a small category C and Lthe smallest Grothendieck topology containing L.

1. The category of separated presheaves on (C ,L) coincides with the cate-gory of separated presheaves on (C ,L).

126


2. The category of sheaves on (C ,L) coincides with the category of sheaveson (C ,L).

Proposition 19.0.8. Let C be a small category. The Grothendieck topology onC constitute a locale.

Theorem 19.0.9. Let (C , T ) be a site. The category of sheaves on (C , T ) isreflective in the category of presheaves on (C , T ) and the reflection preservesfinite limits.

Definition 19.0.10. A Grothendieck topos is a category equivalent to a cate-gory of sheaves on a site.

Proposition 19.0.11. A Grothendieck topos is complete and cocomplete.

Proposition 19.0.12. In a Grothendieck topos, colimits are universal and finitelimits commute with filtered colimits.

Definition 19.0.13. An initial object 0 of a category C is strict when everymorphism C → 0 with codomain 0 is an isomorphism.

Proposition 19.0.14. In a Grothendieck topos, the initial object is strict.

Definition 19.0.15. Let (si : Ai −→∐

i∈I Ai)i∈I be a coproduct in a categoryC . This coproduct is disjoint when

1. for each i ∈ I, si is a monomorphism, and

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2. for each pair i, j of distinct indices, the intersection of Ai, Aj as subobjectsof∐

i∈I Ai exists and is an initial object of C .

Proposition 19.0.16. In a Grothendieck topos, coproducts are disjoint.

Proposition 19.0.17. In a Grothendieck topos, every monomorphism is regular.

Proposition 19.0.18. In a Grothendieck topos, every morphism which is both amonomorphism and an epimorphism is an isomorphism.

Proposition 19.0.19. In a Grothendieck topos, every epimorphism is regular.

Proposition 19.0.20. A Grothendieck topos is a regular category.

Proposition 19.0.21. In a Grothendieck topos, equivalence relations are effec-tive.

Proposition 19.0.22. A Grothendieck topos is locally presentable.

Proposition 19.0.23. A Grothendieck topos is cartesian closed.

Proposition 19.0.24. In a Grothendieck topos, the subsheaves of every sheafconstitute a locale.

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Proposition 19.0.25. A localization of a Grothendieck topos is again a Grothendiecktopos.

Proposition 19.0.26. Let E be a Grothendieck topos. There is a bijective corre-spondence between

1. the localizations of E and

2. the universal closure operations on E .

Theorem 19.0.27. A category E is a Grothendieck topos if and only if the fol-lowing conditions hold:

1. E has a set of generators;

2. E has finite limits;

3. E has coproducts and they are disjoint and universal;

4. every equivalence relation in E has a coequalizer and this coequalizer isuniversal;

5. every equivalence relation in E is effective;

6. every epimorphism in E is regular.

Corollary 19.0.28. Every Grothendieck topos is equivalent to the category ofsheaves on a site (C , T ), where C is a category with finite limits and each rep-resentable functor C (−, C) is a sheaf.

Proposition 19.0.29. Every Grothendieck topos is equivalent to the category ofsheaves on a site (C , T ), where T is the canonical topology on C .

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130

Chapter 20

The Classifying Topos

Proposition 20.0.1. For every Grothendieck topos E , there exists a unique geo-metric morphism Γ : E → Set to the topos of sets.

Definition 20.0.2. A point of a Grothendieck topos E is a geometric morphismf : Set→ E .

Definition 20.0.3. Consider a mathematical theory whose category of models, inevery Grothendieck topos F , can be defined as a full subcategory of the categoryFun(C ,F ) of all functors from C to F , with C a fixed small category. Thistheory admits a classifying topos when there exist

1. a Grothendieck topos E [T ] and

2. a model M : C → E [T ] of the theory T in E [T ],

such that for every Grothendieck topos F , composing with M the left adjointpart of geometric morphisms yields an equivalence of categories

Geom(F ,E [T ]) −→ModF (T )

between the category of geometric morphisms from F to E [T ] and the categoryof models of T in F .

The topos E [T ] is called the classifying topos of the theory, and the model Mthe generic model of the theory.

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CHAPTER 20. THE CLASSIFYING TOPOS

Theorem 20.0.4. Let C be a small category with finite limits. The theory of leftexact functors on C admits as classifying topos the topos Pr(C ) of presheaveson C ; the generic model is the Yoneda embedding.

Proposition 20.0.5. Every Grothendieck topos is the classifying topos of sometheory defined by a sketch.

Proposition 20.0.6. Let S = (S ,P ,F) be a triple, where

• S is a small category,

• P is a set of cones in S ,

• F is a set of families of morphisms (fi : Ti → T )i∈I with the samecodomain in S .

A model of S in a category E is a functor F : S → E which maps the conesof P to limit cones and the families of F to epimorphic families; a morphism ofmodels is a natural transformation. The theory we have just described admits aclassifying topos.

Definition 20.0.7. By a geometric sketch is meant a sketch S = (S ,P , I)where P is a set of finite cones.

Theorem 20.0.8. The theory determined by a geometric sketch has a classifyingtopos. Conversely, every Grothendieck topos is the classifying topos of a theorydetermined by a geometric sketch.

Theorem 20.0.9. Every coherent theory (defined later) has a classifying topos.

Theorem 20.0.10. Let C be a small category and E a Grothendieck topos, withcorresponding geometric morphism Γ : E → Set. There is an equivalence ofcategories

Flat(Γ∗(C ),E ) ∼= Geom(E , P r(C ))

where the left hand side indicates the category of E -valued flat functors on theinternal category Γ∗(C ).

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Chapter 21

Elementary Toposes

Just as abelian categories are the categorical framework for studying those struc-tures which behave like abelian groups, toposes are the categorical frameworkfor studying those structures which behave like sets.

S

s

// 1

t

Aφ// Ω

Definition 21.0.1. Let E be a category with finite limits. By a subobject classi-fier is meant an object Ω together with a monomorphism t : 1 Ω, such thatfor every object A ∈ E and every monomorphism s : S A, there exists aunique morphism φ : A→ Ω such that the above diagram is a pullback.

Proposition 21.0.2. In a category with finite limits, a subobject classifier, whenit exists, is unique up to isomorphism.

Definition 21.0.3. A category E is called a topos when:

1. E has finite limits;

2. E is cartesian closed;

3. E has a subobject classifier.

We shall write t : 1 Ω for the subobject classifier of E . In the example ofsets, an element a of the set A can be identified with a morphism a : 1→ A. Ina topos, this is what is called a ”global element of A”.

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CHAPTER 21. ELEMENTARY TOPOSES

Definition 21.0.4. Let E be a topos. By a global element of an object A ∈ E ismeant a morphism a : 1→ A, where 1 is the terminal object.

Proposition 21.0.5. Let E be a topos. Given two objects A,B of E , the globalelements of BA correspond bijectively with the arrows A→ B.

Proposition 21.0.6. Let E be a topos. Given an object A ∈ E , the globalelements of ΩA correspond bijectively with the subobjects of A.

Proposition 21.0.7. Let E be a topos and f : A → B a morphism of E . Com-posing with

Ωf : ΩB −→ ΩA

yields a mapping from the set of global elements of ΩB to the set of global ele-ments of A, which, in terms of subobjects of B and A, is just the inverse imagealong f .

Some examples of toposes are the following: the category of sets, category offinite sets, category of sheaves on a locale, category of finite sheaves on a finitelocale, category of presheaves on a small category, the category of G-sets for agiven group G, the category of finite presheaves on a finite category, category offinite G-sets, , every Grothendieck topos, and the terminal category.Given an object X of a topos, we write ξX : X → 1 for the unique morphism tothe terminal object and tX : X → Ω for the composite

XξX−→ 1

t−→ Ω

where t is the subobject classifier.

Proposition 21.0.8. In a topos, every monomorphism is regular.

Corollary 21.0.9. In a topos, every morphism which is both a monomorphismand an epimorphism is an isomorphism.

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Proposition 21.0.10. A topos is well-powered.

If A is an object of a topos, we shall write ∆A : A → A × A for the ’diago-nal of A’, i.e. the unique morphism whose composite with both projections isthe identity on A. In particular, from pi ∆A = 1A we deduce that ∆A is amonomorphism.

Definition 21.0.11. The equality on an object A of a topos is the characteristicmorphism

=A: A× A −→ Ω

of the diagonal ∆A : A A× A.

Given two morphisms f, g : B ⇒ A in a topos, we shall write f =A g (or justf = g when no confusion is possible) for the composite

B(fg)−−→ A× A =A−−→ Ω.

Lemma 21.0.12. Given two global elements a, b : 1⇒ A in a topos, the follow-ing are equivalent:

1. a = b;

2. (a =A b) = t.

Definition 21.0.13. For an object A of a topos, the singleton on A is the mor-phism

·A : A→ ΩA

which corresponds to =A: A× A→ Ω by cartesian closedness.

Lemma 21.0.14. Given a global element a : 1 → A in a topos, the globalelement aA of the object ΩA is the element corresponding to the subobjecta : 1 A.

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Proposition 21.0.15. For every object A of a topos, the ”singleton” morphism·A : A→ ΩA is a monomorphism.

Definition 21.0.16. For an object A of a topos, the membership relation ∈A onA is the subobject

∈A A× ΩA

whose characteristic morphism, still written ∈A: A × ΩA → Ω, corresponds tothe identity on ΩA by cartesian closedness.

Given two morphisms f : B → A, g : B → ΩA, we generally write f ∈A g orjust f ∈ g to denote the composite

B(fg)−−→ A× ΩA ∈A−→ Ω.

Lemma 21.0.17. In a topos, consider global elements a : 1 → A and s : 1 →ΩA; write S A for the subobject corresponding to s. The following areequivalent:

1. (a ∈A s) = t;

2. a factors through S.

Definition 21.0.18. In a topos E , a partial morphism from an object A to anobject B is a pair (s, f) where:

1. s : S A is a subobject of A;

2. f : S → B is a morphism of E .

S

s

f// B

ηB

Aφ// B

Theorem 21.0.19. For every object B of a topos, there exist an object B anda monomorphism ηB : B B with the following property. Given an objectA and a partial morphism (s, f) from A to B, there exists a unique morphismφ : A→ B such that the square of the diagram is a pullback.

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Corollary 21.0.20. Let B be an object of a topos. The global elements of B cor-respond bijectively with the partial morphisms (also called the partial elements)from 1 to B.

Corollary 21.0.21. In a topos, the subobject classifier t : 1 Ω coincides withthe monomorphism η1 : 1 1.

Corollary 21.0.22. In a topos, the construction mapping B to B is functorial.

Proposition 21.0.23. In a topos, each object of the form B is injective, and theobject Ω is injective, as well as every object of the form ΩX .

Corollary 21.0.24. A topos is finitely cocomplete.

Theorem 21.0.25. When I is an object of a topos E , the category E /I of arrowsover I is also a topos.

Proposition 21.0.26. In a topos, finite colimits are universal.

Corollary 21.0.27. A topos is a regular category.

Corollary 21.0.28. In a topos, every epimorphism is regular.

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Corollary 21.0.29. In a topos, every morphism factors uniquely as an epimor-phism followed by a monomorphism.

Proposition 21.0.30. Let f : A → B be a morphism in a topos. There existsa morphism ∃f : ΩA → ΩB whose action on global elements, i.e. on actualsubobjects of A,B, reduces to taking the image of f .

Proposition 21.0.31. In a topos, every equivalence relation is effective.

Corollary 21.0.32. A topos is an exact category.

Proposition 21.0.33. In a topos, the initial object is strict.

Corollary 21.0.34. In a topos, the initial object is a subobject of every object.

Proposition 21.0.35. In a topos, the union of two subobjects always exists. Theunion of two subobjects is effective.

Definition 21.0.36. A functor F : E → F between two toposes is called alogical morphism of toposes when:

1. F preserves finite limits;

2. F preserves the cartesian closed structure;

3. F preserves the subobject classifier.

Proposition 21.0.37. A logical morphism of toposes preserves finite colimits.

138

Chapter 22

Internal Logic of a Topos

Definition 22.0.1. Let E be a topos. The language of E consists in the followingdata. For every object A ∈ E , we give ourselves

• one formal symbol, called a constant of type A, for each global element1→ A of A,

• a denumerable set of formal symbols, called variable of type A.

The terms and formulas of our language are then defined inductively from thefollowing rules. First of all, the terms, which come equipped with a type and aset of free variables:

• each constant of type A is a term of type A, without free variables;

• each variable a of typeA is a term of typeA, with a as unique free variable;

• if τ is a term of typeA and f : A→ B is a morphism, the formal expressionf(τ) is a term of type B, with the same free variables as τ ;

• if τ1, . . . , τn are terms of types A1, . . . , An respectively with the same freevariables, the formal expression (τ1, . . . , τn) is a term of typeA1× . . .×Anwith again the same free variables;

• if φ is a formula with free variables x1, . . . , xn, y1, . . . , ym of typesX1, . . . , Xn, Y1, . . . , Ym(each xi being distinct from each yj), the formal expression

(x1, . . . , xn) : φ

is a term of type ΩX1×...×Xn with free variables y1, . . . , ym;

• if τ is a term of type A with free variables x1, . . . , xn and if x1, . . . , xm(m ≥ n) is a bigger set of variables, the formal expression τ(x1,...,xm) isa term of type A with free variables x1, . . . , xm (which in general we stillwrite τ );

139

CHAPTER 22. INTERNAL LOGIC OF A TOPOS

• let τ be a term of typeA with free variables x1, . . . , xn of typesX1, . . . , Xn;let σ1, . . . , σn be terms of typesX1, . . . , Xn with the same free variables butnot containing any bound variable of τ ; the formal expression τ(σ1, . . . , σn)is a term of type A, with, as free variables, the free variables of σ1, . . . , σn.

Next, the formulas, which come equipped with a set of free variables:

• the symbols true and false are formulas without free variables;

• if τ, σ are terms of type A with the same free variables, the formal expres-sion τ = σ is a formula with again the same free variables;

• if τ is a term of type A and Σ is a term of type ΩA, both terms τ,Σ havingthe same free variables, the formal expression τ ∈ Σ is a formula withagain the same free variables;

• if φ is a formula, the formal expression ¬φ is a formula with the same freevariables as φ;

• if φ, ψ are formulas with the same free variables, the formal expressionsφ ∧ ψ, φ ∨ ψ, φ⇒ ψ are formulas with again the same free variables.

• if φ is a formulas with free variables x, y1, . . . , yn (x being distinct fromeach yi), the expressions ∃x φ, ∀xφ are formulas with free variables y1, . . . , yn;

• if φ is a formula with free variables x1, . . . , xn and if x1, . . . , xm (m ≥ n)is a bigger set of variables, the formal expression φ(x1,...,xm) is a formulawith free variables x1, . . . , xm (which in general we still write φ);

• let φ be a formula with free variables x1, . . . , xn of types X1, . . . , Xn; letσ1, . . . , σn be terms of types X1, . . . , Xn with the same free variables butnot containing any bound variable of the formula φ; then φ(σ1, . . . , σn) isa formula with, as free variables, the free variables of σ1, . . . , σn.

Proposition 22.0.2. In a topos, the subobjects of a given object constitute aHeyting algebra.

Definition 22.0.3. In a topos E , a truth table with free variables x1, . . . , xn oftypes X1, . . . , Xn is a morphism X1 × . . .×Xn → Ω

Definition 22.0.4. In a topos, a realization of a term of type A with free vari-ables x1, . . . , xn of respective typesX1, . . . , Xn is a morphismX1× . . .×Xn →A.

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Definition 22.0.5. In a topos, let φ be a formula with free variables x1, . . . , xnof types X1, . . . , Xn. This formula is valid, written |= φ, when the truth table ofφ is the true on X1,× . . .×Xn, i.e. the composite

X1 × . . .×Xn −→ 1t−→ Ω.

Proposition 22.0.6. A logical morphism of toposes preserves the truth table ofevery formula.

Corollary 22.0.7. A logical morphism of toposes preserves the validity of everyformula, and the Heyting algebra structure of Ω.

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