Part III – Category Theory

Based on lectures by Professor P. T. JohnstoneNotes taken by Bhavik Mehta

Michaelmas 2018

Contents

0 Introduction 2

1 Definitions and Examples 2

2 The Yoneda Lemma 9

3 Adjunctions 14

4 Limits 19

5 Monads 30

6 Cartesian Closed Categories 41

7 Toposes 48

Index 59

1

0 Introduction

CategoryLecture 1 theory is like a language spoken by many different people, with many differentdialects. Specifically, different parts of category theory are used in different branches ofmathematics. In this course, we aim to speak the language of category theory, withoutan accent - a broad overview of all aspects of category theory. There will be manyexamples, some of which may not be understandable. As long as some examples makesense, it is not a point of concern that some examples seem unfamiliar.

1 Definitions and Examples

1.1 Definition (Category). A category C consists of

(a) a collection ob C of objects A,B,C, . . .

(b) a collection mor C of morphisms f, g, h, . . .

(c) two operations dom, cod assigning to each f ∈ mor C a pair of objects, its domain

and codomain. We write A Bf

to mean ‘f is a morphism and dom f = Aand cod f = B’.

(d) an operation assigning to each A ∈ ob C a morphism A A1A , called its identity.

(e) a partial binary operation composition (f, g) 7→ fg on morphisms, such thatfg is defined iff dom f = cod g and dom(fg) = dom g, cod(fg) = cod f if fg isdefined.

satisfying

(f) f1A = f = 1Bf for any A Bf

(g) (fg)h = f(gh) whenever fg and gh are defined

1.2 Remark.

(a) This definition is independent of a model of set theory. If we’re given a particularmodel of set theory, we call the category C small if ob C and mor C are sets.

(b) Some texts say fg means ‘f followed by g’, i.e. fg defined ⇐⇒ cod f = dom g.

(c) Note that a morphism f is an identity iff fg = g and hf = h whenever thecompositions are defined. So we could formulate the definition entirely in termsof morphisms.

1.3 Examples.

(a) The category Set has all sets as objects, and all functions between sets as mor-phisms. (Strictly, morphisms A B are pairs (f,B) where f is a set-theoreticfunction.)

(b) The category Gp has all groups as objects, and group homomorphisms as mor-phisms. Similarly, Rng is the category of rings, ModR the category of R-modules.

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(c) The category Top has all topological spaces as objects and continuous functionsas morphisms. Similarly Unif has uniform spaces and uniformly continuous func-tions, and Mf has manifolds and smooth maps.

(d) The category Htpy has the same objects as Top, but morphisms are homotopyclasses of continuous functions. More generally, given C , we call an equivalencerelation ' on mor C a congruence if f ' g =⇒ dom f = dom g and cod f =cod g, and f ' g =⇒ fh ' gh and kf ' kg whenever the composites are defined.Then we have a category C / ' with the same objects as C , but congruence classesas morphisms.

(e) Given C , the opposite category C op has the same objects and morphisms as C ,but dom and cod are interchanged, and fg in C op is gf in C . This leads to theDuality principle: if P is a true statement about categories, so is the statementP ∗ obtained from P by reversing all arrows.

(f) A small category with one object is a monoid, i.e. a semigroup with 1. In partic-ular, a group is a small category with one object, in which every morphism is anisomorphism (f is an isomorphism if ∃g such that fg and gf are identities).

(g) A groupoid is a category in which every morphism is an isomorphism. For atopological space X, the fundamental groupoid π(X) has all points of X as objectsand morphisms x→ y are homotopy classes rel 0, 1 of paths u : [0, 1]→ X withu(0) = x, u(1) = y. (If you know how to prove that the fundamental group is agroup, you can prove that π(X) is a groupoid.)

(h) A discrete category is one whose only morphisms are identities. A preorder isa category C in which, for any pair (A,B) there is at most 1 morphism A → B.A small preorder is a set equipped with a binary relation which is reflexive andtransitive. In particular, a partially ordered set is a small preorder in which theonly isomorphisms are identities.

(i) The category Rel has the same objects as Set, but morphismsA→ B are arbitraryrelations R ⊆ A×B. Given R and S ⊆ B × C, we define

S R = (a, c) ∈ A× C | (∃b ∈ B)((a, b) ∈ R ∧ (b, c) ∈ S) .

The identity 1A : A→ A is (a, a) | a ∈ A .Similarly, the category Part of sets and partial functions (i.e. relations such that(a, b) ∈ R, (a, b′) ∈ R =⇒ b = b′).

(j) Let K be a field. The category MatK has natural numbers as objects, and mor-phisms n → p are (p × n) matrices with entries from K. Composition is matrixmultiplication.

1.4 Definition (Functor). Let C ,D be categories. A functor F : C → D consists of

(a) a mapping A 7→ FA from ob C to ob D

(b) a mapping f 7→ Ff from mor C to mor D

such that dom(Ff) = F (dom f), cod(Ff) = F (cod f), 1FA = F (1A) and (Ff)(Fg) =F (fg) whenever fg is defined.

1.3 Examples (Continued).Lecture 2

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(k) We write Cat for the category whose objects are all small categories, and whosemorphisms are functors between them.

1.5 Examples.

(a) We have forgetful functors Gp SetU , Rng → Set, Top → Set, Rng →AbGp (forgetting ×), Rng→Mon (forgetting +).

(b) Given a set A, the free group FA has the property: given any group G and

any function A UGf

, there’s a unique homomorphism FA Gf

extending

f . F is a functor Set → Gp: given A Bf

, we define Ff to be the unique

homomorphism extending A B UFBf

.

Functoriality follows from uniqueness: given B Cg

, F (gf) and (Fg)(Ff) are

both homoms extending A B C UFCf g

. Call this the free functor.

(c) Given a set A, we write PA for the set of all subsets of A. We can make P into

a functor Set → Set: given A Bf

, we define Pf(A′) = f(a) | a ∈ A′ forA′ ⊆ A. But we also have a functor P∗ : Set → Setop defined on objects by P,but P∗f(B′) = a ∈ A | f(a) ∈ B′ for B′ ⊆ B.

By a contravariant functor C → D , we mean a functor C → Dop (or C op → D).(A covariant functor is one that doesn’t reverse arrows).

(d) Let K be a field. We have a functor ∗ : ModK → ModopK defined by V ∗ =

linear maps V → K and if V Wf

, f∗(θ : W → K) = θf .

(e) We have a functor op : Cat → Cat which is the ‘identity’ on morphisms. (Notethat this is covariant).

(f) A functor between monoids is a monoid homomorphism.

(g) A functor between posets is an order-preserving map.

(h) Let G be a group. A functor F : G → Set consists of a set A = F∗ togetherwith an action of G on A, i.e. a permutation representation of G (where we use ∗to refer to the unique object of the group). Similarly a functor G →ModK is aK-linear representation of G.

(i) The construction of a fundamental group π1(X,x) of a space X with basepointx is a functor Top∗ → Gp where Top∗ is the category of spaces with a chosenbasepoint. Similarly, the fundamental groupoid is a functor Top → Gpd whereGpd is the category of groupoids and functors between them.

1.6 Definition (Natural transformation). Let C ,D be categories and F,G : C Dtwo functors. A natural transformation α : F → G consists of an assignment A 7→ αAfrom ob C to mor D , such that domαA = FA and codαA = GA for all A, and for all

A Bf

in C the square

FA FB

GA GB

Ff

αA αB

Gf

commutes (i.e. αB(Ff) = (Gf)αA).

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1.3 Examples (Continued).

(l) Given categories C ,D , we write [C ,D ] for the category whose objects are functorsC → D , and whose morphisms are natural transformations.

1.7 Examples.

(a) Let K be a field, V a vector space over K. There is a linear map αV : V → V ∗∗

given byαV (v)(θ) = θ(v)

for θ ∈ V ∗. This is the V -component of a natural transformation

1ModK → ∗∗ : ModK →ModK .

(b) For any set A, we have a mapping σA : A→ PA sending a to a.Lecture 3 If f : A→ B,then Pfa = f(a), so σ is a natural transformation 1Set → P.

(c) Let F : Set→ Gp be the free group functor (Examples 1.5(b)) and U : Gp→ Setthe forgetful functor. The inclusions A → UFA form a natural transformation1Set → UF .

(d) Let G,H be groups and f, g : G→ H two homomorphisms. A natural transforma-tion α : f → g corresponds to an element h = α∗ of H such that h.f(x) = g(x).hfor all x ∈ G, or equivalently f(x) = h−1g(x)h, i.e. f and g are conjugate grouphomomorphisms.

(e) Let A,B be two G-sets regarded as functors G→ Set. A natural transformationA→ B is a function f satisfying f(g.a) = g.f(a) for all a ∈ A, i.e. a G-equivariantmap.

1.8 Lemma. Let F,G : C → D be two functors, and α : F → G a natural transforma-tion. Then α is an isomorphism in [C ,D ] iff each αA is an isomorphism in D .

Proof.

=⇒ trivial

⇐= Suppose each αA has an inverse βA. Given f : A→ B in C , we need to show that

GA GB

FA FB

Gf

βA βB

Ff

commutes.

But

(Ff)βA = βBαB(Ff)βA

= βB(Gf)αAβA

= βB(Gf).

1.9 Definition (Equivalent category). Let C ,D be categories. By an equivalencebetween C and D , we mean a pair of functors F : C → D , G : D → C together withnatural isomorphisms α : 1C → GF and β : FG → 1D . We write C ' D if C and Dare equivalent.

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We say a property P of categories is a categorical property if whenever C has Pand C ' D , then D has P .

For instance, being a groupoid or a preorder are categorical properties, but being agroup or a partial order are not.

1.10 Examples.

(a) The category Part is equivalent to the category Set∗ of pointed sets (and basepointpreserving functions). We define F : Set∗ → Part by F (A, a) = A \ a and iff : (A, a)→ (B, b)

Ff(x) =

f(x) if f(x) 6= b

undefined otherwise

and G : Part → Set∗ by G(A) = A+ = A ∪ A and if f : A → B is a partialfunction, we define Gf : A+ → B+ by

Gf =

f(x) if x ∈ A and f(x) defined

B otherwise

The composite FG is the identity on Part, but GF is not the identity, howeverthere’s an isomorphism

(A, a)→ ((A \ a)+, A \ a)

sending a to A \ a and everything else to itself and this is natural.

Note that there can be no isomorphism Set∗ → Part since Part has a 1-elementisomorphism class ∅ and Set∗ doesn’t.

(b) The category FdModK of finite-dimensional vector spaces over K is equivalentto FdModop

K : the functors in both directions are (−)∗ and both isomorphisms arethe natural transformations of Examples 1.7(a).

(c) FdModK is also equivalent to MatK : We define F : MatK → FdModK byF (n) = Kn, and F (A) is the linear map represented by A with respect to thestandard bases of Kn and Kp.

To define G : FdModK → MatK , choose a basis for each finite dimensional

vector space,and define G(V ) = dimV , G(Vf−→ W ) as the matrix representing

f with respect to the chosen bases. GF is the identity, provided we choose thestandard bases for the spacesKn; FG 6= 1, but the chosen basis gives isomorphismsFG(V ) = KdimV → V for each V , which form a natural isomorphism.

1.11 Definition (Faithful, full, essentially surjective). Let C DF be a functor.Lecture 4

(a) We say F is faithful if, given f, f ′ ∈ mor C with dom f = dom f ′, cod f = cod f ′

and Ff = Ff ′ then f = f ′.

(b) We say F is full if, given FA FBg

in D , there exists A Bf

in C withFf = g.

(c) We say F is essentially surjective if, for every B ∈ ob D , there exists A ∈ ob Cand an isomorphism FA→ B in D .

We say a subcategory C ′ ⊆ C is full if the inclusion C ′ → C is a full functor.

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Example. Gp is a full subcategory of Mon, but Mon is not a full subcategory of thecategory Sgp of semigroups.

1.12 Lemma. Assuming the axiom of choice, a functor F : C → D is part of anequivalence C ' D iff it is full, faithful and essentially surjective.

Proof.

=⇒ Given G,α, β as in Definition 1.9, for each B ∈ ob D , βB is an isometry FGB → B,so F is essentially surjective.

Given A Bf

in C, we can recover f from Ff as the composite

A GFA GFB B.αA GFf α−1

B

Hence if A Bf ′

satisfies Ff = Ff ′, then f = f ′.

Given FA FBg

, define f to be the composite

A GFA GFB BαA Gg α−1

B

Then GFf = αBfα−1A = Gg, and G is faithful for the same reason as F , so

Ff = g.

⇐= For each B ∈ ob D , choose GB ∈ ob C and an isomorphism βB : FGB → B in D .Given

B B′g

define Gg : GB → GB′ to be the unique morphism whose image under F is

FGB B B′ FGB′βB g β−1

B′

Uniqueness implies functoriality: given

B′ B′′g′

then note (Gg′)(Gg) and G(g′g) have the same image under F , so they’re equal.

By construction, β is a natural transformation FG→ 1D .

Given A ∈ ob C , define αA : A → GFA to be the unique morphism whose imageunder F is

FA FGFAβ−1FA

αA is an isomorphism, since βFA also has a unique pre-image under F .

Also α is a natural transformation, since any naturality square for α is mappedby F to a commutative square, and F is faithful.

1.13 Definition (Skeleton). By a skeleton of a category C , we mean a full subcategoryC0 containing one object from each isomorphism class. We say C is skeletal if it’s askeleton of itself.

Example. MatK is skeletal, and the image of F : MatK → FdModK of Exam-ples 1.10(c) is a skeleton of FdModK .

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Warning. Almost any assertion about skeletons is equivalent to the axiom of choice.See question 2 on example sheet 1.

1.14 Definition (Monomorphism, epimorphism). Let Af−→ B be a morphism in C

(a) We say f is a monomorphism (or f is monic) if, given any pair C A,f

g

fg = fh implies g = h

(b) We say f is an epimorphism (or epic) if it’s a monomorphism in Cop i.e. ifgf = hf implies g = h.

We denote monomorphisms by A Bf

and epimorphisms by A Bf

Any isomorphism is monic and epic: more generally if f has a left inverse (i.e. ∃gsuch that gf is an identity) then it’s monic. We call such monomorphisms split.

We say C is a balanced category if any morphism which is both monic and epic isan isomorphism.

1.15 Examples.

(a) In Set, mono ⇐⇒ injective ( =⇒ easy; for ⇐= take C = 1 = ∗) and epi⇐⇒ surjective ( =⇒ easy; for ⇐= use two morphisms B → 2 = 0, 1). So Setis balanced.

(b) In Gp mono ⇐⇒ injective (for ⇐= use homoms Z → A) and epi ⇐⇒surjective (⇐= uses free products with amalgamation). So Gp is balanced.

(c) In Rng, mono ⇐⇒ injective (proof much as for Gp) but the inclusion Z→ Q is

an epimorphism, since if Q Rf

gagree on all integers, they agree everywhere.

So Rng isn’t balanced.

(d) In Top, mono ⇐⇒ injective and epi ⇐⇒ surjective (proofs as in Set). ButTop isn’t balanced since a continuous bijection needn’t have a continuous inverse.

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2 The Yoneda Lemma

2.1 Definition (Locally small). WeLecture 5 say a category C is locally small if, for any twoobjects A,B, the morphisms A→ B in C form a set C (A,B).

If we fix A and let B vary, the assignment B 7→ C (A,B) becomes a functor C (A,−) :

C → Set: given B Cf

, C (A, f) is the mapping g 7→ fg. Similarly, A 7→ C (A,B)defines a functor C (−, B) : C op → Set.

2.2 Lemma (Yoneda Lemma). Let C be a locally small category, A ∈ ob C andF : C → Set a functor.

(i) Then natural transformations C (A,−)→ F are in bijection with elements of FA.

(ii) Moreover, this bijection is natural in both A and F .

Proof of Yoneda Lemma(i). Given α : C (A,−)→ F , we define

Φ(α) = αA(1A) ∈ FA.

Given x ∈ FA, we define Ψ(x) : C (A,−)→ F by

Ψ(x)B(Af−→ B) = (Ff)(x) ∈ FB.

Ψ(x) is natural: given g : B → C, we have

Ψ(x)CC (A, g)(f) = Ψ(x)C(gf) = F (gf)(x)

(Fg)Ψ(x)B(f) = (Fg)(Ff)(x) = F (gf)(x).

C (A,B) C (A,C)

FB FC

C (A,g)

Ψ(x)B Ψ(x)C

Fg

We also verify Ψ and Φ are inverse:

ΦΨ(x) = Ψ(x)A(1A) = F (1A)(x) = x.

Given α,

ΨΦ(α)B(f) = Ψ(αA(1A))B(f) = Ff(αA(1A))

= αBC (A, f)(1A) = αB(f)

so ΨΦ(α) = α.

2.3 Corollary. The assignment A 7→ C (A,−) defines a full and faithful functor C op →[C ,Set].

Proof. Put F = C (B,−) in Lemma 2.2(i): we get a bijection between C (B,A) andmorphisms C (A,−)→ C (B,−) in [C ,Set]. We need to verify this is functorial: but itsends f : B → A to the natural transformation g 7→ gf . So functoriality follows fromassociativity.

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We call this functor (or the functor C → [C op,Set]) sending A to C (−, A) theYoneda embedding of C , and denote it by Y .

Proof of Yoneda Lemma(ii). Suppose for the moment that C is small, so that [C ,Set]is locally small. Then we have two functors C × [C ,Set] → Set: One sends (A,F ) toFA, and the other is the composite

C × [C ,Set] [C ,Set]op × [C ,Set] Set

Y×1 [C ,Set](−,−)

Yoneda Lemma(ii) says that these are naturally isomorphic.

We can translate this into an elementary statement, making sense even when C isn’t

small, given Af−→ B and F

α−→ G, the two ways of producing an element of GB from anatural transformation β : C (A,−)→ F give the same result, namely

αB(Ff)βA(1A) = (GF )αAβA(1A)

which is equal to αBβB(f).

2.4 Definition. We say a functor F : C → Set is representable if it’s isomorphic toC (A,−) for some A. By representation of F , we mean a pair (A, x) where x ∈ FA issuch that Ψ(x) is an isomorphism. We also call x a universal element of F .

2.5 Corollary. If (A, x) and (B, y) are both representations of F , then there’s a uniqueisomorphism f : A→ B such that (Ff)(x) = y.

Proof. Consider the composite

C (B,−) F C (A,−)Ψ(y) Ψ(x)−1

By Corollary 2.3, this is of the form Y (f) for a unique isomorphism f : A→ B and thediagram

C (B,−) C (A,−)

F

Y (f)

Ψ(y) Ψ(x)

commutes iff (Ff)x = y.

2.6 Examples.

(a) The forgetful functor Gp→ Set is representable by (Z, 1). Similarly, the forgetfulfunctor Rng→ Set is representable by (Z[x], x) and the forgetful functor Top→Set is representable by (∗, ∗).

(b) The functor P∗ : Setop → Set (defined in Examples 1.5(c)) is representable by(0, 1, 1): this is the bijection between subsets and characteristic functions.

(c) Let G be a group. The unique (up to isomorphism) representable functor G(∗,−) :G→ Set is the Cayley representation of G, i.e. the set UG with G acting by leftmultiplication.

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(d) Let A,B be two objects of a locally small category C . We have a functor C op →Set sending C to C (C,A)×C (C,B). A representation of this, if it exists, is calleda (categorical) product of A and B, and denoted

(A×B, (A×B π1−→ A,A×B π2−→ B)).

This pair has the property that, for any pair (Cf−→ A,C

g−→ B) there’s a unique

Ch−→ A×B with π1h = f and π2h = g.

Products exist in many categories of interest: in Set,Gp,Rng,Top they are ‘just’cartesian products, in posets they are binary meets.

Dually we have the notion of coproduct (A + B, (Aν1−→ A + B,B

ν2−→ A + B)).These also exist in many categories of interest.

(e)Lecture 6 The dual-vector-space functor ModopK → ModK , when composed with the for-

getful functor ModK → Set, is representable by (K, 1K).

(f) Let A Bf

gbe morphisms in a locally small category C . We have a functor

F : C op → Set defined by

F (C) = h ∈ C (C,A) | fh = gh .

A representation of F , if it exists, is called an equalizer of (f, g). It consists of

an objects E and a morphism Ee−→ A such fe = ge, and every h with fh = gh

factors uniquely through e. In Set, we can take E = x ∈ A | f(x) = g(x) ande = inclusion. Similar constructions work in Gp,Rng,Top, . . . .

Dually, we have the notion of coequalizer.

2.7 Remark. If e occurs as an equalizer, then it’s a monomorphism, since any h factorsthrough it in at most one way. We say a monomorphism is regular if it occurs as anequalizer.

Split monomorphisms are regular (c.f. question 6i on sheet 1). Note that regularmono + epi =⇒ iso: if the equalizer e of (f, g) is epic, then f = g, so e ∼= 1cod e.

2.8 Definition (Separating, detecting families). Let C be a category, and G a class ofobjects of C .

(a) We say G is a separating family for C if, given A Bf

gsuch that fh = gh

for all Gh−→ A with G ∈ G , then f = g (i.e. the functors C (G,−), G ∈ G are

collectively faithful).

(b) We say G is a detecting family for C if, given Af−→ B such that every G

h−→ Bwith G ∈ G factors uniquely through f , then f is an isomorphism.

If G = G, we call G a separator/detector.

2.9 Lemma.

(i) If C is a balanced category, then any separating family is detecting.

(ii) If C has equalizers, then any detecting family is separating.

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Proof.

(i) Suppose G is separating and Af−→ B satisfies the condition of Definition 2.8(b).

If B Cg

hsatisfy gf = hf , then gx = hx for every G

x−→ B, so g = h, i.e. f is

epic. Similarly if D Ak

lsatisfy fk = fl, then ky = ly for any G

y−→ D, since

both are factorisations of fky through f . So k = l, i.e. f is monic.

(ii) Suppose G is detecting and A Bf

gsatisfies the condition of 2.8(a) . Then the

equalizer Ee−→ A is an isomorphism, so f = g.

2.10 Examples.

(a) In [C ,Set] the familyC (A,−) | A ∈ ob C

is both separating and detecting (this is just a restatement of Yoneda Lemma.)

(b) In Set, 1 = ∗ is both a separator and a detector since it represents the identityfunctor Set→ Set.

Similarly, Z is both in Gp, since it represents the forgetful functor Gp→ Set.

And 2 = 0, 1 is a coseparator and a codetector in Set, since it represents P∗ :Setop → Set.

(c) In Top, 1 = ∗ is a separator since it represents the forgetful functor Top→ Set,but not a detector. In fact, Top has no detecting set of objects:

For any infinite cardinal κ, let X be a discrete space of cardinality κ and let Y bethe same set with ‘co-< κ’ topology, i.e. F ⊆ Y closed ⇐⇒ F = Y or cardF < κ.The identity X → Y is continuous, but not a homeomorphism.

So if Gi | i ∈ I is any set of spaces, taking κ > cardGi for all i yields an exampleto show that the set is not detecting.

(d) Let C be the category of pointed connected CW-complexes and homotopy classes

of (basepoint-preserving) continuous maps. JHC Whitehead proved that if Xf−→ Y

in this category induces isomorphisms πn(X) → πn(Y ) for all n, then it’s anisomorphism in C . This says that Sn | n ≥ 1 is a detecting set for C .

But PJ Freyd showed there is no faithful functor C → Set, so no separating set :if Gi | i ∈ I were separating, then

x 7→∐i∈I

C (Gi, X)

would be faithful.

Note that any functor of the form C (A,−) preserves monomorphisms, but they don’tnormally preserve epimorphisms.

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2.11 Definition (Projective). We say an object P is projective if, given

P

A B

f

e

there exists Pg−→ A with eg = f . (If C is locally small, this says C (P,−) preserves

epimorphisms).

Dually, an injective object of C is a projective object of C op. Given a class E ofepimorphisms, we say P is E -projective if it satisfies the condition for all e ∈ E .

2.12 Lemma. Representable functors are (pointwise) projective in [C ,Set].

Proof. Take

C (A,−)

F G

β

a

where α is pointwise surjective. By Yoneda Lemma, β corresponds to some y ∈ GA,and we can find x ∈ FA with αA(x) = y. Now if γ : C (A,−) → F corresponds to xthen naturality of the Yoneda bijection yields αγ = β.

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3 Adjunctions

3.1 Definition. LetLecture 7 C and D be two categories and CF−→ D , D

G−→ C two functors.

By an adjunction between F and G we mean a bijection between morphisms FAf−→ B

in D and morphisms Af−→ GB in C which is natural in A and B, i.e. given A′

g−→ A and

Bh−→ B′, we have hf(Fg) = (Gh)fg : FA′ → B′.

We say F is left adjoint to G and write F a G.

3.2 Examples.

(a) The free functor SetF−→ Gp is left adjoint to the forgetful functor Gp

U−→ Set,

since any function f : A→ UB extends uniquely to a homomorphism f : FA→ B.Naturality in B is easy, naturality in A follows from the definition of F as a functor.

(b) The forgetful functor TopU−→ Set has a left adjoint D which equips any set with

the discrete topology and a right adjoint I which equips a set A with the indiscretetopology ∅, A.

(c) The functor ob : Cat → Set has a left adjoint D sending A to the discretecategory with ob(DA) = A and only identity morphisms. It also has a right adjointI sending A to the (indiscrete) category with ob(IA) = A and one morphismx→ y for each (x, y) ∈ A×A. In this case D in turn has a left adjoint π0 sendinga small category C to its set of connected components, i.e. the quotient of ob C bythe smallest equivalent relation identifying dom f with cod f for all f ∈ mor C .

(d) Let M be the monoid 1, e with e2 = e. An object of [M ,Set] is a pair (A, e)where e : A→ A satisfies e2 = e.

We have a functor G : [M ,Set]→ Set sending (A, e) to

x ∈ A | e(x) = x = e(x) | x ∈ A

and a functor F : Set→ [M ,Set] sending A to (A, 1A).

Claim F a G a F : given f : (A, 1A)→ (B, e): it must take values in G(B, e), andany g : (B, e)→ (A, 1A) is determined by its values on the image of e.

(e) Let 1 be the discrete category with one object ∗. For any C , there’s a uniquefunctor C → 1: a left adjoint for this picks out an initial object of C , i.e. anobject I such that there exists a unique I → A for each A ∈ ob C . Dually, a rightadjoint for C → 1 corresponds to a terminal object of C .

(f) Let Af−→ B be a morphism in Set. We can regard PA and PB as posets, and we

have functors

PA PBPf

P∗f

Claim (Pf a P∗f): we have Pf(A′) ⊆ B′ ⇐⇒ f(x) ∈ B′ for all x ∈ A′ ⇐⇒A′ ⊆ P∗f(B′).

(g) Suppose given sets A,B and a relation R ⊆ A×B. We define mappings (−)l, (−)r

between PA and PB by

Sr = y ∈ B | (∀x ∈ S)((x, y) ∈ R) for S ⊆ AT l = x ∈ A | (∀y ∈ T )((x, y) ∈ R) for T ⊆ B.

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These mappings are order-reversing (i.e. contravariant functors) and

T ⊆ Sr ⇐⇒ S × T ⊆ R ⇐⇒ S ⊆ T l.

We say (−)r and (−)l are adjoint on the right.

(h) The functor P∗ : Setop → Set is self-adjoint on the right, since functions A→ PBcorrespond bijectively to subsets of A×B and hence to functions B → PA.

Definition (Comma category). (A ↓ G) is the comma category with objects pairs

(B, f) with Af−→ GB, and morphisms (B, f) → (B′, f ′) are morphisms B

g−→ B′ suchthat

A

GB GB′

f f ′

Gg

commutes.

3.3 Theorem. Let G : D → C be a functor. Then specifying a left adjoint for G isequivalent to specifying an initial object of (A ↓ G) for each A ∈ ob C .

Proof. Suppose we are given F a G. Consider the morphism ηA : A → GFA cor-

responding to FA1−→ FA. Then (FA, ηA) is an object of (A ↓ G). Moreover, given

g : FA→ B and f : A→ GB, the diagram

A

GFA GB

ηA f

Gg

commutes iffFA

FA B

1FA f

g

commutes, i.e. g = f . So (FA, ηA) is initial in (A ↓ G).

Conversely, suppose we are given an initial object (FA, ηA) for each (A ↓ G). Given

Af−→ A′, we define Ff : FA→ FA′ to be the unique morphism making

A GFA

A′ GFA′

f

ηA

GFf

ηA′

commute. Functoriality follows from uniqueness: given f ′ : A′ → A′′, both F (f ′f) and(Ff ′)(Ff) are morphisms (FA, ηA)→ (FA′′, ηA′′f

′f) in (A ↓ G).Lecture 8

To show F a G: given Af−→ GB, we define f : FA→ B to be the unique morphism

(FA, ηA)→ (B, f) in (A ↓ G). This is a bijection with inverse

(FAg−→ B) 7→ (A

ηA−−→ GFAGg−−→ GB)

The latter mapping is natural in B since G is a functor, and in A since, by construction,η is a natural transformation 1C → GF .

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3.4 Corollary. If F and F ′ are both left adjoint to G : D → C , then they are naturallyisomorphic.

Proof. For any A, (FA, ηA) and (F ′A, η′A) are both initial in (A ↓ G), so there’s a uniqueisomorphism αA : (FA, ηA) → (F ′A, η′A). In any naturality square for α, the two waysround are both morphisms in (A ↓ G) where the domain is initial, so they are equal.

3.5 Lemma. Given

C D EF

G

H

K

with (F a G) and (H a K) we have (HF a GK).

Proof. We have bijections between morphisms A → GKC, morphisms FA → KC andmorphisms HFA→ C, which are both natural in A and C.

3.6 Corollary. Given a commutative square

C D

E F

of categories and functors, if the functors all have left adjoints, then the diagram of leftadjoints commutes up to natural isomorphism.

Proof. By Lemma 3.5, both ways round the diagram of left adjoints are left adjoint tothe composite C → F , so by Corollary 3.4 they are isomorphic.

Given an adjunction (F a G), the natural transformation η : 1C → GF emerging inthe proof of Theorem 3.3 is called the unit of the adjunction. Dually, we have a natural

transformation ε : FG → 1D such that εB : FGB → B corresponds to GB1GB−−−→ GB,

called the counit.

3.7 Theorem. Given functors C DF

Gspecifying an adjunction (F a G) is equiv-

alent to specifying natural transformations η : 1C → GF , ε : FG → 1D satisfying thecommutative diagrams

F FGF

F

Fη

1FεF and

G GFG

G

ηG

1GGε

called the triangular identities.

Proof. First suppose given (F a G). Define η and ε as in Theorem 3.3 and its dual; nowconsider the composite

FA FGFA FA.FηA εFA

Under the adjunction this corresponds to

A GFA GFAηA 1GFA

but this also corresponds to 1FA, so εFA · FηA = 1FA. The other identity is dual.

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Conversely, suppose given η and ε satisfying the triangular identities. Given Af−→

GB, let Φ(f) be the composite

FA FGB B,Ff εB

and given FAg−→ B, let Ψ(g) be

A GFA GB.ηA Gg

Then Φ and Ψ are both natural; we need to show that ΦΨ and ΨΦ are identity mappings.But

ΨΦ

(A GB

f)

= A GFA GFGB GBηA GFf GεB

= A GB GFGB GBf ηGB GεB

= f

and dually ΦΨ(g) = g.

3.8 Lemma. Suppose given

C DF

G

and natural isomorphisms α : 1C → GF , β : FG → 1D . Then there are isomorphismsα′ : 1C → GF , β′ : FG → 1D which satisfy the triangular identities, so (F a G) (and(G a F )).

Proof. We define α′ = α and β′ to be the composite

FG FGFG FG 1D .(FGβ)−1 (FαG)−1 β

Note that FGβ = βFG since

FGFG FG

FG 1D

FGβ

βFG β

β

commutes by naturality of β and β is monic. Now (β′F )(Fα′) is the composite

F FGF FGFGF FGF FFα (βFGF )−1 (FαGF )−1 βF

= F FGF FGFGF FGF F(βF )−1

FGFα (FαGF )−1 βF

= F FGF F(βF )−1 βF

= 1F

since GFα = αGF . Similarly (Gβ′)(α′G) is

G GFG GFGFG GFG GαG (GFGβ)−1 (GFαG)−1 Gβ

= G GFG GFGFG GFG G(Gβ)−1

αGFG (GFαG)−1 Gβ

= G GFG G(Gβ)−1 Gβ

= 1G.

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3.9 Lemma. Suppose G : D → C has a left adjoint F with counit ε : FG→ 1D , then

(i) G is faithful iff ε is pointwise epic.

(ii) G is full and faithful iff ε is an isomorphism.

Proof.

(i) Given Bg−→ B′, Gg corresponds under the adjunction to the composite

FGB B B′.εB g

Hence the mapping g 7→ Gg is injective on morphisms with domain B (and speci-fied codomain) iff g 7→ gεB is injective, i.e. iff εB is epic.

(ii) Similarly, G is full and faithful iff g 7→ gεB is bijective. If α : B → FGB is suchthat αεB = 1FGB , then εBαεB = εB , whence εBα = 1B . So εB is an isomorphismfor all B.

3.10 Definition (Reflection). ByLecture 9 a reflection, we mean an adjunction in which theright adjoint is full and faithful (equivalently, the counit is an isomorphism). We say asubcategory C ′ ⊆ C is reflective if the inclusion C ′ → C has a left adjoint.

3.11 Examples.

(a) The category AbGp of abelian groups is reflective in Gp: the left adjoint sendsa group G to its abelianization G/G′, where G′ is the subgroup generated by allcommutators [x, y] = xyx−1y−1, x, y ∈ G. (The unit of the adjunction is thequotient map G 7→ G/G′.)

(b) Given an abelian group A, let At denote the torsion subgroup, i.e. the subgroupof elements of finite order. The assignment A 7→ A/At gives a left adjoint to theinclusion tfAbGp→ AbGp where tfAbGp is the full subcategory of torsion-freeabelian groups. And A 7→ At is right adjoint to the inclusion tfAbGp→ AbGpso this subcategory is coreflective.

(c) Let KHaus ⊆ Top be the full subcategory of compact Hausdorff spaces. Theinclusion KHaus→ Top has a left adjoint β, the Stone-Cech compactification.

(d) Let X be a topological space. We say A ⊆ X is sequentially closed if xn → x∞and xn ∈ A for all n implies x∞ ∈ A. We say X is sequential if all sequentiallyclosed sets are closed. Given a non-sequential space X, let Xs be the same setwith topology given by the sequentially open sets in X; the identity Xs → S iscontinuous, and defines the counit of an adjunction between the inclusion Seq→Top and the functor X 7→ Xs.

(e) If X is a topological space, the poset CX of closed subsets of X is reflective in PX,with reflector given by closure, and the poset OX of open subsets is coreflective,with coreflector given by interior.

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4 Limits

4.1 Definition.

(a) Let J be a category (almost always small, often finite). By a diagram of shapeJ in C we mean a functor D : J → C . The objects D(j), j ∈ ob J are calledvertices of the diagram, and the morphisms D(α), α ∈ mor J are called edges ofD. For example, if J is the category

• •

• •

with 4 objects and 5 non-identity morphisms, a diagram of shape J is a commu-tative square

A B

C D

f

g h

k

If J is• •

• •

a diagram of shape J is a not-necessarily commutative square.

(b) Given D : J → C , a cone over D consists of an object A of C (the apex of the

cone) together with morphisms Aλj−→ D(j) for each j ∈ ob J , such that

A

D(j) D(j′)

λj λ′j

D(α)

commutes for all jα−→ j′ in mor J (the λj are called the legs of the cone).

A

•• •

•D

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Given cones (A, (λj)j∈ob J) and (B, (µj)j∈ob J), a morphism of cones between them

is a morphism Af−→ B such that

A B

D(j)

f

λj µj

commutes for all j.

We write Cone(D) for the category of cones over D.

(c) A limit for D is a terminal object of Cone(D), if this exists. Dually, we have thenotion of cone under a diagram, and of colimit (initial cone under D).

Alternatively if C is locally small and J is small, we have a functor C op → Setsending A to the set of cones with apex A. A limit for D is a representation of thisfunctor.

If ∆A denotes the constant diagram of shape J with all vertices A and all edges 1A,then a cone over D with apex A is the same thing as a natural transformation ∆A→ D.∆ is a functor C → [J,C ], and Cone(D) is the category (∆ ↓ D) (a comma category,reversed). So to say that every diagram of shape J in C has a limit is equivalent tosaying that ∆ has a right adjoint. (We say C has limits of shape J). Dually, C hascolimits of shape J iff ∆ : C → [J,C ] has a left adjoint.

4.2 Examples.

(a) Suppose J = ∅. There’s a unique diagram of shape J in C ; a cone over it is justan object, and a morphism of cones is a morphism of C . So a limit for this emptydiagram is a terminal object of C . (Dually, a colimit for it is an initial object).

(b)Lecture 10 Let J be the category• •

A diagram of shape J is a pair of objects A,B; a cone over it is a span

C

A B

and a limit for it is a product

A×B

A B

π1 π2

Dually, a colimit for it is a coproduct

A B

A+B

ν1 ν2

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(c) More generally, if J is a small discrete category, a diagram of shape J is a J-indexedfamily Aj | j ∈ J , and a limit for it is a product∏

j∈JAj

πj−→ Aj

∣∣∣∣∣∣ j ∈ J .

Dually, a colimit for it is a coproductAjνj−→∑j∈J

Aj

∣∣∣∣∣∣ j ∈ J ,

also written∐j∈J Aj .

(d) Let J be the category• •

A diagram of shape J is a parallel pair

A Bf

g

a cone over this isC

A B

h k

satisfying fh = k = gh, or equivalently a morphism Ch−→ A satisfying fh = gh.

A (co)limit for the diagram is a (co)equalizer.

(e) Let J be the category•

• •

A diagram of shape J is a cospan

A

B C

f

g

a cone over it isD A

B C

p

q r

satisfying fp = r = gq, or equivalently a span (p, q) completing the diagram to acommutative square. A limit for the diagram is called a pullback of (f, g). InSet, the apex of the pullback is the ‘fibre product’

A×C B = (x, y) ∈ A×B | f(x) = g(y)

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Dually, colimits of shape Jop are called pushouts, given

A B

C

f

g

we ‘push g along f ’ to get the right hand side of the colimit square.

(f) Let J be the poset of natural numbers. A diagram of shape J is a direct system

A0 A1 A2 A3 . . .f0 f1 f2 f3

A colimit for this is called a direct limit: it consists of A∞ equipped with mor-

phisms Angn−→ A∞ satisfying gn = gn+1fn for all n, and universal among such.

Dually, we have inverse system and inverse limit.

4.3 Theorem.

(i) Suppose C has equalizers and all finite (resp. small) products. Then C has allfinite (resp. small) limits.

(ii) Suppose C has pullbacks and a terminal object. Then C has all finite limits.

Proof.

(i) Suppose given D : J → C . Form the products

P =∏

j∈ob J

D(j) and Q =∏

α∈mor J

D(codα).

We have morphisms P Qf

gdefined by παf = πcodα, παg = D(α)πdomα for

all α.

Let Ee−→ P be an equalizer of (f, g). The composites λj = πje : E → D(j) form

a cone over D: given α : j → j′ in J ,

D(α)λj = D(α)πje = παge = παfe = πj′e = λj′ .

Given any cone (A, µj | j ∈ ob J ) over D, there’s a unique µ : A → P withπjµ = µj for each j, and

παfµ = µcodα = D(α)µdomα = παgµ

for all α, and hence fµ = gµ, so ∃!ν : A→ E with eν = µ. So (E, λj | j ∈ ob J )is a limit cone.

(ii) It’s enough to construct finite products and equalizers. But if 1 is the terminalobject, then a pullback for

A

B 1

has the universal property of a product A×B, and we can form∏ni=1Ai inductively

as A1 × (A2 × (A3 × · · · (An−1 ×An) · · · )).

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Now, to form the equalizer of A Bf

g, consider the cospan

A

A A×B.

(1A,f)

(1A,g)

A cone over this consists of

P A

A

h

k

satisfying (1A, f)h = (1A, g)k or equivalently, 1Ah = 1Ak and fh = gk, or equiv-

alently a morphism Ph−→ A satisfying fh = gh. So a pullback for (1A, f) and

(1A, g) is an equalizer of (f, g).

Definition (Complete). We say a category C is complete if it has all small limits.(Dually, cocomplete = all small colimits).

Set is complete and cocomplete: products are cartesian products, coproducts are dis-joint unions. Similarly, Gp,AbGp,Rng,ModR, . . . are all complete and cocomplete.Top is also complete and cocomplete.

4.4 Definition. Let F : C → D be a functor.

(a) We say F preserves limits of shape J if, given D : J → C and a limit cone(L, λj | j ∈ ob J ) in C , (FL, Fλj | j ∈ ob J ) is a limit for FD.

(b) We say F reflects limits of shape J if, given D : J → C and a cone (L, (λj)j)such that (FL, (Fλj)j) is a limit for FD, then (L, (λj)j) is a limit for D.

(c) We say F creates limits of shape J if, given D : J → C and a limit (M, (µj)j)for FD, there exists a cone (L, (λj)j) over D whose image under F is isomorphicto the limit cone, and any such that cone is a limit in C .

4.5 Remark.Lecture 11

(a) If C has limits of shape J , F : C → D preserves them and F reflects isomorphisms,then F reflects limits of shape J .

(b) F reflects limits of shape 1 ⇐⇒ F reflects isomorphisms.

(c) If D has limits of shape J and F : C → D creates them, then F both preservesand reflects them.

(d) In any of the statements of Theorem 4.3, we may replace both instances of ‘C has’by either ‘C has and F : C → D preserves’ or ‘D has and F : C → D creates’.

4.6 Examples.

(a) U : Gp → Set (forgetful) creates all small limits: Given a family Gi | i ∈ I of groups, there’s a unique group structure on

∏i∈I UGi making the projections

πi into homomorphisms and this makes it into a product in Gp. Similarly forequalizers.

But U doesn’t preserve coproducts: U(G ∗H) UG t UH.

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(b) U : Top → Set (forgetful) preserves all small limits and colimits, but doesn’treflect them: if L is a limit for D : J → Top, and L is not discrete, there’s anothercone with apex Ld mapping to the limit in Set.

(c) The inclusion functor I : AbGp → Gp reflects coproducts but doesn’t preservethem. The direct sum A ⊕ B (coproduct in AbGp) is not normally isomorphicto the free product A ∗B; A ∗B is not abelian unless either A or B is e, but ifA ∼= e then A ∗B ∼= A⊕B ∼= B.

4.7 Lemma. If D has limits of shape J , then so does the functor category [C ,D ] forany C , and the forgetful functor [C ,D ]→ Dob C creates them.

Proof. Suppose we are given a diagram of shape J in [C ,D ]; think of it as a functorD : J × C → D . For each A ∈ ob C , let (LA, λj,A | j ∈ ob J ) be a limit cone for the

diagram D(−, A) : J → D . Given Af−→ B in C , the composites

Aλj,A−−−→ D(j, A)

D(j,f)−−−−→ D(j, B)

form a cone over D(−, B), since the squares

D(j, A) D(j, B)

D(j′, A) D(j′, B)

D(j,f)

D(α,A) D(α,B)

D(j′,f)

commute. So there’s a unique Lf : LA→ LB making

LA D(j, A)

LB D(j, B)

λj,A

Lf D(j,f)

λj,B

commute for all j.

Uniqueness implies functoriality: given g : B → C, L(gf) and (Lg)(Lf) are factor-izations of the same cone through the limit LC. And this is the unique functor structureon (A 7→ LA) making the λj,− into natural transformations.

The cone (L, λj,− | j ∈ ob J) is a limit: suppose given another cone (M, µj,− |j ∈ ob J), then for each A, (MA, µj,A | j ∈ ob J ) is a cone over D(−, A), so inducesa unique αA : MA → LA. Naturality of α follows from uniqueness of factorisationsthrough a limit. So (M, (µj)) factors uniquely through (L, (λj)).

4.8 Remark. In any category, a morphism Af−→ B is monic iff

A A

A B

1A

1A f

f

is a pullback. Hence any functor which preserves pullbacks preserves monomorphisms.In particular, if D has pullbacks, then monomorphisms in [C ,D ] are just pointwisemonomorphisms.

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4.9 Theorem. Suppose G : D → C has a left adjoint F . Then G preserves all limitswhich exist in D .

Proof 1. Suppose C and D both have limits of shape J . We have a commutative diagram

C D

[J,C ] [J,D ]

F

∆ ∆

[J,F ]

and all the functors in it have right adjoints (in particular [J, F ] a [J,G]). So byCorollary 3.6, the diagram of right adjoints

D C

[J,D ] [J,C ]

G

[J,G]

limJ limJ

commutes up to isomorphism, i.e. G preserves limits of shape J .

Proof 2. Suppose givenD : J → D and a limit cone (L, L λj−→ D(j) | j ∈ ob J ). Given

a cone (A, A αj−→ GD(j) | j ∈ ob J ) over GD the morphisms FAαj−→ D(j) form a cone

over D, so they induce a unique FAβ−→ L such that λj β = αj for all j. Then A

β−→ GLis the unique morphism satisfying (Gλj)β = αj for all j. So (GL, Gλj | j ∈ ob J ) isa limit cone in C .

Lecture 12 The ‘Primeval’ Adjoint Functor Theorem says that the converse of Theorem 4.9 istrue: if D has and G : D → C preserves all limits, then G has a left adjoint.

4.10 Lemma. Suppose D has and G : D → C preserves limits of shape J . Then forany A ∈ ob C , the arrow category (A ↓ G) has limits of shape J , and the forgetfulfunctor U : (A ↓ G)→ D creates them.

Proof. Suppose given D : J → (A ↓ G), write D(j) as (UD(j), fj). Let (L, (λj : L →UD(j))j∈ob J) be a limit for UD; then (GL, (Gλj)j∈ob J) is a limit for GUD.

Since the edges of UD are morphisms in (A ↓ G), the fj form a cone over GUD. Sothere’s a unique h : A→ GL such that (Gλj)h = fj for all j, i.e. there’s a unique h suchthat the λj are all morphisms (L, h)→ (UD(j), fj) in (A ↓ G).

If ((C, k)(µj)j∈ob J) is any cone over D, then (C, (µj)j∈ob J) is a cone over UD, sothere’s a unique l : C → L with λj l = µj for all j. We need to show (Gl)k = h:but (Gλj)(Gl)k = (Gµj)k = fj = (Gλj)h for all j so (Gl)k = h by uniqueness offactorizations through limits.

4.11 Lemma. A category C has an initial object iff 1C : C → C , regarded as a diagramof shape C in C , has a limit.

Proof. Suppose C has an initial object I. Then the unique morphisms

I → A | A ∈ ob C

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form a cone over 1C ; and given any cone C λA−−→ A | A ∈ ob C , then for any A thetriangle

C I

A

λI

λA

commutes, so λI is the unique factorization of λA | A ∈ ob C through the cone

I → A | A ∈ ob C .

Conversely, suppose (I, λA : I → A | A ∈ ob C ) is a limit. Then, for any If−→ A,

the diagram

I I

A

λI

λAf

commutes. In particular, putting f = λA, we see that λI is a factorization of the limitcone through itself, so λI = 1I . Hence every f : I → A satisfies f = λA.

The primeval Adjoint Functor Theorem now follows immediately from Lemma 4.10,Lemma 4.11 and Theorem 3.3. However, it only applies to functors between preorders(cf question 6 on example sheet 2).

4.12 Theorem (General Adjoint Functor Theorem). Suppose that D is locally smalland complete. Then G : D → C has a left adjoint if and only if G satisfies the solutionset condition: the solution set condition says that G preserves all small limits and,

for each A ∈ ob C there exists a set of morphisms A fi−→ GBi | i ∈ I such that every

Ah−→ GC factors as A

fi−→ GBiGg−−→ GC for some i and some g : Bi → C.

Proof. (⇒). If F a G, G preserves limits by Theorem 4.9, and A ηA−−→ GFA is asingleton solution set, by Theorem 3.3.

(⇐). By Lemma 4.10, (A ↓ G) is complete, and it inherits local smallness from D .So we need to show: if A := (A ↓ G) is complete and locally small, and has a weaklyinitial set of objects Bi | i ∈ I , then A has an initial object. (An object is weaklyinitial if it has a (not necessarily unique) morphism to any other object.)

First form P =∏i∈I Bi, then P is weakly initial. Now form the limit of

P P...

(∗)

where edges are all the endomorphisms of P : denote the limit Ii−→ P . I is also weakly

initial in A : suppose given I Cf

g. Form the equalizer E

e−→ I of (f, g), then there

exists Ph−→ E since P is weakly initial. ieh : P → P , and 1P are edges of the diagram

(∗) so i = iehi. But i is monic, so ehi = 1I , so e is split epic, so f = g. Hence I isinitial.

4.13 Examples.

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(a) Consider the forgetful functor U : Gp → Set. By Examples 4.6(a), U creates allsmall limits, so Gp has them and U preserves them. Gp is locally small, givena set A, any f : A → UG factors as A → UG′ → UG, where G′ is the subgroupgenerated by f(x) | x ∈ A and cardG′ ≤ maxℵ0, cardA.Let B be a set of this cardinality; consider all subsets B′ ⊆ B, all group structureson B′, and all mappings A→ B′. These give us a solution set at A.

(b) Consider the category CLat of complete lattices (posets with all meets and joins).Again U : CLat → Set creates all small limits. But A. W. Hales (1964) showedthat, for any cardinal κ, there exist complete lattices of card ≥ κ generated bythree elements, so the solution set condition fails at A = x, y, z, and U doesn’thave a left adjoint.

4.14 Definition (Subobject). ByLecture 13 a subobject of an object A of C , we mean amonomorphism A′ A. Dually, we have quotient objects. The subobjects of A arepreordered by A′′ ≤ A′ if there exists a factorization

A′′ A′

A.

We say C is well-powered if each A ∈ ob C has a set of subobjects Ai A | i ∈ Isuch that every subobject of A is isomorphic to some Ai (e.g. in Set we can take theinclusions A′ → A | A′ ∈ PA).

If Cop is well-powered, we say C is well-copowered (not cowell-powered).

4.15 Lemma. Suppose given a pullback square

P A

B C

h

k f

g

with f monic. Then k is monic.

Proof. Suppose D Px

ysatisfy kx = ky. Then fhx = gkx = gky = fhy, but f is

monic so hx = hy. So x and y are factorizations of the same cone through the limitcone (h, k).

4.16 Theorem (Special adjoint functor theorem). Suppose C and D are both locallysmall, and that D is complete and well-powered and has a coseparating set. Then afunctor G : D → C has a left adjoint iff it preserves all small limits.

Proof. (⇒) by Theorem 4.9.

(⇐). For any A ∈ ob C , (A ↓ G) is complete by Lemma 4.10, locally small, and well-powered since the subobjects of (B, f) in (A ↓ G) are just those subobjects B′ B inD for which f factors through GB′ GB.

Also, if Si | i ∈ I is a coseparating set for D , then the set

(si, f) | i ∈ I, f ∈ C (A,GSi)

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is coseparating in (A ↓ G): given (B, f) (B′, f ′)g

hin (A ↓ G) with g 6= h, there

exists k : B′ → Si for some i with kg 6= kh, and then k is also a morphism (B′, f ′) →(Si, (Gk)f ′) in (A ↓ G).

So we need to show that if A is complete, locally small and well powered, andhas a coseparating set Si | i ∈ I then A has an initial object. Form the productP =

∏i∈I Si. Now consider the diagram

Pi

Pj

P ′ P

whose edges are a representative set of subobjects of P , and form its limit

Pi

Pj

P ′ P

I

By the argument of Lemma 4.15, the legs of this cone are all monic; in particularI P is monic, and it’s a least subobject of P . Hence I has no proper subobjects, so,

given I Af

g, their equalizer is an isomorphism and hence f = g.

Now let A be any object of A ; form the product

Q =∏i∈I

f∈A (A,Si)

Si.

There’s an obvious h : A → Q defined by πi,fh = f ; and h is monic, since the Si are acoseparating set. We also have a morphism k : P → Q defined by πi,fk = πi.

Now form the pullback

B A

P Q

h

k

;

by Lemma 4.15, B → P is monic, so B is a subobject of P . Hence there exists

I B

P

and hence a morphism I → B → A.

4.17 Examples. Consider the inclusion KHausI−→ Top, where KHaus is the full

subcategory of compact Hausdorff spaces. KHaus has and I preserves small products

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(by Tychonoff’s Theorem) and equalizers (since equalizers of pairs X Yf

gwith Y

Hausdorff are closed subspaces). Both categories are locally small and KHaus is well-powered (subobjects of X are all isomorphic to closed subspaces). The closed interval[0, 1] is a coseparator in KHaus by Urysohn’s Lemma. So by Theorem 4.16, I has aleft adjoint β.

4.18 Remark.

(a) Cech’s construction of β: given X form P =∏f :X→[0,1][0, 1] and define h : X → P

by πfh = f . Define βX to be the closure of the image of h.

Cech’s proof that this works is essentially the same as Theorem 4.16.

(b) We could have used General Adjoint Functor Theorem to construct β: we get a

solution set at X by considering all continuous Xf−→ Y with Y compact Hausdorff,

and im f dense in Y and such Y have cardinality ≤ 22cardX

.

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5 Monads

Suppose givenLecture 14

C DF

G

with (F a G). How much of this structure can we describe without mentioning D?

We have the functor T = GF : C → C , and the unit η : 1C → T = GF and thenatural transformation

µ = GεF : TT = GFGF → GF = T.

These satisfy the commutative diagrams

T TT T

T

Tη

1Tµ

ηT

1T

(1) (2)

by triangular identities and

TTT TT

TT T

Tµ

µT (3) µ

µ

by naturality of ε.

5.1 Definition (Monad). A monad T = (T, η, µ) on a category C consists of a functorT : C → C and natural transformation η : 1C → T , µ : TT → T satisfying equations(1)-(3). η and µ are called the unit and multiplication of T.

5.2 Examples.

(a) Any adjunction (F a G) induces both a monad (GF, η,GεF ) on C and a comonad(FG, ε, FηG) on D .

(b) Let M be a monoid. The functor (M × −) : Set → Set has a monad structurewith unit given by ηA(a) = (1M , a) and multiplication µA(m,m′, a) = (mm′, a).The monad identities follow from the monoid ones.

(c) Let C be any category with finite products, A ∈ ob C . The functor (A × −) :C → C has a comonad structure with counit εB : A × B → B given by π2 andcomultiplication δB : A×B → A×A×B given by (π1, π1, π2).

Does every monad arise from an adjunction?

In Examples 5.2(b) we have the category [M,Set]. Its forgetful functor to Set hasa left adjoint, sending A to M × A with M acting by multiplication on the left factor.This adjunction gives rise to the monad of Examples 5.2(b).

5.3 Definition (Eilenberg-Moore category). Let T be a monad on C . A T-algebra is

a pair (A,α) with A ∈ ob C and TAα−→ A satisfying the commutative diagrams

A TA

A

ηA

1Aα

(4)

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TTA TA

TA A.

Tα

µA (5) α

α

A homomorphism f : (A,α)→ (B, β) is a morphism Af−→ B such that

TA TB

A B

Tf

α (6) β

f

commutes. The category of T-algebras is denoted C T, and called the Eilenberg-Moorecategory.

5.4 Lemma. The forgetful functor GT : C T → C has a left adjoint FT and the adjunc-tion induces T.

Proof. We define FTA = (TA, µA) (an algebra by (2) and (3)) and FT(Af−→ B) = Tf

(a homomorphism by naturality of µ).

Clearly GTFT = T , the unit of the adjunction is η. We define the counit

ε(A,a) = α : (TA, µA)→ (A,α)

(a homomorphism by (5)) ε is natural by (6); for the triangular identities, εFA(FηA) =1FA is (1), Gε(A,ε)ηA = 1A is (4).

The monad induced by (FT a GT) has functor T and unit η, and GTεF TA = µA bydefinition of FTA.

Kleisli took a ‘minimalist’ approach: if

C DF

G

induces T, then so does

C D ′F

G|D′

where D ′ is the full subcategory of D on objects FA.

So in trying to construct D , we may assume F is surjective (or indeed bijective)on objects. But then morphisms FA → FB correspond bijectively to morphisms A →GFB = TB in C .

5.5 Definition (Kleisli category). Given a monad T on C , the Kleisli category CThas ob CT = ob C , and morphisms A→ B are morphisms A→ TB in C . The composite

Af−→ B

g−→ C is

A TB TTC TCf Tg µC

and the identity A→ A is AηA−−→ TA.

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To verify associativity, suppose given Af−→ B

g−→ Ch−→ D. Then

A TB TTC TTTD TTD

TC TTD TD

f Tg TTh

µC µTD

TµD

(3) µD

Th µD

commutes by naturality: the upper way round is (hg)f and the lower is h(gf).

The unit laws for the category similarly follow from

A TB TTB

TB

f TηB

1TBµB

(1)

and

A TB

TA TTB TB.

f

ηA1TBηTB

Tf µB

(2)

5.6 Lemma. ThereLecture 15 exists an adjunction

C CTFT

GT

inducing the monad T.

Proof. We define FTA = A,

FT(Af−→ B) = A

f−→ BηB−−→ TB.

FT preserves identities by definition; for composites, consider Af−→ B

g−→ C. We get

A B TB

C TC TTC

TC.

f

g

ηB

Tg

ηC Tηc

1TCµC

(1)

We define GTA = TA,

GT(Af−→ B) = TA

Tf−−→ TTBµB−−→ TB.

GT preserves identities by (1); for composites, consider Af−→ B

g−→ C. We get

TA TTB TTTC TTC

TB TTC TC

Tf TTg

µB (3)

TµC

µTC µC

Tg µC

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We have

GTFTA = TA

GTFTf = µB(TηB)Tf = Tf

so we take η : 1C → T as the unit of (FT a GT). The counit TAεA−→ A is 1TA. To verify

naturality, consider the square

TA TB

A B

FTGTf

εA εB

f

This expands to

TA TTB TB TTB

TB

Tf µB ηTB

1TBµB

(2)

so ε is natural.

GT(TAεA−→ A) = µA, so GT(εA)ηGTA = µA · ηTA = 1TA

and (εFTA)(FTηA) is

A TA TTA

TA

ηA ηTA

1TAµA

(1)

which is 1FTA. Also GT(εFTA) = µA, so (FT a GT) induces T.

5.7 Theorem. Given a monad T on C , let Adj(T) be the category whose objects are

the adjunctions

(C D

F

G

)inducing T, and whose morphisms

(C D

F

G

)→(

C D ′F ′

G′

)are functors H : D → D ′ satisfying HF = F ′ and G′H = G. Then the Kleisli adjunctionis an initial object of Adj(T), and the Eilenberg-Moore adjunction is terminal.

Proof. Let

(C D

F

G

)be an object of Adj(T). We defineK : D → C T (the Eilenberg-

Moore comparison functor) by KB = (GB,GεB) where ε is the counit of (F a G);note this is an algebra by one of the triangular identities for (F a G) and naturality of

ε and K(Bg−→ B′) = Gg (a homomorphism by naturality of ε). Clearly GTK = G and

KFA = (GFA,GεFA) = (TA, µA) = FTA,

KF (Af−→ A′) = Tf = FTf.

So K is a morphism of Adj(T).

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Suppose K ′ : D → C T is another such; then since GTK ′ = G, we know K ′B =(GB, βB) where β is a natural transformation GFG → G. Also, since K ′F = FT, wehave βFA = µA = GεFA.

Now, given any B ∈ ob D , consider the diagram

GFGFGB GFGB

=

GFGB GB

GεFGB βFGB

GFGεB

GεB βB

GεB

Both squares commute so GεB and βB have the same composite with GFGεB . But thisis split epic, with splitting GFηGB , so β = Gε. Hence K ′ = K.

We now define the Kleisli comparison functor L : CT → D by LA = FA,

L(Af−→ B) = FA FGFB FB.

Ff εFB

L preserves identities by one of the triangular identities for (F a G); given Af−→ B

g−→ C,we have

FA FGFB FGFGFC FGFC

FB FGFC FC

Ff FGFg

εFB εFGFC

FGεFC

εFC

Fg εFC

GLA = TA = GTA,

GL(Af−→ B) = (GεFB)(GFf) = µB(Tf).

LFTA = FA,

LFT(Af−→ B) = (εFB)(FηB)(Ff) = Ff = GTf.

Note that L is full and faithful since its effect on morphisms (with given domain andcodomain) is that of transposition across (F a G). Suppose L′ : CT → D is a morphismof Adj(T). We must have L′A = FA, and L′ maps the counit TA → A to the counit

FGFAεFA−−→ FA. For any A

f−→ B, we have f = 1TA(FTf) so L′(f) = εFA · (Ff) =Lf .

IfLecture 16 C has coproducts, then so does CT, since FT preserves them. But in general, ithas few other limits or colimits. In contrast, we have

5.8 Theorem.

(i) The forgetful functor G : C T → C creates all limits which exist in C .

(ii) If C has colimits of shape J , then G : C T → C creates them ⇐⇒ T preservesthem.

Proof.

(i) Suppose given D : J → C T; write D(j) = (GD(j), δj) and suppose

(L, µj : L→ GD(j) | j ∈ ob J )

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is a limit cone for GD. Then the composites

TL TGD(j) GD(j)Tµj δj

form a cone over GD, since the edges of GD are homomorphisms, so they inducea unique λ : TL → L such that µjλ = δj(Tµ) for all j. The fact that λ is aT-algebra structure on L follows from the fact that the δj are algebra structuresand uniqueness of factorisations through limits.

So ((L, λ), µj | j ∈ ob J ) is the unique lifting of the limit cone over GD to acone over D; and it’s a limit, since given a cone over D with apex (A,α), we

get a unique factorisation Af−→ L in C and F is an algebra homomorphism by

uniqueness of factorisations through L.

(ii) (⇒) F : C → C T preserves colimits since it’s a left adjoint, so T = GF preservescolimits of shape J .

(⇐) Suppose given D : J → C T as in (i), and a colimit cone

GD(j)µj−→ L | j ∈ ob J

in C . Then

TGD(j)Tµj−−→ TL | j ∈ ob J

is also a colimit cone, so the composites

TGD(j) GD(j) Lδj µj

induce a unique λ : TL→ L. The rest of the argument is like (i).

5.9 Definition (Monadic adjunction). Given an adjunction

(C D

F

G

), with (F a

G) we say the adjunction (or the functor G) is monadic if the comparison functorK : D → C T is part of an equivalence of categories.

(Note that, since the Kleisli comparison CT → D is full and faithful, it’s part of anequivalence if and only if it (equivalently, F ) is essentially surjective on objects.)

Remark. Given any adjunction (F a G), for each object B of D we have a diagram

FGFGB FGB BFGεB

εFGB

εB

with equal composites. The ‘primeval monadicity theorem’ asserts that C T is charac-terised in Adj(T) by the fact that these diagrams are all coequalizers.

5.10 Definition (Reflexive coequalizers).

(a) We say a parallel pair A Bf

gis reflexive if there exists B

r−→ A such that

fr = gr = 1B . (Note that FGFGB FGBFGεB

εFGBis reflexive, with r = FηGB).

We say C has reflexive coequalizers if it has coequalizers of all reflexive pairs

(equivalently, colimits of shape J where J = • •)

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(b) By a split coequalizer diagram, we mean a diagram

A B Cf

g

t

h

s

satisfyinghf = hg hs = 1C gt = 1B ft = sh.

These equations imply that h is a coequalizer of (f, g); if Bx−→ D satisfies xf = xg

then x = xgt = xft = xsh, so x factors through h, and the factorisation is uniquesince h is split epic. Note that split coequalizers are preserved by all functors.

(c) Given a functor G : D → C , a parallel pair A Bf

gis called G-split if there

exists a split coequalizer diagram

GA GB CGf

Gg

t

h

s

in C .

Note that FGFGB FGBFGεB

εFGBis G-split, since

GFGFGB GFGB GBGFGεB

GεFGB

ηGFGB

GεB

ηGB

is a split coequalizer.

5.11 Lemma. Suppose given an adjunction C DF

Gwhere F a G, inducing a monad

T on C . Then K : D → C T has a left adjoint provided, for every T-algebra (A,α), the

pair FGFA FAFα

εFAhas a coequalizer in D .

Proof. We define L : C T → D by taking FA→ L(A,α) to be a coequalizer for (Fα, εFA).Note that this is a functor C T → D .

Recall that K is defined by KB = (GB,GεB). For any B, morphisms LA →B correspond bijectively to morphisms FA

f−→ B satisfying f(Fα) = f(εFA). These

correspond to morphisms Af−→ GB satisfying

fα = Gf = G(εB(F f)) = (GεB)(T f)

i.e. to algebra homomorphisms (A,α)→ KB. And these bijections are natural in (A,α)and in B.

5.12 Theorem (Precise Monadicity Theorem).Lecture 17 G : D → C is monadic iff G has a leftadjoint and creates coequalizers of G-split pairs.

5.13 Theorem (Refined/Reflexive Monadicity Theorem). Suppose D has and G : D →C preserves reflexive coequalizers and thatG reflects isomorphisms and has a left adjoint.Then G is monadic.

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Proof of Theorem 5.12 ⇒. It is sufficient to show that GT : C T → C creates coequal-izers of GT-split pairs. But this follows from the argument of Theorem 5.8(ii), since if

(A,α) (B, β)f

gis a GT-split pair, the coequalizer of A B

f

gis preserved by T and

TT .

Proof of Theorem 5.12 ⇐ and Theorem 5.13. Let T be the monad induced by (F a G).

For any T-algebra (A,α), the pair FGFA FAFα

εFAis both reflexive and G-split, so has

a coequalizer in D , and hence by Lemma 5.11, K : D → C T has a left adjoint L.

The unit of (L a K) at an algebra (A,α): the coequalizer defining L(A,α) is mappedby K to the diagram

FTTA FTA KL(A,α)

(A,α)

F TA

µA

α ιA,α

and ιA,α is the factorisation of this through the (GT-split) coequalizer α. But either setof hypotheses implies that G preserves the coequalizer defining L(A,α) so ι(A,α) is anisomorphism.

For the counit ζB : LKB → B, we have a coequalizer

FGFGB FGB LKB

B

FGεB

εFGB

εBζB

Again, either set of hypotheses implies that εB is a coequalizer of (FGεB , εFGB) soζB is an isomorphism.

5.14 Examples.

(a) The forgetful functors Gp → Set, Rng → Set, ModR → Set, . . . all satisfythe hypotheses of Theorem 5.13; for the reflexive coequalizers, use question 3 onexample sheet 4 which shows that if

A B Cf

g

h

is a reflexive coequalizer diagram in Set, then so is

An Bn Cn.fn

gn

hn

(b) Any reflection is monadic: this follows from q2 on example sheet 3, but also canbe proved using Theorem 5.12. Let D be a reflective (full) subcategory of C and

suppose a pair A Bf

gin D fits into a split coequalizer diagram

A B Cf

g

h

t s

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in C . Then t and ft = sh belong to D since D is full, and hence s is in D since it’san equalizer of (1B , sh) and D is closed under limits in C . Hence also h ∈ mor D .

(c) Consider the composite adjunction

Set AbGp tfAbGp.F L

G I

The two factors are monadic by (a) and (b) respectively, but the composite isn’t,since the monad it induces on Set is isomorphic to that induced by (F a U).

(d) Consider the forgetful functor TopU−→ Set. This is faithful and has both left and

right adjoints (so preserves all coequalizers), but the monad induced on Set is(1, 1, 1) and the category of algebras is Set.

(e) Consider the composite adjunction

Set Top KHaus.D β

U I

We’ll show that this satisfies the hypotheses of Theorem 5.12. Let

X Y Zf

g

h

t s

be a split coequalizer in Set, where X,Y have compact Hausdorff topologies andf, g are continuous. Note that the quotient topology on Z ∼= Y/R is compact,so it’s the only possible candidate for a compact Hausdorff topology making hcontinuous.

We use the lemma from general topology: if Y is compact Hausdorff, then aquotient Y/R is Hausdorff ⇐⇒ R ⊆ Y × Y is closed. We note

R = (y, y′) | h(y) = h(y′) = (y, y′) | sh(y) = sh(y′) = (y, y′) | ft(y) = ft(y′) .

So if we define S ⊆ X × X = (x, x′) | f(x) = f(x′) then R ⊆ (g × g)(S), but

the reverse inclusion also holds. But S X ×X Yfπ1

fπ2

is an equalizer, Y is

Hausdorff, so S is closed in X × X and hence compact. So R = (g × g)(S) iscompact and hence closed in Y × Y .

5.15 Definition (Monadic tower). LetLecture 18 C DF

Gbe an adjunction, and suppose D has

reflexive coequalizers. The monadic tower of F a G is the diagram

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D

(C T)S

C T

C

G

K

K′

L′

L

GT

F

F T

where T is the monad induced by (F a G), K is as in Theorem 5.7, L as in Lemma 5.11,S is the monad induced by (L a K), and so on.

We say (F a G) has monadic length n if we reach an equivalence after n steps.

For example, the adjunction of Examples 5.14(c) has monadic length 2, and theadjunction of Examples 5.14(d) has monadic length ∞.

5.16 Theorem. Suppose given an adjunction C DR

Land monads T,S on C ,D re-

spectively, and a functor R : DS → C T such that

DS C T

D C

R

GS GT

R

commutes up to isomorphism.

Suppose also that DS has reflexive coequalizers. Then R has a left adjoint L.

Proof. Note that if L exists, we must have LFT ∼= F SL, by Corollary 3.6. So we’dexpect L(A,α) to be a coequalizer of two morphisms

F SLTA F SLAF SLα

?

To construct the second morphism, note first that we can assume wlog GTR = RGS, bytransporting T-algebra structures along the isomorphism GTR(B, β)→ RB. We obtainθ : TR→ RS by

RRι−−→ RS = RGSF S = GTRF S

FTR→ RF S

TR = GTFTRθ−→ GTRF S = RGSF S = RS

Convert it into ϕ : LT → SL by

LT LTRL LRSL SLLTγ LθL δSL

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where γ and δ are the unit and counit of (L a R). Transposing across (F S a GS), we

get F SLTϕ−→ F SL. The pair (F SLα, ϕA) is reflexive, with common splitting F SLη.

It can be verified that the coequalizer of this pair has the universal property werequire for L(A,α).

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6 Cartesian Closed Categories

6.1 Definition (Exponentiable object). Let C be a category with finite products. Wesay A ∈ ob C is exponentiable if the functor (−)×A : C → C has right adjoint (−)A.If every object of C is exponentiable, we say C is cartesian closed.

6.2 Examples.

(a) Set is cartesian closed, with BA = Set(A,B). A function f : C × A → Bcorresponds to f : C → BA.

(b) Cat is cartesian closed, with DC = [C ,D ].

(c) In Top, if an exponential Y X exists, its points must be the continuous mapsX → Y . The compact-open topology on Top(X,Y ) has the universal property ofan exponential iff X is locally compact.

Note that finite products of exponentiable objects are exponentiable: since

(−)× (A×B) ∼= (−×A)×B,

we have (−)A×B ∼= ((−)B)A. However, even if X,Y are locally compact, Y X

needn’t be, so the exponentiable objects don’t form a cartesian closed full subcat-egory.

(d) A cartesian closed poset is called a Heyting semilattice: it’s a poset with finitemeets and a binary operation ⇒ satisfying a ≤ (b⇒ c) iff a∧ b ≤ c. For example,a complete poset is a Heyting semilattice iff it satisfies the infinite distributive law

a ∧∨ bi | i ∈ I =

∨ a ∧ bi | i ∈ I .

For any topological space X, the lattice O(X) of open subsets satisfies this condi-tion, since ∧ and

∨coincide with ∩ and

⋃.

Recall from example sheets that, if B ∈ ob C , we define the slice category C /B tohave objects which are morphisms

A

B

in C and morphisms are commutative triangles

A A′

B.

The forgetful functor C /B → C will be denoted ΣB . If C has finite products, ΣB has

a right adjoint B∗ which sends A to

A×BB

π2

, since morphisms

C A×B

B

(f,g)

g π2

correspond to morphisms C = ΣBgf−→ A.

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6.3 Lemma. If C has all finite limits, then an object B is exponentiable iff B∗ : C →C /B has a right adjoint ΠB .

Proof.

⇐ The composite ΣBB∗ is equal to (−)×B, so we take (−)B to be ΠBB

∗.

⇒ If B is exponentiable, for any Af−→ B we define ΠB(f) to be the pullback

ΠB(f) AB

1 BB .

fB

π2

The morphisms C → ΠB(f) correspond to morphisms C → AB making

C AB

1 BB

fB

π2

i.e. to homomorphisms C ×B → A making

C ×B A

B

π2

f

commute.

6.4 Lemma. SupposeLecture 19 C has finite limits. If A is exponentiable in C , then B∗A isexponentiable in C /B for any B. Moreover, B∗ preserves exponentials.

Proof. Given an object

C

B

f , form the pullback

P CA

B BA.

f(B∗A) fA

π1

Then, for any

D

B

g , morphisms g → fB∗A in C /B correspond to morphisms D

h−→ CA

making

D CA

B BA

h

g fA

π1

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commute, and hence to morphisms D ×A h−→ C making

D ×A C

B

h

gπ1

f

commute. But

D ×A B ×A

D B

g×1A

π1 π1

g

is a pullback in C , i.e. a product in C /B.

For the second assertion, note that if

C

B

f is of the form

B × E

B

π1 , then the pullback

defining fB∗A becomes

B × EA BA × EA

B BA

π1×1

π1 π1

π1

so fB∗A ∼= B∗(EA).

Remark. C /B is isomorphic to the category of coalgebras for the monad structure on(−)×B (Examples 5.2(a)); so the first part of Lemma 6.4 could have been proved usingTheorem 5.16.

6.5 Definition (Locally cartesian closed). We say C is locally cartesian closed if ithas all finite limits and each C /B is cartesian closed. (Note that this includes the factthat C ∼= C /1 is cartesian closed).

6.6 Examples.

(a) Set is locally cartesian closed, since Set/B ∼= SetB for any B.

(b) For any small category C , [C ,Set] is cartesian closed: by Yoneda,

GF (A) ∼= [C ,Set](C (A,−), GF )∼= [C ,Set](C (A,−)× F,G),

so we take the right hand side as a definiition of GF (A) and define GF on mor-

phisms Af−→ B by composition with C (f,−)× 1F . Note that the class of functors

H for which we have

[C ,Set](H,GF ) ∼= [C ,Set](H × F,G)

is closed under colimits; but every functor C → Set is a colimit of representables.

In fact [C ,Set] is locally cartesian closed, since all its slice categories [C ,Set]/Fare of the same form (see question 6 on example sheet 4).

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(c) Any Heyting semilattice H is locally cartesian closed, since H/B ∼= ↓(b), the posetof elements ≤ b, and b∗ = (−) ∧ b is surjective.

(d) Cat is cartesian closed and not locally cartesian closed, since not all strong episare regular (cf question 6 on example sheet 3).

Note that, givenA

B

f in C /B, the iterated slice (C /B)/f is isomorphic to C /A, and

this identifiesf∗ : C /B → (C /B)/f

with the operation of pulling back morphisms along f . So by Lemma 6.3, C is locallycartesian closed iff it has finite limits and f∗ : C /B → C /A has a right adjoint Πf for

every Af−→ B in C .

6.7 Theorem. Suppose C is locally cartesian closed and has reflexive coequalizers.

Then every morphism Af−→ B factors as

I

A B

mq

f

where q is regular epic and m is monic.

Proof. First form the pullback

R A

A B

a

b f

f

and then form the coequalizer

R A I.a

b

q

Since fa = fb, f factors as Aq−→ I

m−→ B.

Suppose given D Ig

hwith mg = mh. Form the pullback

E D

A×A I × I.

n

(k,l) (g,h)

q×q

Since q × q factors as (q × 1I)(1A × q) and both factors are pullbacks of q, it’s anepimorphism, and hence so is n. Now fk = mqk = mgn = mhn = mql = fl, so

∃E p−→ R with ap = k, bp = l. Now qk = qap = qbp = ql, i.e. gn = hn. But n is epic, sog = h. Hence m is monic.

Note that this implies any strong epimorphism Af−→ B is regular, since the monic

part of its image factorization is an isomorphism. In particular, regular epimorphismsare stable under composition.

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6.8 Definition (Cartesian closed functor). If C and D are cartesian closed categoriesand F : C → D preserves finite products, then for each pair of objects A,B of C we geta natural morphism

θ : F (BA)→ FBFA,

namely the transpose of

F (BA)× FA ∼= F (BA ×A)F (ev)−−−→ FB

where ev is the counit of ((−)× A a (−)A). We say F is a cartesian closed functorif θ is an isomorphism for every pair (A,B).

Note that the second part of Lemma 6.4 says that if C is locally cartesian closed

then f∗ : C /B → C /A is a cartesian closed functor for any Af−→ B.

6.9 Theorem. Let C and D be cartesian closed categories and F : C → D a functorhaving a left adjoint L. Then F is cartesian closed iff the canonical morphism

L(B × FA) LB × LFA LB ×A(Lπ1,Lπ2) 1LB×εA

is an isomorphism for all A ∈ ob D , B ∈ ob C .

This condition is called Frobenius reciprocity. Note that if C is locally cartesianclosed, then f∗ : C /B → C /A has a left adjoint Σf given by composition with f andit’s easy to verify that Σf (g × f∗h) ∼= Σfg × h.

Proof.

(⇒)Lecture 20 Given an inverse for θ : F (CA)→ FCFA, we define ϕ−1A,B to be the composite

LB ×A L((B × FA)FA)×A L(FL(B × FA)FA)×A

L(B × FA) L(B × FA)A ×A LF (L(B × FA)A)×A

Lλ×1 L(ηFA)×1

Lθ−1×1

ev ε×1

(⇐) Given an inverse for ϕ, we define θ−1 to be

F ((L(FCFA)×A)A) FL(FCFA) FCFA

F ((L(FCFA × FA))A) F ((LFC)A) F (CA).

F (ϕ−1A)

Fλ η

F((L(ev))A) F (εA)

We omit checking that these work.

6.10 Corollary. Suppose C and D are cartesian closed, and F : C → D has a leftadjoint L which preserves finite products. Then F is cartesian closed ⇐⇒ F is fulland faithful.

Proof.

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(⇒) L preserves 1, so if we substitute B = 1 in the definition of ϕ, we get

LFAεA−→ A.

But ε an isomorphism ⇐⇒ F is full and faithful by Lemma 3.9.

(⇐) If L preserves binary products and ε is isomorphic, then both factors in the defi-nition of ϕ are isomorphisms.

6.11 Definition (Exponential ideal). Let C be a cartesian closed category. By anexponential ideal in C , we mean a class of objects (or a full subcategory) E such thatB ∈ E =⇒ BA ∈ E for all A ∈ ob C .

6.12 Examples.

(a) We say A is subterminal if A→ 1 is monic. In any cartesian closed category C ,the class SubC (1) of subterminal objects is an exponential ideal, since

A subterminal ⇐⇒ ∃ ≤ 1 B → A, for any B

=⇒ ∃ ≤ 1 C ×B → A, for any B and C,

⇐⇒ ∃ ≤ 1 C → AB

⇐⇒ AB subterminal

More generally, if C is locally cartesian closed, then SubC (A) is an exponentialideal in C /A, for any A. If C also satisfies the hypotheses of Theorem 6.7, thenSubC (A) is reflective in C /A.

(b) Let X be a topological space. By a presheaf on X, we mean a functor

O(X)op F−→ Set,

where O(X) is the poset of open subsets of X. So F has sets F (U) for each openU , and restriction maps x 7→ x|V : F (U) → F (V ) whenever V ⊆ U . We say F isa sheaf if, whenever U =

⋃i∈I Ui and we’re given xi ∈ F (Ui) for each i such that

xi|Ui∩Uj = xj |Ui∩Uj

for all (i, j), there exists a unique x ∈ F (U) such that x|Ui = xi for all i.

We write Sh(X) ⊆ [O(X)op,Set] for the full subcategory of sheaves. We’ll showSh(X) is an exponential ideal:

Given presheaves F,G, GF (U) is the set of natural transformations

F ×O(X)(−, U)→ G

or equivalently the set of natural transformations F |U → G|U , where

F |U : O(U)op → Set

is the presheaf obtained by restricting F to open sets ⊆ U .

Now suppose G is a sheaf, suppose U =⋃i∈I Ui and suppose given αi : F |Ui →

G|Ui for each, such thatαi|Ui∩Uj = αj |Ui∩Uj

for all i, j. Given x ∈ F (V ) where V ⊆ U , write Vi = V ∩ Ui, then V =⋃i∈I Vi.

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The elements x|Vi , i ∈ I, satisfy the compatibility condition, and hence so do theelements

(αi)Vi(x|Vi) ∈ G(Vi).

So there’s a unique y ∈ G(V ) such that y|Vi = (αi)Vi(x|Vi) for all i, and we definethis to be αV (x). This defines a natural transformation α : F |U → G|U , and it’sthe unique transformation where restriction to Ui is αi, for each i.

Note that, since Sh(X) is closed under finite products (and in fact all limits) in[O(X)op,Set], it is itself cartesian closed.

6.13 Lemma. Suppose C is cartesian closed, D ⊆ C is a (full) reflective subcategory,with reflector L : C → D . Then D is an exponential ideal ⇐⇒ L preserves binaryproducts.

Proof.

(⇒) Suppose A,B ∈ ob C , C ∈ ob D . Then we have bijections

A×B → C

A→ CB

LA→ CB

LA×B → C

B → CLA

LB → CLA

LA× LB → C

so LA× LB has the universal property of L(A×B).

(⇐) Suppose B ∈ ob D , A,C ∈ ob C . We have bijections

C → BA

C ×A→ B

LC × LA ∼= L(C ×A)→ B

L(LC ×A)→ B

LC ×A→ B

LC → BA

So every C → BA factors through C → LC, hence BA ∈ ob D .

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7 Toposes

GrothendieckLecture 21 (around 1963) introduced toposes as categories of ‘generalized sheaves’.J. Giraud gave a characterization of such categories by (set-theoretic) categorical prop-erties.

F. W. Lawvere and M. Tierney (1969-70) investigated the elementary categoricalproperties of these categories and came up with the elementary definition. In facta Grothendieck topos is exactly a Lawvere-Tierney topos which is (co)complete andlocally small, and has a separating set of objects.

7.1 Definition (Subobject classifier, topos).

(a) Let E be a category with finite limits. A subobject classifier for E is a monomor-

phism Ω′> Ω such that for every monomorphism A′

m A in E , there is a unique

χm : A→ Ω for which there is a pullback square

A′ Ω′

A Ω

m >χm

Note that, for any A, there’s a unique A → Ω which factors through Ω′> Ω, so

the domain of > is actually a terminal object.

If E is well-powered, we have a functor

SubE (−) : E op → Set

sending A to the set of (isomorphism classes of) subobjects of A and acting onmorphisms by pullback, and a subobject classifier is representation of this functor.

(b) A topos is a category which has finite limits, is cartesian closed and has a subob-ject classifier.

(c) If E and F are toposes, a logical functor F : E → F is one which preservesfinite limits, exponentials and the subobject classifier.

7.2 Examples.

(a) Set is a topos, with Ω = 0, 1 and > = 1 : 1→ 0, 1. Here

χm(a) =

1 if a ∈ A′

0 if a /∈ A′.

So also is the category Setf of finite sets, or the category Setκ of sets of cardinality< κ, where κ is an infinite cardinal such that λ < κ =⇒ 2λ < κ.

(b) For any small category C , [C op,Set] is a topos: we’ve seen that it’s cartesianclosed, and Ω is determined by Yoneda Lemma:

Ω(A) ∼= [C op,Set](C (−, A),Ω) ∼= subfunctors of C (−, A)

So we define Ω(A) to be the set of sieves on A, i.e. sets of morphisms with cod = A,such that f ∈ R =⇒ fg ∈ R for any g.

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Given Bf−→ A and a sieve R on A, we define f∗R to be the set of g with codomain B

such that fg ∈ R. This makes Ω into a functor C op → Set, as follows. > : 1→ Ωis defined by >A(∗) = all morphisms with codomain A. Given a subfunctor

F ′m F , we define χm : F → Ω by

(χm)A(x) = f : B → A | Ff(x) ∈ F ′(B) .

This is the unique natural transformation making

F ′ 1

F Ω

m >

a pullback.

(c) For any space X, Sh(X) is a topos. It’s cartesian closed by Examples 6.12(ii); forthe subobject classifier, we take

Ω(U) = V ∈ O(X) | V ⊆ U ,

Ω(U ′ → U) is the map V 7→ V ∩U ′ and Ω is a sheaf since if we have U =⋃i∈I Ui,

and Vi ⊆ Ui such that Vi ∩ Uj = Vj ∩ Ui for each i, j, then V =⋃i∈I Vi is the

unique open subset of U with V ∩ Ui = Vi for each i.

If F ′m F is a subsheaf, then for any x ∈ F (U), the sieve V ⊆ U | x|V ∈ F ′(V )

has a greatest element since F ′ is a sheaf, so we define χm : F → Ω to send x tothis element.

(d) Let C be a group G. The topos structure on [G,Set] is particularly simple: BA

is the set of all G-equivariant maps A×G f−→ B, but such an f is determined byits values at elements of the form (a, 1) since f(a, g) = g · f(g−1 · a, 1), and thisrestriction can be any mapping A×1 → B. So we can take BA to be the set offunctions A→ B, with G acting by (g · f)(a) = g(f(g−1 · a)), and Ω = 0, 1 withtrivial G-action.

So the forgetful functor [G,Set] → Set is logical, as is the functor which equipsa set A with trivial G-action. Moreover, even if G is infinite, [G,Setf ] is a topos,and the inclusion [G,Setf ] → [G,Set] is logical. Similarly, if G is a large group,then [G,Set] is a topos.

(e) Let C be a category such that every slice C /A is equivalent to a finite category.Then [C op,Setf ] is a topos; similarly if C is large but all C /A are small then[C op,Set] is a topos. In particular, [Onop,Set] is a topos, but it’s not locallysmall.

7.3 Lemma.Lecture 22 Suppose E has finite limits and a subobject classifier. Then every monomor-phism in E is regular. In particular, E is balanced.

Proof. The universal monomorphism 1> Ω is split and hence regular. But any pullback

of a regular monomorphism is regular: if f is an equalizer of (g, h) then k∗(f) is anequalizer of (gk, hk). The second assertion follows since regular monomorphism togetherwith epimorphism implies isomorphism.

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Given an object A in a topos E , we write PA for the exponential ΩA, and 3APA× A for the subobject corresponding to PA× A ev→ Ω. This has the property that,

for any B and any Rm B ×A there is a unique pmq : B → PA such that

R 3A

B ×A PA×A

m

pmq×1A

is a pullback.

7.4 Definition (Power object, weak topos). By a power object for A in a categoryE with finite limits, we mean an object PA equipped with 3A PA×A satisfying theabove.

We say E is a weak topos if every A ∈ ob E has a power-object. Similarly, we sayF : E → F is weakly logical if F (3A) F (PA)× FA is a power-object for FA, forevery A ∈ ob E .

7.5 Lemma. P is a functor E op → E . Moreover, it is self-adjoint on the right.

Proof. Given Af−→ B, we define PB

Pf−−→ PA to correspond to the pullback

Ef 3B

PB ×A PB ×B.1×f

For any Cpmq−−→ PB, it’s easy to see that (Pf)pmq corresponds to (1C × f)∗(m); hence

f 7→ Pf is functorial. For any A,B, we have a bijection between subobjects of A × Band of B×A, given by composition with (π2, π1) : A×B → B×A; this yields a (natural)bijection between morphisms A→ PB and B → PA.

We write A : A→ PA for the morphism corresponding to A A×A(1A,1A).

7.6 Lemma. Given Af−→ B, Bf corresponds to A A×B(1A,f)

and (Pf)B corre-

sponds to A B ×A(f,1A).

Proof. The square

A B

A×B B ×B

f

(1,f) (1,1)

f×1

is a pullback. Similarly for the second assertion.

7.7 Corollary.

(i) : A→ PA is monic.

(ii) P is faithful.

Proof.

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(i) If f = g, then (1A, f) and (1A, g) are isomorphic as subobjects of A × B,which forces f = g.

(i) Similarly, if Pf = Pg then (Pf) = (Pg), so we again deduce f = g.

Given a monomorphism Af−→ B in E , we define ∃f : PA → PB to correspond to

the composite

3A PA×A PA×B1×f

Then, for any Cpmq−−→ PA, (∃f)pmq corresponds to

R C ×A C ×B.m 1×f

So f 7→ ∃f is a functor Mono(E )→ E .

7.8 Lemma (Beck-Chevalley condition). Suppose

D A

B C

h

k f

g

is a pullback with f monic. Then the diagram

PA PC

PD PB

∃f

Ph Pg

∃k

commutes.

Proof. Consider the diagram

En 3A

PA×D PA×A

PA×B PA× C

1×h

1×k 1×f

1×g

The lower square is a pullback, so the upper square is a pullback if and only if thecomposite is a pullback.

7.9 Theorem (Pare). The functor P : E op → E is monadic.

Proof. It has a left adjoint P : E → E op by Lemma 7.5. It’s faithful by Corollary 7.7(ii)and hence reflects isomorphisms by Lemma 7.3. E op has coequalizers, since E hasequalizers. Suppose

A B

f

gr

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is a coreflexive pair in E ; then f and g are (split) monic, and the equalizer Ee−→ A

makesE A

A B

e

e g

f

a pullback square since any cone over

A

A B

g

f

has both legs equal. So by Lemma 7.8 we have (Pf)(∃g) = (∃e)(Pe); but we also have(Pg)(∃g) = 1PA since

A A

A B

1

1 g

g

is a pullback, and similarly (Pe)(∃e) = 1PE . So

PB PA PEPf

Pg

∃g

Pe

∃e

is a split coequalizer, and in particular a coequalizer. Hence by Theorem 5.13, P ismonadic.

7.10 Corollary.

(i) A weak topos has finite colimits. Moreover, if it has any infinite limits, then it hasthe corresponding colimits.

(ii) If a weakly logical functor has a left adjoint, then it has a right adjoint.

Proof.

(i) P creates all limits which exist, by Theorem 5.8.

(ii) By definition, if F is weakly logical then

E op F op

E F

F

P P

F

commutes up to isomorphism. So this follows from Theorem 5.16.

7.11 Lemma.Lecture 23 Let E be a category with finite limits, suppose A ∈ ob E has a powerobject PA. Then for any B, B∗(PA) is a power object for B∗A in E /B.

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Proof. GivenC

B

g , we have a pullback square

C ×A B ×A

C B

g×1

π1 π1

g

So ΣB(g × B∗A) ∼= C × A. Hence SubE/B(g × B∗A) ∼= SubE (C × A), but if Ch−→ PA

corresponds toR

C ×A, then the upper square of the diagram

R B× 3A

C ×A B × PA×A

B

(g,h)×1A

gπ1 π1

is a pullback. SoB × PA

B

π1 , equipped with B∗(3A) B∗(PA×A) is a power object for

B∗A.

7.12 Theorem. Suppose E is a weak topos. Then for any B ∈ ob E , E /B is a weaktopos and B∗ : E → E /B is weakly logical.

Proof. The second assertion follows from Lemma 7.11. For the first, we need to construct

a power object for an arbitraryA

B

f in E /B. Then the pullback

ΣB(g × f) A

C B

f

g

is a subobject of C ×A, namely the equalizer of

C ×A B.fπ2

gπ1

Define ∧ : PA×PA→ PA to correspond to the intersection of π∗13(3A PA×A) and

π∗23(3A PA×A) and define P1A PA×PA to be the equalizer of PA× PA PA∧

π1

.

Then, for any C,

C(pmq,pnq)−−−−−−→ PA× PA

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factors through P1A iff m ≤ n in SubE (C ×A). Now form the pullback

Q P1A

PA×B PA× PB PA× PA.

(h,k)

1× 1×Pf

Given any

C

B

g , morphisms gl−→ k in E /B correspond to morphisms C

hl−→ PA such that

the subobject named by hl is contained in that named by (Pf)()g. But the latter isindeed ΣB(g × f) C ×A. So k is a power-object for f in E /B.

7.13 Corollary. A weak topos is locally cartesian closed (in particular, it’s a topos).

Proof. For any f : A → B in E , we can identify (E /B)/f with E /A, and f∗ : E /B →E /A with pullback along f . Hence all such functors are weakly logical. But f∗ has aleft adjoint Σf , so by Corollary 7.10(ii) it has a right adjoint Πf . Hence by Lemma 6.3,E /B is cartesian closed for any B.

Remark. It can be shown that a weakly logical functor is cartesian closed (and hencelogical).

7.14 Corollary.

(i) Any epimorphism in a topos is regular.

(ii) Any Af−→ B in a topos factors uniquely up to isomorphism as

A I B.q m

Proof. For a topos E , it is locally cartesian closed by Corollary 7.13 and has coequalizersby Corollary 7.10(i), so by Theorem 6.7 every f factors uniquely as regular epi + mono.If f itself is epic, then the monic part of this factorization is isomorphic by Lemma 7.3,so f is regular epic.

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*Sheaves and local operators

Recall that Sh(X) ⊆ [O(X)op,Set] is a full subcategory closed under limits: in factit’s reflective and the reflector L : [O(X)op,Set] → Sh(X) preserves finite limits. Thissuggests considering reflective subcategories D ⊆ E for which the reflector preservesfinite limits (equivalently, pullbacks).

7.15 Lemma. Given such a reflective subcategory, and a monomorphism A′ A inE , define c(A′) A by the pullback diagram

c(A′) LA′

A LA.ηA

Then A′ 7→ c(A′) is a closure operation on SubE (A), and commutes with pullbackalong a fixed morphism of E . (A closure operation is an order-preserving inflationaryidempotent operator.)

Proof. Since

A′ LA′

A LA.

ηA′

ηA

commutes, we have A′ ≤ c(A′) and A′ ≤ A′′ in Sub(A) implies LA′ ≤ LA′′ in Sub(LA)and hence c(A′) ≤ c(A′′). Since Lη is an isomorphism,

LA′ LLA′

LA LLA.

LηA′

LηA

is a pullback, and since L preserves pullbacks we deduce Lc(A′) ∼= LA′ in Sub(LA).Hence c(c(A′)) ∼= c(A′). For stability under pullback, suppose

A′ B′

A Bf

is a pullback. Then in the cube

c(A′) LA′

c(B′) LB′

A LA

B LB

ηA

f

Lf

ηB

the front, back and right faces are pullbacks, whence the left face is too.

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7.16 Definition (Local operator). Let E be a topos. By a local operator on E , wemean a morphism j : Ω→ Ω satisfying the commutative diagrams

1 Ω Ω

Ω

>

>j

jand

Ω1 Ω1

Ω× Ω Ω× Ωj×j

where Ω1 is the order-relation in Ω, defined as in Theorem 7.12.

Given a closure operator on subobjects as in Lemma 7.15, define J Ω to be the

closure of 1 Ω> and j : Ω → Ω to be the classifying map of J Ω. Then for anyA′ A with classifying map χm : A→ Ω, the composite jχm classifies c(A′) A.

GivenLecture 24 a pullback-stable operator c on subobjects, we say A′ A is dense if c(A′)A is an isomorphism, and closed if A′ c(A′) is an isomorphism.

7.17 Lemma. Suppose given a commutative square

B′ A′

B A

f ′

n m

f

with n dense and m closed. Then there is a unique Bg−→ A′ with mg = f (and gn = f ′).

Proof. We have n ≤ f∗(m) in Sub(B), so

1B ∼= c(n) ≤ f∗(c(m)) ∼= f∗(m).

So we define g as B∼=−→ f∗(A′)→ A′.

Note that c(A′) may be characterised as the unique (up to isomorphism) subobjectA′′ such that A′ A′′ is dense and A′′ A′ is closed.

7.18 Lemma. Suppose c is induced as in Lemma 7.15 by a reflector L : E → Dpreserving finite limits. Then an object A of E belongs to D (up to isomorphism) iff,given any diagram

B′ A

B

m

f ′

with m dense, there exists a unique Bf−→ A such that fm = f ′.

Proof. Note first that m is dense ⇐⇒ Lm is an isomorphism: ⇐ follows from thedefinition; ⇒ follows, since by the proof of Lemma 7.15, we know L(B′) and L(c(B′))are isomorphic in Sub(B).

Given this, if A is in D then the given diagram extends uniquely to

B′ LB′ A.

B LB

ηB′

∼=ηB

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Conversely, suppose A satisfies the condition. Let

R Aa

b

be the kernel-pair of AηA−−→ LA and d : A R the factorisation of (1A, 1A) through

(a, b). Since LηA is an isomorphism and L preserves pullbacks, Ld is an isomorphism,so d is dense. This forces a = b, so ηA is monic. And ηA is dense, so we get a uniquer : LA→ A with rηA = 1A. Now ηArηA = ηA and since LA satisfies the condition, wehave ηAr = 1LA.

We say A is a sheaf (for c, or for j) if it satisfies the condition in Lemma 7.18. Givena local operator j on E , we write shj(E ) for the full subcategory of j-sheaves in E .

7.19 Lemma. shj(E ) is closed under limits in E , and is an exponential ideal.

Proof. The first assertion follows since the definition involves only morphisms with

codomain A. For the second, note if B′ Bm is dense then so is B′ × C B × C(m,1)

for any C (since it is π∗1(m)) so if A is a sheaf then any morphism B′ → AC extendsuniquely to a morphism B → AC .

7.20 Lemma. If A is a sheaf, then a subobject A′ Am in E is a sheaf iff it is closed.

Proof.

⇐ Immediate by Lemma 7.17.

⇒ ConsiderA′ c(A′) A

p q

with p dense, so if A is a sheaf, we get a unique r : c(A′)→ A′ with rp = 1A′ . Butc(A′) is a sheaf and prp = p, so deduce pr = 1c(A′).

We define Ωj Ω to be the equalizer of

Ω Ω.j

1

Then, for any A, morphisms A → Ωj correspond to closed subobjects of A (a ‘closedsubobject’ classifier).

7.21 Lemma. Ωj is a j-sheaf.

Proof. We have to show that if B Am is dense, then the pullback along m yields abijection from closed subobjects of A to closed subobjects of B. If A′ An is closed,then in the pullback

B′ A′

B A

m′

n′ n

m

m′ is dense, so A′ A is the closure of B′ B A.

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It remains to show that if B′ B is closed, it is isomorphic to the pullback of itsclosure in A. But (writing A′ A for the closure) we have a factorisation B′ → f∗A′

which is dense since B′ → A′ is dense, and closed since B′ → B is closed.

7.22 Theorem. For any local operator j on E , shj(E ) is a topos. Moreover, it’sreflective in E and the reflector preserves finite limits.

Proof. We have that shj(E ) is cartesian closed by Lemma 7.19 and has subobject clas-sifier Ωj by Lemma 7.20 and Lemma 7.21. To construct the reflector, consider thecomposite

f : A ΩA ΩAj . jA

This corresponds to the closureA A×A(a,b)of the diagonal subobjectA A×A(1A,1A)

.

Claim: A Aa

bis the kernel-pair of f (proof omitted). Hence any morphism g :

A→ B for sheaf B satisfies ga = gb. So if we form the image

A I ΩAjq m

of f , any such g factorises uniquely through q.

Now ΩAj is a sheaf by Lemma 7.19 and Lemma 7.21 so if we form the closure

LA ΩAj of m, we get a morphism A → LA, through which any morphism fromA to a sheaf factors uniquely. Hence L becomes a functor E → shj(E ), left adjoint toinclusion.

By Lemma 6.13, we know L preserves finite products. In fact it preserves equalizersas well (omitted).

7.23 Theorem. Let E be a category. The following are equivalent:

(i) E is a topos, complete, locally small with a separating set of objects.

(ii) ∃ a small category C and a local operator on [C op,Set] such that

E ∼= shj([Cop,Set]).

Proof. (ii) ⇒ (i): since shj([C op,Set]) has given properties. (i) ⇒ (ii): take C the full

subcategory of E on the separating set and consider EY−→ [E op,Set]→ [C op,Set].

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Index

adjoint, 14Adjoint Functor Theorem

Primeval, 25adjunction, 14algebra

homomorphism, 31apex, 19

balanced, 8

cartesian closed, 41functor, 45locally, 43

categorical property, 5category, 2

axioms, 2balanced, 8discrete, 14equivalence, 5indiscrete, 14small, 2

*closed, 56*closure, 55codomain, 2colimit, 20comma category, 15complete, 23composition, 2cone, 19coproduct, 11cospan, 20counit, 16creates limits, 23

*dense, 56detecting family, 11diagram, 19domain, 2duality, 3

Eilenberg-Moore category, 31epimorphism, 8equivalent, 5exponential, 41exponential ideal, 46

free, 4Frobenius reciprocity, 45functor, 3

contravariant, 4

essentially surjective, 6faithful, 6forgetful, 4free, 4full, 6

has limits, 20Heyting semilattice, 41

identity, 2initial object, 14isomorphism, 3

Kleisli category, 31

limit, 20created, 23preserved, 23reflected, 23

*local operator, 56locally cartesian closed, 43locally small, 9

monad, 30monadic length, 39monadic tower, 38monoid, 3monomorphism, 8

regular, 11split, 8

morphism, 2epi-, 8iso-, 3mono-, 8

natural transformations, 4

object, 2opposite, 3opposite category, 3

power object, 50preserves limits, 23presheaf, 46product, 11projective, 13

reflection, 18reflects limits, 23reflexive, 35regular, 11

59

representable, 10representation, 10restriction maps, 46

separating family, 11sheaf, 46slice category, 41small, 2small category, 2solution set condition, 26span, 20split, 8split coequalizer, 36subobject, 27subobject classifier, 48subterminal, 46

T-algebra, 30terminal object, 14topos, 48

unit, 16universal element, 10

weak topos, 50weakly initial, 26weakly logical, 50well-powered, 27

Yonedaembedding, 10lemma, 9

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