FU B B

’ ’ ’ ’ FUB ’ ’ B

’’’

FUFUB ’ ’ ’ ’ ’’ (21)

’’’

FUB

is a coequalizer for every object B of B.

3.3 Tripleability 117

For all objects B and B of B,

Lemma 8.

HomB (F U B, B ) ∼ HomC T (¦F U B, ¦B )

=

where ¦: B ’ C T is the comparison functor. (In other words, “¦ is full and

’

faithful on arrows out of free objects”).

Proof. We have,

HomB (F U B, B ) ∼ HomC (U B, U B )

=

∼ HomC (U B, U T (U B , U B ))

=

∼ HomC T (F T (U B), (U B , U B ))

=

∼ HomC T ((T U B, µU B), (U B , U B ))

=

∼ HomC T (¦F U B, ¦B )—¦

=

Theorem 9. [Beck] ¦ is full and faithful if and only if B is a regular epi for

all objects B of B.

Proof. U B is the structure map of an algebra in C T by Exercise 1, and so by

Proposition 4 is a coequalizer of a parallel pair with domain in the image of F T ,

hence in the image of ¦. Since ¦( B) = U B, it follows from Exercise 9 that if

¦ is full and faithful, then B is a regular epi.

Conversely, suppose B is a regular epi. If f, g: B ’ B in B and U f = U g,

’

then F U f = F U g and this diagram commutes:

F U fE

E FUB

FUB

FUg

B B

f

c Ec

˜

B EB

g

Thus since B is epi, f = g. Hence ¦ is faithful.

Under the hypothesis, (21) is a U -contractible coequalizer diagram by Corol-

laries 5 and 7. Since U T —¦ ¦ = U , applying ¦ to (21) gives a U T -contractible

coequalizer diagram. It follows that the horizontal edges of the diagram below

are equalizers; the top row homsets are computed in B and the bottom row in

C T . The vertical arrows are those induced by ¦; by Lemma 8, the middle and

118 3 Triples

right one are isomorphisms. Thus the left one is an isomorphism, too, proving

the Proposition.

E

E Hom(F U B, B )

Hom(B, B ) E Hom(F U F U B, B )

c c c

E

E Hom(¦F U B, ¦B )

Hom(¦B, ¦B ) E Hom(¦F U F U B, ¦B )

Theorem 10. [Beck™s Precise Tripleability Theorem] U : B ’ C is tripleable

’

if and only if

(i) U has a left adjoint.

(ii) U re¬‚ects isomorphisms.

(iii) B has coequalizers of re¬‚exive U -contractible coequalizer pairs and U

preserves them.

Proof. If U is tripleable it has a left adjoint F by de¬nition and it satis¬es (ii)

and (iii) by Propositions 1 and 3. (Note that in fact B has and U preserves the

coequalizers of all U -contractible parallel pairs, not merely re¬‚exive ones”that

is the way Beck originally stated the theorem).

Now, to do the other direction, we know (16) is a re¬‚exive U -contractible

coequalizer pair, so by (iii) it has a coequalizer B . Since B coequalizes (16)

(because is a natural transformation), we know that there is an arrow f making

F U BE

EB

FUFUB E FUB

FU B d

Bd f

d

dc

‚

B

commute. However, as observed in the proof of Corollary 5, U B is coequalizer

of U of this diagram, so U f is an isomorphism. Hence f is an isomorphism, so

B is a regular epi, so that ¦ is full and faithful by Theorem 9.

The argument just given that f is an isomorphism can easily be used to show

that in fact any functor U satisfying (ii) and (iii) must re¬‚ect coequalizers of

re¬‚exive U -contractible coequalizer pairs (Exercise 1).

For any object (C, c) of C T , we must ¬nd an object B of B for which ¦(B) ∼ =

(C, c). Now ¦ of the following diagram is (13),

Fc

FUFC ’’ ’ FC

’ ’’

’’ ’

’ ’’

FC

3.3 Tripleability 119

so it is a re¬‚exive U -contractible coequalizer. Thus by assumption there is a

coequalizer B for which the sequence underlying

Fc

FUFC ’’ ’ FC ’ B

’ ’’ ’

’’ ’

’ ’’

FC

is

UFc

UFUFC ’ ’ ’ ’ UFC ’ UB

’’’ ’

’’ ’’

’’’

U FC

By Proposition 4, this last diagram is U T of a coequalizer diagram in C T with

coequalizer (C, c). Since U T re¬‚ects such coequalizers and U T —¦ ¦ = U , it follows

that ¦(B) ∼ (C, c), as required.

=

Theorem 10 is the precise tripleability theorem as distinct from certain theo-

rems to be discussed in Section 3.5 which give conditions for tripleability which

are su¬cient but not necessary. Theorem 10 is commonly known by its acronym

“PTT”.

Other conditions su¬cient for tripleability are discussed in Section 9.1.

We extract from the last paragraph of the proof of the PTT the following

proposition, which we need later. This proposition can be used to provide an

alternate proof of the equivalence of C T and B in the PTT (see Exercise 12 of

Section 1.9).

Proposition 11. If U : B ’ C has a left adjoint and B has coequalizers of

’

re¬‚exive U -contractible coequalizer pairs, then the comparison functor ¦ has a

left adjoint.

Proof. The object B constructed in the last paragraph of the proof of Theorem 10

requires only the present hypotheses to exist. De¬ne Ψ(C, c) to be B. If g: (C, c)

’ (D, d), then the diagram

’

FcE

E ΨC

FUFC E FC

FC

FUFg Fg Ψg

c F dE c c

E ΨD

FUFD E FD

FD

commutes serially, the top square because c and d are structure maps and the

bottom one because is a natural transformation. Thus it induces a map Ψ(g).

It is straightforward to check that this makes Ψ a functor which is left adjoint to

the comparison functor ¦.

120 3 Triples

Compact Hausdor¬ spaces

We illustrate the use of the PTT by proving that compact Hausdor¬ spaces

are tripleable over Set. This fact was actually proved by F. E. J. Linton before

Beck proved the theorem, using an argument which, after suitable generalization,

became Duskin™s Theorem of Section 9.1.

The underlying set functor U from the category CptHaus of compact Hausdor¬

spaces and continuous maps to Set has a left adjoint β, where βX is the Stone-

˜

Cech compacti¬cation of the set X considered as a discrete space.

Proposition 12. U is tripleable.

Proof. The statement that U re¬‚ects isomorphisms is the same as the statement

that a bijective continuous map between compact Hausdor¬ spaces is a home-

omorphism, which is true. A pair d0 , dl : C ’ C of continuous maps between

’

compact Hausdor¬ spaces has a coequalizer in T which is necessarily preserved

op

by the underlying set functor since that functor has a right adjoint (the functor

which puts the indiscrete topology on a set X).

The quotient is compact and will be Hausdor¬ if and only if the kernel pair is

closed. Thus we will be ¬nished if we show that the kernel pair of this coequalizer

is closed. That kernel pair is the equivalence relation generated by R (the relation

which is the image of C in C — C). By Exercise 2(c) below, the kernel pair is

R —¦ Rop . Now R is closed in C — C (it is the image of a map of a compact space

into a Hausdor¬ space), and so is compact Hausdor¬. Hence the ¬ber product

R —C Rop is compact Hausdor¬, so is closed in C — C — C. R —¦ Rop is the image

of that space in the Hausdor¬ space C — C and so is closed, as required.

The functor part of this triple takes a set to the set of ultra¬lters on it. The

functor which takes a set to the set of ¬lters on it is also part of a triple, the

algebras for which are continuous lattices (Day [1975], Wyler [1981]). Continuous

lattices are also algebras for a triple in the category of topological spaces and

elsewhere. Another example of this last phenomenon of being the category of

algebras for triples in di¬erent categories is the category of monoids, which is

tripleable over Set and also (in three di¬erent ways) over Cat (Wells [1980]).

Exercises 3.3

1. Show that a functor U : B ’ C which re¬‚ects isomorphisms has the property

’

that if a diagram D in B has a colimit and X is a cocone from D in B for

which U X is a colimit of U D, then X is a colimit of D. Do the same for limits.

(These facts are summarized by the slogan: “A functor that re¬‚ects isomorphisms

3.3 Tripleability 121

re¬‚ects all limits and colimits it preserves.” This slogan exaggerates the matter

slightly: The limits and colimits in question have to be assumed to exist.)

2. (Suggested in part by Barry Jay.) A parallel pair d0 , dl : X ’ X in Set

’

determines a relation R on X , namely the image of (d0 , dl ): X ’ X — X.

’

Conversely, a relation R on a set S de¬nes a parallel pair from R to S (the two

projection maps).

(a) Show that a relation R in Set determines a re¬‚exive parallel pair if and

only if it contains the diagonal.

(b) Show that a relation R in Set determines a contractible pair if and only if

each equivalence class [x] in the equivalence a relation E generated by R contains

an element x— with the property that xEx if and only if xRx— and x Rx— .

(c) Show that the parallel pair determined by an ordering on a set is con-

tractible if and only if every connected component of the coequalizer ordered set

has a maximum element.

(d) Show that an equivalence relation in Set is always contractible.

(e) Show that if R is the relation determined by any contractible parallel pair

in Set , then E = R —¦ Rop is the equivalence relation generated by R.

3. Show that the algebra map c constructed in the proof of Proposition 3 is the

only structure map T C ’ C which makes d an algebra map.

’

4. Prove that Diagram (19) is a contractible coequalizer.

5. Show that the Eilenberg-Moore comparison functor is the only functor ¦: B

’ C T for which U T —¦ ¦ = U and ¦ —¦ F = F T . (Hint: Show that for B ∈ ObB,

’

¦(B) must be (U B, b) for some arrow b: U F U B ’ U B), and then consider this

’

diagram:

UBE

UFUB UB

·U F U B ·U B

c c

E UFUB

UFUFUB

UFb

U FUB b

c c

E UB

UFUB

b

122 3 Triples

6. Let U be a functor with left adjoint F . Then the comparison functor for

the induced triple is an isomorphism of categories (not merely an equivalence)

if and only if U re¬‚ects isomorphisms and creates coequalizers for re¬‚exive U -

contractible coequalizer pairs. (Compare Exercise 22 of Section 1.7.)

7. A subcategory CO of a category C is re¬‚ective (or re¬‚exive) if the inclu-

sion functor has a left adjoint. Show that the inclusion functor of a re¬‚ective

subcategory is tripleable.

8. Show that an equivalence of categories re¬‚ects isomorphisms.

9. Show that if H is a full and faithful functor and Hf is the coequalizer of a

parallel pair with domain in the image of H, then f is a coequalizer.

3.4 Properties of Tripleable Functors

In this section we describe various properties a tripleable functor must have.

Some of these are useful in the development of topos theory, and as necessary