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Corollary 7.

FU B B
в€’ в€’ в€’ в†’ FUB в€’ в€’ B
в€’в€’в€’
FUFUB в€’ в€’ в€’ в†’ в€’в†’ (21)
в€’в€’в€’
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 ) в€ј HomC T (О¦F U B, О¦B )
=

where О¦: B в€’ C T is the comparison functor. (In other words, вЂњО¦ is full and
в†’
faithful on arrows out of free objectsвЂќ).
Proof. We have,

HomB (F U B, B ) в€ј HomC (U B, U B )
=
в€ј HomC (U B, U T (U B , U B ))
=
в€ј HomC T (F T (U B), (U B , U B ))
=
в€ј HomC T ((T U B, ВµU B), (U B , U B ))
=
в€ј HomC T (О¦F U B, О¦B )в—¦
=

Theorem 9. [Beck] О¦ 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 О¦. Since О¦( B) = U B, it follows from Exercise 9 that if
О¦ is full and faithful, then B is a regular epi.
Conversely, suppose B is a regular epi. If f, g: B в€’ B in B and U f = U g,
в†’
then F U f = F U g and this diagram commutes:
F U fE
E FUB
FUB
FUg

B B
f
c Ec
вЂ˜
B EB
g
Thus since B is epi, f = g. Hence О¦ is faithful.
Under the hypothesis, (21) is a U -contractible coequalizer diagram by Corol-
laries 5 and 7. Since U T в—¦ О¦ = U , applying О¦ 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 О¦; 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(О¦F U B, О¦B )
Hom(О¦B, О¦B ) E Hom(О¦F U F U B, О¦B )

Theorem 10. [BeckвЂ™s Precise Tripleability Theorem] U : B в€’ C is tripleable
в†’
if and only if
(i) U has a left adjoint.
(ii) U reп¬‚ects isomorphisms.
(iii) B has coequalizers of reп¬‚exive U -contractible coequalizer pairs and U
preserves them.
Proof. If U is tripleable it has a left adjoint F by deп¬Ѓnition and it satisп¬Ѓes (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 reп¬‚exive onesвЂ”that
is the way Beck originally stated the theorem).
Now, to do the other direction, we know (16) is a reп¬‚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
В‚
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 О¦ 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 reп¬‚ect coequalizers of
reп¬‚exive U -contractible coequalizer pairs (Exercise 1).
For any object (C, c) of C T , we must п¬Ѓnd an object B of B for which О¦(B) в€ј =
(C, c). Now О¦ of the following diagram is (13),
Fc
FUFC в€’в€’ в€’ FC
в€’ в€’в†’
в€’в€’ в€’
в€’ в€’в†’
FC
3.3 Tripleability 119
so it is a reп¬‚exive U -contractible coequalizer. Thus by assumption there is a
coequalizer B for which the sequence underlying
Fc
FUFC в€’в€’ в€’ FC в€’ B
в€’ в€’в†’ в†’
в€’в€’ в€’
в€’ в€’в†’
FC
is
UFc
UFUFC в€’ в€’ в€’ в†’ UFC в€’ UB
в€’в€’в€’ в†’
в€’в€’ в€’в†’
в€’в€’в€’
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 reп¬‚ects such coequalizers and U T в—¦ О¦ = U , it follows
that О¦(B) в€ј (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 suп¬ѓcient but not necessary. Theorem 10 is commonly known by its acronym
вЂњPTTвЂќ.
Other conditions suп¬ѓcient 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 в€’ C has a left adjoint and B has coequalizers of
в†’
reп¬‚exive U -contractible coequalizer pairs, then the comparison functor О¦ has a
Proof. The object B constructed in the last paragraph of the proof of Theorem 10
requires only the present hypotheses to exist. Deп¬Ѓne ОЁ(C, c) to be B. If g: (C, c)
в€’ (D, d), then the diagram
в†’
FcE
E ОЁC
FUFC E FC
FC
FUFg Fg ОЁg

c F dE c c
E ОЁD
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 ОЁ(g).
It is straightforward to check that this makes ОЁ a functor which is left adjoint to
the comparison functor О¦.
120 3 Triples
Compact Hausdorп¬Ђ spaces

We illustrate the use of the PTT by proving that compact Hausdorп¬Ђ 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 DuskinвЂ™s Theorem of Section 9.1.
The underlying set functor U from the category CptHaus of compact Hausdorп¬Ђ
spaces and continuous maps to Set has a left adjoint ОІ, where ОІX is the Stone-
Л˜
Cech compactiп¬Ѓcation of the set X considered as a discrete space.
Proposition 12. U is tripleable.
Proof. The statement that U reп¬‚ects isomorphisms is the same as the statement
that a bijective continuous map between compact Hausdorп¬Ђ spaces is a home-
omorphism, which is true. A pair d0 , dl : C в€’ C of continuous maps between
в†’
compact Hausdorп¬Ђ 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 Hausdorп¬Ђ if and only if the kernel pair is
closed. Thus we will be п¬Ѓnished 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 Г— C). By Exercise 2(c) below, the kernel pair is
R в—¦ Rop . Now R is closed in C Г— C (it is the image of a map of a compact space
into a Hausdorп¬Ђ space), and so is compact Hausdorп¬Ђ. Hence the п¬Ѓber product
R Г—C Rop is compact Hausdorп¬Ђ, so is closed in C Г— C Г— C. R в—¦ Rop is the image
of that space in the Hausdorп¬Ђ space C Г— C and so is closed, as required.
The functor part of this triple takes a set to the set of ultraп¬Ѓlters on it. The
functor which takes a set to the set of п¬Ѓlters on it is also part of a triple, the
algebras for which are continuous lattices (Day , Wyler ). 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 diп¬Ђerent categories is the category of monoids, which is
tripleable over Set and also (in three diп¬Ђerent ways) over Cat (Wells ).

Exercises 3.3

1. Show that a functor U : B в€’ C which reп¬‚ects isomorphisms has the property
в†’
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: вЂњA functor that reп¬‚ects isomorphisms
3.3 Tripleability 121
reп¬‚ects all limits and colimits it preserves.вЂќ 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 в€’ X in Set
в†’
determines a relation R on X , namely the image of (d0 , dl ): X в€’ X Г— X.
в†’
Conversely, a relation R on a set S deп¬Ѓnes a parallel pair from R to S (the two
projection maps).
(a) Show that a relation R in Set determines a reп¬‚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 xв€— with the property that xEx if and only if xRxв€— and x Rxв€— .
(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 в—¦ 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 в€’ C which makes d an algebra map.
в†’

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

5. Show that the Eilenberg-Moore comparison functor is the only functor О¦: B
в€’ C T for which U T в—¦ О¦ = U and О¦ в—¦ F = F T . (Hint: Show that for B в€€ ObB,
в†’
О¦(B) must be (U B, b) for some arrow b: U F U B в€’ U B), and then consider this
в†’
diagram:
UBE
UFUB UB

О·U F U B О·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 reп¬‚ects isomorphisms and creates coequalizers for reп¬‚exive U -
contractible coequalizer pairs. (Compare Exercise 22 of Section 1.7.)

7. A subcategory CO of a category C is reп¬‚ective (or reп¬‚exive) if the inclu-
sion functor has a left adjoint. Show that the inclusion functor of a reп¬‚ective
subcategory is tripleable.

8. Show that an equivalence of categories reп¬‚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
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