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F C2 в€’ F C1 в€’ LB
в†’ в†’
в€’в†’

is as well. The universal mapping property of coequalizers gives, for any object
B of B the diagram below in which both lines are equalizers,
E
E Hom(F C1 , B)
Hom(LB , B) E Hom(F C2 , B)
в€ј в€ј
= =
c c
E
E Hom(F C1 , W B)
Hom(B , W B) E Hom(F C2 , W B)

from which the adjointness follows.
Theorem 3. [Butler] In the situation

WE
B B
T T
F U F U
cVEc
C' C
G
(a) Suppose:
(i) F is left adjoint to U ,
(ii) F is left adjoint to U ,
(iii) W в—¦ F в€ј F в—¦ V ,
=
(iv) G is right adjoint to V ,
(v) U is of descent type, and
(vi) B has and W preserves coequalizers of U -contractible coequalizer pairs.
Then W has a right adjoint.
(b) Suppose:
(i) F is left adjoint to U ,
(ii) F is left adjoint to U ,
(iii) V в—¦ U в€ј U в—¦ W ,
=
(iv) G is left adjoint to V ,
(v) U is of descent type, and
140 3 Triples
(vi) B has coequalizers.
Then W has a left adjoint.
.
Proof.
(a) Apply Theorem 2(a) to the diagram

W EB
B
d
sd В
В
dd В
В
F d dU F V В  GU
В
dd В В
dВ
dВ
C
(b) Apply Theorem 2(b) to the diagram

W EB
B
d В
d
s В

dd В
В
F Gd dV U F В  U
В
dd В
В
dВ
dВ
C

By an adjoint triple in a category C , we mean
(i) A triple T = (T, О·, Вµ) in C ,
(ii) A cotriple G = (G, , Оґ) in C , for which
(iii) T is left adjoint to G.
We say in this case that T is left adjoint to G.
A functor U : B в€’ C is adjoint tripleable if it is tripleable and cotripleable
в†’
(the latter means that U op : B op в€’ C op is tripleable). Theorem 5 below implies,
в†’
among other things, that an adjoint tripleable functor results in an adjoint triple.
3.7 Adjoint Triples 141
Proposition 4. Let U : B в€’ C a functor, and suppose B has either U -
в†’
contractible equalizers or U -contractible coequalizers. Then U is adjoint tripleable
if and only if it has left and right adjoints and reп¬‚ects isomorphisms.
Proof. We п¬Ѓrst prove a weakened version of this proposition which is suп¬ѓcient
to prove Theorem 5 below. Then we will use Theorem 5 to prove the version as
stated.
Assume that B has both U -contractible equalizers and U -contractible coequal-
izers. The existence of both adjoints implies that they are preserved, so that the
(weakened) proposition follows from PTT.
Theorem 5. Let T be a triple in C and suppose that T has a right adjoint G.
Then G is the functor part of a cotriple G in C for which C T is equivalent to CG
and the underlying functor U T has left and right adjoints which induce T and G
respectively.
Conversely, let U : B в€’ C be a functor with right adjoint R and left adjoint
в†’
L. Let T = (T, О·, Вµ) be the triple induced by L and U and G = (G, , Оґ) the cotriple
in C induced by U and R. Then T is left adjoint to G and the category C T of
T-algebras is equivalent to the category CG of G-coalgebras.
Proof. To prove the п¬Ѓrst statement, let G be right adjoint to T and consider the
diagram
UT EC
T
C
d В
d
s 
В
dd В В
F Td dU T TВ  G
В
dd В В
dВ
dВ
C
Theorem 1 implies that U T has a right adjoint RT : C в€’ C T for which U T в—¦
в†’
T = G. By the weak version of Proposition 4, U T is tripleable and cotripleable.
R
Hence C T is equivalent to CG .
To do the converse, the adjunction between T and G is seen from the calcu-
lation
Hom(U LC, C ) в€ј Hom(LC, RC ) в€ј Hom(C, U RC )
= =
Now the п¬Ѓrst half of the theorem yields a right adjoint RT to U T which induces
G, so C T is equivalent to CG .
We now complete the proof of Proposition 4. Assume that B has U -contractible
coequalizers. (The proof in the case that B has U -contractible equalizers is dual.)
The existence of a right adjoint to U means that U preserves them, so that U is
tripleable by PTT. Hence B is equivalent to C T . The second part of Theorem 5
142 3 Triples
then implies that B is equivalent to C G , so that B must have U -contractible
equalizers.
We saw in Section 3.1 that for any monoid M , the functor M Г— (в€’) is the
functor part of a triple in Set. This functor has the right adjoint Hom(M, в€’), so
is part of an adjoint triple, and the underlying functor SetM в€’ Set is adjoint
в†’
tripleable. Analogously, if K is a commutative ring and R a K-algebra, then
R вЉ—K в€’: ModK в€’ ModK has a right adjoint HomK (R, в€’) and so gives rise to
в†’
an adjoint triple. The algebras for this triple are the modules over the K-algebra
R (R modules in which the action of K commutes with that of R). If K = Z, we
just get R-modules.
In Section 6.7 we will make use of the fact that in a topos, functor categories
(functors from a category object to the topos) are adjoint tripleable. Exercise 1
asks you to prove this for Set. The general situation is complicated by the problem
of how to deп¬Ѓne that functor category in a topos.

Exercises 3.7

1. Let C be a small category and E the category of functors from C to Set.
There is an underlying functor E в€’ Set/Ob(C ). Show that this functor is
в†’
adjoint tripleable. (Hint: One way to approach this is to use the Yoneda lemma
to determine what the left or right adjoint must be on objects of the form 1
в€’ Ob(C ) and then use the fact that a set is a coproduct of its elements and left
в†’
and right adjoints preserve colimits and limits respectively.)

2. Deduce the Yoneda lemma from Exercise 1.

3.8 Historical Notes on Triples
It is very hard to say who invented triples. Probably many scientiп¬Ѓc discoveries
are like that. The п¬Ѓrst use of them was by Godement  who used the п¬‚abby
sheaf cotriple to resolve sheaves for computing sheaf cohomology. He called it
the вЂњstandard constructionвЂќ and presumably intended by that nothing more than
a descriptive phrase. It seems likely that he never intended to either create or
name a new concept.
Nonetheless Huber  found these constructions useful in his homotopy
theory and now did name them standard constructions. He also provided the
proof that every adjoint pair gave rise to one, whatever it was called. He com-
mented later that he proved that theorem because he was having so much trouble
3.8 Historical Notes on Triples 143
demonstrating that the associative identity was satisп¬Ѓed and noticed that all his
standard constructions were associated with adjoints.
As remarked in Section 3.2, Kleisli  and independently Eilenberg-Moore
 proved the converse. Although Hilton had conjectured the result, it was
Kleisli  who had an application. He wanted to show that resolutions using
resolvent pairs (essentially pairs of adjoint functors) and those using triples give
the same notion of resolution. HuberвЂ™s construction gave the one direction and
KleisliвЂ™s gave the other.
Eilenberg and Moore also gave them the name by which they are known here:
triples. Although we do not regard this name as satisfactory we do not regard the
proposed substitutes as any better. In this connection, it is worth mentioning
that when asked why they hadnвЂ™t found a better term, Eilenberg replied that
they hadnвЂ™t considered the concept very important and hadnвЂ™t thought it worth
investing much time in trying to п¬Ѓnd a good name. (By contrast, when Cartan-
Eilenberg  was composed, the authors gave so much thought to naming
their most important concept that the manuscript had blanks inserted before the
п¬Ѓnal preparation, when they п¬Ѓnally found the exact term.)
At the same time, more or less, Applegate  was discovering the con-
nection between triples and acyclic models and Beck  (but the work was
substantially п¬Ѓnished in 1964) was discovering the connection with homology.
In addition, Lawvere  had just found out how to do universal algebra by
viewing an algebraic theory as a category and an algebra as a functor. Linton
was soon to connect these categories with triples. In other words triples were
beginning to pervade category theory but it is impossible to give credit to any
one person. The next important step was the tripleableness theorem of BeckвЂ™s
which in part was a generalization of LintonвЂ™s results. Variations on that theo-
rem followed (Duskin , ParВґ ) and acquired arcane names, but they
e
all go back to Beck and Linton. They mostly arose either because of the failure
of tripleableness to be transitive or because of certain special conditions.
ButlerвЂ™s theoremsвЂ”Theorem 3 above includes somewhat special cases of two
of themвЂ”are due to a former McGill University graduate student, William Butler.
They consisted of a remarkable series of 64 theorems, 12 on the existence of
adjoints and 52 on various technical results on tripleableness and related questions
such as when a functor is of descent type. These theorems have never been
published and, as a matter of fact, have remained unveriп¬Ѓed, except by Butler,
since 1971. Within the past two years, they have been independently veriп¬Ѓed and
substantially generalized in his doctoral thesis: , by another student, John
A. Power, who found a few minor mistakes in the statements.
4
Theories
In this chapter, we explicate the naive concept of a mathematical theory, such as
the theory of groups or the theory of п¬Ѓelds, in such a way that a theory becomes
a category and a model for the theory becomes a functor based on the category.
Thus a theory and a model become instances of mathematical concepts which are
widely used by mathematicians. This is in contrast to the standard treatment of
the topic (see Shoenп¬Ѓeld , Chang and Keisler ) in which вЂњtheoryвЂќ is
explicated as a formal language with rules of deduction and axioms, and a model
is a set with structure which corresponds in a speciп¬Ѓc way with the language and
satisп¬Ѓes the axioms. Our theories should perhaps have been called вЂњcategorical
theoriesвЂќ; however, the usage here is now standard among category theorists.
Our theories are, however, less general than the most general sort of theory
in mathematical logic.
We will construct a hierarchy of types of theories, consisting of categories with
various amounts of structure imposed on them. For example, we will construct
the theory of groups as the category with п¬Ѓnite products which contains the
generic group object, in the sense to be deп¬Ѓned precisely in Section 4.1. (The
deп¬Ѓnition of group object using representable functors mentioned in Section 1.7
does not require that the category have п¬Ѓnite products but we do not know how
to handle that more general type of theory.) On the other hand, a theory of
п¬Ѓelds using only categories with п¬Ѓnite products cannot be given, so one must
climb further in the hierarchy to give the generic п¬Ѓeld.
In this chapter we consider the part of the hierarchy which can be developed
using only basic ideas about limits. In the process we develop a version of Ehres-
mannвЂ™s theory of sketches suitable for our purposes. This chapter may be read
immediately after Chapter 1, except for Theorem 5 of Section 4.3. The theories
higher in the hierarchy (in particular including the theory of п¬Ѓelds) require the
machinery of Grothendieck topologies and are described in Chapter 8.
A brief description of this hierarchy and its connections with diп¬Ђerent types
of logical systems has been given by Lawvere . Makkai and Reyes 
provide a detailed exposition of the top of the hierarchy. AdВґmek and RosiЛ‡ky
a c
 present much modern material on theories and their model categories not
covered here. Barr and Wells  is more elementary and gives many examples
of the use of sketches in computing science.
144
4.1 Sketches 145
4.1 Sketches
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