’ ’

’’

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)

∼ ∼

= =

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 —¦ F ∼ F —¦ 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 —¦ U ∼ U —¦ 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

dd

F d dU F V GU

dd

d

d

‚©

C

(b) Apply Theorem 2(b) to the diagram

W EB

B

d

d

s

dd

F Gd dV U F U

dd

d

d

‚©

C

Adjoint triples

By an adjoint triple in a category C , we mean

(i) A triple T = (T, ·, µ) in C ,

(ii) A cotriple G = (G, , δ) 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 ’ C is adjoint tripleable if it is tripleable and cotripleable

’

(the latter means that U op : B op ’ C op is tripleable). Theorem 5 below implies,

’

among other things, that an adjoint tripleable functor results in an adjoint triple.

3.7 Adjoint Triples 141

Proposition 4. Let U : B ’ C a functor, and suppose B has either U -

’

contractible equalizers or U -contractible coequalizers. Then U is adjoint tripleable

if and only if it has left and right adjoints and re¬‚ects isomorphisms.

Proof. We ¬rst prove a weakened version of this proposition which is su¬cient

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 ’ C be a functor with right adjoint R and left adjoint

’

L. Let T = (T, ·, µ) be the triple induced by L and U and G = (G, , δ) 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 ¬rst statement, let G be right adjoint to T and consider the

diagram

UT EC

T

C

d

d

s

dd

F Td dU T T G

dd

d

‚©

d

C

Theorem 1 implies that U T has a right adjoint RT : C ’ C T for which U T —¦

’

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 ) ∼ Hom(LC, RC ) ∼ Hom(C, U RC )

= =

Now the ¬rst 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 — (’) is the

functor part of a triple in Set. This functor has the right adjoint Hom(M, ’), so

is part of an adjoint triple, and the underlying functor SetM ’ Set is adjoint

’

tripleable. Analogously, if K is a commutative ring and R a K-algebra, then

R —K ’: ModK ’ ModK has a right adjoint HomK (R, ’) and so gives rise to

’

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 de¬ne 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 ’ Set/Ob(C ). Show that this functor is

’

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

’ Ob(C ) and then use the fact that a set is a coproduct of its elements and left

’

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 scienti¬c discoveries

are like that. The ¬rst use of them was by Godement [1958] who used the ¬‚abby

sheaf cotriple to resolve sheaves for computing sheaf cohomology. He called it

the “standard construction” 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 [1961] 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 satis¬ed and noticed that all his

standard constructions were associated with adjoints.

As remarked in Section 3.2, Kleisli [1965] and independently Eilenberg-Moore

[1965] proved the converse. Although Hilton had conjectured the result, it was

Kleisli [1964] 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. Huber™s construction gave the one direction and

Kleisli™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 hadn™t found a better term, Eilenberg replied that

they hadn™t considered the concept very important and hadn™t thought it worth

investing much time in trying to ¬nd a good name. (By contrast, when Cartan-

Eilenberg [1956] was composed, the authors gave so much thought to naming

their most important concept that the manuscript had blanks inserted before the

¬nal preparation, when they ¬nally found the exact term.)

At the same time, more or less, Applegate [1965] was discovering the con-

nection between triples and acyclic models and Beck [1967] (but the work was

substantially ¬nished in 1964) was discovering the connection with homology.

In addition, Lawvere [1963] 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 Beck™s

which in part was a generalization of Linton™s results. Variations on that theo-

rem followed (Duskin [1969], Par´ [1971]) 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.

Butler™s theorems”Theorem 3 above includes somewhat special cases of two

of them”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 unveri¬ed, except by Butler,

since 1971. Within the past two years, they have been independently veri¬ed and

substantially generalized in his doctoral thesis: [1984], 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 ¬elds, 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 Shoen¬eld [1967], Chang and Keisler [1973]) in which “theory” is

explicated as a formal language with rules of deduction and axioms, and a model

is a set with structure which corresponds in a speci¬c way with the language and

satis¬es the axioms. Our theories should perhaps have been called “categorical

theories”; 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 ¬nite products which contains the

generic group object, in the sense to be de¬ned precisely in Section 4.1. (The

de¬nition of group object using representable functors mentioned in Section 1.7

does not require that the category have ¬nite products but we do not know how

to handle that more general type of theory.) On the other hand, a theory of

¬elds using only categories with ¬nite products cannot be given, so one must

climb further in the hierarchy to give the generic ¬eld.

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-

mann™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 ¬elds) require the

machinery of Grothendieck topologies and are described in Chapter 8.

A brief description of this hierarchy and its connections with di¬erent types

of logical systems has been given by Lawvere [1975]. Makkai and Reyes [1977]

provide a detailed exposition of the top of the hierarchy. Ad´mek and Rosiˇky

a c

[1994] present much modern material on theories and their model categories not

covered here. Barr and Wells [1999] is more elementary and gives many examples

of the use of sketches in computing science.

144

4.1 Sketches 145

4.1 Sketches