¬nal among all ways of factoring T as an adjoint pair. We will describe how this

works with C T and leave the other part to you (Exercise 5).

Suppose we have F : C ’ B, U : B ’ C , with F left adjoint to U and unit

’ ’

and counit ·: id ’ U F , : F U ’ id, with T = (T = U F, ·, U F ). Let U T : C T

’ ’

T : C ’ C T be the adjoint pair given by Construction 2 of the proof of

’ C, F

’ ’

Theorem 1. The Eilenberg-Moore comparison functor is the functor ¦: B

’ C T which takes B to (U B, U B) and f to U f . It is easy to see that this really

’

is a functor, and in fact the only functor for which U T —¦ ¦ = U and ¦ —¦ F = F T .

This says C T is the terminal object in the category of adjoint pairs which induce

T (Exercise 5).

The Eilenberg-Moore functor is in many important cases an isomorphism or

equivalence of categories, a topic which is pursued in Sections 3.3 and 3.4.

Coalgebras for a cotriple

If G = (G, , δ) is a cotriple in a category C , the construction of the Eilenberg

Moore category of algebras of G regarded as a triple in C op yields, when all arrows

are reversed, the category CG of coalgebras of G. Precisely, a G-coalgebra is a

pair (A, ±) with ±: A ’ GA for which the following diagrams commute:

’

A Ga

A' G2 A '

GA GA

d

s T T T

d (3)

a a

idA δA

d

d

GA ' a

A A

3.2 The Kleisli and Eilenberg-Moore Categories 107

A morphism f : (A, ±) ’ (B, β) is an arrow f : A ’ B for which Gf —¦± = β —¦f .

’ ’

We will prove in Section 3.5 that when a functor has both a left and a right

adjoint, the corresponding categories of algebras and coalgebras are isomorphic.

Our major use of cotriples in this book will be based on the fact that the category

of coalgebras for a left exact cotriple (meaning the functor is left exact) in a topos

is itself a topos.

Exercises 3.2

1. Let T = (T, ·, µ) be a triple, A and B be objects of the underlying category.

(a) Show that (T A, µA) is an algebra for T. (Such algebras are called free.)

(b) Show that for any f : A ’ B, T f is an algebra map from (T A, µA) to

’

(T B, µB).

(c) Show that µA is an algebra morphism from (T T A, µT A) to (T A, µA).

2. (Manes) Let C be a category. Show that the following data:

(i) A function T : Ob(C ) ’ Ob(C );

’

(ii) for each pair of objects C and D of C a function Hom(C, T D) ’ ’

Hom(T C, T D), denoted f ’ f — ;

(iii) for each object C of C a morphism ·C: C ’ T C;

’

subject to the conditions:

(i) For f : C ’ T D, f = ·T D —¦ f — ;

’

(ii) for any object C, (·C)— = idT C ;

(iii) for f : C ’ T D and g: D ’ T E, (g — —¦ f )— = g — —¦ f — ;

’ ’

are equivalent to a triple on C . (Hint: An elegant way to attack this exercise

is to use the data to de¬ne the Kleisli category for the triple, using the pointwise

adjunction construction (Theorem 1 of Section 1.9) to get the adjoint pair whose

corresponding triple is the one sought.)

3. Show that if T is the group triple, the Eilenberg-Moore comparison functor

¦: Grp ’ SetT is an isomorphism of categories.

’

4. Let T be a triple in a category C . Let (C, c) be an algebra for T and let B be

a subobject of C. Show that a map b: T B ’ B is an algebra structure on B for

’

108 3 Triples

which inclusion is an algebra map if and only if

E TC

TB

c

b

c c

EC

B

commutes, and that there cannot be more than one such map b. (This says in

e¬ect that B “is” a subalgebra if and only if it is “closed under the operations””

in other words, c(T B) ⊆ B. In Section 6.4, we give a condition for a subobject

of a coalgebra of a left exact cotriple in a topos to be a subcoalgebra.)

5. For a given triple T in a category C , let E be the category in which an object

is a category B together with an adjoint pair of functors F : C ’ B, U : B ’ C

’ ’

which induces the triple T via Theorem 1 of Section 3.1, and in which an arrow

from (B, F, U ) to (B , F , U ) is a functor G: B ’ B for which U —¦ G = U and

’

G —¦ F = F . Show that K (T) is the initial object in E and C T is the terminal

object.

6. Show that the coalgebras for the cotriple de¬ned in Exercise 6 of Section 3.1

form a category isomorphic to the category of sets acted on on the right by M .

7. Show that for a triple T in a category C each of the following constructions

give a category K isomorphic to the Kleisli category.

(a) K is the full subcategory of C T whose objects are the image of F T , i.e.

all objects of the form (F T C, µC), C an object of C .

(b) K op is the full subcategory of Func(C T , Set) whose objects are of the

form HomC (C, U T (’)), C an object of C .

Linton used the second de¬nition, in which F T does not appear, to study

algebraic theories in the absence of a left adjoint to the underlying functor.

8. (Linton) Let T = (T, ·, µ) be a triple in C . Let K be the Kleisli cate-

gory of T and FT : C ’ K be the left adjoint to UT : K ’ C . Let H: C T

’ ’

’ Func(K op , Set) denote the functor which takes (C, c) to the restriction of

’

HomC T (’, (C, c)) to K , where K is regarded as a subcategory of C T as in

Exercise 7(a) above. Prove that the diagram

HE

CT op

Func(K , Set)

op

Func(FT , Set)

UT

c c

E Func(C op , Set)

C

Yoneda

3.3 Tripleability 109

is a pullback.

3.3 Tripleability

In this section we will be concerned with the question of deciding when a func-

tor U : B ’ C with a left adjoint has the property that the Eilenberg-Moore

’

category for the corresponding triple is essentially the same as B.

To make this precise, a functor U which has a left adjoint for which the corre-

sponding Eilenberg-Moore comparison functor ¦ is an equivalence of categories

is said to be tripleable. If ¦ is full and faithful, we say that U is of descent

type (and if it is tripleable, that it is of e¬ective descent type). We will often

say B is tripleable over C if there is a well-understood functor U : B ’ C . Thus

’

Grp is tripleable over Set, for example (Exercise 3 of Section 3.2).

In this section we state and prove a theorem due to Beck giving conditions on

a functor U : B ’ C which insure that it is tripleable. Variations on this basic

’

theorem will be discussed in Sections 3.4 and 9.1. Before we can state the main

theorem, we need some background.

Re¬‚ecting isomorphisms

A functor U re¬‚ects isomorphisms if whenever U f is an isomorphism, so

is f . For example, the underlying functor U : Grp ’ Set re¬‚ects isomorphisms”

’

that is what you mean when you say that a group homomorphism is an iso-

morphism if and only if it is one to one and onto. (Warning: That U re¬‚ects

isomorphisms is not the same as saying that if U X is isomorphic to U Y then X

is isomorphic to Y ”for example, two groups with the same number of elements

need not be isomorphic). Observe that the underlying functor U : T ’ Set

op ’

does not re¬‚ect isomorphisms.

Proposition 1. Any tripleable functor re¬‚ects isomorphisms.

Proof. Because equivalences of categories re¬‚ect isomorphisms, it is su¬cient to

show that, for any triple T in a category C , the underlying functor U : C T ’ C

’

re¬‚ects isomorphisms. So let f : (A, a) ’ (B, b) have the property that f is an

’

110 3 Triples

isomorphism in C . Let g = f ’1 . All we need to show is that

TgE

TB TA

(1)

a

b

c c

EA

B g

commutes. This calculation shows that:

a —¦ Tg = g —¦ f —¦ a —¦ Tg = g —¦ b —¦ Tf —¦ Tg

= g —¦ b —¦ T (f —¦ g) = g —¦ b —¦ T (id) = g —¦ b—¦

Contractible coequalizers

A parallel pair in a category is pair of maps with the same domain and

codomain:

d0

A ’’ B

’’ (2)

’’

’’

1

d

The parallel pair above is contractible (or split) if there is an arrow t: B

’ A with

’

d0 —¦ t = id

and

d1 —¦ t —¦ d0 = d1 —¦ t —¦ d1

A contractible coequalizer consists of objects and arrows

d0 d

d1 ‚ (3)

‚

EB

A C

s

st s

for which

(i) d0 —¦ t = id,

(ii) d1 —¦ t = s —¦ d,

(iii) d —¦ s = id, and

(iv) d —¦ d0 = d —¦ d1 .

We will eventually see that any Eilenberg-Moore algebra is a coequalizer of a

parallel pair which becomes contractible upon applying U T .

3.3 Tripleability 111

Proposition 2.

(a) A contractible coequalizer is a coequalizer.

(b) If (3) is a contractible coequalizer in a category C and F : C ’ D is any

’

functor, then

F d0 Fd

1‚ (4)

‚

Fd E

FA FB FC

s Fs

s Ft

is a contractible coequalizer.

(c) If

d0 d

A ’’ B ’’ C

’’ ’ (5)

’’

’’

1

d

is a coequalizer, then the existence of t making

’’

A ’’ B

←’ (6)

t

a contractible pair forces the existence of s making (5) a contractible coequalizer.

Proof. To show (a), let f : B ’ D with f —¦ d0 = f —¦ d1 . The unique g: C ’ D

’ ’

required by the de¬nition of coequalizer is f —¦ s (“To get the induced map, compose

with the contraction s”). It is straightforward to see that f = g —¦ d, and g is

unique because d is a split epimorphism. Statement (b) follows from the fact

that a contractible coequalizer is de¬ned by equations involving composition and

identities, which functors preserve. A coequalizer which remains a coequalizer

upon application of any functor is called a absolute coequalizer; thus (a) and

(b) together say that a contractible coequalizer is an absolute coequalizer.

As for (c), if t exists, then by assumption, d1 —¦ t coequalizes d0 and d1 , so there

is a unique s: C ’ B with s —¦ d = d1 —¦ t. But then

’

d —¦ s —¦ d = d —¦ d1 —¦ t = d —¦ d0 —¦ t = d

and d is epi, so d —¦ s = id.

If U : B ’ C , a U -contractible coequalizer pair is a pair of morphisms

’

as in (2) above for which there is a contractible coequalizer

U d0 d

U d1 ‚ (7)