An extended comparison functor for triples

C AHIERS DE

TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES

R OBERT J. P ERRY

An extended comparison functor for triples

Cahiers de topologie et géométrie différentielle catégoriques, tome 18, n

^o

1 (1977), p. 61-65

<http://www.numdam.org/item?id=CTGDC_1977__18_1_61_0>

© Andrée C. Ehresmann et les auteurs, 1977, tous droits réservés.

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AN

EXTENDED

COMPARISON FUNCTOR

FOR

TRIPLES

by

^Robert

^J.

^PERRY

CAHIERS DE TOPOLOGIE ET GEOMETRIE DIFFERENTIELLE

Vol. X VIII -1 (1977)

ABSTRACT

(B, C)

generates ^a

triple

^on

S

when

a

is a coreflective

subcateg-

ory

of (2 and $3

is a reflective

subcategory of C.

^{For a}

given triple

^{T on}

Ens ,

^a category

A ( T )

^is constructed with

( Ens T , A ( T ) ) generating ^T.

The

Eilenberg - Moore comparison functor rP: 93 -+ ^EnsT

^{is then extended to}

a

functor Y : C -&#x3E; A(T) satisfying

^similar

uniqueness

conditions.

If

F : C -&#x3E; 93

^and

U : C -&#x3E; ^S

^are reflector and coreflector

functors, respectively,

^with

the

corresponding

reflection and coreflection _maps, set

then

P ROPOSITION 1.

( 1 )

^F

| Q is

the

left adjoint of U I 93 ;

(2) T - ( T, n,03BC ) is

^a

triple

^on

^N.

PROOF. It follows

easily

that TJ is ^a natural transformation. If

f : A -&#x3E; U( B )

then

t = F (F ( B ) f )

^{is the}

unique

map

satisfying U(t)n(A) = f. ^//

We restrict our attention to the case

d

⁼ Ens .

DEFINITION 1. We say that the

pair (B, C) generates

^the

triple

^{T on Ens}

and we call

( B , C )

^a

generating p air. / /

The observation that

( Comp ^{Haus ,} Top)

generates ^the ultrafilter

triple

B motivated our results.

In [1]

^{Barr showed}

R (B ) = Top ,

^where

R T )

^{is the} category ^of T-relational

algebras; Manes [3] proved

It follows

easily

^{from a theorem of Manes}

[4]

^that

(EnsT , ^{R(T ))}

gener-

ates T _{for any}

triple

^{T on}

^{Ens .}

An additional

example

^{of a}

generating pair

^is

(

^{see Kennison}

[2] ) ;

^{Ens is} coreflective in

[ 9, ^Ens] provided 9

^{is con-}

nected.

DEFINITION 2. Let

(A, p) E EnsT

^{and let}

f :

K -&#x3E; A . Set

DEFINITION 3. If

U : R(T ) - Ens ,

^let

A ( T )

^{be the full}

subcategory

^of

the comma category

( U, U EnsT) with objects

We construct a

functor Vi : C - A ( T )

^{that extends the}

comparison

functor of

Eilenberg - Moore

^{as indicated in the}

following diagram:

where

Moreover, Y

^is

unique

with respect ^{to these} conditions.

We consider an alternative view of the situation. Given ^a functor

G : C -&#x3E; C’ ,

^where

( B , C )

^and

( B’ , C’ )

^are

generating pairs

^{for a fixed}

triple

^{T on}

Ens ,

^the

following properties

^{are essential to G’s}

preservation

of the

generation

process :

DEFINITION 4. If G has

properties ( 1)-(4),

^{we call G an} admissible

func- tor.

The category ^of

generating pairs

^and admissible functors will be de- noted

by G(T). //

DEFINITION

5. (1) Let (B, C),E G(T)

^with

U: C - Ens

^{faithful. We let}

e*

be the full

subcategory

^of

(U, U |B)

^with

objects

Through

^{I we can}

regard C ^{C C * .}

PROPOSITION 2.

are the reflection and coreflection _maps,

respectively. / /

Since V

^is

faithful, ( EnsT , A ( T ) ) E G ( T ) .

^{The existence of}

rfr

with the

required properties

^is

equivalent

^to

showing ( EnsT , A ( T ) )

^{is a}

terminal

object

ⁱⁿ

G( T ) .

^Then

rfr

will be the

unique

admissible functor from

(B,C)

^to

(EnsT , A (T)).

DEFINITION 6.

P ROP O SIT IO N 3’.

implying j

^and

it follows that

, since

they

^are

coequalized buy

ⁱ

Thu s 1

and

^{is re-}

flexive.

since

they

^{are T-homo-}

morphisms equalized by

^{then there}

exists 2 E

T U ( C )

^with

and also there exists

wET (À( C))

^with

and Thus

Whence

^is

transitive,

^and

DEFINITION 7. For define

PROPOSITION 4.

PROOF.

Range

^follows

by observing

^that

The

remaining

^{details are}

straightforward given

^the observation that

EXAMPLES.

By

Lemma 2 of

[5],

it follows that

(K, f-1(p))

^{is a}

^complet-

ely regular

relational

algebra.

^Thus

f -1 ( p )

^{is the}

largest completely

reg- ular relation ^on K

making f

admissible.

( 1 )

^{If T is the}

identity triple,

^then

( A , p ) -

^{A and}

f -1 ( p )

^{is the}

equivalence

^{relation on K induced}

by f .

( 2 ) In A (B), (A, p)

^{is a} compact Hausdorff space, while

(K, f -1 (p))

has the smallest

completely regular topology

^{on K}

making f

continuous.

Descriptions

^{of the}

completely regular

relational

algebras

^{of addi-}

tional

triples

^are

given

ⁱⁿ

[5].

REFERENCES

1. BARR, M., Relational algebras, ^{Lecture Notes in Math.} 137, Springer ( 1970 ), 39-55.

2. KENNISON, J. F., ^On

limit-preserving

Functors, Ill. J. ^Math. 12 (

1968),

^616-619.

3. MANES, E., ^A triple ^theoretic construction of compact algebras, ^{Lecture Notes}

in Math. 80,

Springer

( 1969), 91- 118.

4. MANES, E., Relational Models I, preprint, ^1970.

5. PERRY, ^R. J., Completely regular relational algebras, ^Cahiers

Topo.

^{et Géo.}

Dif.

^{XVII- 2} ( 1976), 125- 133.

Department of Mathematics Worce ster State

College

WORCESTER, ^{Mass. 01602} U. S. A.