Coherent Extensions and Relational Algebras

Coherent Extensions and Relational Algebras Author(s): Marta C. Bunge Reviewed work(s): Source: Transactions of the American Mathematical Society, Vol. 197 (Oct., 1974), pp. 355-390 Published by: American Mathematical Society Stable URL: http://www.jstor.org/stable/1996942 . Accessed: 03/09/2012 09:09 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp

. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].

.

American Mathematical Society is collaborating with JSTOR to digitize, preserve and extend access to Transactions of the American Mathematical Society.

http://www.jstor.org

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 197, 1974

COHERENT EXTENSIONS AND RELATIONAL ALGEBRAS(1) BY

MARTA C. BUNGE ABSTRACT. The notion of a lax adjoint to a 2-functoris introducedand some aspects of it are investigated, such as an equivalent definitionand a correspondingtheoryof monads. This notion is weaker than the notion of a 2-adjoint (Gray) and may be obtained fromthe latter by weakening that of 2functorand replacing the adjointness equations by adding 2-cells satisfying coherence conditions. Lax monads are induced by and resolve into lax adjoint pairs, the latter via 2-categories of lax algebras. Lax algebras generalize the relational algebras of Barr in the sense that a relational algebra fora monad in S&ek is precisely a lax algebra forthe lax monad induced in RU. Similar considerations allow us to recover the T-categories of Burronias well. These are all examples of lax adjoints of the "normalized" sort and the universal propertythey satisfy can be expressed by the requirementthat certain generalized Kan extensions exist and are coherent. The most important example of relational algebras, i.e., topological spaces, is analysed in this new light also with the purpose of providinga simple illustration of our somewhatinvolved constructions.

Introduction.Ever since Kan [9] introducedadjoint functors,several variants of this notionhave appeared in the literature. One such is the generalization achieved by replacing the category of sets and mappingsby any monoidal category (or "multiplicative category", cf. Be'nabou [21) and by relativizing to it all the ingredientsenteringinto the description of an adjoint situation. We have shown in [3] that the theoryof monads (Huber [8] and Eilenberg and Moore [6], therein called "triples") carries over to the relative case. In particular,this applies to 2-monads (or "strong" monads) in 2-categories, as these are the notions relative

to Ca. Weakertypes of adjointness for2-functorshave also been considered. Thus, Gray [7] defines "2-adjointness" by weakening the notion of a natural transformation and applies it to the fibredcategoryconstruction. Received by the editors April 2, 1973. AMS (MOS) subject classifications (1970). Primary18D05, 18C99, 18A40; Secondary 08A15. Key words and phrases. 2-category,lax functor,lax adjoint, lax monad, relational algebra, Kan extension, lifting,topological spaces. (1) TMis workwas initiated in Corsica duringthe summerof 1971 and completed at the E. T. H. Zu7richduringthe 1972-73 academic year, on leave fromMcGill University. It has been partially supported by a grantfromthe National Research Council of Canada. 0 1974,American Copyright Mathematical Society

355

356

M. C. BUNGE

In this paper we introducea notion of "lax adjointness" which encompasses those of strongadjointness and of 2-adjointness. From a formalpoint of view, but also the functors we obtain it by weakening not just natural transformations involved and by replacing the adjointness identities by adding appropriate2-cells in theirplace-all of this temperedby the presence of fourcoherence conditions. Froma universal point of view a special instance called "normalized lax adjointness" has a nice interpretation:it is completelydeterminedby giving a familyof generalized Kan extensions which behave coherently. We arrivedat the above definitionsnot out of a merewish to generalize but ratherout of a desire to incorporateinto the theoryof 2-categories the notion of a relational algebra due to Barr [1]. Motivatedby the same example, Burroni[51 introducedthe notion of a "T-category", a moregeneral structurethan the relational algebras and liable to a variety of interestingapplications. We show here that any lax monadresolves into a lax adjoint pair by means of a category of lax and is induced by algebras. If the lax monad lies in Span X forsome category 5X a monadin X, its lax algebras are none otherthan the T-categories. This supplies us, in principle, with manymoreinstances of lax adjointness thanthose originallyenvisaged. The details of these applications will not, however,be given here. The contents of the paper are, briefly,as follows. In ?I, we define the notion of a familyof V-cells in a 2-category( being coherentlyclosed for U. The motivatingexample inextensions, where U is a given 2-functor963 volves topological spaces (the relational algebras over the monadof ultrafilters in SvA&, as proved in [11) and is shown in detail to be part of an instance of the universal property. In ?2, lax monads and the corresponding2-categoryof lax algebras are defined. In ?3 formallax ad joints come in as a way to resolve lax monads; theyalso induce them. In ?4 it is shown that any familyof coherent Uextensions, in the sense of ?1, determinesa lax adjoint to U. The converse holds if the lax ad joint is "normalized". It is then pointed out that such is the case with the available applications. Lax functorsoccur in Be'nabou [21 with a reversal of 2-cells and underthe name "morphismsof bicategories". We assume, however,that the bicategories are 2-categories. (Recall that the pseudo-functorsintroducedby Grothendieckto correspondto arbitraryfibrationsare of this kind.) Lax naturaltransformations, called "2-natural" in [7] and "quasi-natural" in [4], are responsible for "2adjointness" and are due to Gray. The "lax" terminologyhas been borrowed fromStreet [12]. Our lax functors,however, are dual to those of [12]; our lax are those which there have been labelled "right". Aside fromthe transformations occurs throughoutthe paper, a reason for fact that only one type of transformation

COHERENT EXTENSIONS AND RELATIONAL ALGEBRAS

357

avoiding labels is the relationship which these notions bear to extensions and liftings. Thus, fora 2-functorU, a family11X:X - UFX with the leftextension propertymakes F a lax functorand 71a rightlax transformation.But also, a family ex: UFX -- X with the left liftingpropertymakes F into a lax functorand e

into a left transformation.

1. Coherent U-extensions. We start by giving a definitionthat generalizes the notion of a left Kan extension, as in Mac Lane [10]. The generalization is two-fold: first,extensions take place in an arbitrary2-categoryratherthan in &zt; secondly, extensions are required to be relative to U in some sense. KX:

(1.1) Definition.Let (?, i3be 2-categoriesand U: i3 UX andf: X - UY be 1-cells of (i.

X-

L a 2-functor.Let

The (left) U?extensionof f along KX is given by a pair (f; QtA)consisting of a 1-cell /: X ) Y and a 2-cell tfA:/ -- Ur * Kx, i.e., as in the diagram K

x

x

oUX

Xtul~~u

UY

Y satisfyingthe followinguniversal property:forany otherpair (g; 0) withg: X g such that the diagramof and ?: / -) Ug * Kx, there exists a unique 0: I/ the 2-cells in KX x

UX

X f

Ug

Uf-

I~~~

c ommute s. This s ays , exactly, that ~[(Uk) KX] Note that the usual notion is recovered with d 2-functor. Assume now that foreach XE

.

i/rf

=

= &d and U the identity ~J3

fjwe are given K: X - UXforsomeXcFI a is Note then that there diagram for any /: X --+ UY, obtained bycomposing the U-extension of (Kcuy-f) withthe U-extensionof 1 uy', assuming these exist.

358

M. C. BUNGE x

-

ux

/

U(KUYI) 0(KUY

U y_

SiUY

* f)

\

,~~ U(i JY)

KUY~U

IUYXtU(uy)

Also note that the U-extensionof ,X along itself, if it exists, is a pair

X;

r

)

as in KX

K

u(

f

U()

UX (1.2) Definition. A familyof 1-cells Kx: X -- UX}, indexed by the objects of (, is said to be coherentlyclosed for U-extensionsiffthe followinghold: (i) forevery f: X -l LJY,the pair (/, fl) exists; (ii) I u y* (Kuy *= (iii) RX = 1lgand (X

/ and [U(1 7j)If(K

)l;

(

I (KX)

(1.3) Example. Recall the description of the monadi3 in &Se, whose algebras are the compact T2'spaces (Manes [11]). For a set X, OX is the set of all ultrafilterson X and a basis forthe topology on f3X(makingit into a compact T2'space) is given by all sets of the formA = tI' e3X :A e IU forsubsets A


359

of X. The unit of the monad,71: I , -, 8 assigns, to a point x c X, the principal ultrafilteron x, i.e., x =A C X :x E Al. Finally, if c fiX, the monad , -- f3has the effectthat = IA C X :A C C f X. multiplication g: fJ,f 15x(S) a a In [il, Barr showed that topological space is relational algebra forthe monad f8. If Y is a topological space, let 0: flY - Y denote the relation on y (i.e., SRconfY x Y which is determinedby the condition: (U, y) C 0 iff1 verges to y). - fRe,where fRe Let us extend the functorf3: &A& -& over to a fi: ?ReC is the 2-categoryof sets, relations and inclusions of theirgraphs as 2-cells. This is done in [11 as follows: given a relation r :X Y, decompose it as X

d-r I

rr

c

r,yt

where rr C X x Y is the graph of the relation and where dr, cr are the domain and codomain functions. Define fi(r):fiX- fY as the composite -

18X

-

Ul)

-,

08Y.

If we recall that, fora functionf: X -X Y, fi px-, fY assigns to an ultrafllter

U C pX the filtergenerated by sets of the formfA forA e U, the latterdenoted f[U]and automaticallyan ultrafilter,we have now the followingdescription of fir: let (U, 2) C pX x Y. Then, (U, 2) C fr iffthereexists 1 e f(]r) such that dr[S] = Uand c.r[2n= Recall also Barr's observation that, in general, forcomposable relations r s and one only has fi(r- s) < f(r) - (s). This will later on be called a "lax functor". (1.3.1) Definition. A relation r: X -* Y, where (X, 4) and (Y, 0) are topological spaces, is called a lax morphismof topological spaces iffthe following holds:

px

fr

>

X

g

0

Y

0 Pr rt- e. By the above, this means that, given U c fX and y C Y, 1f = U and crUl]s y, thenthereexists thereexists 1 C i(Lr)such thatdr[12]

i.e.,

x

*

-*x. Xsuch that (x, y)C rand We make some remarkson this notion. First, it follows fromthe characteri-

M. C. BUNGE

360

zation of continuous functionsgiven in [1] that an inverse of a function,i.e., r= g withg: Y -- X, is a lax morphismof topological spaces iffg is a conY a continuous tinuous function. This suggests that we call a relation r: X 0 < r namely, O3r. relation wheneverthe reverse inequality holds, e Basil Rattraypointed out to us that any lax morphismf: X -- Y withf a functionis always a closed mapping,as it is easy to prove. He also called our attentionto the followingobservations. There exist closed mappingswhich do not satisfy the condition of (1.3.1). For example, a constant mapping X -+ Xy satisfies the condition iffin X every ultrafilterconverges. (Needless to say, the conditionalways holds forcontinuous functionsbetween compact spaces.) The above example shows that continuous or open mappingsare in the same predicamentwithrespect to the condition. But also, the conditiondoes not implycontinuity. An example is the following: let f: X -p Y and g: Y -- X be inverse functionswith /continuous and g closed but not conversely. Then g satisfies the conditionand is not continuous. Denote by SRXee3the 2-categoryof topological spaces, lax morphismsand usual orderingbetween relations. Let U: ReKJJ-Ref be the forgetful2functor. K4XjX, (1.3.2) Proposition. The family tX

indexed by all sets, is coherently

closed for U-extensions. Y withX a set and Y a topological space, define F: K Y as follows: (U, y) C r iffthereexists !2 E ,BY such that (1U, 2) E Or and BX such that ?2- y. We show now that F is a U-extensionof r along ix (note that 2-cells need not be specified in this example). First, we show that r < F - 7xThis statementsays: given (x, y) C r it follows that (, y) C F. In orderto see Proof. Given r: X

that this is so, we only need to observe that, since y - y in any topology, (x, y) C fr. Now, let S be the principal ultrafilteron (x, y) in rF. Clearly

d Ml = x whilecrIS]=

y.

Next, we wish to show that r: OX.3 Y satisfies the condition (1.3.1), i.e., < rF- x holds. To do so, endow F-r witha topology 4 in the cathat 06 .F nonicalway so as to have both diagrams below commutative;

d-~~~~~~~~F

F C

3X

,BX*-~

rF

r

361

COHERENT EXTENSIONSAND RELATIONALALGEBRAS

Once this is done, the result follows fromthe way the left-handside commutative square is affectedif one inverts dr and /(dr7). In general, forfunctionsa, b, c, c, as the reader mayeasily d, it follows froma d = c * b that d * b-1

bf

77X

, UFh UFg UFf.-X

Next, observe that anotherapplication of (2.2.2) for77 delivers: (UFh. UFg

77f)* (UFh * 7g f) * (77b gf) =(UFbh-UFg r -f) *(Ucp,

Calling this 2-cell

*

y*f)h

)

13,uniqueness yields (Fhb cF f)(c

gf) =

(cFg *

Ff)

gc C

,

Proof of (2.1.4). Observe that 7f * (7Xt 1f= (U(lFf)* X) f forany f:X ' X', trivially.Hence,F(l) =df (rvX, lf) * 7f 1Ff, byuniqueness. Proof of (2.1.5). Let a: f - f', a': f' unique so that 7lf,,('lX,(a'o a))= (UF(a'a)7

-

f". By definition,F(a' - a) is r7X)r7f Since also

* (7x i(a a)) = 7lfit(r1x' * a') (77x, * a) = (UFa' * .x)(77f

) (7xi - a)

= (UFa . 27x)(UFa * 7x) * 'f/= (U(Fa' * Fa) .7x) * * 7)f.

one has that F(a' - a) = Fa'

Fa.

Proof of (2.1L6). Let a: f -* composable.

f', b: g -

Consider the commutativediagrams

g' and g, fcomposable, g',

f'


7z * gf 7Z ba |

- ef

77z* g'f'*f

TlgI

JUF(ba). rX

fU

I UFg

1

UFg'. 71y. f'-

71x

F(g'f

Uc

(2.2.2)

I

77x

UF(gf)

(2.2.3)

379

1 77/ - > UFg

.1

IX

* UFf ' 71x

and f

gf

71Z-

j,

(2.2.2)

UFg b

-

-

1

UFg * UFf - 71x |UFg- UFa 77x

(2.2.3)

.

77

f

J

UFg* 77fI

-

UFg j UFf I*77x UFb - UFf' *-lX

UFb- rqy*j

71

UFg.

71x

jUcFf.x

Fg* 71 n

UFg * -qy . f UFg -y -a|

-

UF(gf)

f_

UFg'.

1UFf'

77x

Next, we observe that I f )(71z (77gt

(U Fg' * 71f)

=

since 71z *g779f

t

UFg * 7y f

- ba)

(UFg'* 71fI)(UFb 77Z*g a

71Y f')(UFg *71Y * a)(ig* 77z*b *f

* 7z g

I

'79'gf

UFg r7y a

--

I UFg 77yf

(2.2.3) UFb r77y*'

-

f)

g *f

|~~~~~~~~~~ 177gt*fi' ,

UFg'

77y

f

M. C. BUNGE

380

. F(ba)= (Fb Fa) * Cgf The uniqueness now gives: cg,F (4.1.2). The rest of the data, i.e., E, (Lx), (RY). So far we have only used the fact that thereare U-extensionsalong the 71. This gave us F a lax functor, UF a lax naturaltransformation.The coherence conditions (ii) and 71:1 (iii) of (1.2), the definitionof a coherentlyclosed familyfor U-extensions, are indispensable forestablishing lax adjointness.

Let s =

177and RY UY-

Note that bothare well defined, as we have

3/r(1Y). 7u

--FUY

UU

*

)

UY+lU'?U

GU(ly)

UY

P forany f: X - UY and By coherence (ii) and the above definitions,f -y Ff = = -f (Usy * 7f)- (R * f). In particular, rIx = EFx * FX and (7 ) (UEFX * 77X ) (RFX * 77X). One thendeduces the existence of a unique Lx I-,

FX

*

1FX' satisfying U(Lx)

F77X

*V(,q )= 1

) Before reducing,

let us translate: the above is exactly condition (3.1.2*) of a lax adjoint. It says precisely (U(LX)

* 7lx)(UEFx

RFX

7(77X))

7/X

I(7X)

Let us now bringin the coherence (i) into this picture. It says that =X But then, Lx = identity. This will be one of the 1FX and that t/(%X) = I (7)* conditions on an arbitraryformallax ad joint when we attemptto recapture the universal property. Note somethingelse. x

The diagram

FX

x UP77

IA

XX

-

x

U(Tx)

UFX

71x

says also that 71X= FOlX), by definition,and that 'fr(77 )= 77(1x)- Since 1 is unique so that 1',x one has F(lx)= 1Fx' and, since e=1F=

7X

381


and snc since 77lX> f f 1 - 71),one 7lX an 7x)*77(X) must have -X 7X df'(x 7X identityas well. This will be anotherconditionwhich will allow us to recapture (Ue

the universal property. We have not finisheddefiningthe data. We need to choose, foreach g: Y Z e J3,some Eg: Ez

*FUg

-.

By coherence, cz * FUg of the form

g * y. =

Ug. Thus, we only need some appropriatediagram UFUY

UY

-

y) l~~~~~~~~~~U(g

\

Uz

* and letting Eg9 -we obtain as By taking f3 Ug R Y: Ug - Ug UEy 77uy a result a characterizationof Eg as the only 2-cell which makes UgU

\S

>

tU~~~~~~~~~~(J(g)*7U

s

x,

U(g * Ey) *

UY

commutativewhich, upon translationwith VAug= (UEz * 7ug) - (Rz * Ug) by coherence (ii), gives precisely the condition (3.1.1) on lax ad joints, with g: Y -4 Z. (4.1.3) c: FU -. 1s is a lax natural transformation. Proofof (2.2.1). By definition,E(1 ) = U(1 Y) * Ry

=' -r (

)

(CYY This, of course, is true of any Vf. On the otherhand we have shown that coherence implies that eX = 1FX. Thus the result. Proof of (2.2.2). By definition,Ebg = U(hg) * Ry forany pair g: Y Z, W of 1-cells of -3. This says that Ebg is the only 2-cell forwhich h: Z (U(Cbg)

'

uiY)(Uew.

77U(bg))(RW U(Mg))= U(hg) . Ry.

Thus, in orderto establish (2.2.2) all we have to do is show that the composite U )EW*C EW . FU(hg)

EC. e

FUh FUh.-* FUg

-

* .hFeU cg h hEZ

h Ey . FUg --h.g.E *

*g

382

M. C. BUNGE

satisfies the same conditionas Ehg above. This is done below:

U(hg)R

) ,UEW.7uw'wU(hg) UEW* 'U( g) UgW. jUEW.77YUh*Ug

(3.1.1)

|~~~~~c

UEw. UFUh Tluz Ug

UFU(hg). 77uy UcwEUcF

(2.2.2) wUFUb

.,rUg

I

Ucw* UFUh* UFUg. 77UY Uh* UFUg 7nUy

jUh '77UZ Ug UbhRz- Ug UhbU

Ub Ug

+

UCg

Ub *Ucz *77uz*Ug

tUgb

g

*nU

'

,Ub*VCz UFUg *77uy

Uc.77Uy(3.1.1)

Rb.

7Uy

g 7UY

Ubh Ug ? UEy *? UY Notethatcondition(3.1.1)was establishedin (4.1.2). Proofof(2.2.3). Let b: g g' in 53,gg ': Y -Z. Wewanttoshowthat the 2-cells g = CZ FUg and g. C bY, g * Cz . FUb

iEg

I

Y2 = cZ * U Z*FU * sy -+ are equal. To do so we mustfindsome ,B: Ug U(g'. cy). 7kuyforwhichit is thecase that (U(yi)

- 7uy) (Ucz * 7Ug) (Rz * Ug) = 3, fori = 1, 2. Ub ,Ug'-Ry I

Weclaim thatthis is so withf = Ug Ug'g ficationis givenin thediagramsbelow: U

RZ- Ug

UEZ

.

iz*

-

U Ug UUz

UbI

(3.1.1)

Ug

Ucy

I

77uy The veri-

* UFUg * 7Uy UEg.77UY

Ug- Ry!

Ug'

Ug * Ucy * Uy

Ug'

Ub Ucy Ry

Ug' UEyItu Y

7Uy


383

and

Ubj

Ug'-

*4UgB u

UgRz R

Ug

|UEZ '-uzg

6z

U

uz

Z

--

UEz - UFUg 71uy

Ub

TJcz. UFUb . Uy

(2.2.3)

* ug ,uy UEZ.7U

*

UgUZ

Ug

UFUg *

U U

Ug Y

(4.1.4). The remainingconditions on lax adjointness hold. We have already established (3.1.2*) and (3.1.1), both in (4.1.2). We need to prove (3.1.1*) and (3.1.2). Proofof (3.1.2). We want to show that the composite

Y Ey * F(luy)

Cy, F(Ry)

-

y * F(Ucy * U y)

Cy I CF

-+Y * FUcy * FIuy

* FFuy

-

y* F71uy

is the identity. Indeed, this is all that remains of (3.1.2) afterthe identifications with identity2-cells have been made. of eF and L The followingobservations will guide us to findthe correctdiagram. First, since by the coherence conditions imposed, it followed note that y: cy -y that F(l u Y

Ry - (1

1FUY and that cuy

), Ry = 1

* F =uy - 1 FUY.

and thereforethat y - 1,

Secondly, note that, since

will immediatelyfollow if we

could establish the equation (*) Note also that

(Uy * luy) * Ry -= 1(r)

Ry.

has been established in (4.1.2) using coherence

(i), and finally, note that (3.1.2* )reduces, afterall the identificationswith identity2-cells, to the equation (3Wu.2*) (UtFX v

7e x)(RFX of7

7be

We use the above remarksin the proofof (*) below:

384

M. C. BUNGE Ry

1u

~Uly

Y U'fy

I

_

y 77u

UyUy* Ry

Itu

f ucy 'NUY

yi

Uy

UY

.

* r/ Y Ue6YU 7uEyRpuy7?

r

*Y

UEFuy

UF(Ucy' 7UY).l?UY

*U7Y)UY.

77uyU*

Uy

UYUFUY.*UFJIUY*

6

7UY

U

Ufy

UEUF,UEy

Y/7

77UFUY

RyUy

(2.2.2)

7?UY

U(j'r )*UFUY

U(YU

(UCY

uy Y

L.y *UFUy I 17UFUY *

yU

(2.23)

Ry

U(y |J7 (3.1.Y 1)

I

. rUY

-Uy

( U

UeY UFUY

I

(EEY)*UIFUWFUY ." FUY -UF77uy 7uJy YUEY

(3.1.2*)

UWy l7luy

Proofof (3.1.1*). This amounts, afterreducingthe 2-cells which are identities, to showing the equation: X I* cF :

F 7f'I)-F

* CF)( (FXt*

(CFf * F77X) (FXI

77F X

. F(77XI

f

Ff.

Let yl= pFX, cF and Y2 be the otherside of the equation given above. The for i= 1, 2. proofwill be achieved by showing that (U(yi) - 7x)((7X f))= 7f, Recall that, since

rx,

007X

Let y = Y1 = X./f

U

*c

UFXI

UFX',

-

(u

I* f)

eFx

RpFXI* X#*'f

- f: X

F

FX I'

{X

Xf FXT

Let y = Y2= (EFI Fx)* (x, mutes and is the following:

UF(FX,)*X

(2.2.2) *

UExI

*UF1xI*1xI *f/ 77x'

*7(?7X )

UFX /UpXf.

1.

7UFX'

*FX UEFXI

f)

The diagramis given below and commutes:

.

(3.1.2*)

' 7X'

77(77XI- f ))(RFX

UFX UC .X

jUXI.U

':UFX'*

UF7,x,*UFJ *x

UFf -7X

,f C

) * (EFX

*

F7f1). The relevant diagramcom-


.qxt

RFX' '

*f -

tXF*XJ

W111gUEFXg

l7f

jUEFx

\

\

T

UFf* RFX-

X.

7 UF7X#

'f

I 77UFX ,f

X' 1)

UEFpXa7(Fl

7UFI71X UIJFXI

U1FX,

-:)\

(JFI~~RFX'iix UFf * RFX * 7X \ UFI ' 7X

UFUFf

*

tUFX

UFIf 77FX.7X I}UF/ ?urx.71X

UFI - UEFX *7UFX

I7X

, f).71x

UF(UFf-?

*X

*

x

UUfFI, UFF *7

(22.2)

\~~~~~~~~~~~~UEX *

*UFx'

kUFX

{EFXI ' UF/7I .7X

(2.2.3)

UFX* UF . 7x

77X UFX.

fFIf - RFX- -UF

-*

.UFX.

385

JLUFt. UFf - U"FX "7, x '7X f3.1.2*)

jUFX UCF. 77X

L FX'

- UFUFf

. a UF77X

U-X UF7lX -7lX 7x 1|UEFf.UFTx

UFf* U,FX*

UF7X

^ qX

11

J UFf

77X

This completes the proofof the theorem. O be a 2-functorwith (F, 77,c, (Lx), (Ry)) a (4.2) Definition. Let U:$ ' lax formal adjoint to U. Say that it is a normalized lax adjoint providedthe following 2-cells are all identities: forall X; (i) e F: F(lx) - 1 (ii) cF>

F(l7x

f)

-F7x

- Ff, forall f: X'X;

UF7TX* 77X forall X; (iv) Lx: EFX FTIx FX' forall X. We point out that this list can be expanded since (i) implies that also (iii) -q(c?i 71UFX-IX

(v) n()

E(

; c,1

and cl

are all identities,and (iii) and (iv) yield

(vi) RFX: 1UFX UEFX * 77UFX is the identity,forall X. Note however that nothingin the worldgives arbitraryRy to be 'the identityand that, in general, the 77Tand Eg need not be so either. Same forthe arbitrarycgfF In otherwords, the matterdoes not trivialize. be a 2-functor, and (F, TI, E, (Lx), (RY)) a (4.3) Theorem. Let U: S3 normalizedlax adjoint. Then, the family 177x:X UFXI is coherentlyclosed for U-extensions. Proof. Given any f: X - UY, define T: FX Y to be f = cy* Ff, and _ f : *I 7x to be Vf f I Of1= [(U )771i * [Ry f]. Assumingwe have shown the extension property,let us establish the coherence, independently. Here, the normalizationis the key. (4.3.1) Coherence of the extensions. The conditions (ii) say that one should have

M. C. BUNGE

386

f=

and

y . 77uy

f- [U(lUy) * (?y-f)*

[V(luY)

By definition,I uY =Cy * F(1uy) = Cy and 7uy * f = EFUY* F67Uy 0 f) Ff= Ff. Note that the above uses that: eFy, cCF ,and Lx are

EFUY * F2uy

identities. Therefore,it is truethat 1 uy

/= f 7Uy * f

Also by definition,tAq7UYi)= (UEFUY * 7(7UY-f)) - (RFUY - 7UY -f)

77(,

aand 77

.fi)= (UF77uy)r7f,since CF

identity. Thus,

(77

rff.

)[R y * 1uY]=

[(UEy)r(l

As for A(1

Ry, since 77(,

But

are identities;also, RFUY= ) =

), by definition'A(1

) is the identity. Therefore,their

[R y * f] = dfOf . composite yields [(UEy7)r7f] We now verifythe coherence condition (i), which says that 7x =.1 FX and = 1 t77X) 0(77X )

By definition,7x = EFX * .X- On the otherhand, since Lx is the identity, the latter is 1FX. Also, by definition,'fAx=: [(UEFX)7r17X] [RFX ./X] identityby conditions (iii) and (vi) of a normalized lax adjoint. (4.3.2) Extension propertyofthe pairs (f; Of). Let g: FX E * FfJ- g as follows: Ug *.7x be given. Define then Cy: 3=c

Ff

E . F3

Ey

F(Ug * 7X)

y . CF -Cy

Eg*F1X

-4

Y and /3:f -

* FUg * F77X F

g* EFX * IlX.

The diagram on the next page establishes that (U8)7qx - 'Af= /3. of this diagram,we did not take Note that, when verifyingthe commutativity advantage of the fact that some of the 2-cells are identities. However, this made it easier to identifythe coherence conditions involved. The only thingthat remains is to establish the uniqueness of /3. Thus, In assume y: y * F - g satisfies the equation (Uy)77x-A,= /8. Claim: =y. of /3 /3 definition down the write * replacing by (UY)r7x 'Af orderto prove it let us in it. Whatresults, priorto the usual reductions, is the composite: Y

CY F(Rf

Cy' F(U7x) -________

_eY CEY

C F.Uy

. y .F

cU *g77) X)Ey F(Ug.-

Cy,

.

UE*

E

y F(UE.

FF77 FUg. F7)x g -+g

F

7X)

.

F-qX.

387

COHERENT EXTENSIONSANDRELATIONALALGEBRAS --rV) Uey * UFf -

/1.

.x

lIEy * 7U Y ' f

I

frY

Ry.(U

tJg-X

'7X)

u.7f

UEy~(Ug -7y)

g.* 4U1y * U*Ug .y

UEy- UFUg

Vf

* UFUg -,UX

-UF(Ug ' -X)--X

(2.2.2)

UEy'r Ug*XX

(3.i.i)

UcyUFI3)7X

(2.2.3)

cp

7n

-q

,

* U* UFUg UFn q V

| UEg h Fx IT rx U.1X Ug - Ug(FX Vg 'UaFX

/5 lig

'

'

UFX

.EXUx7X

Ug ULX yrX

(3.1.2*)

FX x

UF71X X

J

ijg- utFX

VX

IX

'

UgX

Firstreduction.The diagrambelowis commutative: EY

Cy*F(Ry' f) ( U F FfCY

y17uy' f) Y F(UEy

/) E' Cy, F(USyX)

ty *Ft Jcy* UFf- 7x) -

---ty

* tUg -nx)

Y

g

111

Ey' FUcy. FUFf-F?lx CY

Cy'

(2. i.6)

CF

FU(Qy*F/ *'Fiix

F-

t(Ey. Ff)* FX Cy .Ff.EFx'Plx

Cy

h

Ff

in o

'p'F' 7

fg .Iq X

(2.2.3) 8

tFX

-F

F177X XF

to oa

t

dsr

rsl

Combineth'iswiththefollowingi'norderto obtaini thedesiredresult:

M. C. BUNGE

388

Second reduction. The followingdiagramcommutes:

L~~~~~~~~~~~~. ~ ~ ~ LL.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i

LL

>

*i

x

X~~~~~~

L

t

N

X~~~~~~~~~~~~~~~~L

>

Li.

'--%

*-

C)

.-. S

:1;.

Z)

i _4 LiLl.

Li

Li.

X

S~~~~~~~~~~~~~~~~I

___

,

O. L

__ ,

_

_

LI. sLI.

Li

a

q

>

:!

X

a

o~-

_

_

_

_

_

_

_

_

_

_

L


389

This completes the proofof the theorem. o (4.4) How to recover the original example. Note that any monadin Sev induces a lax monadin i4-just follow the indications given by Barr [1]. That the example (1.3), shown directly in (1.3.2) to have the universal propertyof (1.2), is a consequence of (4.3) follows fromthe observation that any such resolves (as in (3.3)) into a 2-functorUT and a norinduced lax monadT on RKef malized lax adjoint FT and therefore,by (4.3), it satisfies the universal property of (1.2). This observation consists of the followingremark,which the reader can find in [1]. First,thatif T: ef- Re is induced fromT: SPdi -0 &A& in the manner thereinindicated, one always has T(lx) = 1TX, trivially. This gives (i) of (4.2). Next, if r is a function(or if s is an inverse function)then equality holds in T(r * s) < T(r) - T(s). Apply this to the pair Vx, f in orderto obtain (ii) of (4.2), i.e., use that 71x is a function. Thirdly,observe that lax natural a: T - t Tlt i.e., having the propertythat if r: X - Y is a relation then ay . T(r) < T1(r). ax, becomes naturalrelative to functions-clearly, if r: X -4 Y is a function,the above inequality becomes an equality. This gives (iii) for77since it is there requiredthat the 2-cell 7(7x) should be the identity. As for(iv) it is immediate: it says that FX FriX< 1FX should be an equality where here F = FT. Note forthis that EFX = -lx, Fl1x= T7lx and that equality lx -T7- X = 'TX holds for the original monad. Or, should we obtain a 2-cell ,aX * T77X< 'TX fromformal considerations, note that an inequality between relations which are functions mustbe an equality. This completes the proofthat all conditions of (4.2) hold. (4.5) Remarkabout continuous relations. We may ask now whetherit is also possible to recover the continuous relations in a similar way. The recipe is this. Let the functor&e?P -* Rief(/ H f l) act on the monad/8in Sd. There results a comonad (,3, 77-1, t',-1) with/3lax but 77-1, p- 1 dual (or "left") lax.

The lax coalgebras are again the topological spaces (view Xe as a coalgebra via e- 1: X - /X) but the morphismsare continuous relations, as desired. The universal propertychanges; explicitly it is the following. Given a relation r: X - Y where Xf is a topological space and Y is a set, there exists a continuous relation 7: X -3Y (in fact, F = /8r 4-1) * such that r