Directory UMM :Journals:Journal_of_mathematics:TAC:

ENRICHED LAWVERE THEORIES

Dediated to Jim Lambek

JOHN POWER

ABSTRACT. Wedene thenotionofenrihed Lawveretheory,for enrihmentover

amonoidalbilosedategoryV that isloallynitely presentable asalosedategory.

We provethat theategoryof enrihed Lawvere theoriesis equivalent tothe ategory

of nitary monads on V. Moreover,the V-ategoryof models of aLawvere V-theory

is equivalent to the V-ategory of algebrasfor the orresponding V-monad. This all

extendsroutinelytoloalpresentabilitywithrespettoanyregularardinal. Wenally

onsider thespeial asewhereV isCat, andexplainhowtheorrespondene extends

to pseudomapsofalgebras.

1. Introdution

Inseekingageneralaountofwhathavebeenallednotionsofomputation[13℄,onemay

onsideranitary2-monadT onCat, andaT-algebra(A;a),thenmakeaonstrutionof

aategoryB andanidentityonobjetsfuntorj :A !B. Inmakingthatonstrution,

one is allowed touse the struture on A determinedby the 2-monad, and that is all. For

instane, given the 2-monadforwhihanalgebrais asmallategory witha monadonit,

then one possible onstrution would be the Kleisli onstrution. For another example,

given the 2-monad for whih an algebrais a smallmonoidal ategory A together with a

speied objet S, then a possible onstrution is that for whih B(a;b) is dened tobe

A(Sa;Sb), with the funtor j :A !B sending amap h to Sh.

Weneedapreisestatementof whatwemeanbysayingthatthese onstrutionsonly

use struturedetermined bythe 2-monad. Ingeneral,one mayobtainone suhdenition

by asserting that B(a;b) must be of the form A(fa;gb) for endofuntors f and g on A

generatedbythe2-monad; similarlyfordeningompositioninB. Thusweneedtoknow

what exatly we mean by endofuntors onA and natural transformationsbetween them

that are generated by the 2-monad. This paper is motivated by a desire to make suh

notions preise.

In order to make preise the notion of funtor generated by a nitary 2-monad T on

a T-algebra (A;a), we rst generalise from onsideration of a nitary 2-monad on Cat

to a nitary V-monad on V for any monoidal bilosed ategory that is loally nitely

Thiswork hasbeendonewiththesupport ofEPSRCgrantsGR/J84205: Frameworksfor

program-minglanguagesemantisandlogiandR34723: TheStrutureofProgrammingLanguages: syntaxand

semantis.

Reeivedbytheeditors1998Deember14and, inrevisedform,1999August27.

Publishedon1999November30.

1991MathematisSubjetClassiation: 18C10,18C15,18D05.

Keywordsandphrases: Lawveretheory,monad.

presentable as a losed ategory: we shall make that denition preise in the next

se-tion. We moveto this generalityprimarily for simpliitybut also beausethe omputing

appliation willneed it later, for instane onsidering not mere ategories but ategories

enrihed inPoset or inthe ategory of !-po's.

Tosupportourmaindenition,weproveatheorem: wedenethenotionofaLawvere

V-theory,and we prove thattogiveaLawvere V-theory isequivalenttogivinganitary

V-monadonV. So,foranitaryV-monadT onV, thereisaLawvereV-theoryL(T)for

whihtheV-ategoryof modelsofL(T),whihwedene,isequivalenttothe V-ategory

T-Alg of algebras for the monad. Conversely, given any Lawvere V-theory T, there is

a orresponding nitary V-monad M(T); and these onstrutions yield an equivalene

between the ategory of nitary V-monadson V and that of Lawvere V-theories.

Onewehavesuhaorrespondene, wehavethe denitionweseek: givena2-monad

T and aT-algebra(A;a), anendofuntor f onA is generatedby T whenever itis inthe

imageofL(T)inthemodelofL(T)determinedbythealgebra(A;a). Similarly,anatural

transformation : f ) g is is said to be generated by T whenever it is the image of a

2-ell of L(T) inthe modelof L(T) determined by the algebra (A;a).

For ognosenti of enrihed ategories, the denitions and results here, at least

as-suming our enrihment is over a symmetri monoidal ategory, should ome as nogreat

surprise. However, they are a little subtle. One's rst guess for a denition of Lawvere

V-theorymaybeasmallV-ategorywithniteproduts,subjettoaonditionasserting

single-sortedness. But that is too rude: not only does itnot yield our theorem relating

Lawvere theories and nitary monads, as in the ase for Set (see [1℄), but it does not

agreewiththeknownandveryuseful equivalenesbetweennitarymonadsanduniversal

algebraas in [9℄, [6℄, and in appliationto omputationin [11℄ and [10℄, with anaount

for omputer sientists in [14℄. So we onsider something a little more subtle: if V is

symmetri,we dene aLawvere V-theory tobe asmall V-ategory withnite otensors,

subjet toasingle-sortedness ondition. Amodelis thereforeanite otensor preserving

V-funtorintoV. In ordertoextend tononsymmetriV, asforinstane if V is the

ate-goryof smallloallyordered ategories withthe laxGray tensorprodut[11℄, weneed to

introdueatwist: soaLawvereV-theoryisasmallV

t

-ategorywithniteotensors,

sub-jettoasingle-sortednessondition,andamodelisaniteotensorpreserving V

t

-funtor

into V

t

: that allows us to speak of the V-ategory of models, and prove our equivalene

with nitary V-monads on V and their V-ategories of algebras. There has been

on-siderable development of ategories enrihed in monoidalbilosed ategories [4, 5, 6, 11℄

reently: the main problemwith extending the usual theory is that funtor V-ategories

need not exist ingeneral; but funtor V-ategories with base V

t

do, and that suÆes for

ourpurposeshere. Ifoneisonlyinterestedinthease ofsymmetriV here, onemay

sim-ply ignore all subsripts t, and one will have orretstatements; of ourse, exponentials,

i.e., terms of the form [X; ℄

l

and [X; ℄

r

, may alsobeabbreviated to [X; ℄.

The speial ase that V = Cat also ontains a mild surprise. Given a 2-monad T

stritly. Suhmapsareanalysedin[2℄,withanexplanationforomputersientistsin[12℄.

One might imagine that, extending the above orrespondene in this ase, pseudo maps

of algebraswouldorrespondexatlyto pseudonaturaltransformationsbetween the

or-responding models of the orresponding Lawvere 2-theory. But the position is a little

more deliate than that: pseudo natural transformations allow too muh exibility to

yield abijetion. However, given a pair of algebras, one still has an equivalene between

the ategory of pseudo maps of algebras between the two algebras and that of pseudo

naturaltransformationsbetween theorrespondingmodelsof the orrespondingLawvere

2-theory. Hene one has a biequivalenebetween T-Alg

p

and the 2-ategory ofmodels of

the Lawvere theory, with pseudonatural transformations. We explainthat inSetion5.

I amsurprised I havenot been able tond our main results,inthe ase of symmetri

V, inthe literatureonenrihedategories. There hasbeen workonenrihedmonads,for

instane[9℄and[6℄,andtherehas beenworkonenrihednitelimittheories,primarily[8℄

and[5℄. ButthelosestworktothisofwhihIamawareisbyDubu[3℄,andthatdoesnot

aountforthenitarynatureofLawveretheories,whihisentralforus. Ofourse,what

we say here isnot expliitly restrited tonitariness: one ouldeasily extend everything

toardinality< for any regular.

Thepaperisorganizedasfollows. InSetion2,wereallthebasifatsaboutenrihed

ategories we shall use to dene our terms. In Setion 3, we dene Lawvere V-theories

and their models, and prove that every nitary V-monad on V gives rise to a Lawvere

V-theory with the same V-ategory of models. And in Setion 4, we give the onverse,

i.e., toeah Lawvere V-theory,wedisoveranitaryV-monadwith thesame V-ategory

of models. We also prove the equivalene between the ategories of Lawvere V-theories

and nitary V-monads on V. Finally, in Setion 5, we explain the more fundamental

pseudo maps of algebras when V = Cat, and show how they relate to maps of Lawvere

2-theories.

For symmetri V, the standard referene for all basi strutures other than monads

is Kelly's book [7℄. For nonsymmetri V, a reasonable referene is [11℄, but the entral

results were proved in the more general setting of ategories enrihed in biategories as

in [4, 5, 6℄. For 2-ategories, the best relevant referene is [2℄, and for an explanation

direted towards omputer sientists, see [12℄.

2. Bakground on enrihed ategories

AmonoidalategoryV isalledbilosed ifforeveryobjetXofV,both X: V

Æ !V

Æ

and X : V

Æ

!V

Æ

have right adjoints,denoted [X; ℄

r

and [X; ℄

l

respetively. For

monoidal bilosed loally small V, it is evident how to dene V-ategories, V-funtors

andV-naturaltransformations,yieldingthe 2-ategoryV-Catof smallV-ategories. The

ategoryV

Æ

enrihes toaV-ategorywithhomgivenby[X;Y℄

r

. Notethat[X;Y℄

r

annot

be replaed by [X;Y℄

l

here, using the usual onventions of Kelly's book [7℄. One an

speakof representable V-funtorsand thereis aneleganttheoryof V-adjuntions, seefor

speak of the Eilenberg-Moore V-ategory for a V-monad. If V is also oomplete, there

is an elegant theory of limits and olimits in V-ategories generalising the situation for

symmetrimonoidallosedV, seefor instane [5℄. Notethatfor aV-ategoryC,there is

aonstrution of whatshould learly be alledC op

, but C op

isa V

t

-ategory,not apriori

a V-ategory.

In general, there is no denition of a funtor V-ategory. However, if V is omplete,

forasmallV-ategoryC, onedoeshaveafuntorV-ategoryof theform[C op

;V

t

℄, whose

objets are V

t

-funtorsfrom C op

into V

t

, and with homs given by the usual onstrution

for symmetri V. Details appear in [5℄, where it is denoted PC. This also agrees with

Street's onstrution[16℄,but the latteris formulated interms of modules.

A V-monad on a V-ategory C onsists of a V-funtor T: C ! C and V-natural

transformations : 1 ) T and : T 2

) T satisfying the usual three axioms. The

Eilenberg-Moore V-ategory T-Alg has as objets the T

Æ

-algebras, where T

Æ

is the

or-dinary monad underlying T, on the ordinary ategory C

Æ

underlying C. Given algebras

(A;a)and(B;b),thehomobjetT-Alg((A;a);(B;b))istheequaliserinV ofthediagram

C(A;B)

C(a;B)

-C(T(A);B)

T R

C(T(A);b)

C(T(A);T(B))

(1)

Composition in T-Alg is determined by that in C. Moreover, there is a forgetful V

-funtor U: T-Alg !C, whihhas aleft adjoint. IfC isomplete, thensoisT-Alg,and

U preserves limits.

Given monads T and S on C, amap of monads from T to S isa V-natural

transfor-mation : T ) S that ommutes with the unit and multipliation data of the monads.

With the usual omposition of V-natural transformations, this yields the ordinary

ate-goryMnd(C) of monads onC.

AmonoidalbilosedategoryV isalledloallynitelypresentableasalosedategory

if V

Æ

is loallynitely presentable, if the unit I of V is nitely presentable, and if xy

isnitely presentable whenever xand y are. Heneforth,allmonoidalbilosedategories

towhih we referwillbe assumed tobe loally nitely presentable aslosed ategories.

AV-ategoryC issaidtohavenitetensorsifforeverynitelypresentableobjetxof

V,andforeveryobjetAofC,[x;C(A; )℄

r

:C !V isrepresentable,i.e., ifthereexists

an objet xA of C together with a naturalisomorphism [x;C(A; )℄

r

=

C(xA; ).

The V-ategory C has nite otensors if it satises the dual ondition that for every

nitely presentable x in V and every A in C, the V

t

-funtor [x;C( ;A)℄

l :C op ! V t is

representable, where V

t

is the dual of V; i.e., there exists an objet A x

of C together

with a natural isomorphism [x;C( ;A)℄

l

=

C( ;A x

). A V-ategory C is oomplete

whenever C

Æ

is oomplete,C has nite tensors,and x :C

Æ

!C

Æ

C

Æ

is loally nitely presentable, C has nite otensors, and ( ) :C

Æ

! C

Æ

preserves

ltered olimits for all nitely presentable x. In general, a V-funtor is alled nitary if

the underlying ordinary funtor is so, i.e., if it preserves ltered olimits. We therefore

denote the full subategory of Mnd(C) determined by the nitary monads on C by

Mnd

f

(C). In [6℄, nitary monads on an lfp V-ategory are haraterized in terms of

algebraistruture.

For any V that is loally nitely presentable as a losed ategory, V is neessarily

a loally nitely presentable V-ategory. The otensor A X

is given by [X;A℄

l

. Given a

small nitely otensored V

t

-ategory C, we write FC(C;V

t

) for the full sub-V-ategory

of [C;V

t

℄ determined by those V-funtors that preserve nite otensors. It follows from

Freyd's adjointfuntortheorem, andfromthe fatthatthe inlusionpreservesotensors,

that FC(C;V

t

) is a full reetive sub-V-ategory of [C;V

t

℄, i.e., the inlusion has a left

adjoint. It follows immediatelyfromthe denitions that the inlusionis alsonitary.

Notethat,inallthatfollows,ifV issymmetri,onemaysimplydisregardallsubsripts

t, and one will have orretstatements, with orret proofs: if V is symmetri, then the

symmetri monoidal ategory V

t

is isomorphi to V, and of ourse [X; ℄

l

and [X; ℄

r

agree.

3. Enrihed Lawvere theories

Weseek adenition of a Lawvere V-theory. In order tovalidate that denition,we shall

prove that the V-ategory of models of a Lawvere V-theory is nitarily monadi over

V; and for any nitary V-monad T on V, the V-ategory T-Alg is the V-ategory of

models of a Lawvere theory. More elegantly, we shall establish an equivalene between

theordinaryategoryofnitary V-monadsonV andthe ategoryofLawvere V-theories.

First we shall give our denition of Lawvere V-theory. Observe that the full

subat-egory V

f

of V determined by (the isomorphism lasses of) the nitary objets of V has

nite tensors given asinV and issmall.

3.1. Definition. A Lawvere V-theory onsists of a small V

t

-ategory T with nite

otensors,together withanidentityonobjetsstritlyniteotensorpreserving V

t

-funtor

:V op

f

!T.

It is immediate that if V = Set, this denition agrees with the lassial one. We

extend the usual onvention for Set by informallyreferring to T as a Lawvere V-theory,

leavingthe identityon objets V

t

-funtorimpliit. A map of Lawvere V-theories fromT

toS isa stritnite otensor preserving V

t

-funtorfromT toS that ommuteswith the

V

t

-funtors from V op

f

. Together with the usual omposition of V

t

-funtors, this yields a

ategory we denoteby Law

V .

We nowextend the usualdenition of a modelof a Lawvere theory.

3.2. Definition. A model of a Lawvere V-theory T is a nite otensor preserving V

t

-funtorfromT toV

t

. TheV-ategoryofmodelsofT isFC(T;V

t

),thefullsub-V-ategory

3.3. Example. LetT beaV-monadonV. Then,oneanspeakoftheKleisliV-ategory

Kl(T) for T, and there is a V-funtor j : V ! Kl(T) that is the identity on objets,

has a rightadjoint, and satises the usual universal property of Kleisli onstrutions, as

explained in Street's artile [15℄. Sine j has a right adjoint, it preserves tensors. So

if we restrit j to the nitely presentable objets of V, we have an identity on objets

(stritly)nite tensorpreserving V-funtorfromV

f

tothe full sub-V-ategoryKl(T)

f of

Kl(T) determined by the objets of V

f

. Dualising, we have a Lawvere V-theory, whih

wedenotebyL(T). ThisonstrutionextendstoafuntorL:Mnd(V) !Law

V

. Sine

our primary interest is in nitary V-monads, we also use L to denote its restrition to

Mnd

f (V).

3.4. Theorem. Let T bea nitary V-monadon V. ThenFC(L(T);V

t

)isequivalent to

T-Alg.

Proof. There is a omparison V-funtor from Kl(T) to T-Alg, and it preserves tensors

sine the anonial funtors from V into eah of Kl(T) and T-Alg do. By omposition

with the inlusion of Kl(T)

f

into Kl(T), we have a V-funtor : Kl(T)

f

! T-Alg,

and this yields a V-funtor ~:T-Alg ![Kl(T) op

f ;V

t

℄ sending a T-algebra(A;a) to the

V

t

-funtor T-Alg(( );(A;a)). This V-funtor fators through FC(Kl(T) op

f ;V

t

) sine

preserves nite tensors and sine representables preserve nite otensors. Moreover, it is

fully faithful sine every objet of T-Alg is a anonial olimit of a diagram in Kl(T)

f .

Soit remainsto showthat ~is essentially surjetive.

Suppose h : Kl(T) op

f

! V

t

preserves nite otensors. Let A = h(I), where I is the

unit of V. The behaviour of h on all objets is fully determined by its behaviour on I

sine h preserves niteotensors and everyobjetof Kl(T) op

f

, i.e., every objet of V op

f , is

a nite otensor of I. The behaviour of h on homs gives, for eah nitely presentable x,

a map in V from Kl(T)(I;x) to [A x

;A℄

l

, or equivalently, sine otensors in V are given

by a left exponential, and by the usual propertiesof maps in Kleisliategories and maps

from units, a map in V from [x;A℄

l

Tx to A. This must all be natural in x, thus, by

nitariness of T, we have a map a : TA ! A. Funtoriality of h, together with the

V

t

-funtor from V op

f

into Kl(T) op

, fore a to be an algebra map. It is routine to verify

that ~((A;a)) is isomorphito h.

4. The onverse

Inthis setion,westart byprovingaonverse toTheorem3.4. This involvesonstruting

a nitary monad M(T) from a Lawvere V-theory T. Having proved that result, we

establishthatLtogetherwithM formanequivaleneofategoriesbetweentheategories

of nitary V-monadson V and Lawvere V-theories, for any monoidalbilosed V that is

loally nitely presentable asa losed ategory.

First we reall Bek's monadiity theorem. Given a funtor U : C ! D, a pair of

arrows h

1 ;h

2

: X ! Y in C is alled a U-split oequaliser pair if there exist arrows

splits h,j splits h

1

, and Uh

2

j =ih. It followsthat h isa oequaliserof Uh

1

and Uh

2 ,

and that oequaliser is preserved by all funtors. Bek's theorem says

4.1. Theorem. A funtor U :C !D is monadi if and only if

U has a left adjoint

U reets isomorphisms, and

C has and U preserves oequalisers of U-split oequaliser pairs.

For a detailed aount of Bek's theorem, see Barr and Wells' book [1℄. Now we are

ready to prove our main result.

4.2. Theorem. LetT beanarbitraryLawvereV-theory. Thenthereisanitarymonad

M(T) on V suh thatM(T) Alg is equivalent to FC(T;V

t ).

Proof. ReallfromthedenitionofLawvereV-theory,wehaveaniteotensorpreserving

V

t

-funtor from V op

f

to T. By omposition, this yields a V-funtor U : FC(T;V

t

) !

FC(V op

f ;V

t

),butthelatterisequivalenttoV: oneproofofthisisbyapplyingTheorem3.4

to the identity monad. The V-funtor U is given by evaluation at I. Moreover, it has a

left adjoint sine the inlusionof FC(T;V

t

) into [T;V

t

℄ has a left adjoint, as mentioned

in Setion 2, and sine evaluation funtors from V-presheaf ategories have left adjoints

given by tensors. Moreover, itis nitary sine ( ) x

preserves lteredolimits fornitely

presentable x. It remainsto showthat U is monadi.

We shall apply Bek's monadiity theorem. It is routine to verify that U reets

isomorphisms, and one an readily alulate that U preserves the oequalisers of U-split

oequaliser pairs: sine FC(T;V

t

) is a full reetive sub-V-ategory of [T;V

t

℄, it is

o-omplete; and the required oequaliser lifts to all objets as they are all nite otensors

of the unit I,so one need merely apply the assoiated otensor tothe given map and its

splitting. ThatprovesmonadiityoftheunderlyingordinaryfuntorofU;extendingthat

tothe enrihed funtorfollows immediatelyfrom the fatthat U respets otensors.

Our onstrution M extends routinely to a funtor M : Law

V

!Mnd

f

(V). Reall

from the previous setion that we also have a funtor L : Mnd

f

(V) ! Law

V

. In fat

we have

4.3. Theorem. The funtors M : Law

V

! Mnd

f

(V) and L : Mnd

f

(V) ! Law

V

form an equivalene of ategories.

Proof. Given a nitaryV-monadT onV,itfollows fromTheorem 3.4andthe

onstru-tionofM thatML(T)isisomorphitoT. Fortheonverse, givenaLawvereV-theoryT,

we need tostudy the onstrutionof M(T). It is given by taking the left adjoint to the

funtor fromFC(T;V

t

)to V given by evaluationat the unit I. The left adjoint,applied

toanitelypresentableobjetxofV,yieldstherepresentablefuntorT(x; ):T !V

t :

5. 2-Monads on Cat

WenowrestritourattentiontotheasethatV =Cat. Soweonsidernitary2-monads

on Cat. An exampleis that there is a nitary 2-monadfor whih the algebras are small

monoidalategorieswith aspeiedobjetS,asmentionedintheintrodution. Another

nitary 2-monad on Cat has an algebra given by a small ategory with a monad on it,

again as in the introdution. There are many variants of these examples: see [2℄ or [12℄

for many moreexamples and more detail.

When one does have a2-monadT,the maps of primaryinterestare the pseudo maps

of algebras. These orrespond tothe usual notion of struture preserving funtor. They

are dened asfollows.

5.1. Definition. Given T-algebras (A;a) and (B;b), a pseudo map of algebras from

(A;a) to (B;b) onsists of a funtor f :A !B and a naturalisomorphism

TA

Tf

-TB

+

f

A a

?

f

-B b

?

suh that

T 2

A T

2

f

-TB T

2

A T

2

f

-TB

+ T

f

TA Ta

?

Tf

-TB Tb

?

= TA

A

?

Tf

-TB

B

?

+

f

+

f

A a

?

-B b

?

A a

?

-B b

and

A

f

-B A

f

-B

= TA

A

?

Tf

-TB

B

?

+

f

A id

A

?

f

-B id

B

?

A a

?

f

-B b

?

A 2-ell between pseudo maps (f;

f) and (g;g) is a natural transformation between f

and g that respets

f and g.

ByusingtheompositionofCat,itfollowsthatT-algebras,pseudomapsofT-algebras,

and 2-ells forma 2-ategory,whihwe denote by T-Alg

p

. This agrees with the notation

in some of the relevant literature, and this situation has undergone extensive study, in

partiular in [2℄; and for anaount direted towards omputer sientists, see [12℄.

Similarly, given a small2-ategory C, one has the notion of pseudo natural

transfor-mation between 2-funtors h;k : C !Cat: one has isomorphisms where the denition

ofnaturaltransformationhas ommutingsquares, andthose isomorphismsare subjet to

two oherene onditions, expressing oherene with respet to omposition and

identi-ties inC. Thus, forany Lawvere 2-theoryT,we havethe 2-ategoryFC

p

(T;Cat),given

by nite otensor preserving 2-funtors,pseudonatural transformations,and the evident

2-ells. If anitary 2-monad T orresponds tothe Lawvere 2-theoryT, one mightguess

that the 2-equivalene between T-Alg and FC(T;Cat) would extend to a 2-equivalene

between T-Alg

p

and FC

p

(T;Cat), but it does not!

5.2. Example. Let T be the identity 2-monad on Cat. Then T-Alg

p

is 2-equivalent

to Cat. But FC

p

(T;Cat) is not, as it ontains more maps. First, all funtors lie in

FC

p

(T;Cat)viatheinlusionofCat,whihis2-equivalenttoFC(T;Cat),inFC

p

(T;Cat).

Butalso,onemayvaryanyomponentofapseudonaturaltransformation ,say

2

,byan

isomorphism,leavingeveryother omponentxed, and varythe strutural isomorphisms

of by onjugation,and one stillhas a pseudo naturaltransformation.

Sothe orrespondene we seekisalittlemore subtle. The notionof 2-equivalene

be-tween 2-ategories isquite rare. More ommonly,one has a biequivalene. This amounts

to relaxing fully faithfulness to an equivalene on homategories, and similarly for

es-sential surjetivity. In our ase, we already have essential surjetivity. So we ould ask

whether our 2-funtor ~ : T-Alg ! FC(T;Cat) extends to giving, for eah pair of

pseudo naturaltransformations between ~((A;a)) and ~((B;b)), hene a biequivaleneof

2-ategories. Infat, we have

5.3. Theorem. The 2-funtor ~ : T-Alg ! FC(T;Cat) extends to a biequivalene

between T-Alg

p

and FC

p

(T;Cat).

Proof. Ifone routinely follows the argumentfor essentialsurjetivity of ~asinthe proof

of Theorem 3.4, and ones uses the 2-dimensional property of the olimit dening TA,

one obtains a bijetion between pseudo maps of T-algebras from (A;a) to (B;b), and

those pseudo natural transformations between ((A;~ a)) and ~((B;b)) that respet nite

otensors stritly, i.e., up to equality, not just isomorphism. Now, sine otensors are

pseudolimits,theyareautomatiallybilimits,andsoeverypseudonaturaltransformation

is isomorphi to one that respets nite otensors stritly. Loal fully faithfulness is

straightforward.

Referenes

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