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COMPARING COEQUALIZER AND EXACT COMPLETIONS

Dediated to Joahim Lambek

on the oasion of his 75th birthday.

M. C. PEDICCHIO AND J. ROSICK

Y

ABSTRACT. We haraterizewhen the oequalizer and the exatompletion of a

ategoryCwithnitesumsandweaknitelimitsoinide.

Introdution

Ouraimistoompare twowellknownompletions: the oequalizerompletionC

oeq of a

smallategoryCwithnitesums(see[P℄) andtheexat ompletionofasmallategoryC

withweaknitelimits(see[CV℄).ForaategoryCwithnitesumsandweaknitelimits,

C

ex

is always a full subategory of C

oeq

. We haraterize when the two ompletions are

equivalent-itturnsoutthat thisorresponds toanitenessondition expressed interms

of reexive and symmetrigraphs in C .

1. Two ompletions

ForasmallategoryCwithnitesums,theoequalizerompletionofCisaategoryC

oeq

with nite olimits together with a nite sums preserving funtor G

C

: C ! C

oeq suh

that,foranynitesumspreservingfuntorF :C!Xintoanitelyoompleteategory,

there isaunique nite olimitspreserving funtor F :C

oeq

!Xwith F G

C

=F. This

onstrution has been desribed by Pitts (f. [BC℄).

For a small ategory C with weak limits, the exat ompletion E

C

: C ! C

ex an

be haraterized by a universal property as well (see [CV℄). In a speial ase when C

has nite limits, E

C

is a nite limits preserving funtor into an exat ategory C

ex suh

that, for any nite limitspreserving funtor F : C !X into an exat ategory, there is

a unique funtor F 0

: C

ex

! X whih preserves nite limits and regular epimorphisms

suh that F 0

E

C

= F. Following [HT℄, C

ex

an be desribed as a full subategory of

Set C

op

and E

C

as the odomainrestrition of the Yoneda embedding Y :C! Set C

op

.

To explainit, wereall that afuntor H :C op

!Set is weakly representable if it admits

Seondauthorpartiallysupported bytheGrantAgeny oftheCzehRepubli underthegrantNo.

201/99/0310andbytheItalianCNR.

Reeivedbytheeditors1999January12and, inrevisedform,1999August27.

Publishedon1999November30.

1991MathematisSubjetClassiation: 18A99.

Keywordsandphrases: exatompletion,oequalizerompletion,variety.

a regular epimorphism : YC ! H from a representable funtor. Then C

ex

onsists of

those weakly representable funtors H admitting :YC!H whose kernel pair

K

/

/

/

/

YC

/

/

H

(1)

has K weaklyrepresentable.

We willstart by showing that C

oeq

an bepresented asa full subategoryof Set C

op

too, with G

C

being the odomain restrition of Y. The full subategory of Set C

op

onsisting of all nite produts preserving funtors will be denoted by FP(C op

). It is

well known that FP(C op

) isa variety (see [AR℄ 3.17).

1.1. Lemma. Let C be a ategory with nite sums. Then C

oeq

is equivalent to the full

subategory of FP(C op

) onsisting of nitely presentable objets in FP(C op

).

Proof. By the universal property of C

oeq

, we get

FP(C op

)Lex ((C

oeq )

op

)

where, on the right, there is the full subategory of Set (Coeq)

op

onsisting of all nite

limitspreserving funtors. The result thus follows from [AR℄ 1.46.

1.2. Proposition. LetC be aategory withnitesums andweaknite limits. ThenC

ex

is equivalent to a full subategory of C

oeq .

Proof. Let H 2 C

ex

and onsider the orresponding diagram (1). There is a regular

epimorphismÆ :YD!K (beauseK isweakly representable) and weobtain a

oequal-izer

YD

/

/

/

/

YC

/

/

H

(2)

where = Æ and =Æ. Sine YD is a regularprojetive in Set C

op

, the graph ( ;)

is reexive (itmeans the existeneof ':YC ! YD with '='=id

Y(C)

). Following

[PW℄, (2) is a oequalizer in FP(C op

). Therefore, H is nitely presentable in FP(C op

)

(f. [AR℄1.3). Hene, using Lemma 1.1, H belongs toC

oeq .

When C has nite sums, objets of C are preisely nitely generated free algebras in

the variety FP(C op

). The ondition of havingweak nite limitstoo,isa veryrestritive

one. Wegiveanother formulationof it.

1.3. Proposition. Let Chave nite sums. ThenChas weak nitelimits inite limits

of objets of C in FP(C op

) are nitely generated.

Proof. LetD:D!Cbeanitediagramand(Æ

d

:A!YD

d )

d2D

itslimitinFP(C op

).

AssumethatAisnitelygenerated. Thenthereisaregularepimorphism :YC!A

where C 2C . Consider a one (f

d

:X !D

d )

d2D

in C. There is a unique ': YX !A

projetive) ' fatorizes through and therefore (Æ

d

: YC ! YD

d )

d2D

is a weak limit

of YD inY(C). Hene D has a weak limitinC .

Conversely, assumethatDhasaweaklimit(

d

:C !D

d )

d2D

inC. Thereisaunique

:YC ! A with Æ

d

=Y

d

for alld 2D. Consider ':YX ! A, X 2 C. There exists

': X ! C suh that Y(

d

)= Æ

d

' for all d2 D. Hene '= , whih implies that

is a regular epimorphism(beauseYX;X 2C are nitely generated free algebrasin the

variety FP(C op

)). Hene A is nitely generated.

2. When do they oinide?

2.1. Constrution. Let C be a ategory with weak nite limits. Let r

0 , r

1 :C

1 ! C

0

bea reexiveand symmetrigraph inC. It meansthat thereare morphismsd:C

0 !C

1

and s:C

1 !C

1

with r

0 d=r

1 d=id

C

0 and r

1 s=r

0 , r

0 s=r

1

. We form aweak pullbak

C

2

r0

~

~}}

}}

}}

}}

r1

A

A

A

A

A

A

A

A

C

1

r

1

A

A

A

A

A

A

A

A

C

1

r0

~

~}}

}}

}}

}}

C

0

(3)

By taking r 2

i =r

i r

i

, i=0;1,we get the graph

C

2 r

2

0

/

/

r 2

1

/

/

C

1

This graph is reexive: d 2

: C

0 ! C

2

is given by r

0 d

2

= r

1 d

2

= d. It is also symmetri:

s 2

: C

2 ! C

2

is given by r

0 s

2

=sr

1

and r

1 s

2

= sr

0

. By iterating this proedure, we get

reexive and symmetrigraphs

C

n r

n

0

/

/

r n

1

/

/

C

1

for n=1;2;:::.

2.2. Definition. Let C have weak nite limits. We say that a reexive and symmetri

graph r

0 ;r

1 : C

1 !C

0

has a bounded transitive hull if there is n 2 N suh that, for any

m>n, there exists f

m :C

m !C

n

with r n

0 f

m =r

m

0

and r n

1 f

m =r

m

1 .

It is easy to hek that the denition does not depend on the hoie of weak nite

limits. Evidently, if C has nite limits, Denition 2.2 means that the pseudoequivalene

generated by the graphr

0 ;r

1 :C

1 !C

0

is equalto r n

0 ;r

n

1 :C

n !C

0 .

2.3. Theorem. Let C have nite sums and weak nite limits. ThenC

ex

is equivalent to

Proof. I. Let r 0 ;r 1 : C 1 ! C 0

be a reexive and symmetri graph in C . Consider the

oequalizer YC 1 Yr 0

/

/

Yr1

/

/

YC

0

/

/

H in Set C op

. Put H

1

=YC

1 , H

0

=YC

0

and

i = Yr

i

for i = 0;1. Let n 0 , n 1 :H n ! H 0

be iterations of the graph (

0 ;

1

) onstruted as before, by using pullbaks in Set C

op

.

There are morphisms

n : YC

n ! H n suh that 1 = id H0 and n i n+1 = n Y(r

n

i ) for

i=1;2 and n =0;1;::::

YC n+1 Yr n 0

z

zuu

uu

uu

uu

uu

uu

uu

uu

uu

uu

n+1

Yr n 1

$

$

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

I

YC n n

"

"

E

E

E

E

E

E

E

E

H n+1 n 0

{

{ww

ww

ww

ww

w

n 1

#

#

G

G

G

G

G

G

G

G

G

YC n n

|

|yy

yy

yy

yy

H n n 1

#

#

H

H

H

H

H

H

H

H

H

H n n 0

{

{vv

vv

vv

vv

v

H 0

Byindution,wewillprovethat

n

are regularepimorphismsinSet C

op

. Assumethat

n

is aregular epimorphismand onsider the pullbak

G 0

|

|yy

yy

yy

yy

y

1

"

"

E

E

E

E

E

E

E

E

E

YC n n 1 n

"

"

E

E

E

E

E

E

E

E

YC n n 0 n

|

|yy

yy

yy

yy

H 0 inSet C op

. Therearemorphisms':YC

n+1

!Gand :G!H

n+1

suhthat '=

n+1 ,

%

i

'=Yr n i and n i = n % i

fori=0;1. Sine

n

isaregularepimorphisminSet C

op

, is

a regular epimorphismin Set C

op

. Furthermore,it follows from the proof of Proposition

1.3that 'is a regularepimorphismin Set C

op

too. Hene

n+1

is aregular epimorphism

in Set C

op

.

Let I be the relation in Set C

op

given by the regular epi-monopairfatorization H 1 0

/

/

1

/

/

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

@

H 0 I 0

?

?

~

~

~

~

~

~

~

~

~

~

~

~

~

~

~

~

1

?

?

~

~

~

~

~

~

~

~

~

~

~

~

~

~

~

~

in Set C op . Let n 0 ; n 1 : I n ! H 0

be the omposition of n opies of I. Then I n

is the

relation generated by the graph n 0 ; n 1 : H n ! H 0 . Sine n : YC

n ! H

n

is a regular

epimorphism,I n

isalsotherelationgeneratedbythegraphYr n

0 ;Yr

n

1 :YC

n

!YC

0 =H

0 .

II.Now, assumethatChasboundedtransitivehullsofreexiveandsymmetrigraphs.

We are going to prove that C

ex C

oeq

. Let H : C op

!Set belong to C

oeq

. Following

Proposition 1.2, it suÆes to prove that H belongs to C

ex

. Sine FP(C op

) is a variety

and, followingLemma 1.1, H isnitely presentable, H is presented by a oequalizer

H 1 0

/

/

1

/

/

H 0

/

/

H (4)

of a reexive and symmetri graph in FP(C op

) where H

1

and H

0

are free algebras in

FP(C op

)overnitelymanygenerators(f. [AR℄,Remark3.13). Sinenitelypresentable

freealgebrasinFP(C op

)arepreiselynitesumsofrepresentablefuntorsandG

C :C!

C

oeq

preserves nitesums,the funtorsH

0

and H

1

are representable,H

i

=YC

i

,i=0;1.

Hene we get a reexive and symmetri graph r

0 ;r 1 :C 1 ! C 0

in C suh that

i =Yr

i ,

i =0;1. Sine (4) is a oequalizer in Set C

op

too (by [PW ℄), it suÆes to show that the

kernel pair K

/

/

/

/

YC

0

/

/

H

has K weaklyrepresentable.

Sine the graph (r

0 ;r

1

) has a bounded transitive hull, there is n suh that, for any

m>n,thereexists agraphmorphismYf

m :YC

m

!YC

n

fromthe graph(Yr m

0 ;Yr

m

1 )to

thegraph(Yr n

0 ;Yr

n

1

). FollowingI., theyinduemorphismsI m

!I n

ofthe orresponding

relations. Hene I n

is anequivalenerelation and, onsequently, ityieldsa kernelpair of

I n n 0

/

/

n 1

/

/

H 0

/

/

H Hene K = I n

and sine I n

is a quotient of YC

n

, K is weakly representable.

III. Conversely, let C

ex C

oeq

and onsider a reexive and symmetri graph r

0 ;r 1 : C 1 !C 0

in C. Take aoequalizer

YC 1 Yr0

/

/

Yr

/

/

YC

0

/

inSet C

. Following[PW ℄,itisaoequalizerinFP(C op

)aswelland, usingLemma1.1,

we get that H 2C

oeq

. Hene H 2C

ex

and therefore the kernel pair

K

/

/

/

/

YC

0

/

/

H

has K weakly representable. Hene K is nitely generated in Set C

op

(see [AR℄ 1.69).

Sine K is a union of the hain of ompositions I n

, n = 0;1;:::, there is n suh that

K

= I

n

. Hene I m

= I

n

for all m n. Following I., I m

is the relation generated by

the graph (Yr m

0 ;Yr

m

1

). Sine YC

m

are regular projetives, there are graph morphisms

YC

m

! YC

n

for all m > n. Hene, there are graph morphisms f

m : C

m ! C

n

for all

m > n. We have proved that C has bounded transitive hullsof reexive and symmetri

graphs.

2.4. Example. 1)LetVbeavarietyinwhihnitelygeneratedalgebrasarelosedunder

nite produts and subalgebras(likesets, vetor spaes orabelian groups). Let Cbe the

full subategoryof V onsisting of nitely generated freealgebras. Then C

ex C

oeq .

Atrst, following[AR℄ 3.16,V

=

FP(C op

)and Y :C!FP(C op

)orresponds tothe

inlusion C V. Consider a reexive and symmetri graph r

0 ;r

1 : C

1 ! C

0

in C. The

equivalene relationK C

0 C

0

determinedby it isnitely generated (as asubalgebra

of C

0 C

0

). FollowingIII. ofthe proof of2.3, the graph (r

0 ;r

1

) has abounded transitive

hull.

Remark that Chas weak nitelimits(followingProposition 1.3).

2) On the other hand, it is easy to nd examples of a small ategory C suh that

C

ex C

oeq

does not hold. It suÆes to onsider the ategory C of ountable sets (and

the innite path asa reexive and symmetrigraph in it)and touse Theorem 2.3.

Referenes

[AR℄ J. Adamek and J. Rosik y, Loally presentable and aessible ategories, Cambridge University

Press1994

[BC℄ M.BungeandA.Carboni,Thesymmetritopos,Jour. PureAppl. Algebra105(1995),233-249

[CV℄ A. Carboni and E. M. Vitale, Regular and exat ompletions, Jour. Pure Appl. Algebra 125

(1998),79-117

[HT℄ H.Hu andW. Tholen,A noteonfreeregularandexatompletions andtheirinnitary

general-izations,TheoryandAppliationsofCategories,2(1996),113-132

[PW℄ M.C.PedihioandR.Wood,Asimpleharaterizationtheoremoftheoriesofvarieties,toappear

[P℄ A.PittsThelexreetionofaategorywithniteproduts,unpublishednotes1996

Universityof Trieste MasarykUniversity

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Email: pedihiuniv.trieste.it and rosikymath.muni.z

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pureategorytheory,inludinghigherdimensionalategories;appliationsofategorytheorytoalgebra,

geometry and topology and other areas of mathematis; appliations of ategorytheory to omputer

siene,physisandothermathematialsienes;ontributionstosientiknowledgethatmakeuseof

ategorialmethods.

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Editorial board.

JohnBaez,UniversityofCalifornia,Riverside: baezmath.ur.edu

MihaelBarr,MGillUniversity: barrbarrs.org

LawreneBreen,UniversitedeParis13: breenmath.univ-paris13.fr

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RobertPare,DalhousieUniversity: paremss.dal.a

AndrewPitts,UniversityofCambridge: apl.am.a.uk

RobertRosebrugh,MountAllisonUniversity: rrosebrughmta.a

JiriRosiky,MasarykUniversity: rosikymath.muni.z

JamesStashe, UniversityofNorthCarolina: jdsharlie.math.un.edu

RossStreet, MaquarieUniversity: streetmath.mq.edu.au

WalterTholen,YorkUniversity: tholenmathstat.yorku.a

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RobertF.C.Walters,UniversityofSydney: walters bmaths.usyd.edu.au