Directory UMM :Journals:Journal_of_mathematics:TAC:

Theory and Appli
ations of Categories,

Vol. 6, No. 6, pp. 77{82.

COMPARING COEQUALIZER AND EXACT COMPLETIONS

Dedi
ated to Joa
him Lambek
on the o

asion of his 75th birthday.


M. C. PEDICCHIO AND J. ROSICKY
ABSTRACT. We
hara
terize when the
oequalizer and the exa
t

ompletion of a
ategory C with nite sums and weak nite limits
oin
ide.
Introdu
tion

Our aim is to
ompare two well known
ompletions: the
oequalizer
ompletion C
oeq of a
small
ategory C with nite sums (see [P℄) and the exa
t
ompletion of a small
ategory C
with weak nite limits (see [CV℄). For a
ategory C with nite sums and weak nite limits,

Cex is always a full sub
ategory of C
oeq . We
hara
terize when the two
ompletions are
equivalent - it turns out that this
orresponds to a niteness
ondition expressed in terms
of re
exive and symmetri
graphs in C.
1. Two
ompletions

For a small
ategory C with nite sums, the
oequalizer
ompletion of C is a
ategory C

oeq
with nite
olimits together with a nite sums preserving fun
tor GC : C ! C
oeq su
h
that, for any nite sums preserving fun
tor F : C ! X into a nitely
o
omplete
ategory,
there is a unique nite
olimits preserving fun
tor F : C
oeq ! X with F
GC = F . This
onstru
tion has been des
ribed by Pitts (
f. [BC℄).
For a small

ategory C with weak limits, the exa
t
ompletion EC : C ! Cex
an
be
hara
terized by a universal property as well (see [CV℄). In a spe
ial
ase when C
has nite limits, EC is a nite limits preserving fun
tor into an exa
t
ategory Cex su
h
that, for any nite limits preserving fun
tor F : C ! X into an exa
t
ategory, there is
a unique fun
tor F : Cex ! X whi

h preserves nite limits and regular epimorphisms
su
h that F
EC = F . Following [HT℄, Cex
an be des
ribed as a full sub
ategory of
op
op
SetC and EC as the
odomain restri
tion of the Yoneda embedding Y : C ! SetC .
To explain it, we re
all that a fun
tor H : C op ! Set is weakly representable if it admits
0

0

Se
ond author partially supported by the Grant Agen

y of the Cze
h Republi
under the grant No.
201/99/0310 and by the Italian CNR.
Re
eived by the editors 1999 January 12 and, in revised form, 1999 August 27.
Published on 1999 November 30.
1991 Mathemati
s Subje
t Classi
ation: 18A99.
Key words and phrases: exa
t
ompletion,
oequalizer
ompletion, variety.
M. C. Pedi

hio and J. Rosi
ky 1999. Permission to

opy for private use granted.



77

Theory and Appli
ations of Categories, Vol. 6, No. 6

78

a regular epimorphism
: Y C ! H from a representable fun
tor. Then Cex
onsists of
those weakly representable fun
tors H admitting
: Y C ! H whose kernel pair


K



//
//

YC


//

H

(1)

has K weakly representable.
op
We will start by showing that C
oeq

an be presented as a full sub
ategory of SetC
op
too, with GC being the
odomain restri
tion of Y . The full sub
ategory of SetC
onsisting of all nite produ
ts preserving fun
tors will be denoted by F P (C op ). It is
well known that F P (C op ) is a variety (see [AR℄ 3.17).

Let C be a
ategory with nite sums. Then C
oeq is equivalent to the full
sub
ategory of F P (C op )
onsisting of nitely presentable obje
ts in F P (C op ).

1.1.

Lemma.

Proof.

By the universal property of

C
oeq , we get

F P (C op )  Lex((C
oeq )op )
where, on the right, there is the full sub
ategory of Set(C
oeq )
onsisting of all nite
limits preserving fun
tors. The result thus follows from [AR℄ 1.46.
op

Let C be a
ategory with nite sums and weak nite limits. Then Cex
is equivalent to a full sub
ategory of C
oeq .
Proof. Let H 2 Cex and
onsider the
orresponding diagram (1). There is a regular
epimorphism Æ : Y D ! K (be
ause K is weakly representable) and we obtain a
oequal-

1.2.

Proposition.

izer

YD




//

//

YC



// H

(2)

where  = Æ and  = Æ . Sin
e Y D is a regular proje
tive in SetC , the graph (;  )
is re
exive (it means the existen
e of ' : Y C ! Y D with ' = ' = idY (C ) ). Following
[PW℄, (2) is a
oequalizer in F P (C op ). Therefore, H is nitely presentable in F P (C op )
(
f. [AR℄ 1.3). Hen
e, using Lemma 1.1, H belongs to C
oeq .
op

When C has nite sums, obje
ts of C are pre
isely nitely generated free algebras in
the variety F P (C op ). The
ondition of having weak nite limits too, is a very restri
tive
one. We give another formulation of it.
1.3. Proposition. Let C have nite sums. Then C has weak nite limits i nite limits
of obje
ts of C in F P (C op ) are nitely generated.
op
Proof. Let D : D ! C be a nite diagram and (Æd : A ! Y Dd )
d2D its limit in F P (C ).
Assume that A is nitely generated. Then there is a regular epimorphism  : Y C ! A
where C 2 C. Consider a
one (fd : X ! Dd )d2D in C. There is a unique ' : Y X ! A
with Æd ' = Y fd for all d 2 D. Sin
e  is a regular epimorphism (and Y X regular

Theory and Appli
ations of Categories, Vol. 6, No. 6

79

!

proje
tive) ' fa
torizes through  and therefore (Æd  : Y C
Y Dd )d2D is a weak limit
of Y D in Y (C). Hen
e D has a weak limit in C.
Conversely, assume that D has a weak limit (
d : C
Dd )d2D in C. There is a unique
 : YC
A with Æd  = Y
d for all d D. Consider ' : Y X
A, X C. There exists
':X
C su
h that Y (
d ) = Æd ' for all d D. Hen
e ' =  , whi
h implies that 
is a regular epimorphism (be
ause Y X; X C are nitely generated free algebras in the
variety F P (C op )). Hen
e A is nitely generated.

!
!

2

2

!

2

!

2

2. When do they
oin
ide?

!
!

2.1. Constru
tion. Let C be a
ategory with weak nite limits. Let r0 , r1 : C1
C0
be a re
exive and symmetri
graph in C. It means that there are morphisms d : C0 C1
and s : C1 C1 with r0 d = r1 d = idC0 and r1 s = r0 , r0 s = r1 . We form a weak pullba
k

!

r0 }}}}

AA r
AA 1
AA
A

}}
~~}}

C1 A

AA
AA
r1 AAA

(3)

C2 A

r0 }}}}

C0

C1

}}
~~}}

By taking ri2 = ri ri , i = 0; 1, we get the graph
C2

!

r02
r

2
1

C1
//
//

This graph is re
exive: d2 : C0
C2 is given by r0 d2 = r1 d2 = d. It is also symmetri
:
s2 : C2
C2 is given by r0 s2 = sr1 and r1 s2 = sr0 . By iterating this pro
edure, we get
re
exive and symmetri
graphs

!

Cn

r0n
r1n

//

//

C1

for n = 1; 2; : : :.
2.2. Definition. Let C have weak nite limits. We say that a re
exive and symmetri
graph r0 ; r1 : C1 C0 has a bounded transitive hull if there is n N su
h that, for any
m > n, there exists fm : Cm
Cn with r0nfm = r0m and r1nfm = r1m .

!

2

!

It is easy to
he
k that the denition does not depend on the
hoi
e of weak nite
limits. Evidently, if C has nite limits, Denition 2.2 means that the pseudoequivalen
e
generated by the graph r0 ; r1 : C1 C0 is equal to r0n ; r1n : Cn C0 .

!

!

Theorem. Let C have nite sums and weak nite limits. Then Cex is equivalent to
C
oeq i any re
exive and symmetri
graph in C has a bounded transitive hull.

2.3.

Theory and Appli
ations of Categories, Vol. 6, No. 6

I. Let r0 ; r1 : C1
oequalizer

Proof.

!C

0

80

be a re
exive and symmetri
graph in C. Consider the
Y r0

Y C1

//

Y r1



Y C0
//

//

H

!

in SetC . Put H1 = Y C1 , H0 = Y C0 and
i = Y ri for i = 0; 1. Let
0n ,
1n : Hn
H0
op
be iterations of the graph (
0 ;
1 )
onstru
ted as before, by using pullba
ks in SetC .
There are morphisms n : Y Cn
Hn su
h that 1 = idH0 and
in n+1 = n Y (
rin ) for
i = 1; 2 and n = 0; 1; : : ::
op

!

Y Cn+1I

II
II
II
II Y r1n
II
n+1
II
II
II
II
I$$


u
uu
uu
u
n
Y r0 uuu
u
uu
uu
u
u
uu
zz u
u

Hn+1G

Y CnE

EE
EE
E
n EE""

0n www
w

GG
n
GG 1
GG
GG
##

HH
HH
H
n
1 HHH##

vv
vv
v
vv n
{{vv
0

w
ww
{{ww

Hn H

H0

Y Cn

Hn

yy
yy
y
y
||yy n

op

By indu
tion, we will prove that n are regular epimorphisms in SetC . Assume that
n is a regular epimorphism and
onsider the pullba
k
0 yyy

y

G EE

EE 1
EE
EE
E""

y
yy
y|| y

Y CnE

EE
EE
E
1n n EE""

op

Y Cn

H0

!

yy
yy
y
y n
||yy
0 n

!

in SetC . There are morphisms ' : Y Cn+1 G and : G Hn+1 su
h that ' =
 ,
op n+1
n
n
C
%i ' = Y ri and
i = n %i for i = 0; 1. Sin
e n is a regular epimorphism in Set
, is
op
C
it follows from the proof of Proposition
a regular epimorphism in Set . Furthermore,
op
C
1.3 that op
' is a regular epimorphism in Set
too. Hen
e n+1 is a regular epimorphism
C
in Set .
op
Let I be the relation in SetC determined by the graph (
0 ;
1). It means that I is

Theory and Appli
ations of Categories, Vol. 6, No. 6

81

given by the regular epi-monopair fa
torization
0
1

//
//?? H0
~
??
@@
~
~~~~
@@
~
@@
0 ~~~~~~~
@@
~
@
~~~~~1
 @@
~
@@
~~ ~
@@ ~~~~~
 ~~~

H1 @

I

in SetC . Let n0 ; n1 : I n ! H0 be the
omposition of n
opies of I . Then I n is the
relation generated by the graph
0n ;
1n : Hn ! H0 . Sin
e n : Y Cn ! Hn is a regular
epimorphism, I n is also the relation generated by the graph Y r0n ; Y r1n : Y Cn ! Y C0 = H0 .
op

II. Now, assume that C has bounded transitive hulls of re
exive and symmetri
graphs.
We are going to prove that Cex  C
oeq . Let H : C op !Set belong to C
oeq . Following
Proposition 1.2, it suÆ
es to prove that H belongs to Cex . Sin
e F P (C op ) is a variety
and, following Lemma 1.1, H is nitely presentable, H is presented by a
oequalizer
H1

0
1

H0
//
//



H
//

(4)

of a re
exive and symmetri
graph in F P (C op ) where H1 and H0 are free algebras in
F P (C op ) over nitely many generators (
f. [AR℄, Remark 3.13). Sin
e nitely presentable
free algebras in F P (C op ) are pre
isely nite sums of representable fun
tors and GC : C !
C
oeq preserves nite sums, the fun
tors H0 and H1 are representable, Hi = Y Ci , i = 0; 1.
Hen
e we get a re
exive and symmetri
graph
r0 ; r1 : C1 ! C0 in C su
h that
i = Y ri ,
op
C
i = 0; 1. Sin
e (4) is a
oequalizer in Set
too (by [PW℄), it suÆ
es to show that the
kernel pair
K




//

//

Y C0



//

H

has K weakly representable.
Sin
e the graph (r0 ; r1 ) has a bounded transitive hull, there is n su
h that, for any
m > n, there exists a graph morphism Y fm : Y Cm ! Y Cn from the graph (Y r0m ; Y r1m ) to
the graph (Y r0n; Y r1n ). Following I., they indu
e morphisms I m ! I n of the
orresponding
relations. Hen
e I n is an equivalen
e relation and,
onsequently, it yields a kernel pair of


In

n0
n
1

H0
//
//


//

H

Hen
e K 
= I n and sin
e I n is a quotient of Y Cn , K is weakly representable.
III. Conversely, let Cex  C
oeq and
onsider a re
exive and symmetri
graph r0 ; r1 :
C1 ! C0 in C. Take a
oequalizer
Y C1

Y r0
Y r1

//

//

Y C0


//

H

Theory and Appli
ations of Categories,

Vol. 6, No. 6

82

in SetC . Following [PW℄, it is a
oequalizer in F P (C op ) as well and, using Lemma 1.1,
we get that H 2 C
oeq . Hen
e H 2 Cex and therefore the kernel pair
op

K




//
//

Y C0


//

H

has K weakly representable. Hen
e K is nitely generated in SetC (see [AR℄ 1.69).
Sin
e K is a union of the
hain of
ompositions I n , n = 0; 1; : : :, there is n su
h that
K 
= I n. Hen
e I m 
= I n for all m  n. Following I., I m is the relation generated by
the graph (Y r0m ; Y r1m ). Sin
e Y Cm are regular proje
tives, there are graph morphisms
Y Cm ! Y Cn for all m > n. Hen
e, there are graph morphisms fm : Cm ! Cn for all
m > n. We have proved that C has bounded transitive hulls of re
exive and symmetri
graphs.
2.4. Example. 1) Let V be a variety in whi
h nitely generated algebras are
losed under
nite produ
ts and subalgebras (like sets, ve
tor spa
es or abelian groups). Let C be the
full sub
ategory of V
onsisting of nitely generated free algebras. Then Cex  C
oeq .
At rst, following [AR℄ 3.16, V 
= F P (C op ) and Y : C ! F P (C op )
orresponds to the
in
lusion C  V. Consider a re
exive and symmetri
graph r0 ; r1 : C1 ! C0 in C. The
equivalen
e relation K  C0  C0 determined by it is nitely generated (as a subalgebra
of C0  C0 ). Following III. of the proof of 2.3, the graph (r0 ; r1 ) has a bounded transitive
hull.
Remark that C has weak nite limits (following Proposition 1.3).
2) On the other hand, it is easy to nd examples of a small
ategory C su
h that
Cex  C
oeq does not hold. It suÆ
es to
onsider the
ategory C of
ountable sets (and
the innite path as a re
exive and symmetri
graph in it) and to use Theorem 2.3.
op

Referen
es

[AR℄ J. Adamek and J. Rosi
ky, Lo
ally presentable and a

essible
ategories, Cambridge University
Press 1994
[BC℄ M. Bunge and A. Carboni, The symmetri
topos, Jour. Pure Appl. Algebra 105 (1995), 233-249
[CV℄ A. Carboni and E. M. Vitale, Regular and exa
t
ompletions, Jour. Pure Appl. Algebra 125
(1998), 79-117
[HT℄ H. Hu and W. Tholen, A note on free regular and exa
t
ompletions and their innitary generalizations, Theory and Appli
ations of Categories, 2 (1996), 113-132
[PW℄ M. C. Pedi

hio and R. Wood, A simple
hara
terization theorem of theories of varieties, to appear
[P℄

A. Pitts The lex re
e
tion of a
ategory with nite produ
ts, unpublished notes 1996

University of Trieste
Ple Europa 1
34100 Trieste, Italy
Email: pedi

hiuniv.trieste.it

Masaryk University
Jan
a
kovo n
am. 2a
66295 Brno, Cze
h Republi

and rosi
kymath.muni.
z

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