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AN ALGEBRAIC DESCRIPTION OF

LOCALLY MULTIPRESENTABLE CATEGORIES

JIRI ADAMEK, JIRI ROSICKY

Transmitted by Robert Pare

Abstract. Locally nitely presentable categories are known to be precisely the

cate-gories of models of essentially algebraic theories, i.e., catecate-gories of partial algebras whose domains of denition are determined by equations in total operations. Here we show an analogous description of locally nitely multipresentable categories. We also prove that locally nitely multipresentable categories are precisely categories of models of sketches with nite limit and countable coproduct specications, and we present an example of a locally nitely multipresentable category not sketchable by a sketch with nite limit and nite colimit specications.

Introduction.

We have shown in AR1] how each locally nitely presentable category

Kin the sense of

GU] can be described by an essentially algebraic theory. This means that there exists a (nitary, many-sorted) signature = t psuch that

K is equivalent to the category

of partial -algebrasA such that

(i) each operation A with

2

t is total (i.e., everywhere dened)

(ii) for each 2

p a nite set Def of equations in signature t is given and A(

a 1

:::a

n) is dened i all equations of Def are fullled in ( a

1 :::a

n).

and

(iii) a set of equations is given which all partial algebras of the given category satisfy. In the present paper we discuss locally nitely multipresentable categories, as introduced by Y. Diers D]. Recall that a category is locally nitely multipresentable i it has

(a) connected limits (or, equivalently, multicolimits)

Financial support of the Grant Agency of the Czech Republic under the grant no.201/93/0950 is gratefully acknowledged.

Received by the Editors 11 May 1995, and in revised form 22 June 1996. Published on 5 July 1996.

1991 Mathematics Subject Classication: 18C99.

Key words and phrases: locally multipresentable category, sketch.

and

(b) a set of nitely presentable objects A such that each object is a directed colimit

of A-objects.

Examples: elds and homomorphisms,linearly ordered sets and strictly increasing maps. We describe a natural generalization of the concept of essentially algebraic theory, called essentially multialgebraic theory, and we show that a category is locally nitely multi-presentable i it is equivalent to the category of models of such a theory. Here we again work with partial algebras of signature = t p(where t are the total operations).

But rather than having a nite set of equations for each element of p, we assume that

p is decomposed into a doubly indexed union p = S 2;

S i2I

i and for each

2; a

nite set Def of equations in the signature t is given. In each model

A the equations

of Def determine the domain of denition of all operations of S i2I

i as follows: if all

equations of Def are fullled in ( a

1

:::a

n) then there exists precisely one i2I

such

that A( a

1 :::a

n) are dened for all 2

i, whereas all (a

1 :::a

n) for

2

;

i

are undened.

We consider next sketches describing locally nitely multipresentable categories: we have shown in AR1] that a category is locally nitely multipresentable i it can be

sketched by a (nite limit, coproduct)-sketch. We now prove that, assuming the non-existence of measurable cardinals, we can restrict ourselves to (nite limit, countable coproduct)-sketches. On the other hand, nite coproducts are not sucient: a locally nitely multipresentable category K is exhibited which cannot be sketched by a (nite

limit, nite coproduct)-sketch. Curiously, K can be sketched by a (nite limit, nite

colimit)-sketch.

I. Essentially Multialgebraic Theory

Throughout the paper we work with partial -algebras, where is a many-sorted (ni-tary) signature. That is, if S is the set of sorts under consideration, then is a set of

operation symbols with prescribed arities :s

1

s k

!s:

This means that a partial -algebra A consists of an S-sorted set, i.e., a collection

(A

s)s2S of sets, together with partial maps A:

A s

1

A s

k !A

s

for each :s 1

s k

!s. Ahomomorphism from a partial -algebraAinto a partial

-algebra B is an S-sorted map h = (h

s)s2S :

A ! B (i.e., h s :

A s

! B

s are maps)

preserving the operations in the strong sense: whenever A( a

1 :::a

k) is dened in A, it

follows that B( h

s 1(

a 1)

:::h s

k( a

k)) is dened in

Band has the valueh( A(

a 1

:::a k)).

This results in the category PAlg of partial algebras and homomorphisms.

Definition. By anessentially multialgebraic theory is understood a quadruple

T = (E;Def)

where

is a signature

E is a set of equations (in the signature )

; is a set of collections = (

i)

i2I of nite subsets i

such that

i are

pairwise disjoint for 2;, i2I

Def is a map assigning to each 2; a nite set Def

of equations in the signature

t def

= ;

2;i2I

i

such that variables appearing in the equations of Def are the same as those

used by any operation of S i2I

i.

Remark. (1) The collection can also be empty, i.e., the case I =

is not excluded.

(2) The last clause concerning variables is only needed because the setS i2I

i can

be innite.

Definition. By a model of an essentially multialgebraic theory T we understand a

partial -algebra A such that

(1) A satises all equations in E

(2) A is a total operation for each

2 t

(3) Given 2;, then whenever A(

a 1

:::a

n) is dened for some 2

S i2I

i then

all equations of Def hold in the

n-tuple (a 1

:::a n)

(4) Given 2;, then when all equations of Def

hold in some

n-tuple (a 1

:::a n),

then there exists a unique i 0

2 I

such that A( a

1 :::a

n) is dened for each 2

i 0.

The category of all models of T and homomorphisms is denoted by ModT.

Remark. In case I =

, a modelA of T has the property that for no n-tuple ofA all

equations of Def are satised.

Examples. (1)Fields. We start with the algebraic theory ( t

E

t) of rings with a unit

element. We introduce two unary operation symbols oandi(whereo(x) will be dened

precisely when x= 0 and i(x) is the inverse of x for all x6= 0). We put

= t

foig

; =f 0

where

=ffogfigg

0 =

Def =

Def 0 =

f0 = 1g E =E

t E

0

where E

0 are the following four equations: i(x)x= 1 xi(x) = 1 o(x) = 1 xo(x) = 0:

Thus, a model of this theory is a ring with a unit and with two partial unary operations such that for each element x exactly one of i(x) and o(x) is dened, and 0 nonequal

1. Moreover, if i(x) is dened then i(x) = x

;1 and if

o(x) is dened, then x = 0 (and o(0) = 1). This precisely describes all elds.

(2) Graphs. The category of graphs, i.e., sets with a binary relation, and

homo-morphisms is locally nitely presentable. Thus, it can be described by an essentially algebraic theory. The following theory was presented in AR1]: we have sorts edge and

vertex. We put =f g where

: edge!vertex

are the operations of source and target, respectively, and

: edgeedge!edge

is an auxiliary operation to express the implication

(x) = (y)^(y) =(y) !x =y:

Thus, t =

f g and

Deffg =

f (x) = (y) (x) =(y)g

whereas E consists of two equations

(xy) = x

(1)

(xy) = y:

A model of this theory is a pair of sets A edge,

A

vertex together with two operations A

A:

A edge

!A vertex

satisfying the above implications. This can be identied with the graph on the setA vertex

with vertices pq connected by an edge ip= (x) and q =(x) for somex2A edge.

(3)Linearly ordered sets.We will now model the full subcategory of the category

of graphs consisting of all strictly linearly ordered sets. We extend the above essentially algebraic theory as follows:

(i) Linearity. We introduce two binary operations 1

2 of arity

vertexvertex!edge

and one binary operation

3 of arity

vertexvertex!vertex:

We put

=ff i

gg i=123

Def =

and add the following equations to E:

1(

uv) =u and 1(

uv)=v

(3)

2(

uv) =v and 2(

uv)=u

(4)

3(

uv) =u and 3(

uv) =v:

(5)

Thus, in a model A, given vertices u and v precisely one of i(

uv) is dened for i = 123. In case i = 1 we have u < v (since

1(

uv) is the edge u !v), if i = 2 we

havev <u, and in case i= 3 we haveu =v.

(ii) Irreexivity. We introduce 0 =

with

Def 0 =

f (x) =(x)g:

(iii) Transitivity. Choose an operation : edgeedge!edge and put

00 =

ffgg

Def 0 0 =

f(x) = (y)g

and add to E the equations

(xy) = (x) and (xy) =(y):

The theory of signature =

f

123 g

with the equation (1)-(6) formingE, with ; =f

000 g

and with the above Def axiomatizes linearly ordered sets.

Remark. We recall from AR1] 4.30 a characterization of locally nitely

multipresent-able categories as precisely the!-cone-orthogonalityclasses in locally nitely presentable categories:

LetLbe a locally nitely presentable category. For each setMof conesm= (Dm

dmj

! Dmj)j2Jm in

L we denote by M

? the full subcategory of

L consisting of all objects L such that given m 2 M then L is orthogonal to m, i.e., given a morphism f : Dm ! L, there exists a unique j 2 Jm such that f factorizes through dmj, and also

the factorization is unique (i.e., there exists a unique morphism f0 : D

mj ! L with f = f0

dmj). We call M

? a cone-orthogonality class, and in case each D

m and Dmj

is nitely presentable in L, M

? is called an !-cone-orthogonality class. The latter

subcategories ofLare locally nitely multipresentable. Conversely, every locally nitely

multipresentable category is equivalent to an!-cone-orthogonality class of some locally nitely presentable category.

Proposition 1. For every essentially multialgebraic theory TTT the category ModT is locally nitely multipresentable.

Proof. Denote by L the full subcategory of PAlg which consists of all partial

-algebras which satisfy all equations inE and in which all t-operations are everywhere

dened. It is easy to see that L is a locally nitely presentable category. We are now

going to present a set M of cones such that M

? is equal to Mod T.

For each element 2 ; we construct a cone (F

f _i

! Fi)i 2I in

L as follows. Let x0:::xn;1 be all variables which appear in Def. (Since the same variables appear in

operations of S

i2I

i, we can consider each of these operations as n-ary, and write

(x0:::xn;1). This is done for technical convenience only since we can simply forget

those variables on which does not depend.) Denote byF0

the free algebra generated by

fx

0:::xn;1

gin the equational class of all (total) t-algebras satisfying the equations

of Def. We consider F0

as a partial -algebra by leaving all operations of ;t

nowhere dened. Then F0

has a reection in L, say, r : F 0

of generators and Def are nite sets, it follows that F is nitely presentable in L.

Next, for each i 2 I

we form a free algebra F 0

i generated by fx

0:::xn;1

g

i in

the above equational class, and we consider F0

i as a partial -algebra as follows: for

any 2

i we dene

F 0

i(x

0]:::xn;1]) = ]

(where ] denotes the inclusion of generators to F0

i) and leave F

i undened in all

other n-tuples for any 2 ;(

t

i) the operation F

i is nowhere dened. Let ri :F

0 i

!F

i be a reection ofF 0

i in

L again, F

i is nitely presentable in

L. The

inclusion of the sets of generators extends to a -homomorphismf0 i :F

0

!F 0

i whose

reections is denoted by f i :F

!F

i. Put m = (F

f i ! F i) i2I :

If an algebra A2L is orthogonal to the cone m

then it has the property that

() for eachn-tuple (a

0:::an;1) in which all Def-equations are satised there exists

precisely one i2I

such that A(a0:::an;1) is dened for each

2

i.

In fact, since the n-tuple is satised by all equations of Def, we have a unique

-homomorphismh0 :F0

!A with h 0(x

k]) =ak for k= 0:::n

;1 let h:F

!A be

the -homomorphism with h0 = h r

. We have a unique i 2 I

such that h factorizes

throughf

i. Let us show that, then,

A(a0:::an;1) is dened for each 2

i. Given

a -homomorphism h:F i

!Awithh = hf

i, then since F

0 i(x

0]:::xn;1]) = ],

and since f0

i(x

k]) = xk] for k = 0:::n

;1, we obtain A(h

r

i(x

0]):::h r

i(x

n;1])) = h r

i( ])

thus

A(a0:::an;1) = h r

i( ]):

Conversely, ifj2I

has the property that all A(a0:::an;1) with

2

i are dened,

then i =j because the unique t-homomorphism h 0 : F0

i

!A with h 0(x

k]) =ak and

h0( ]) =

A(a0:::an) for all

2

j then is, obviously, a homomorphism h 0 :F0

j

! A of partial -algebras. Let ~h : Fj

! A be the -homomorphism with h 0 = ~h

r j,

then we have

hr

(xk]) = ~h r j f 0

j(x

k]) for k = 0:::n ;1

thushr = ~h

r j f 0

j = ~h f

j

r

, from which it follows that h= ~h f

j. Therefore, h factorizes throughf

j, which proves i=j. Thus, (

Conversely, let A2Lhave the property (), thenA is orthogonal to m

. In fact, for

each homomorphismh:F

!A the n-tuple a k =h

r

(xk]), k = 0:::n

;1 satises

all the equations of Def. Thus, there is a unique i 2 I

with all A(a0:::an;1), 2

i, dened. It follows that hfactorizes throughf

i: let h 0 :F0

i

! Abe the unique

t-homomorphism with h 0(x

k]) = ak and h

0( ]) =

A(a0:::an;1) for all

2

i,

then h0 =F0 i

!A is a homomorphism of PAlg, thus, h 0 = h

r

i for a homomorphism

h :F i

!A it follows that h = hf

i. Moreover, this factorization is unique: suppose h = h

f

i, then we prove h

r

i = h r

i (which implies h

= h) by showing that

these homomorphisms agree on all the generators of F0

i. In fact, h

r

i(x

k]) =h r i f 0

i(x k]) =h f i r

(xk])

=hr

(xk])

=ak

and for ], where 2

i, the equality F

0 i(x

0]:::xk ;1]) = ] implies h

r

i( ]) = A(h

r

i(x

0]):::h

r

i(x n;1]))

= A(a0:::an;1):

This shows that h factorizes uniquely through f

i. Conversely, if h factorizes through f

i, we show that j =i by verifying that

A(a0:::an;1) is dened for all

2

j {

this is performed precisely as in the above proof that h r

i( ]) =

A(a0:::an;1).

We conclude that

ModT =fm

2;g ?:

Since all domains and codomains appearing inm are nitely presentable (for all 2;),

we conclude by the preceding Remark that ModT is locally nitely multipresentable.

Remark. Locally nitely multipresentable categories were also characterized in AR1]

4.30 as multireective subcategories of Alg (categories of total -algebras) closed under directed colimits. More precisely, a category is locally nitely multipresentable i there exists a nitary signature and a full subcategory K of Alg equivalent to the

given category, which has the following properties:

(i) Every -algebra A has a multireection in K, i.e., a cone (A r

i ! A

i)i2I such that

all Ai lie in

K and any homomorphism h : A ! B with B in K factorizes through a

uniqueri, i

2I, and the factorization h

0 with h=h0r

i is also unique

(ii) K is closed under directed colimits, or, equivalently, a -algebra K lies in K

whenever it is orthogonal to the multireection of each nitely presentable -algebra

A. If (A r i ! A

i)i2I is such a multireection, then it is easy to verify that Ai are also

Theorem 1. A category is locally nitely multipresentable i it is equivalent to the category of models of some essentially multialgebraic theory.

Proof. Due to the preceding Proposition and Remark, it is sucient to show that

any full subcategory Kof Alg satisfying (i) and (ii) above is equivalent to ModT for

some essentially multialgebraic theory T. The signature

of the theory

T we dene

now will be an extension of the given signature (where the operations of will be precisely the total operations, =

t). Let (Ai)i2I be a set of representatives of all

nitely presentable -algebras w.r.t. isomorphism. For each i 2 I we have, since Ai is

nitely presentable, a nite set of variables x1:::xk (of sorts t1:::tk, respectively)

and a nite set called Defi of equations in those variables such that Ai is presented by

those variables and equations. That is, Ai can be identied with the quotient algebra

of the free -algebra F(x1:::xk) generated by fx

1:::xk

g modulo the congruence i generated by Defi. We also form a multireection (Ai

mij

! Aij)j

2Ji and, since Aij

is nitely presentable, we can choose a nite set Gij of generators of Aij for each

j 2 Ji (containing mij(xn]), n = 1:::k where xn] is the congruence class of xn in Ai =F(x1:::xk)=

i).

We now extend to by adding, for eachi

2I, j 2Ji and each element a of sort s(a) in Aij, an operation

ija :t1

tk!s(a):

(The informal idea of ija in models K is the following: the dening equations Defi

guarantee that if ija (x1:::xk) is dened, then the interpretation of the variables x1:::xk yields a homomorphismh : Ai

! K. Then h factors as h = h 0m

ij and the

result of ija will be the image ofaunderh0.) Thus, we put =

f ijai2I j 2Ji

and a2Gijg and

ij =f ijaa2Gijg:

We dened Defi above for each i2I. It remains to dene the setE of equations in the

signature . These will be of two types:

(a) xn = ijx

n(x1:::xk) for all i

2 I, j 2Ji, n= 1:::k where x

1:::xk are the

variables of the given presentation of Ai, andx

n =mij(xn]).

(b) Since Aij is generated by Gij = fg

1:::gm

g, we can choose, for each element a 2 Aij, a -term

0

a(x1:::xm) such that ( 0

a)Aij(g1:::gm) = a we denote by a =0

a( ijg1::: ijgm) the corresponding

-term. For each

2 and each instance

of computation of A, say,A(a1:::an) =a, we add to E the equation (a1:::an) =a:

We thus dened an essentially multialgebraic theory

T= (

E;Def)

where ; = ff

ijgj 2Ji

gi

2I. We will prove that

K is equivalent to ModT. For each

-algebra K in K we dene a partial

-algebra H(K) as follows. Giveni

then fora 2G

ij we dene ( ija)K in a tuple (b1:::bk) i there exists a homomorphism h :Aij

!K with (hm ij)t

n(x

n]) =bn, and then

( ija)K(b1:::bk) =h

s(a)(a):

Let us verify H(K) is a model of T. It satises the equations of E:

(a) If ( ijx n)

K is dened in (b1:::bk), then we have h : Aij

! K with

(hm ij)t

n(x

n]) =bn and

( ijx n)

K(b1:::bn) =htn(x n) =h

tn(mij)tn(xn]) =bn:

(b) The equations A(a

1::: a

n) =

a are satised since h is a -homomorphism.

Furthermore, if all equations of Defi are satised in (b1:::bk), then the

homo-morphism f : F(x1:::xk)

! K determined by f t

n(x

n) = bn, n = 1:::k,

fac-torizes through Ai = F(x1:::xk)=

i, i.e., we have a homomorphism g : Ai

! K

with gt n(x

n]) = bn. Since K

2 K, there exists a unique j such that g = hm ij for

a unique -homomorphism h : Aij

! K. Since (h m

ij)tn(xn]) = bn, we conclude

that ( ija)K(b1:::bk) is dened for each a 2 G

ij. Conversely, given j

0 such that

( ij 0

a 0)

K(b1:::bk) is dened for each a 0

2 G ij

0, it follows that a homomorphism h0 : A

ij 0

! K exists with h 0

m ij

0 = h m

ij { consequently, j = j

0 and h = h0.

(In fact, dene h0 by h0

s(a)(a) = ij

0 a

0(b

1:::bk) the equations in (b) guarantee that h 0

is a -homomorphism and those of (a) guarantee h=h0 m

ij 0.)

We conclude that H(K) is a model of T. We obtain a functor H : K!ModT

dened on homomorphismsbyH(h) =h. To show thatH is an equivalence of categories, it is sucient to verify that whenever a partial -algebra B is a model of

T then its

-reductB0, i.e., the -algebra obtained by forgeting the operations in

;, lies inK

{ it then follows that B=H(B0). We can assume, without loss of generality, thatB0 is

a nitely presentable -algebra. In fact, sinceB is a model of T, for each -subalgebra C of B0 the restriction of the partial operations of B to C denes a partial

-algebra C0 which is also a model of

K. Suppose that we know already that this implies that C (the reduct of C0) lies in

K, then B

0 also lies in

K since K is closed under directed

colimits (and B0 is a directed colimit of all suchC.) Thus,B0 can be assumed to beAi

for some i 2I. The equations of Def

i are all satised in the tuple (x1]:::xk]) of B,

thus, there exists j2 J

i such that ( ija)B are dened in that tuple for each element a

of Gij (because Bis a model of

T) and thus, also (

a)B(x1]:::xn]) are dened. This

allows us to dene

k(a) = ( ija)B(x1]:::xk]) for all a 2G

ij:

The equations of E guarantee that k is a -homomorphismk :Aij

!B satisfying km

In fact, let Aij(a1:::an) = a for some : s1

sn ! s in , then from the

equation in (b) above we conclude that

k(a) =k((0

a)Aij(g1:::gm))

= (0

a)B(k(g1):::k(gm))

= (0

a)B(( ijg1)B(x

1]:::xk]):::( ijgm)B(x1]:::xk])

= (a)B(x1]:::xk])

=B((a1)B(x

1]:::xk]):::(an)B(x1]:::xk]))

=B((0

a1)B(( ijg

1)B(x

1]:::xk]):::( ijgn)B(x1]:::xk])):::)

=B((0

a1)B(k(g

1):::k(gm)):::(an)B(k(g1):::k(gm))

=B(k(a1):::k(an))

thus,k is a homomorphisms. To verify kmij =idAi, it is thus sucient to prove that k mij(xn]) = xn] for all n = 1:::k, and this follows from the equation (a). This

shows that mij is an isomorphism: from

(mijk)mij =mij =idAij mij

and from Aij 2K we get, by the denition of multireection, thatmijk =idAij, thus k =m;1

ij . This proves that B0 = Ai

= Aij lies in K. Consequently, H :K! ModT is

an equivalence of categories.

Remark. The above theorem has an immediate generalization to innitary algebras

given a regular cardinal, we can introduce essentially multialgebraic theories of -ary partial algebras { here Def is a set of less than equations and ; is a collection of

subsets of of cardinality less than , otherwise the denition is quite analogous. A category is locally -multipresentable i it is equivalent to the category of models of some essentially multialgebraic-ary theory.

II. Finitary Sketches for Locally Multipresentable categories

Recall that a sketch is a quadrupleS = (ALC ) whereA is a small category,L and C are sets of diagrams in A and assigns to each diagram D 2 L a cone (D) and

to each diagram D 2 C a cocone (D). A model of S is a functor F : A ! Set such

that for each diagramD 2L the image of (D) underF is a limit of FD and for each

diagram D2C the image of (D) is a colimit ofFD.

A category is said to be sketchable byS if it is equivalent to the full subcategory of

SetA consisting of all models of

S. It is proved in AR

1] 4.32 that a category is locally

nitely multipresentable i it is sketchable by a (nite limit, coproduct)-sketch (i.e. a sketch with all diagrams inLnite and all diagrams inCdiscrete). We will now present

Example. (4) The following locally nitely multipresentable category is not sketchable

by a (nite limit, nite coproduct) - sketch. Objects: 0 and all primesp>2.

Morphisms: (a) a unique morphism 0!p for each object p

(b) hom(pp) is the cyclic group of order p

(c) no other morphisms except those in (a), (b).

This category has only trivial directed colimits, thus, each object is nitely presentable. It has connected limits: an equalizer of any pair of distinct morphisms in hom(pp) is

0 ! p, analogously with pullbacks. Thus, the category is locally nitely

multipresent-able.

Any (nite limit, nite coproduct)-sketchable category K an be axiomatized by a

rst-order theory. That is, there exists a many-sorted, nitary signature and a theory

T of the rst-order logic of -structures such that K is equivalent to the category of

all T-models and -homomorphisms { see MP]. Thus, it is sucient to prove that our

category K cannot be axiomatized. In fact, suppose that E is an equivalence between K and the category ModT of all models of T. It is well-known that ModT is closed

under ultraproducts in the category Str of all -structures. LetU be a free ultralter

on the set of all objects of K. Since the ultraproduct Q

U

E(p) lies in ModT, there

exists an object q with E(q) isomorphic to Q

U

E(p). To obtain the desired

contradic-tion, we will show that the above ultraproduct has innitely many endomorphisms, although hom(E(q)E(q))

= hom(qq) is nite. In fact, for each element f of the

product Q

p2K

hom(pp) we obtain an endomorphismf

0 of the ultraproduct, dened by f

0(

xp]) = Efp(xp)], and for two such elements fg we have f 0 =

g 0 i

U contains the

setfp fp =gpg. It is sucient to choose an innite set M Q

hom(pp) such that for fq2M with f 6=q the set of allp

0s with

fp =gp is always nite, thenff 0

f 2Mg is

an innite set of endomorphisms of Q U

E(p). This proves that K is not axiomatizable

in rst-order logic.

Theorem 2. Assuming the non-existence of measurable cardinals, each locally nitely

multipresentable category can be sketched by a ( nite limit, countable coproduct)-sketch.

Proof. LetKbe a locally nitely multipresentable category. By AR

1] 4.32 there exists

a (nite limit, coproduct)-sketch S with K equivalent to ModS. Let be a cardinal

such that each coproduct-specication of S has less than objects. We will construct

a (nite limit, countable coproduct)-sketchS

with Mod S

= ModS .

Since no cardinal is measurable, for each cardinal there exists a basic theory T

which is one-sorted, has constants ci (i 2 ) in the signature, and has, up to

iso-morphism, a unique model B such that (ci)B 6= (cj)B for i 6= j and f(ci)Bgi 2 is

the underlying set of B. In fact, T is explicitely described in AJMR]. By inspecting

that description, it can be easily seen that all disjunctions used in all sentences of T

are actually disjoint. This means that whenever p_q was used we can write, instead,

(p_q)^(p^q !?) and where W

n2!

pn was used, we can write W

n2! pn^

V

n6=m

Here ?denotes the formula false. It means that T

uses only

(a) nite conjunctions (b) nite quantications and

(c) nite or countable disjunctions which are disjoint.

We now apply the procedure of MP] 3.2.5 associating with each basic theory a sketchS

with ModSequivalent to the category of all models of that theory. LetS

be the sketch

associated to T

. Due to (a){(c), S

is a (nite limit, countable coproduct)-sketch. S

contains the objectA

describing the unique sort, and each of the constants c

i gives us

a morphismc i : 1

!A

such that the (essentially unique) model of S

has the above

properties.

II. The rest of the proof is analogous to that of Theorem 12 of AJMR]. Let K be a

locally nitely multipresentable category. There exists a (nite limit, coproduct)-sketch

S sketchingK, see AR

1] 4.32. For each uncountable coproduct-specication

(B i

b i !B)

i2 (

a uncountable cardinal)

ofS we take a copy of the above sketchS

and we denote by

S the disjoint union of the

sketchS and all these sketchesS . Let

S

be the sketch we obtain from

Sby deleting

all of the uncountable coproduct specications and, instead, by adding to S

(a) a formal new morphismf :B!A

(b) the following pullback-specications for all i2

B i

b i

>B t

i _

_ f

1 c

i

> A

where t i =

t

f b i.

It is easy to see that the sketches S, S, and S

have equivalent categories of models.

Remark. The statement of the Theorem 2 is actually logically equivalent to the non-existence of measurable cardinals. That is, assuming that a measurable cardinal

ex-ists, there are locally nitely multipresentable categories not sketchable by (nite limit, countable coproduct)-sketch. In fact, not sketchable by -ary sketches, i.e., sketches in

Example. (5) If is a measurable cardinal, the following category is locally nitely

multipresentable, but it is not sketchable by a -ary sketch: Objects: all ordinalsi < .

Morphisms: (a) a unique morphism 0!i for each object i

(b) hom(ii) is a group of cardinality @i for all i >0

(c) no other morphisms except those in (a), (b).

The proof of local nite multipresentability is analogous to that in Example (4). Also the proof that the category is not sketchable by a -ary sketch is quite analogous: by MP], any -ary sketchable category can be axiomatized by innitary rst-order theory of the -ary logic L. Thus, if our category K would be sketchable by a-ary sketch,

there would exist an equivalence E between K and ModT for some rst-order theory

in a-ary signature . Then E(K) would be closed in Mod under ultraproducts over

all -complete ultralters. Since is measurable, there exists a nontrivial -complete ultralterU on the set of all objects of K. The proof is then concluded by showing that

the ultraproduct Q

U E(i) has

@ pairwise distinct endomorphisms (whereas hom(ii)

has cardinality @i <@ for each object i of K).

Remark. The category K of Example (4) has the following remarkable property: we

have seen above that

(a) K is sketchable by a (nite limit, coproduct)-sketch

and

(b) K is not sketchable by a (nite limit, nite coproduct)-sketch.

We now show that, nevertheless,

(c) K is sketchable by a (nite limit, nite colimit)-sketch.

In fact, as proved in Theorem 3 of AJMR], for (c) it is sucient to nd an axiomati-zation of K by a -coherent theory in the logic L!

1!. A -coherent theory consists of

sentences

(8x

1:::xn)(' !)

where'andare formulas built from atomic formulas by nite conjunctions, countable disjunctions, and nite existential quantication. Now observe that K is equivalent to

the category of all graphs (i.e. -structures where consists of one binary relation symbol R) isomorphic either to G0, the empty graph, or Gp, the circuit of length p

(with vertices 01:::p;1 and edgesi!i+ 1 for all i < p;1 and p;1!0) for all

primesp >2. The latter category has the following -coherent axiomatization in which

R2(xy) is a shorthand for (

9z)(R(xz)^R(zy)), analogously with R

3R4::::

(1) (8x)(9!y)R(xy)

(2) (8y)(9!x)R(xy)

(3) (8x)(Rnm(xx) !Rn(x)_Rm(x))

(5) (8xy) W

n2! R

2m(

xy) (R 0(

xy) meansx=y).

In fact, each Gn is a model of (1){(5). Let G 6= G

0 be a model of (1){(5). By (5),

each pair of vertices can be connected by a path of even length. By (1) and (2),

G is a circuit or an innite path, but (5) excludes the latter and forces the length P of the circuit to be larger than 2. Finally, by (3) we know that p is a prime. Thus, G

=Gp, andGp is a model of (1){(5).

References

AJMR] J. Adamek, P. T. Johnstone, J. Makowsky, J. Rosicky, Finitary sketches, to appear in J. Symb. Logic.

AR1] J. Adamek and J. Rosicky, Locally presentable and accessible categories, Cambridge Univ.

Press, Cambridge 1994.

AR2] J. Adamek and J. Rosicky, Finitary sketches and nitely accessible categories, Math. Str. in

Comp. Sci., accepted for publication.

D] Y. Diers, Categories localement multipresentables, Arch. Math. 34 (1980), 344{356.

GU] P. Gabriel and F. Ulmer, Lokal Prasentierbare Kategorien, Lect. Notes in Math. 221, Springer-Verlag, Berlin 1971.

MP] M. Makkai and R. Pare, Accessible categories: the foundations of categorical model theory, Contemporary Math. 104, Amer. Math. Soc., Providence 1989.

J. Adamek J. Rosicky

Technical University Braunschweig Masaryk University

Postfach 3329 Janackovo nam. 2a, 662 95 Brno

38023 Braunschweig, Germany Czech Republic

E-mail: adamek@iti.cs.tu-bs.de and rosicky@math.muni.cz

This article may be accessed via WWW at http://www.tac.mta.ca/tac/ or by anonymous ftp

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