(Recall that a pretopos is a regular category with e¬ective equivalence relations

and disjoint universal ¬nite sums.) Note that you can still force a particular sieve

to go to a sum by adding ¬ber products forced to be zero as in the preceding

section.

A pre-geometric theory is a left exact category together with a class of

sieves. As above, a model is a left exact functor which takes the given sieves to

regular epimorphic families. A geometric sketch induces, in an obvious way, the

structure of a sketch on its associated left exact theory.

8.3 Geometric Theories 283

A geometric theory is simply a Grothendieck topos. A model is a functor

which is the left adjoint part of a geometric morphism. Using the special ad-

joint functor theorem, it is not hard to show that a left exact functor between

Grothendieck toposes which takes covers to covers has a right adjoint, so that

provides an equivalent de¬nition of geometric morphism. A model in Set is also

known as a point. We have:

Theorem 1. Every geometric sketch has a model in a universal geometric theory

which induces an equivalence of model categories.

Proof. It is clear that there is an equivalence of categories between the models

of the geometric sketch and models of the associated pregeometric theory. The

rest is similar to the proof of Corollary 4 of Section 8.1.

An important variation on the notion of geometric theory is that of a coherent

sketch. A coherent sketch is a sketch in which all the sieves are ¬nite. A model

is required to be a model of the associated LE-theory which takes the sieves to

regular epimorphic families. However, the models are permitted to take values in

an arbitrary pretopos.

A coherent theory is the same thing as a pretopos. We leave to the reader

the task of modifying the constructions given above to ¬nd the coherent theory

associated to a coherent sketch and of proving the analog of Theorem 1.

The model of the sketch in the universal geometric theory is called the generic

model. The generic topos is called the classifying topos for the theory. Of

course, as we have de¬ned things, the classifying topos for the geometric theory

is the theory. However, there are many kinds of theories besides geometric (FP,

LE, Regular, FS, coherent) and they all have classifying toposes.

The classifying topos of a pre-geometric theory is often constructed directly;

that is one adds directly all the necessary disjoint sums, quotients of equivalence

relations, etc. necessary to have a topos. This bears about the same relation to

our construction as does the construction of the free group using words does to

the argument using the adjoint functor theorem. In each case, both constructions

are useful. The one is useful for getting information about the detailed internal

structure of the object while the other is useful for the universal properties. Less

obvious is the fact that the syntactic construction provides a convenient locus

for the semantic one to take place in. Did you ever wonder what x was in the

polynomial ring k[x] (which is the free k-algebra on a singleton set)? Even if x

is de¬ned, what is a “formal sum of powers of x”?

For more about classifying toposes, see Johnstone [1977], Makkai and Reyes

[1977], Tierney [1976] and Mac Lane and Moerdijk [1992].

284 8 Cocone Theories

Exercises 8.3

1. Let S be a single sorted FP-sketch with generic object (generating sort) B.

S generates a geometric theory G by taking E to be empty. Let Th = LE(S ).

(a) Show that SetA is the geometric theory associated to S , where A = Thop

is the category of ¬nitely presented algebras. (See Theorem 5 of Section 4.1.)

(b) Show that the generic model of S in G takes B to the underlying functor

in SetA .

2. Let S be an LE-sketch and G be the geometric theory generated by S , taking

E to be empty.

(a) Show that G is a full subcategory of SetLE(S ) .

(b) Show that the generic model of S in G takes each object B of S to

the functor from LE(S ) to Set which evaluates at B, and each arrow of S to a

natural transformation between such functors.

(c) Show that if f is a model of S in a topos E , then the induced model

—

f : G ’ E is determined uniquely by what it does to the objects and arrows

’

of S . (This is the semantic explication of the generic nature of the geometric

theory of S , as opposed to the syntactical one of Theorem 1. Note that a similar

exercise can be done for FS- and geometric sketches, with the functors from

LE(S ) replaced by sheaves).

3. Show that there is a coherent theory whose models correspond to in¬nite

sets. (Hint: For any set S and any natural number n, let S n denote, as usual, all

functions from n to S. If E is an equivalence relation on n, let S E ⊆ S n denote

all functions whose kernel pair is exactly E. Then S n is the disjoint union of all

the S E over all the equivalence relations. Moreover, S is in¬nite if and only for

all n and all equivalence relations E in n, S E is non-empty, meaning its terminal

map is epi.)

4. Construct a geometric theory which classi¬es the category of dense linear

orderings and strictly increasing maps. (See Exercise 1 of Section 8.2.) Show

that the topos is Boolean.

8.4 Properties of Model Categories

In this section we raise and partly answer questions of recognizing categories

as categories of models of di¬erent sorts of theories. The answers we give are

in the form of properties that categories of models have, so that any category

8.4 Properties of Model Categories 285

lacking them is not such a category. For example, the category of categories is

not a regular category and hence cannot be the category of algebras of any triple

(Corollary 4 of Section 3.4 and Theorem 5 of Section 4.3) or even the category

of models of any FP-theory (see Theorem 1 below).

In the following theorem we refer to these properties of a category C of models

in Set of a sketch S . “Underlying functors” are functors which take a model to

the set corresponding to a given object B of the sketch.

(L:) C has all limits and the underlying functors preserve them.

(FC:) C has all ¬ltered colimits and the underlying functors preserve them.

(R:) C is regular.

(EE:) C has e¬ective equivalence relations and the underlying functors pre-

serve their coequalizers.

Other properties to which we refer require de¬nitions.

For the purpose of the de¬nitions that follow we must give a more general

de¬nition of a regular epimorphic sieve in the case that the ambient category

lacks the relevant pullbacks. In Exercise 4 you are invited to show that the two

de¬nitions agree in the presence of pullbacks.

A sieve {fi : Ai ’ A} is said to be regular epimorphic if whenever B is any

’

object and for each i there is given a morphism gi : Ai ’ B with the property

’

that for any object C of the category and pair of maps d : C ’ Ai , d1 : C ’ Aj ,

0

’ ’

0 1 0 1

fi —¦ d = fj —¦ d implies gi —¦ d = gj —¦ d , then there is a unique h: A ’ B such

’

that gi = h —¦ fi for all i. The class C of objects may be replaced without loss of

generality by a generating set.

An object G of a category is a regular projective generator if

(i) It is regular projective: for any regular epi A ’ B, the induced map

’

Hom(G, A) ’ Hom(G, B)

’

is surjective, and

(ii) It is a regular generator: the singleton {G} is a regular generating family

(Section 6.8).

More generally, a set G of objects is a regular projective generating set

if it is a regular generating family in which each object is a regular projective.

A ¬lter is a subset f of a lower semilattice (a poset with ¬nite meets) for

which (i) if a ∈ f and a ¤ b then b ∈ f, and (ii) if a ∈ f and b ∈ f then the meet

a § b ∈ f. An ultra¬lter on a set I is a ¬lter f (under inclusion) of subsets of I

with the property that for each set J, either J or its complement belongs to f.

It follows that if J ∪ K ∈ f, then at least one of J or K must be (Exercise 1). In

286 8 Cocone Theories

a category C with all products, an ultraproduct is an object constructed this

way: Begin with a family {(Ci )}, i ∈ I of objects and an ultra¬lter f. For each

set J in the ultra¬lter we de¬ne an object N (J) = i∈J (Ci ). If K ⊆ J, then the

universal property of products induces an arrow N (J) ’ N (K). This produces

’

a ¬ltered diagram and the colimit of that diagram is the ultraproduct induced

by (Ci ) and f. However, the concept of ultraproduct does not itself have any

universal mapping property. Note that since an ultraproduct is a ¬ltered colimit

of products, L and FC together imply that a category has ultraproducts.

In the theorem below, we list a number of properties of categories of models.

We would emphasize that the categories may well and often do permit other limit

and colimit constructions. The ones mentioned in the theorem below are limited

to those which are preserved by the functors which evaluate the models (which

are, after all, functors) at the objects of the sketch. Equivalently, they are the

constructions which are carried out “pointwise”, meaning in the category of sets

where the models take values.

Theorem 1.

(a) If S is a single sorted FP-sketch, then C has L, FC, R, EE and a regular

projective generator.

(a) If S is an FP-sketch, then C has L, FC, R, EE and a regular projective

generating set.

(b) If S is an LE-sketch, then C has L and FC.

(c) If S is a regular sketch, then C has FC and all products.

(d) If S is a coherent theory, then C has FC and all ultraproducts.

(e) If S is a geometric theory, then C has FC.

Note that we give no properties distinguishing FS- and coherent theories.

Proof. The category of models in sets of an LE-sketch has all limits and ¬ltered

colimits by Theorem 4 of Section 4.4. It is trivial to see that the same is true of

models of FP-sketches.

To show that FP-theories have e¬ective equivalence relations and are regular,

we need a lemma.

8.4 Properties of Model Categories 287

Lemma 2. Given the diagram

'E E C02

C00 E C01

T

cc Ec c cc

E C12

C10 C11

E

c Ec c

E C22

C20 E C21

which is serially commutative and in which all three rows and columns are co-

equalizers and the top row and left column are re¬‚exive, the induced

C00 ’ C11 ’ C22

’ ’

’’

is also a coequalizer.

Proof. Exercise 2.

Let M be a model, E a submodel of M which is an equivalence relation, and

C the quotient functor in SetFP(S ) , which we will prove to be a model. This last

claim is equivalent to the assertion that C preserves products. Let A and B be

objects of FP(S ). We get the diagram

'E E CA — EB

E M A — EB

EA — EB

T

cc cc cc

E E CA — M B

E MA — MB

EA — M B

c c c

E E CA — CB

E M A — CB

EA — CB

The rows and columns are coequalizers because products in a topos have a right

adjoint. The lemma then implies that the diagonal is a coequalizer. But since E

and M preserve products, this implies that C does also.

This not only shows that the model category has coequalizers of equivalence

relations, but that the evaluation functors preserve them. In view of the Yoneda

Lemma, that is just the assertion that the objects of FP(S ) are regular projective.

To see that they generate, observe that the maps from representable functors form

an epimorphic family in the functor category. But that category is a Grothendieck

topos, so those maps form a regular epimorphic family. It is easy to see that they

288 8 Cocone Theories

continue to do so in any full subcategory. If the FP-theory is single sorted, that set

may be replaced by the generic object to get a single regular projective generator.

Any surjective natural transformation in a Set-valued functor category is a

coequalizer. Since C is closed under limits, this means that maps between models

are regular epis if and only if they are surjective. It follows from this and the fact

that the pullback of a surjective map in Set is surjective that C is regular. This

takes care of (a) and (b). (Note that in fact C has all colimits, but they are not

necessarily preserved by the evaluations.)

Let S be a regular sketch and R = Reg(S ) be the regular theory associated

to it. If {Mi } is a family of models then we claim that the pointwise product M

is a model. Models must preserve ¬nite limits and take regular epis to regular

epis. But a product of ¬nite limits is a ¬nite limit and a product of regular epis

is a regular epi (since it is in Set and we carry out these constructions pointwise).

As for ¬ltered colimits, the argument is similar. Filtered colimits commute with

¬nite limits in Set and in any category with regular epis. This completes the

proof of (d).

Let R denote a coherent theory and C be the category of models. Let I be an

index set, {Mi } an I-indexed family of models and f an ultra¬lter on I. A product

of models is not a model, but it is regular. Since this property is preserved by

¬ltered colimits (Exercise 3), an ultraproduct of models also preserves it. Thus

it su¬ces to show that the ultraproduct preserves ¬nite sums.

Let A and B be objects of the theory and suppose ¬rst that for all i ∈ I, Mi A =

… and Mi B = …. Let M = Mi . Then if N is the ultraproduct, the canonical

morphism M ’ N is surjective at A, B and A + B. Consider the diagram

’

E M (A + B)

MA + MB

c c

c c

E N (A + B)

NA + NB

The vertical arrows are quotients and the horizontal arrows the ones induced

by the properties of sums. We want to show that the bottom arrow is an iso-

morphism. Let x ∈ N (A + B) and choose a y ∈ M (A + B) lying over it. Since

M (A + B) is a colimit of the products over sets in f, there is a set J ∈ f such that

y = (yi ), i ∈ J is an element of (Mi A + Mi B), the product taken over the i ∈ J.

Let K = {i | yi ∈ Mi A} and L = {i | yi ∈ Mi B}. Since K ∪ L = J ∈ f either K

or L belongs to f. If K does, then (yi ), i ∈ K is an element of M A whose image

in N A goes to x ∈ N (A + B).

To ¬nish the argument, let IA = {i ∈ I | Mi A = …} and IB = {i ∈ I |

Mi B = …}. We observe that if IA © IB ∈ f, the proof above may be repeated

8.4 Properties of Model Categories 289

with IA © IB replacing I. If neither IA nor IB is in f, then it is clear that

N A = N B = N (A + B) = …. Finally, suppose that one of the two, say IA , is in

f and the other one isn™t. Then we have