involves only a ¬nite number of objects and morphisms and can be taken as

being in some countable subtopos. We have already illustrated the technique in

Section 7.6.

276 7 Representation Theorems

One of the main thrusts of categorical logic is the exploitation of the insight

that each pretopos corresponds to a theory in the sense of model theory (a lan-

guage, a set of deduction rules and a set of axioms), and vice versa. Under

this equivalence, embedding theorems correspond to completeness theorems”

theorems to the e¬ect that if a statement made in the language is true in every

model of a certain type, then it follows from the axioms. In particular, Deligne™s

theorem is an easy consequence of G¨del™s completeness theorem for ¬nitary ¬rst

o

order logic. In fact it is equivalent to that theorem for the case of ¬nitary geo-

metric theories. Barr™s theorem can be interpreted as saying that if something

follows by classical logic from the axioms, then it follows by intuitionistic logic.

See Makkai, Reyes [1977] and Lambek-Scott [1986] for more details.

8

Cocone Theories

In this chapter, we consider a general type of theory which is left exact and

which in addition has various types of cocones included in the structure. In

the kinds of theories considered here, the family of cocones which is part of the

structure is always induced by the covers of a Grothendieck topology according

to a construction which we will now describe.

Let S = {Ai ’ A} be a sieve indexed by I in a left exact category C . Form

’

a graph I whose objects are I + (I — I), with arrows of the form

l r

i ←’ (i, j) ’ ’ j

’ ’

We de¬ne a diagram D: I ’ C which takes i to Ai and (i, j) to the ¬ber

’

product Ai —A Aj . The cocone induced by S is the cocone from D to A whose

arrows are the arrows of the sieve S.

The reason for this restriction is not that this is the only conceivable kind of

cocone theory, but this the only kind of theory for which there is a generic topos

(classifying topos).

8.1 Regular Theories

A regular sketch R = (G , U, D, C, E) is a sketch (G , U, D, C) together with a

class E of arrows in G . A model of R is a model of the sketch which satis¬es the

additional condition that every arrow in E is taken to a regular epimorphism.

A preregular theory is a left exact theory Th together with a class E of

arrows. A model of a preregular theory is a left exact functor which takes every

arrow in E to a regular epimorphism. It is clear that if R is a regular sketch, then

the left exact theory generated by R in the sense of Section 4.4 is a preregular

theory, which we will denote PR(R), with the “same” class E of arrows.

The sieves are, of course, the single arrow sieves and the corresponding cocones

consist of diagrams

e

A —B A ’ A ’ ’ B

’ ’

’’

for e ∈ E. If a functor M is left exact, then it is immediate that M takes (—) into

a colimit if and only if M (e) is regular epi. Thus models are indeed characterized

277

278 8 Cocone Theories

by the properties of being left exact and taking the corresponding cocones to

colimits.

A regular theory Th is a regular category, i.e., a left exact category in which

regular epis are stable under pullbacks. A model of a regular theory is a regular

functor which we will always suppose to take values in a regular category. This

is the same as the model of the underlying regular sketch which has all regular

epis as its distinguished class of morphisms.

In this section, we show how to begin with a regular sketch R and construct

the regular theory induced by R. In regular categories, it will have the same

models as R.

An example of such a theory which arises in real life is the theory of regular

rings. (The coincidence of terminology is purely accidental). A regular ring is a

ring A in which for every element a there is an element b such that aba = a. This

condition can be rewritten as follows: De¬ne B = {(a, b) | aba = a} ⊆ A — A.

Then A is a regular ring if and only if the composite B ’ A — A ’ A (the last

’ ’

map is the ¬rst projection) is surjective.

The regular sketch to describe regular rings can be constructed as follows.

Begin with the sketch for rings in which A denotes the ring and add an object B

together with an arrow B ’ A2 and a cone that forces the image of B in any

’

model to be [(a, b) | aba = a]. Then E consists of the single arrow B ’ A that

’

corresponds to the ¬rst projection.

Notice that B and the map from B to A can be de¬ned in an arbitrary left

exact category (B is an equalizer). We say that a ring object in an arbitrary

regular category is a regular ring object if the map B ’ A is a regular epi. Of

’

course, other de¬nitions of regular ring object in a category are conceivable; e.g.,

one could ask that the map be a split epi. However, an attempt to formalize

this would almost surely lead to the introduction of a splitting map de¬ned by

equations as part of the structure. This would lead to a non-full subcategory

of the category of rings. In the case of commutative rings and some classes of

non-commutative rings, this equational de¬nition is actually equivalent to the

existential one.

Another example of a regular theory is the theory of groups in which every

element is an nth power for some ¬xed n > 1 (see Exercise 1).

Regular theories from regular sketches

If R is a regular sketch, then as remarked above, it generates a preregular

theory PR(R) with the same models by following the process of Section 4.4 and

taking for the class of arrows the image of the given class in the regular sketch.

8.1 Regular Theories 279

This preregular theory PR(R) generates a site by closing the class E under

pullbacks and composition to obtain a topology. The covers in this topology each

consist of only one arrow. This site is also a preregular theory, with E consisting

of all arrows which are covers in the resulting topology.

Proposition 1. A model in a regular category of PR(R) is the same as a model

for the site generated by R.

Proof. This follows from the fact that a model preserves pullbacks and compo-

sition, and a pullback of a regular epi in a regular category is a regular epi.

Now given a regular sketch R, we de¬ne Reg(R), the regular theory associated

to R, to be the full image of the composite

Y L

op

A ’’’ SetA ’ ’ Sh(Th)

’

where A is the site generated by R, Y is the Yoneda embedding and L is shea¬-

¬cation. Observe that Reg(A ) = Reg(R).

The induced map A ’ Reg(A ) will be denoted y. Note that y is left

’

exact. If A is already a regular category, then by Proposition 4 of Section 6.7,

A = Reg(A ).

Proposition 2. In the notation of the preceding paragraph, the covers in A

become regular epis in Reg(A ).

Proof. Let f : B ’ A be a cover. In a left exact category to say that yf is a

’

regular epi is to say that

y(B —A B) ∼ yB —yA yB ’ yB ’ yA

’ ’

= ’’

is a coequalizer. This is equivalent to saying that for every sheaf F ,

Hom(yA, F ) ’ Hom(yB, F ) ’ Hom(y(B —A B), F )

’

’ ’’

is an equalizer. Now yA = LY (A) where L is left adjoint to the inclusion of the

sheaf category, so

Hom(y(’), F ) ∼ Hom(Y (’), F ) ∼ F (’)

= =

Hence

F A ’ F B ’ F (B —A B)

’

’ ’’

must be an equalizer, which is exactly the condition that F be a sheaf.

280 8 Cocone Theories

Proposition 3. Let A and B be preregular theories and f : A ’ B a left

’

exact functor which takes arrows in the distinguished class of A to arrows in the

distinguished class of B. Then there is a unique regular functor Reg(f ): Reg(A )

’ Reg(B) for which

’

f EB

A

y y

c c

E Reg(B)

Reg(A )

Reg(f )

commutes.

Proof. The condition on f is equivalent to the assertion that it is a morphism of

the associated sites. The map f# constructed in Theorem 2 of Section 7.3 clearly

takes yA into yB, and so takes Reg(A ) into Reg(B).

Corollary 4. If B in the diagram above is a regular theory and A is the site

associated to some regular sketch R, then there is an equivalence of categories

between models of R in B and regular morphisms from Reg(R) = Reg(A ) to B.

In other words, every regular sketch R has a model in a universal regular

theory Reg(R) which induces an equivalence of model categories.

Exercise 8.1

1. (a) Let n > 1 be an integer. Say that a group G is n-divisible if for any

a ∈ G there is a b ∈ G for which bn = a. Show that the category of n-divisible

groups (and all group homomorphisms between them) is the category of models

for a regular theory.

(b) Show that the group of all 2n roots of unity is 2-divisible. This group is

denoted Z2∞ .

(c) Show that in the category of 2-divisible groups, the equalizer of the zero

map and the squaring maps on Z2∞ is Z/2Z, which is not divisible. Conclude that

there is no left exact theory which has this category as its category of algebras

and for which the underlying set functor is represented by one of the types.

8.2 Finite Sum Theories

If we wanted to construct a theory of ¬elds, we could clearly start with the sketch

for commutative rings, let us say with an object F representing the ring. Since

8.2 Finite Sum Theories 281

the inverse is de¬ned only for the subset of nonzero elements, we need to add an

object Y to be the nonzero elements and a map Y ’ Y together with equations

’

forcing this map to be the inverse map. All this is clear, except how to force Y

to be the nonzero elements. We will see in Section 8.4 that this cannot be done

in an LE or even a regular theory.

The approach we take is based on the observation that a ¬eld is the sum

(disjoint union) of Y and a set Z = {0}. Thus to the LE-sketch of commutative

rings we add objects Y and Z and arrows

Z’ F← Y

’ ’

Besides this, we need an arrow from Y to Y which takes an element to its inverse,

and a diagram forcing any element of Z to be zero (remember zero is already given

by an arrow from the terminal object to F in the LE-theory of commutative rings).

This can be done by techniques of Chapter 4 so we will not give details here.

A ¬eld is a model of this theory which takes (1) to a sum diagram. This

construction suggests that we vary the concept of regular theory de¬ned in Sec-

tion 8.1 to allow more general classes of covers. In this section, we develop the

idea of a ¬nite-sum theory.

A ¬nite-sum sketch or FS-sketch S = (G , U, D, C, E) is a sketch (G , U, D, C)

with a class E of ¬nite sieves. A model of S is a LE-model of the sketch

(G , U, D, C) which takes each sieve to a sum diagram. In this book, the models

will be in left exact categories with disjoint ¬nite universal sums.

A pre-FS-theory is an LE-theory together with a distinguished class of

sieves. Again, a model of the theory is an LE model which takes the sieves

to sums. An FS-sketch clearly induces a pre-FS-theory by taking the theory

to be the LE completion of the sketch and taking as distinguished sieves those

corresponding to the distinguished sieves of the original sketch.

An FS-theory is a left exact category with ¬nite disjoint universal sums. It

will be regarded as a pre-FS-theory by taking all ¬nite sums as distinguished

sieves. A model of one FS-theory in another is a left exact functor that preserves

¬nite sums.

An FS-sketch S induces an FS-theory FS(S ) in the following way. Begin

with the LE-theory and take as covers the images of all the sieves in the original

sketch. To this add to the covers, for any sieve {Ai ’ A} in S , the sieves {Ai

’

’ Ai —A Ai } (using the diagonal map) and for any i = j, the empty sieve with

’

vertex Ai —A Aj . Then add all sieves obtained from the above by pullbacks and

composition. Finally, take as FS(S ) the smallest subcategory closed under ¬nite

sums of the category of sheaves for that topology which contains the image of S .

282 8 Cocone Theories

Proposition 1. Let S be an FS-sketch. Any model of S in a left exact category

with ¬nite disjoint universal sums extends to a model of the associated FS-theory.

Proof. It is an easy exercise to show that in a left exact category with disjoint

(¬nite) sums, a cocone {Ai ’ A} is characterized as a sum by the following

’

assertions:

(i) Ai ’’ A

’

(ii) Ai ’’ Ai —A Ai

’

(iii) 0 ’’ Ai —A Aj for i = j

’

Theorem 2. Any FS-sketch has a model in a universal FS-theory which induces

an equivalence of model categories.

Proof. Essentially the same as Corollary 4 of Section 8.1 (the covers correspond

to the regular epis there).

Exercise 8.2

1. Prove that total orderings and strictly increasing maps are models of a ¬nite-

sum theory. (Hint: express the order as a strict order and consider trichotomy).

8.3 Geometric Theories

A local ring is charactized as a ring A in which for each a ∈ A either a or 1 ’ a

has an inverse. Both may be invertible, however, so that {a | a is invertible }

and {a | 1 ’ a is invertible } are not necessarily disjoint. We will describe in this

section a more general kind of theory in which such predicates may be stated.

A geometric sketch S = (G , U, C, D, E) is a sketch (G , U, C, D) togeth-

er with a class E of sieves. A model of a geometric sketch is a model of the