. 51
( 60 .)


are not a signi¬cant limitation on results used for diagram-chasing. Any diagram
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
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.
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
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
278 8 Cocone Theories
by the properties of being left exact and taking the corresponding cocones to
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

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 (’)
= =

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

y y
c c
E Reg(B)
Reg(A )
Reg(f )
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

(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


. 51
( 60 .)