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’ ’
that E±—¦(f i)x, j = E± —¦(f i )x , j . But naturality implies that E±—¦f i = f j —¦D±
and E± —¦ f i = f j —¦ D± , so this equation becomes f j —¦ D±x = f j —¦ D± x . Since
f j is monic, this means that D±x = D± x so that x, i = x , i .
1.11 Notes to Chapter I 71
Theorem 3. In the category of models of a ¬nitary equational theory, every
object is a ¬ltered colimit of ¬nitely presented objects.

Proof. We will do this for the category of groups. We could do abelian groups,
except it is too easy because a ¬nitely generated abelian groups is ¬nitely pre-
sented. So let G be a group. For each ¬nite set of elements of i ∈ G, let F i be the
free group generated by i. For each ¬nite set of relations j that are satis¬ed by
the elements of i, let D(ij) be F i modulo those relations. Make the set of pairs
ij into a graph in which there is a single arrow ij ’ i j if i ⊆ i and j ⊆ j .

This is obviously a poset, so write ij ¤ i j when there is such a map. If there
is, then the inclusion induces an inclusion F i ’ F j and since j ⊆ j , there is an

induced map (not necessarily injective) D(ij) ’ D(i j ). Since the union of two

¬nite sets is ¬nite and there is at most one path between any two nodes of the
graph, D is a ¬ltered diagram in the category of groups. It is left to the reader
to verify that G is its colimit.


1.11 Notes to Chapter I
Development of category theory

Categories, functors and natural transformations were invented by S. Eilen-
berg and S. Mac Lane (announced in “The general theory of natural equivalences”
[1945] in order to describe the connecting homomorphism and the long exact se-
ˇ
quence in Cech homology and cohomology. The problem was this: homology was
de¬ned in the ¬rst instance in terms of a cover. If the cover is simple, that is if
every non-empty intersection of a ¬nite subset of the cover is a contractible space
(as actually happens with the open star cover of a triangulated space), then that
homology in terms of the cover is the homology of the space and that is the end
ˇ
of the matter. What is done in Cech theory, in the absence of a simple cover, is to
form the direct limit of the homology groups over the set of all covers directed by
re¬nement. This works ¬ne for de¬ning the groups but gives no information on
how to de¬ne maps induced by, say, the inclusion of a subspace, not to mention
the connecting homomorphism. What is missing is the information that homol-
ogy is natural with respect to re¬nements of covers as well as to maps of spaces.
Fortunately, the required condition was essentially obvious and led directly to the
notion of natural transformation. Only, in order to de¬ne natural transformation,
one ¬rst had to de¬ne functor and in order to do that, categories.
The other leading examples of natural transformations were the inclusion of
a vector space into its second dual and the commutator quotient of a group.
72 1 Categories
Somewhat surprisingly, in view of the fact that the original motivation came
from algebraic topology, is the fact that the Hurewicz homomorphism from the
fundamental group of a space to the ¬rst homology group of that space was not
recognized to be an example until later.
Later, Steenrod would state that no paper had in¬‚uenced his thinking more
than “The general theory of natural equivalences”. He explained that although
he had been searching for an axiomatic treatment of homology for years and that
he of course knew that homology acted on maps (or vice versa, if you prefer) it
had never occurred to him to try to base his axiomatics on this fact.
The next decisive step came when Mac Lane [1950] discovered that it was
possible to describe the cartesian product in a category by means of a universal
mapping property. In fact, he described the direct sum in what was eventually
recognized as an additive category by means of two mapping properties, one
describing it as a product and the other describing it as a sum. Mac Lane also
tried to axiomatize the notion of Abelian category but that was not completely
successful. No matter, the universal mapping property described by Mac Lane
had shown that it was possible to use categories as an aid to understanding.
Later on, Grothendieck succeeded in giving axioms for Abelian categories [1957]
and to actually prove something with them”the existence of injectives in an
Abelian category with su¬cient higher exactness properties. Thus Grothendieck
demonstrated that categories could be a tool for actually doing mathematics
and from then on the development was rapid. The next important step was the
discovery of adjoint functors by Kan [1958] and their use as an e¬ective tool in
the study of the homootopy theory of abstract simplicial sets.
After that the mainstream of developments in category theory split into
those primarily concerned with Abelian categories (Lubkin [1960], Freyd [1964],
Mitchell [1964], which are interesting but tangential to our main concerns here,
and those connected with the theories of triples and toposes of which we have
more to say later.

Elements

Although the thrust of category theory has been to abstract away from the use
of arguments involving elements, various authors have reintroduced one form or
another of generalized element in order to make categorical arguments parallel to
familiar elementwise arguments; for example, Mac Lane [1971, V111.4] for Abelian
categories and Kock [1981, part II] for Cartesian closed categories. It is not clear
whether this is only a temporary expedient to allow older mathematicians to argue
in familiar ways or will always form a permanent part of the subject. Perhaps
1.11 Notes to Chapter I 73
elements will disappear if Lawyers succeeds in his goal of grounding mathematics,
both in theory and in practice, on arrows and their composition.
An altogether deeper development has been that of Mitchell [1972] and others
of the internal language of a topos (developed thoroughly in a more general setting
by Makkai and Reyes [1977]). This allows one to develop arguments in a topos
as if the objects were sets, speci¬cally including some use of quanti¬ers, but with
restricted rules of deduction.

Limits

Limits were originally taken over directed index sets”partially ordered sets
in which every pair of elements has a lower bound. They were quickly generalized
to arbitrary index categories. We have changed this to graphs to re¬‚ect actual
mathematical practice: index categories are usually de¬ned ad hoc and the com-
position of arrows is rarely made explicit. It is in fact totally irrelevant and our
replacement of index categories by index graphs re¬‚ects this fact. There is no
gain”or loss”in generality thereby, only an alignment of theory with practice.
2
Toposes
A topos is, from one point of view, a category with certain properties charac-
teristic of the category of sets. A topos is not merely a generalized set theory,
but the very elementary constructions to be made in this chapter are best under-
stood, at least at ¬rst, by looking at what the constructions mean in Set. From
another point of view, a topos is an abstraction of the category of sheaves over a
topological space. This latter aspect is described in detail in this chapter.
Other treatments of toposes and sheaves are given by Johnstone [1977], Mac Lane
and Moerdijk [1992] and McLarty [1993].


2.1 Basic Ideas about Toposes
De¬nition of topos

We will take two properties of the category of sets”the existence of all ¬nite
limits and the fact that one can always form the set of subsets of a given set”as
the de¬ning properties for toposes.
For a ¬xed object A of a category E with ¬nite limits, ’ — A is a functor from
E to itself; if f : B ’ B , then f — A is the arrow (f —¦ p1 , p2 ): B — A ’ B —A. By
’ ’
composition, we then have a contravariant functor Sub(’ — A): E ’ Set. The ’
power object of A (if it exists) is an object PA which represents Sub(’ — A),
so that HomE (’, PA) Sub(’ — A) naturally. This says precisely that for
any arrow f : B ’ B, the following diagram commutes, where φ is the natural

isomorphism.
U φ(A, B)
E
HomE (B, PA) Sub(B — A)

HomE (f, PA) Sub(f — A) (1)
c c
E Sub(B — A)
HomE (B , PA)
φ(A, B )
The de¬nition of PA says that the “elements” of PA de¬ned on B are es-
sentially the same as the subobjects of B — A. In Set, a map f from B to the
74
2.1 Basic Ideas about Toposes 75
powerset of A is the same as a relation from B to A (b is related to a if and
only if a ∈ f (b)), hence the same as a subset of B — A. When B is the terminal
object (any singleton in Set), the “elements” of PA de¬ned on B are the subsets
of A 1 — A; thus PA is in fact the powerset of A.
In general, if the category has a terminal object 1 and P(1) exists, then Sub
is represented by P(1), since 1 — A ∼ A. This object P(1) is studied in detail in
=
Section 2.3.
De¬nition. A category E is a topos if E has ¬nite limits and every object
of E has a power object.
We will assume that PA is given functionally on ObE (it is determined up
to isomorphism in any case). This means that for each object A of E , a de¬nite
object PA of E is given which has the required universal mapping property.
The de¬nition of toposes has surprisingly powerful consequences. (For ex-
ample, toposes have all ¬nite colimits.) Probably the best analogy elsewhere in
mathematics in which a couple of mild-sounding hypotheses pick out a very nar-
row and interesting class of examples is the way in which the Cauchy-Riemann
equations select the analytic functions from all smooth functions of a complex
variable.
The properties of toposes will be developed extensively in this Chapter and
in Chapters 5 and 6. However, the rest of this section and the next are devoted
to examples.

Examples of toposes

(i) The category Set is evidently a topos. As we have already pointed out, if
X is a set, we can take PX to be the set of all subsets of X, but that does not
determine a unique topos structure on Set since we have a choice of φ in diagram
(1). The natural choice is to let φ: Hom(1, PB) ’ Sub(B) be the identity map

(thinking of an arrow from 1 to PB as an element of PB), but we could be
perverse and let φ of an element of the powerset be its complement.
(ii) To see a more interesting example, let G be a group and let G-Set be
the category of all sets on which G acts. The morphisms are equivariant G
homomorphisms. The existence of ¬nite (in fact, all) limits is an easy exercise.
They are calculated “pointwise”. If X is a G-set, let PX denote the set of all
subsets of X with G action given by gX0 = {gx | x ∈ X0 }. Note that a global
element of PX is a G-invariant subset of X.

Actually, the category of actions by a given monoid, with equivariant maps,
is a topos. That will follow from the discussion of functor categories below, since
76 2 Toposes
such a category is the same as a Set-valued functor category from a monoid
regarded as a category with one object.

Functor categories

An important example of toposes are Set-valued functor categories. In order to
prove that these categories are toposes, a number of elementary facts about them
are needed. A guiding principle in this development is the fact that Func(C , D)
inherits most of its properties from D (Exercise 25 of Section 1.7).
In this section, C is a ¬xed small category and E = Func(C op , Set). We will
outline the proof that E is a topos. Of course, everything we say in this section
is true of Func(C , Set), but because of the applications we prefer to state it this
way.
Each object C of C determines an evaluation map »C: E ’ Set, where

»C(F ) = F C and for γ: F ’ G, »C(γ) = γC.

Proposition 1. For each object C of C , the evaluation preserves all limits and
colimits. I.e.,“limits and colimits in E are computed pointwise”. In particular,
E is complete and cocomplete.
In other words, if D: I ’ E is a diagram in E , then (lim D)(C) = lim(D(C)).

The proof is in Exercise 25 of Section 1.7.
For a ¬xed object E, the functor ’ — E: E ’ E commutes with
Corollary 2. ’
all colimits.
Proof. The property claimed for this functor is valid when E = Set by Exercise 6,
and the Proposition allows one to extend it to an arbitrary functor category.
The fact that Sub is representable in a topos (by P(1)) means that it takes
colimits to limits. (Exercise 14 of Section 1.7). In particular, Sub( Ai ) =
Sub(Ai ) and if
p1
’ ’ B ’c C
’’
A ’’ ’’ (2)
’’
p2
is a coequalizer in a topos, then
Sub p1 Sub c
← ’ ’ ’ Sub(B) ← ’ ’ Sub(C)
’’’
Sub(A) ← ’ ’ ’ ’’’ (3)
’’’
Sub p2
is an equalizer. The ¬rst is easy to see in Set, but a direct proof of the second fact
in Set, not using the fact that Sub is representable, is surprisingly unintuitive.
As a step toward proving that E is a topos, we prove the fact just mentioned
for E .
2.1 Basic Ideas about Toposes 77
If D: I ’ E is a diagram in E , then
Proposition 3. ’
Sub(colim D) lim(Sub(D))
Proof. We use repeatedly the fact that the result is true in Set because there
Sub is representable by the two-element set. Let F = colim F D. For an object
i of I , the cocone F Di ’ F gives a cone Sub(F ) ’ Sub(F Di) which in turn
’ ’
gives Sub(F ) ’ lim Sub(F Di). Now to construct an arrow going the other way,

let EDi ⊆ F Di be a compatible family of subobjects, meaning that whenever i
’j

E F Dj
F Di
T T


E Edj
Edi
is a pullback. Let E = colim EDi. Since colimits preserve monos (by Propo-
sition 1, since they do so in Set), E is a subfunctor of F . This gives a map
lim Sub(F Di) ’ Sub(F ). Finally, to see that both composites are the identity,

it su¬ces to see that all those constructions are identical to the ones carried out
in SetOb(C ) , for which the result follows from Proposition 1 since it is true in Set.

E is a topos.
Theorem 4.
Proof. Finite limits, indeed all limits, exist by Proposition 1. As for P, for a
functor E, let PE be de¬ned by letting PE(A) be the set of subfunctors of
Hom(’, A) — E. It is straightforward to verify that PE is a functor.
We must show that subfunctors of F — E are in natural one to one corre-

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