c c

HomE (C, PA) ' HomE (C, PB)

commutes, so that (iii) commutes by de¬nition of ∃a.

2.5 Notes to Chapter 2 95

Exercises 2.4

1. Show that for any object A in a topos, ∃idA = idPA .

2. Show that for any monic a: A0 ’ A in a topos, Pa —¦ ∃a = idA0 .

’

2.5 Notes to Chapter 2

The development of topos theory resulted from the con¬‚uence of two streams of

mathematical thought from the sixties. The ¬rst of these was the development

of an axiomatic treatment of sheaf theory by Grothendieck and his school of

algebraic geometry. This axiomatic development culminated in the discovery by

Giraud that a category is equivalent to a category of sheaves for a Grothendieck

topology if and only if it satis¬es the conditions for being what is now called a

Grothendieck topos (section 6.8).The main purpose of the axiomatic development

was to be able to de¬ne sheaf cohomology. This purpose was amply justi¬ed by

Deligne™s proof of the Weil conjectures [1974].

The second stream was Lawvere™s continuing search (which, it is probably

only a slight exaggeration to state, had characterized his career to that date) for

a natural way of founding mathematics (universal algebra, set theory, category

theory, etc.) on the basic notions of morphism and composition of morphisms. All

formal (and naive) presentations of set theory up to then had taken as primitives

the notions of elements and sets with membership as the primitive relation. What

Lawvere had in mind for set theory was to take sets and functions as the primitives

(and you don™t really need the sets if you are interested in reducing the number of

primitives to a minimum”see Exercise 1 of Chapter 1.1) and the partial operation

of composition as the basic relation.

In a formal way it is clear that this can always done by de¬ning the terminal

object 1 and then an element as a morphism with domain 1. Subobjects and

membership can be readily de¬ned and it is clear that set theory can be recovered.

However, Lawvere did not have in mind a slavish translation of Zermelo-Fraenkel

set theory into categorical language, but rather a treatment in which functions

were clearly the fundamental notion. See Lawvere [1965] for an example of this.

The closest he had come prior to 1969 was the notion of a hyperdoctrine which

is similar to that of a topos except that PA is a category rather than an object

of the ambient category.

The foundation of mathematics on the concept of function or arrow as prim-

itive is revolutionary, but no more revolutionary than the introduction of set

96 2 Toposes

theory was early in the century. The idea of constructing a quotient space with-

out having to have an ambient space including it, for example, was made possible

by the introduction of set theory, in particular by the advent of the rather dubi-

ous idea that a set can be an element of another set. There is probably nothing

in the introduction of topos theory as foundations more radical than that.

In the fall of 1969, Lawvere and Tierney arrived together at Dalhousie Uni-

versity and began a research project to study sheaf theory axiomatically. To be a

possible foundation for set theory, the axioms had to be elementary”which Gi-

raud™s axioms were not. The trick was to ¬nd enough elementary consequences

of these axioms to build a viable theory with.

The fact that a Grothendieck topos has arbitrary colimits and a set of gener-

ators allows free use of the special adjoint functor theorem to construct adjoints

to colimit-preserving functors. Lawvere and Tierney began by assuming explic-

itly that some of these adjoints existed and they and others pared this set of

hypotheses down to the current set.

They began by de¬ning a topos as a category with ¬nite limits and colimits

such that for each f : A ’ B the functor

’

f — : E /A ’ E /B

’

gotten by pulling back along f has a right adjoint and that for each object A

, the functor E op ’ Set which assigns to B the set of partial maps B to A is

’

representable. During the year at Dalhousie, these were reduced to the hypothesis

that E be cartesian closed (i.e. that

’ — A: E ’ E /A

’

have a right adjoint) and that partial functions with codomain 1 (that is, sub-

objects) be representable. Later Mikkelsen showed that ¬nite colimits could be

constructed and Kock that it was su¬cient to assume ¬nite limits and power

objects.

The resulting axioms, even when the axioms for a category are included,

form a much simpler system on which to found mathematics than the Zermelo-

Fraenkel axioms. Moreover, they have many potential advantages, for example

in the treatment of variability.

It has been shown that the topos axioms augmented by axioms of two-

valuedness and choice give a model of set theory of power similar to that of

Zermelo-Fraenkel, but weaker in that all the sets appearing in any axiom of the

replacement schema must be quanti¬ed over sets rather than over the class of all

sets. See Mitchell [1972] and Osius [1974, 1975].

3

Triples

From one point of view, a triple is an abstraction of certain properties of algebraic

structures. From another point of view, it is an abstraction of certain properties

of adjoint functors (Theorem 1 of Section 3.1). Triple theory has turned out to be

an important tool for studying toposes. In this chapter, we develop those parts

of the theory we need to use in developing topos theory. In Chapter 9, we present

additional topics in triple theory.

3.1 De¬nition and Examples

A triple T = (T, ·, µ) on a category C is an endofunctor T : C ’ C together with

’

two natural transformations ·: idC ’ T , µ: T T ’ T subject to the condition

’ ’

that the following diagrams commute.

T µE T · E 2 ' ·T

T3 T2 T T T

d

d

µ µ (1)

= =

µT d

d

dc

c c ‚ ©

ET

T2 T

µ

In these diagrams, T n means T iterated n times. As explained in Section 1.3,

the component of µT at an object X is the component of µ at T X, whereas the

component of T µ at X is T (µX); similar descriptions apply to ·.

The terms “monad”, “triad”, “standard construction” and “fundamental con-

struction” have also been used in place of “triple”.

Examples

The reader will note the analogy between the identities satis¬ed by a triple

and those satis¬ed by a monoid. In fact, the simplest example of a triple involves

monoids:

97

98 3 Triples

(i) Let M be a monoid and de¬ne T : Set ’ Set by T X = M — X. Let ·X: X

’

’ M — X take x to (1M , x) and µX: M — M — X ’ M — X take (m, n, x) to

’ ’

(mn, x). Then the associative and unitary identities follow from those of M .

(ii) In a similar way, if R is a commutative ring and A an associative unitary

R-algebra, there is a triple on the category of R-modules taking M to A — M .

The reader may supply · and µ.

(iii) A third example is obtained by considering an object C in a category C

which has ¬nite sums, and de¬ning T : C ’ C by T X = X + C. Take ·X: X

’

’ X + C to be the injection into the sum and µX: X + C + C ’ X + C to be

’ ’

idX + , where is the codiagonal”the map induced by idC .

(iv) If C is a category with arbitrary products and D is an object of C , we

can de¬ne a triple T = (T, ·, µ) on C by letting T C = DHom(C,D) . To de¬ne T

on arrows, as well as to de¬ne · and µ, we establish some notation which will

be very useful later. For u: C ’ D, let u : T C ’ D be the corresponding

’ ’

projection from the product. Then for f : C ’ C, we must de¬ne

’

T f : DHom(C ,D) ’ DHom(C,D)

’

The universal mapping property of the product is such that the map T f is

uniquely determined by giving its projection on every coordinate. So if v: C

’ D, de¬ne v —¦ T f = v —¦ f . The proof of functoriality is trivial. We de¬ne

’

·C: C ’ T C by u —¦ ·C = u and µ: T 2 C ’ T C by u —¦ µC = u . We

’ ’

could go into more detail here, but to gain an understanding of the concepts, you

should work out the meaning of the notation yourself. Once you have facility with

the notation, the identities are trivial to verify, but they were mind-bogglingly

hard in 1959 using elements. You might want to try to work these out using

elements to see the di¬culty, which comes in part because the index set is a set

of functions.

(v) More generally, if C is a category with arbitrary products and D is a set

of objects of C , let

T C = {D | D ∈ D, f : C ’ D} ’

This de¬nes T on objects. The remainder of the construction is similar to the

one above and is left to the reader.

(vi) An example of a di¬erent sort is obtained from the free group construc-

tion. Let T : Set ’ Set take X to the underlying set of the free group generated

’

by X. Thus T X is the set of equivalence classes of words made up of symbols x

and x’1 for all x ∈ X; the equivalence relation is that generated by requiring that

any word containing a segment of the form xx’1 or of the form x’1 x be equivalent

to the word obtained by deleting the segment. We will denote the equivalence

3.1 De¬nition and Examples 99

class of a word w by [w], and we will frequently say “word” instead of “equiva-

lence class of words”. The map ·X takes x to [x], whereas µX takes a word of

elements of T X, i.e., a word of words in X, to the word in X obtained by drop-

ping parentheses. For example, if x, y and z are in X, then [xy 2 z ’1 ] and [z 2 x2 ]

are in T X, so w = [[xy 2 z ’1 ][z 2 x2 ]] ∈ T T X, and µX(w) = [xy 2 zx2 ] ∈ T X. There

are many similar examples based on the construction of free algebraic structures

of other sorts. Indeed, every triple in Set can be obtained in essentially that way,

provided you allow in¬nitary operations in your algebraic structures.

Sheaves

In this section we describe the ¬rst triple ever explicitly considered. It was

produced by Godement, who described the construction as the standard con-

struction of an embedding of a sheaf into a “¬‚abby” sheaf (faisceau ¬‚asque).

In Example (2) of Section 2.2, we described how to construct a sheaf “ given

any continuous map p: Y ’ X of topological spaces. If each ¬ber of p (that is,

’

’1

each set p (x) for x ∈ X) is endowed with the structure of an Abelian group

in such a way that all the structure maps +: Y —X Y ’ Y , 0: X ’ Y (which

’ ’

’1

assigns the 0 element of p (x) to x), and ’: Y ’ Y are continuous, then the

’

sheaf of sections becomes in a natural way an Abelian group. In the same way,

endowing the ¬bers with other types of algebraic structure (rings or R-modules

are the examples which most often occur in mathematical practice) in such a way

that all the structure maps are continuous makes the sheaf of sections become

an algebraic structure of the same kind. In fact a sheaf of Abelian groups is an

Abelian group at three levels: ¬bers, sections, and as an Abelian group object in

the category Sh(X). (See Exercise 11).

If Y is retopologized by the coarsest topology for which p is continuous (so

all ¬bers are indiscrete), then every section of p is continuous. In general, given

any set map p: Y ’ X, using the coarsest topology this way produces a sheaf

’

Rp which in fact is the object part of a functor R: Set/|X |’ Sh(X), where

’

|X | is the discrete space with the same points as X. (See Exercise 9). The

resulting sheaf has the property that all its restriction maps are surjective. Such

a sheaf is called ¬‚abby, and Godement was interested in them for the purpose

of constructing resolutions of objects to compute homology groups with.

For each sheaf F , Godement constructed a sheaf T F (which turns out to be

¬‚abby) which de¬nes the object map of the functor part of a triple, as we will

describe. Let Y be the disjoint union of the stalks of F , and let p: Y ’ X take

’

the stalk Fx to x. Topologize Y by the coarsest topology for which p is continuous

and let T F be the sheaf of sections of Y . (Compare the construction of LF in

Section 2.2).

100 3 Triples

Evidently, T F U = {Fx | x ∈ U }. De¬ne ·F : F ’ F by requiring that

’

·F (s) be the equivalence class containing s in the stalk at x. Then ·F is monic

because of the uniqueness condition in the de¬nition of sheaf (Exercise 8). De¬n-

ing µ is complicated. We will postpone that until we have shown how adjoint

pairs of functors give rise to triples; then we will factor T as the composite of a

pair of adjoints and get µ without further work.

If F is an R-module, then so is T F (Exercise 10). In this case, iterating T

gives Godement™s standard resolution.

Adjunctions give triples

The di¬culty in verifying the associative identity for µ in examples like the

group triple, as well as the fact that every known triple seemed to be associated

with an adjoint pair, led P. Huber [1961] to suspect and prove:

Theorem 1. Let U : B ’ C have a left adjoint F : C ’ B with adjunction

’ ’

morphisms ·: id ’ U F and : F U ’ id. Then T = (U F, ·, U F ) is a triple on

’ ’

C.

Proof. The unitary identities

·U F E UF·

UFUF '

UF UF

d

d

(2)

=d =

UF

d

d

dc

‚ ©

UF

are just ·U —¦ U = id evaluated at F (see Exercise 15 of Section 1.9) and U applied