Table Of ContentTopos Theory
I. Moerdijk and J. van Oosten
Department of Mathematics
Utrecht University
2007
Contents
1 Presheaves 1
1.1 Recovering the category from its presheaves? . . . . . . . . . 8
1.2 The Logic of Presheaves . . . . . . . . . . . . . . . . . . . . . 9
1.2.1 First-order structures in categories of presheaves . . . 11
1.3 Two examples and applications . . . . . . . . . . . . . . . . . 14
1.3.1 Kripke semantics . . . . . . . . . . . . . . . . . . . . . 14
1.3.2 Failure of the Axiom of Choice . . . . . . . . . . . . . 16
2 Sheaves 19
2.1 Examples of Grothendieck topologies . . . . . . . . . . . . . . 27
2.2 Structure of the category of sheaves . . . . . . . . . . . . . . 28
2.3 Application: a model for the independence of the Axiom of
Choice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.4 Application: a model for \every function from reals to reals
is continuous" . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3 The E(cid:11)ective Topos 41
3.1 Some subcategories and functors . . . . . . . . . . . . . . . . 43
3.2 Structure of Eff . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.2.1 Finite products . . . . . . . . . . . . . . . . . . . . . . 44
3.2.2 Exponentials . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.3 Natural numbers object . . . . . . . . . . . . . . . . . 46
3.2.4 Finite Coproducts . . . . . . . . . . . . . . . . . . . . 47
3.2.5 Finite limits . . . . . . . . . . . . . . . . . . . . . . . . 48
3.2.6 Monics and the subobject classi(cid:12)er . . . . . . . . . . . 49
3.3 Intermezzo: interpretation of languages and theories in toposes 53
3.4 Elements of the logic of Eff . . . . . . . . . . . . . . . . . . . 60
4 Morphisms between toposes 64
5 Literature 69
i
1 Presheaves
We start by reviewing the category SetCop of contravariant functors from C
to Set. C is assumed to be a small category throughout. Objects of SetCop
are called presheaves on C.
We have the Yoneda embedding y : C ! SetCop; we write its e(cid:11)ect on
objects C and arrows f as y , y respectively. So for f : C ! D we have
C f
y : y ! y . Recall: y (C0) = C(C0;C), the set of arrows C0 ! C in
f C D C
C; for (cid:11) : C00 ! C0 we have y ((cid:11)) : y (C0) ! y (C00) which is de(cid:12)ned by
C C C
composition with (cid:11), so y ((cid:11))(g) = g(cid:11) for g : C0 ! C. For f : C ! D we
C
have y : y !y which is a natural transformation with components
f C D
(yf)C0 :yC(C0) !yD(C0)
given by (yf)C0(g) = fg. Note, that the naturality of yf is just the associa-
tivity of composition in C.
Presheaves of the form y are called representable.
C
The Yoneda Lemma says that there is a 1-1 correspondence between
elements of X(C) and arrows in SetCop from y to X, for presheaves X and
C
objectsC ofC,andthiscorrespondenceisnaturalinbothX andC. Toevery
element x 2 X(C) corresponds a natural transformation (cid:22) : y ! X such
C
that ((cid:22)) (id ) = x; and natural transformations from y are completely
C C C
determined by their e(cid:11)ect on id . An important consequence of the Yoneda
C
lemma is that the Yoneda embedding is actually an embedding, that is: full
and faithful, and injective on objects.
Examples of presheaf categories
1. A (cid:12)rst example is the category of presheaves on a monoid (a one-
object category) M. Such a presheaf is nothing but a set X together
with a right M-action, that is: we have a map X (cid:2)M ! X, written
x;f 7! xf, satisfying xe = x (for the unit e of the monoid), and
(xf)g = x(fg). There is only one representable presheaf.
2. Ifthecategory C isaposet(P;(cid:20)), forp2P wehavetherepresentable
y with y (q) = f(cid:3)g if q (cid:20) p, and ; otherwise. So we can identify the
p p
representable y with the downset #(p) =fqjq (cid:20) pg.
p
3. The category of directed graphs and graph morphisms is a presheaf
category: it is the category of presheaves on the category with two
objects e and v, and two non-identity arrows (cid:27);(cid:28) : v ! e. For a
presheaf X on this category, X(v) can be seen as the set of vertices,
1
X(e) the set of edges, and X((cid:27));X((cid:28)) : X(e) ! X(v) as the source
and target maps.
4. A tree is a partially ordered set T with a least element, such that for
any x 2 T, the set #(x) = fy 2 T jy (cid:20) xg is a (cid:12)nite linearly ordered
subset of T. A morphism of trees f : T ! S is an order-preserving
function wth the property that for any element x 2 T, the restriction
of f to #(x) is a bijection from #(x) to #(f(x)). A forest is a set of
trees; a map of forests X ! Y is a function (cid:30) : X ! Y together with
an X-indexed collection (f jx 2 X) of morphisms of trees such that
x
f : x ! (cid:30)(x). The category of forests and their maps is just the
x
category of presheaves on !, the (cid:12)rst in(cid:12)nite ordinal.
Recall the de(cid:12)nition of the category y#X (an example of a ‘comma cat-
egory’ construction): objects are pairs (C;(cid:22)) with C an object of C and
(cid:22) : y ! X an arrow in SetCop. A morphism (C;(cid:22)) ! (C0;(cid:23)) is an arrow
C
f :C !C0 in C such that the triangle
yC yf //yC0
BB(cid:22)BBBBBB!! }}{{{{{(cid:23){{{
X
commutes.
Note that if this is the case and (cid:22) : y ! X corresponds to (cid:24) 2 X(C)
C
and (cid:23) : yC0 !X corresponds to (cid:17) 2X(C0), then (cid:24) = X(f)((cid:17)).
There is a functor U : y#X ! C (the forgetful functor) which sends
X
(C;(cid:22)) to C and f to itself; by composition with y we get a diagram
y(cid:14)U : y#X !SetCop
X
Clearly, there is a natural transformation (cid:26) from y(cid:14)U to the constant
X
functor (cid:1) from y#X to SetCop with value X: let (cid:26) = (cid:22) : y ! X. So
X (C;(cid:22)) C
there is a cocone in SetCop for y(cid:14)U with vertex X.
X
Proposition 1.1 The cocone (cid:26):y(cid:14)U )(cid:1) is colimiting.
X X
Proof. Suppose (cid:21) : y(cid:14)U ) (cid:1) is another cocone. De(cid:12)ne (cid:23) : X ! Y by
X Y
(cid:23) ((cid:24)) = ((cid:21) ) (id ), where (cid:22) : y ! X corresponds to (cid:24) in the Yoneda
C (C;(cid:22)) C C C
Lemma.
2
Then (cid:23) is natural: if f : C0 ! C in C and (cid:22)0 : yC0 ! X corresponds to
X(f)((cid:24)), the diagram
yC0 yf //yC
CC(cid:22)C0CCCCC!! }}|||||(cid:22)|||
X
commutes, so f is an arrow (C0;(cid:22)0) ! (C;(cid:22)) in y#X. Since (cid:21) is a cocone,
we have that
yC0 yf //yC
(cid:21)(C0;C(cid:22)C0C)CCCCC!! ~~||||(cid:21)||(|C;(cid:22))
Y
commutes; so
(cid:23)C0(X(f)((cid:24))) = ((cid:21)(C0;(cid:22)0))C0(idC0) =
((cid:21)(C;(cid:22)))C0((yf)C0(idC0)) = ((cid:21)(C;(cid:22)))C0(f) =
Y(f)(((cid:21) ) (id )) = Y(f)((cid:23) ((cid:24)))
(C;(cid:22)) C C C
It is easy to see that (cid:21) : y(cid:14)U ) (cid:1) factors through (cid:26) via (cid:23), and that the
X Y
factorization is unique.
Proposition1.1isoftenreferredtobysayingthat\everypresheafisacolimit
of representables".
LetusnotethatthecategorySetCop iscompleteandcocomplete,andthat
limits and colimits are calculated ‘pointwise’: if I is a small category and
F :I !SetCop is a diagram, then for every object C of C we have a diagram
F : I ! Set by F (i) = F(i)(C); if X is a colimit for this diagram in
C C C
Set, there is a unique presheaf structure on the collection (X jC 2 C )
C 0
making it into the vertex of a colimit for F. The same holds for limits.
Some immediate consequences of this are:
i) an arrow (cid:22) : X !Y in SetCop is mono (resp. epi) if and only if every
component (cid:22) is an injective (resp. surjective) function of sets;
C
ii) the category SetCop is regular, and every epimorphism is a regular epi;
iii) the initial object of SetCop is the constant presheaf with value ;;
iv) X is terminal in SetCop if and only if every set X(C) is a singleton;
v) for every presheaf X, the functor ((cid:0))(cid:2)X : SetCop !SetCop preserves
colimits.
3
Furthermore we note the following fact: the Yoneda embedding C ! SetCop
is the ‘free colimit completion’ of C. That is: for any functor F : C ! D
where D is a cocomplete category, there is, up to isomorphism, exactly one
colimit preserving functor F~ :SetCop !D such that the diagram
C F //D
E <<
EEyEEEEE"" yyyyyF~yyy
SetCop
commutes. F~(X) is computed as the colimit in D of the diagram
y#X U!X C !F D
The functor F~ is also called the ‘left Kan extension of F along y’.
We shall now calculate explicitly some structure of SetCop. Exponentials
can be calculated using the Yoneda Lemma and proposition 1.1. For YX,
we need a natural 1-1 correspondence
SetCop(Z;YX)’ SetCop(Z (cid:2)X;Y)
In particular this should hold for representable presheaves y ; so, by the
C
Yoneda Lemma, we should have a 1-1 correspondence
YX(C) ’ SetCop(y (cid:2)X;Y)
C
which is natural in C. This leads us to de(cid:12)ne a presheaf YX by: YX(C)=
SetCop(y (cid:2)X;Y), and for f :C0 !C we let YX(f) :YX(C) !YX(C0) be
C
de(cid:12)ned by composition with yf (cid:2)idX : yC0 (cid:2)X !yC (cid:2)X. Then certainly,
YX is a well-de(cid:12)ned presheaf and for representable presheaves we have the
natural bijection SetCop(y ;YX) ’ SetCop(y (cid:2) X;Y) we want. In order
C C
to show that it holds for arbitrary presheaves Z we use proposition 1.1.
Given Z, we have the diagram y(cid:14)U : y#Z ! C ! SetCop of which Z is a
Z
colimit. Therefore arrows Z ! YX correspond to cocones on y(cid:14)U with
Z
vertex YX. Since we have our correspondence for representables y , such
C
cocones correspond to cocones on the diagram
y#Z U!Z C !y SetCop ((cid:0)!)(cid:2)X SetCop
with vertex Y. Because, as already noted, the functor ((cid:0))(cid:2)X preserves
colimits, these correspond to arrows Z (cid:2)X !Y, as desired.
4
It is easy to see that the construction of YX gives a functor ((cid:0))X :
SetCop ! SetCop which is right adjoint to ((cid:0))(cid:2) X, thus establishing that
SetCop iscartesian closed. Theevaluation map ev :YX(cid:2)X !Y isgiven
X;Y
by
((cid:30);x) 7! (cid:30) (id ;x)
C C
Exercise 1 Show that the map ev , thus de(cid:12)ned, is indeed a natural
X;Y
transformation.
Exercise 2 Prove that y : C ! SetCop preserves all limits which exist in C.
Prove also, that if C is cartesian closed, y preserves exponents.
Another piece of structure we shall need is that of a subobject classi(cid:12)er.
Suppose E is a category with (cid:12)nite limits. A subobject classi(cid:12)er is a
monomorphism t : T ! (cid:10) with the property that for any monomorphism
m : A ! B in E there is a unique arrow (cid:30) : B ! (cid:10) such that there is a
pullback diagram
A //T
m t
(cid:15)(cid:15) (cid:15)(cid:15)
B //(cid:10)
(cid:30)
We say that the unique arrow (cid:30) classi(cid:12)es m or rather, the subobject rep-
resented by m (if m and m0 represent the same subobject, they have the
same classifying arrow). In Set, any two element set fa;bg together with a
speci(cid:12)c choice of one of them, say b (considered as arrow 1 !fa;bg) acts as
a subobject classi(cid:12)er: for A(cid:26) B we have the unique characteristic function
(cid:30) :B !fa;bg de(cid:12)ned by (cid:30) (x) = b if x2 A, and (cid:30) (x) =a otherwise.
A A A
It is no coincidence that in Set, the domain of t : T ! (cid:10) is a terminal
object: T is always terminal. For, for any object A the arrow (cid:30) : A ! (cid:10)
which classi(cid:12)es the identity on A factors as tn for some n: A !T. On the
other hand, if k : A!T is any arrow, then we have pullback diagrams
A k //T idT //T
idA idT t
(cid:15)(cid:15) (cid:15)(cid:15) (cid:15)(cid:15)
A //T //(cid:10)
k t
so tk classi(cid:12)es id . By uniqueness of the classifying map, tn = tk; since t
A
t
is mono, n = k. So T is terminal. Henceforth we shall write 1 ! (cid:10) for the
subobject classi(cid:12)er, or, by abuse of language, just (cid:10).
5
t
Note: if 1 ! (cid:10) is a subobject classi(cid:12)er in E then we have a 1-1 corre-
(cid:30)
spondencebetween arrowsA !(cid:10)andsubobjects ofA. Thiscorrespondence
is natural in the following sense: given f : B ! A and a subobject U of A;
by f](U) we denote the subobject of B obtained by pulling back U along f.
Then if (cid:30) classi(cid:12)es U, (cid:30)f classi(cid:12)es f](U).
First a remark about subobjects in SetCop. A subobject of X can be
identi(cid:12)ed with a subpresheaf of X: that is, a presheaf Y such that Y(C)(cid:18)
X(C) for each C, and Y(f) is the restriction of X(f) to Y(cod(f)). This
follows easilyfromepi-monofactorizations pointwise, andthecorresponding
fact in Set.
Again, we use the Yoneda Lemma to compute the subobject classi(cid:12)er
in SetCop. We need a presheaf (cid:10) such that at least for each representable
presheaf y , (cid:10)(C) is in 1-1 correspondence with the set of subobjects (in
C
SetCop) of y . So we de(cid:12)ne (cid:10) such that (cid:10)(C) is the set of subpresheaves of
C
y ; for f :C0 !C we have (cid:10)(f) de(cid:12)nedby the action of pulling back along
C
y .
f
What do subpresheaves of y look like? If R is a subpresheaf of y then
C C
R can be seen as a set of arrows with codomain C such that if f : C0 ! C
is in R and g : C00 ! C0 is arbitrary, then fg is in R (for, fg = y (g)(f)).
C
Such a set of arrows is called a sieve on C.
Under the correspondence between subobjects of y and sieves on C,
C
the operation of pulling back a subobject along a map y (for f : C0 ! C)
f
sends a sieve R on C to the sieve f(cid:3)(R) on C0 de(cid:12)ned by
f(cid:3)(R) = fg :D !C0jfg 2 Rg
So(cid:10)canbede(cid:12)nedasfollows: (cid:10)(C)isthesetofsievesonC,and(cid:10)(f)(R)=
f(cid:3)(R). The map t: 1!(cid:10) sends, for each C, the unique element of 1(C) to
the maximal sieve on C (i.e., the unique sieve which contains id ).
C
Exercise 3 Suppose C is a preorder (P;(cid:20)). For p 2 P we let #(p) = fq 2
P jq (cid:20) pg. Show that sieves on p can be identi(cid:12)ed with downwards closed
subsets of #(p). If we denote the unique arrow q ! p by qp and U is a
downwards closed subset of #(p), what is (qp)(cid:3)(U)?
Let us now prove that t : 1 ! (cid:10), thus de(cid:12)ned, is a subobject classi(cid:12)er
in SetCop. Let Y be a subpresheafof X. Then for any C and any x2 X(C),
the set
(cid:30) (x) = ff :D !CjX(f)(x) 2 Y(D)g
C
6
is a sieve on C, and de(cid:12)ning (cid:30) : X ! (cid:10) in this way gives a natural trans-
formation: for f : C0 !C we have
(cid:30)C0(X(f)(x)) = fg :D !C0jX(g)(X(f)(x)) 2Y(D)g
= fg :D !C0jX(gf)(x) 2Y(D)g
= fg :D ! C0jfg 2 (cid:30) (x)g
C
= f(cid:3)((cid:30) (x))
C
= (cid:10)(f)((cid:30) (x))
C
Moreover, if we take the pullback of t along (cid:30), we get the subpresheaf of X
consisting of (at each object C) of those elements x for which id 2(cid:30) (x);
C C
that is, we get Y. So (cid:30) classi(cid:12)es the subpresheaf Y.
On the other hand, if (cid:30) : X ! (cid:10) is any natural transformation such
that pulling back t along (cid:30) gives Y, then for every x 2 X(C) we have that
x 2Y(C) if and only if id 2 (cid:30) (x). But then by naturality we get for any
C C
f :C0 ! C that
X(f)(x) 2 Y(C0) , idC0 2 f(cid:3)((cid:30)C(x)) , f 2 (cid:30)C(x)
which shows that the classifying map (cid:30) is unique.
Combiningthesubobjectclassi(cid:12)erwiththecartesianclosedstructure,we
obtain power objects. In a category E with (cid:12)nite products, we call an object
A a power object of the object X, if there is a natural 1-1 correspondence
E(Y;A) ’ Sub (Y (cid:2)X)
E
The naturality means that if f : Y !A and g :Z !Y are arrows in E and
f corresponds to the subobject U of Y (cid:2)X, then fg : Z ! A corresponds
to the subobject (g(cid:2)id )](U) of Z (cid:2)X.
X
Power objects are unique up to isomorphism; the power object of X,
if it exists, is usually denoted P(X). Note the following consequence of
the de(cid:12)nition: to the identity map on P(X) corresponds a subobject of
P(X)(cid:2)X which we call the \element relation" 2 ; it has the propertythat
X
whenever f : Y ! P(X) corresponds to the subobject U of Y (cid:2)X, then
U = (f (cid:2)id )](2 ).
X X
Convince yourself that power objects in the category Set are just the
familiar power sets.
In a cartesian closed category with subobject classi(cid:12)er (cid:10), power objects
exist: let P(X) = (cid:10)X. Clearly, the de(cid:12)ning 1-1 correspondence is there.
P(X)(C) = Sub(y (cid:2)X)
C
with action P(X)(f)(U) = (y (cid:2)id )](U).
f X
7
Exercise 4 Show that P(X)(C) = Sub(y (cid:2)X) andthat, forf :C0 !C,
C
P(X)(f)(U) = (y (cid:2)id )](U). Prove also, that the element relation, as a
f X
subpresheaf 2 of P(X)(cid:2)X, is given by
X
(2 )(C) = f(U;x) 2 Sub(y (cid:2)X)(cid:2)X(C)j(id ;x) 2 U(C)g
X C C
De(cid:12)nition 1.2 A topos is a category with (cid:12)nite limits, which is cartesian
closed and has a subobject classi(cid:12)er.
1.1 Recovering the category from its presheaves?
In this short section we shall see to what extent the category SetCop deter-
minesC. Inotherwords,supposeSetCop andSetDop areequivalentcategories;
what can we say about C and D?
De(cid:12)nition 1.3 In a regular category an object P is called (regular) projec-
tive if for every regular epi f : A ! B, any arrow P ! B factors through
f. Equivalently, every regular epi with codomain P has a section.
Exercise 5 Prove the equivalence claimed in de(cid:12)niton 1.3.
De(cid:12)nition 1.4 An object X is called indecomposable if whenever X is a
coproduct U , then for exactly one i the object U is not initial.
i i i
Note, t‘hat an initial object is not indecomposable, just as 1 is not a
prime number.
In SetCop, coproducts are stable, which means that they are preserved by
pullback functors; this is easy to check. Another triviality is that the initial
object is strict: the only maps into it are isomorphisms.
Proposition 1.5 In SetCop, a presheaf X is indecomposable and projective
if and only if it is a retract of a representable presheaf: there is a diagram
i r
X !y !X with ri= id .
C X
Proof. Check yourself that every retract of a projective object is again pro-
jective. Similarly, a retract of an indecomposable object is indecomposable:
i r
if X !Y ! X is such that ri= id and Y is indecomposable, any presen-
X
tation of X as a coproduct U can be pulled back along r to produce, by
i i
stability of coproducts, a pr‘esentation of Y as coproduct iVi such that
‘
Vi //Y
r
(cid:15)(cid:15) (cid:15)(cid:15)
Ui //X
8
