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VOL. 7 ( 1 9 7 2 ) , 1-76.
Aspects of topoi
Peter Freyd
After a review of the work of Lawvere and Tierney, i t i s shown
that every topos may be exactly embedded in a product of topoi
each with
1
as a generator, and near-exactly embedded in a
power of the category of sets .
derived.
Several metatheorems are then
Natural numbers objects are shown to be characterized
by exactness properties, which yield the fact that some topoi can
not be exactly embedded in powers of the category of s e t s ,
indeed that the "arithmetic" arising from a topos dominates the
exactness theory.
Finally, several, necessarily non-elementary,
conditions are shown to imply exact embedding in powers of the
category of sets.
The development of elementary topoi by Lawvere and Tierney strikes
this writer as the most important event in the history of categorical
algebra since i t s creation.
The theory of abelian categories served as the
"right" generalization for the category of abelian groups.
for - no less - the category of sets.
For each the motivating examples
were categories of sheaves, abelian-valued sheaves for the
set-valued sheaves for the second.
So topoi serve
first,
But topoi are far richer than abelian
categories (surely foreshadowed by the fact that abelian-valued sheaves are
just the abelian-group objects in the category of set-valued sheaves).
Whereas abelian categories, nice as they are, appear in various contexts
only with the best of luck, topoi appear at the very foundation of
mathematics.
The theory of topoi provides a method to "algebraicize" much
Received 29 December 1971- Prepared from a series of lectures
delivered at the Conference on Categorical Algebra, University of New South
Wales, 13-17 December 1971. The author was in Australia as a Fulbright
Fellow.
I
2
Peter Freyd
of mathematics.
In this work, we explore the exact embedding theory of topoi.
the subject is richer than that for abelian categories.
Again
It is not the case
that every small topos can be exactly embedded in a power of the category
of sets or even a product of ultra-powers (Theorem 5-23).
It is the case
that every topos can be exactly embedded in a product of well-pointed topoi
(Theorem 3.23) ("well-pointed" means
"l
is a generator"), and
near-exactly embedded in a power of the category of sets (Theorem 3.2U)
("near-exact" means "preserves finite limits, epimorphisms, and
coproducts" ).
We thus gain several metatheorems concerning the exactness
theory of topoi (Metatheorem 3-3l).
The obstructions to the existence of exact functors lie in the
"arithmetic" of topoi (Proposition 5-33, Theorem 5.52).
No set of
elementary conditions can imply exact embeddability into a power of the
category of sets (Corollary 5.15), hut rather simple, albeit
non-elementary, conditions do allow such (§5.6).
The easiest to state:
a
countably complete topos may be exactly embedded in a power of the category
of sets.
The most impressive use of the metatheorems is that certain exactness
conditions imply that something is a natural numbers object (Theorem 5.i+3)A consequence is that a topos has a natural numbers object iff it has an
object
A
such that
1 + A - A
(Theorem 5.UU).
We begin with a review of the work of Lawvere and Tierney (through
Corollary 2.63).
All the definitions and theorems are theirs, though some
of the proofs are new.
Sections 4.1 and 5.1 and Propositions 5-21, 5-22
are surely also theirs.
It is easy to underestimate their work:
it is not
Just that they proved these things, it's that they dared believe them
provable.
1. Cartesian closed categories
A cartesian closed category is a finitely bicomplete category such
that for every pair of objects
representable.
A, B
the set-valued functor
(-M,
B)
is
This is the non-elementary (in the technical sense of
"elementary") definition.
We shall throughout this work tend to give first
Aspects of topoi
3
a definition in terms of the representability of some functor and then show
how such can be replaced with an elementary definition - usually, indeed,
with an essentially algebraic definition.
The translation from representability to elementary is the
Mac Lane-notion of "universal element". For cartesian-closedness (forgive,
oh Muse, but "closure" is just not right) we obtain the following:
A
A
For every A, B there is an object Er and a map a x A -»• B
(the "evaluation map") such that for any X x A -*• B there
exists a unique f : X -*• a such that
• B
.
A
Holding A fixed,
f
: a -»• U
U
becomes a functor on B : given
/ : B •*• C ,
i s the unique map such that
B 4 x A ^ X 1 > (^ x A
»• C = S 4 x A
>• B -$-+ C .
A
Holding
B fixed,
B^ becomes a contravariant functor on A : given
g : A' -*• A , £r : a •* a
/
is the unique map such that
XA,
One can easily verify that
that is, e
is a bifunctor.
Recall that a group may be defined either as a semigroup satisfying a
couple of elementary conditions or as a model of a purely algebraic theory
(usually with three operators: multiplication, unit, and inversion). It
is important that groups may be defined either way: .there are times when
groups are best viewed as special kinds of semigroups, and there are times
when they thrive as models of an algebraic theory.
So it is with lattices,
4
Peter Freyd
and every elaboration of the notion of lattice;
and so it is with
cartesian-closed categories.
To begin with, a category may be viewed as a model of a two-sorted
partial algebraic theory.
The two sorts are "maps" and "objects".
We are
given two unary operators from maps to objects, "domain" and "codomain",
and one coming back, "identity map";
and we are given a partial binary
operator from maps to maps, the domain of which is given by an equation in
the previous operators.
Partial algebraic theories for which such is the
case, namely those such that the partial operators may be ordered and the
domain of each is given by equations on the previous, are better than just
partial algebraic theories.
We shall call them essentially algebraic.
A
critical feature of essentially algebraic theories is that their models are
closed, in the nicest way, under direct limits.
Finite bicompleteness becomes algebraic.
called the "terminator") is a constant,
operation,
The terminal object (better
1 , together with a uninary
t , from objects to maps, such that
domain ( t ) = A ,
codomain(t.) = 1 ,
h-h •
B -±+ 1 = tA .
The equations that stipulate domain and codomain are conventionally
absorbed in the notation, thus:
t{A) = A -*• 1 .
For binary products we have four binary operations, one from objects
to objects denoted
/I
x A^ , two from objects to maps denoted
one from maps to maps denoted
and one unary operation from objects to maps:
A
•A x A .
The
Aspects of topoi
equations:
A x A —^
]
A = 1A ,
i = 1, 2 ,
A
x B 2 = ^ - 2 * yl xyl
±1
2S
_B
x B^
.
For equalizers we have partial binary operators from maps, one to objects,
one to maps.
The domain of each operator is given by the equations
domain(/) = domain(g) , codomain(/) = codomain(g) .
We may denote the pairs
f,g
in the domain by
object-valued operator is denoted
denoted
A ? '9+ B . The
E{f, g) , the map-valued operator is
E(f, g) -* A .
We have a third partial operator from triples
maps.
(h, f, g) of maps to
The equations that define the domain are:
codomain(?i) = domain(/) = domain(g) ,
codomain(/) = codomain(g') ,
hf = hg .
Given
X
• A I'$ > B the value of this operator is denoted
X -^+ E{f, g) . The equations:
£•(/, g) + A -£+ B = £ ( / , g) •* A - 2 - B ,
X-^
X
JU
E{f, g)
.
E(f, g) * A = h ,
X
( X - W . g ) * ) . E(f, g) .
We can make cartesian-closedness essentially algebraic by taking two
binary operators:
one 'from objects to objects denoted
a
, one from
A
objects to maps denoted
Zr x A -*• B ; and a quaternary partial operator
from quadruples (X, A, B, f) such that domain(/) = X x A ,
codomain(/) = S , valued as a map X -*->• a . The equations:
Peter Freyd
B = / ,
S 4 = (X, A, B, X
Such is the nearest approximation to the previous elementary
definitions.
It is usually more convenient to ask that
a
is a bifunctor
covariant in the lower variable, contravariant in the upper, equipped with
natural transformations
e
A , B •• ** *
A
+
B
e
>
A,B
:
such that
x
A
The existence of elementary definitions should not, in itself, oblige
us to give elementary proofs.
The great technical tour de force in Godel's
incompleteness proof, namely that primitive recursive functions (a
second-order notion if there ever was one) are all first-order definable
(indeed Gode I-recursive) does not oblige us, but allows us to stop worrying
about primitive recursive functions.
Certainly it is worth knowing when
things are elementary - we shall use the elementary nature of topoi (for
example, Corollary 5.15, Theorem 5-23) and their essentially algebraic
nature (Theorem 3.21).
Pirst-order logic is surely an artifice, albeit
one of the most important inventions in human thought.
thinks in a first-order language.
But none of us
The predicates of natural dialectics
are order-insensitive (one moment's individuals are another's equivalenceclass) and our appreciation of mathematics depends on our ability to
interpret the words of mathematics.
The interpretation itself is not
first-order.
The reduction of a subject to an elementary one - in other than the
formal method of set theory - usually marks a great event in mathematics.
The elementary axioms of topoi are a testament to the ingenuity and insight
of human genius.
I will refuse to belittle this triumph of mind over
matter by taking it as evidence that mind is matter.
Aspects of Topoi
PROPOSITION 1.11 for cartesian-closed categories. The following
natural maps are isomorphisms:
0 =0 xA ,
(A*C) + (B*C) - (A+B) x C ,
1
=1 ,
(A*B)C - AC x BC ,
AB+C ~-AB*AC
A -A
ABXC
Proof.
,
,
« (^ S ) C .
Each is a result of the existence of an adjoint.
have a right-adjoint
coproducts.
Dually
For
-*A
to
i A\
[- j , it must be cocontinuous, hence it preserves
(- ) , having a left-adjoint, preserves limits.
A~
is adjoint to itself on the right, hence carries colimits to limits.
Elementary proofs would go like this:
unique
0 x A •* X
coterminator.
because there exists unique
X
0 •+ X
there exists
, hence
0 x A
is a
D
Caution:
0
PROPOSITION
then A = 0 .
Proof.
for each
need not be
0 .
(indeed
0
= 1 .)
1.12 for cartesian-closed categories. If A •* o exists
The existence of
A -*• 0
yields a map
A •*• A x 0 , and we
obtain
A+A*O->-A
I?
0
As always,
0 -»• i4 ->• 0 = 1_ .
= l. .
A
O
A degenerate category is one with just one object and map.
finite bicompleteness implies non-empty.)
(Note that
8
Peter Freyd
PROPOSITION 1.13.
there exists 1 -*• 0 .
PROPOSITION
A aartesian-closed category is degenerate iff
D
1.14.
For any small
A the category of
set-valued
Aop
contravariant
Proof.
denoted
functors
S
H
Aop
J.
S
is cartesian-closed.
has a generating set, the representable functors (herein
For any
T ,
-xT : S
Aop
Aop
-*• S
is easily seen to preserve
colimits because such are constructabie "object-wise" and
preserves colimits.
Hence for any
X
T, S ,
{- T, S)
-*TA : S •* S
carries colimits into
limits and the special adjoint functor theorem says that it is
representable.
We can, of course, construct fir more directly.
s{A)
of
- (H.XT,
A
- \H.,
S\
A
K
)
S (A) .
object-wise:
n : H
S)
and we could use the latter as the definition
The evaluation map
given
x T •* S
s
x
T -*• 5
and an element
x £ T{A) , define
S ->• (S * T)
, let
e*{x) € (5 x T)T(.A) as the transformation
is such that
is easily constructed
< !"|, x > i (ST x T) (A ) , that is, a transformation
For the co-evaluation map
n
We know that
r\. (l.) = x .
e{r\, x) = ^ ( l . , ^) •
x € S{A)
r\*l:
and define
HA*T-»-SxT
where
All the equations are directly verifiable. D
H
Note that if
5 U)
A
is a monoid
representations of
, where
If
by
has finite products, then
is the set of transformations from
If
S
A
M
M
M , S
and
£T
A° P
T
5
to
A
= S(Ax-) , and hence
S(A*-) .
may be viewed as the discrete
is the set of homomorphisms from
M x T
hi is used to denote the "regular" representation.
is a group, then a homomorphism
f{l, x) , x € T , and given any function
/"(a, x) = ag(a~ x ) .
Thus
5
f : M x y •+ 5
g : T •* S
is determined
we may define
is the set of all functions from
T
to
to
Aspects of topoi
S
S
.
ag
•* S
is the function
-
(ag)(x) = a\g(oT x)\ •
The forgetful functor
preserves exponentiation.
1.2 Heyting algebras.
A Heyting algebra is a cartesian-closed category in which for every
A, B ,
(A, B) \j {B, A)
has at most one element.
The latter condition
says, of course, that we are dealing with a partially ordered set.
finite bicompleteness says that it is a lattice with
0
and
1 .
The
The
cartesian-closedness says that there is an operation on the objects such
that
(-4, BC) + 0
We switch notation:
and
y
is denoted
operation" as
lattice with
(AxC, B) * 0 .
the objects are lower-case
existence of a map from
x
iff
x
y
x
x v y , the "Heyting
x = x A y :
O v x = x ,
x = x ,
xAj/
x 5 y , the product of
We recall that the following equations give us a
defined as
l A x = x ,
A
x, y, z, ... , the
is stated with
x A y , the coproduct as
x -*• y .
x - y
to
x v x = x ,
= j/Ax,
x v y = y vx ,
x A (j/As) = (a;Ay) A z , x v (yvz) = (x</y) v z ,
x
Hon-equationally,
z Ax 5 # .
with
x
x •+ y
A {yvx) = x = (xAy) v x .
is characterized by
That is, x -* y
is dominated by
PROPOSITION
z 2 (x •* y)
y .
1.21 for Heyting algebras.
0 = 0 A X,
(xAt/) v (xAz) = x A {yvz) ,
x -" 1 = 1 ,
x
iff
is the largest element whose intersection
-»• (t/Az) =
( x •*• y)
A
(x •*• z)
,
0 -»• x = 1 ,
(xvy) •*• z =
(x •* z) A
{y •* z) ,
x = 1 ->• x ,
•* z
=
x
•*
(y
•*• z)
.
10
Peter Freyd
Proof.
Translate Proposition 1.11.
PROPOSITION
1.22
•
for Heyting algebras.
We can characterize x •*• y
with the following equations:
x
•* x
= 1
,
y A (x -»• y) = y ,
x •* (y A z) = (x •* y)
Proof.
z A x = s A
Given
z £ (x •* y) , that is,
(x ->• y) Ax
= zAxAy<y
we note first that the fact that
.
is,
x •*• (z Ax)
z £ x ->• z .
z = z A ( X •*• y) , then
Conversely, given
f(u) = x •* u
implies that it preserves order, and
hand
A (x •+ z) .
x-+(aAx)±x-*-y.
= (x -*• x) A (x •* s) = x -»• z
Hence
that
s £ ~l x
iff
x .
z A X = 0 , that is , ~1 x
A complement of
x v y = 1 .
"1 x :
x
is an element
In a Heyting algebra, if
because
PROPOSITION
xAy
1.23
On the other
and
z A (x •+ z) = z , that
3£x-*-3 = x-*-(xAs)£x->-j/.
We define the negation of an element, denoted
disjoint from
z A X £ y ,
preserves intersections
= 0=>ys~\x
y
x
n
~\ x , as
x -»• 0 .
Note
is the largest element
such that
x A y = 0
and
has a complement it must be
and
for Heyting algebras, x £ ~l ~l x ;
if x s y then
Proof. The f i r s t two statements are immediate. (~l x) £ ~1 ~l (~l x) by
the f i r s t statement, ~~\ ("1 ~\ x) £ ~~\ x by the second applied to the f i r s t . D
A boolean algebra i s a Heyting algebra which s a t i s f i e s the further
equation x = ~l ~1 x .
In a boolean algebra, negation i s thus an order-reversing involution
and De Morgan's laws are easy consequences: ~\(xAy)=~\xv~\y
. (Note
t h a t in any Heyting algebra
Hence
xv~lx=~l~l (xv~lx)=~l(~lxAx)=~IO = l
and every element
Aspects of topoi
has a complement. Conversely if for all
l l l = l l l A
I I
x , x v "I x = 1
then
( l V 1 l ) = O ~1 X A x ) V (~l ~1 x A ~1 x ) = ~ l ~ l x A x
It is easy to verify that in a boolean algebra
= X .
I + J ="li v J .
A complete lattice has a Heyting algebra structure, by the adjoint
functor theorem, iff
if for any set
f(n) = x A n
{y .} .
preserves all conjunctions, that is,
, x A V y . = V [x A y .) .
a complete distributive lattice.
Such is usually called
The lattice of open sets in a topological
space is therefore a Heyting algebra.
Consider the unit interval, and let:
V ~l X = I- >' ' ,
x
~1 ~1 x = x ,
1 (x v 1 x) = 0 ,
•"i ~i (x v n x) = i ,
(x v "1 x) t 1 ~1 (x v 1 x ) .
In any space, negation yields the interior of the set-theoretic complement.
Double negation yields the interior of the closure.
Another ready example of a complete Heyting algebra is the lattice of
left-ideals in any monoid.
ordered set with
0
and
For a non-complete example, take any linearly
1
but otherwise not complete and define
(l
(x *y)
if
x S y ,
=
[y if y < x .
In any Heyting algebra, define
z 2 (x «-»• y)
which meets
iff
x
and
x •*-*• y = (x •*• y) A (y •+ x ) .
z A x = 2 A y , that is, x •<->• y
j/ in the same way.
Then
is the largest element
We can reverse things to obtain
the symmetric definition of Heyting algebras, namely the operations and
equations of a lattice together with a binary operator satisfying:
l-t->-x = x = x-i->-l
( l i s a unit),
12
Peter Freyd
x
•*-*• x = 1
(a; is a n inverse of x ) ,
(a; «-»• y) A z = (x A z -<->• y A z) A z
Then
(almost distributive).
(x •*->• y) A x = (x *-*• x A y) A x = (l «-+• y ) A x = y Ax
(x-<->-y)Ay=xAy.
Hence if 3 < (a: -*-»• i/)
.
Similarly
then
z A x = s A (a; •*-•• i/) Ax =
zAyAx
and
z
that
i s
A y = s A (x •*-*• y)
z A x = z A y
Conversely, i f
A S =
( x A 3 - < - » - y A 3 )
A
S= 1 A 3
= 3 ,
z £ (a; •<-*• j/) .
x •*• y
as
a; •*-*• x A y
and v e r i f y
z S (x -»• jy)
3 Ax S y .
+-»-
i s a symmetric b i n a r y operation with a u n i t and i n v e r s e s .
not a s s o c i a t i v e i n g e n e r a l , for
A s s o c i a t i v i t y implie s t h a t
"1 x = x •*-*• 0
++
i f we can find
and
It is
~1 ~~\ x = (x •*-+ 0) *-*• 0 .
"l~la: = a;-t-+ (0-^-»-0) = x •*-*• 1 = x .
Conversely, i n boolean a l g e b r a s ,
comment:
•*-»• i s a s s o c i a t i v e .
An orthogonal
i s a loop o p e r a t i o n only in a boolean a l g e b r a , for given
y
such t h a t
Given a congruence
set
,
,
We may t h e n d e f i n e
iff
= z A x A y
z A x = z A y , then
(a: « - » - ! / )
that i s ,
Ay
F = {x\x = 1}
=
y •*->• 0 = x
then
x
a; = ~ l j / = ~ i n ~ l j / = " l ~ l x
on a Heyting a l g e b r a i t i s easy t o see t h a t t h e
has t h e p r o p e r t i e s :
1 €-F ,
i f
f ' l v i / f
F ,
We call such a set a filter. We can recover the congruence from
iff
(x -»-»• y) € F
because if x = y
(x *-+ y) £ F ; if (x •*->• y) = 1
Moreover:
then
then
F :x = y
{x •>-*• x) = (x •*-*• y) and
•
Aspects of topoi
PROPOSITION
relation
=
Proof.
suppose
1.24
for Heyting algebras.
defined by
=
13
x = y
iff
If
F
(x •*-*• y) 6 f
is clearly reflexive and symmetric.
(x *-*- y), (y <->• 3) € F .
For transitivity
(This is the only use of the fact that
F
And for that inequality we need only show
{x *-* y) A (y *-+ z) A x = (x •*->• y) A {y *-* z) A s
For
is a congruence.
It suffices to show that
(x «-• y) A (y *->• z) 5 (x •<-*• 3) .
is closed under intersection.)
is a filter then the
xSy=>xAz=yAz
, an easy matter.
i t s u f f i c e s t o show
(x •«-*• y) 5 (x A 2) +-* (y A 3) , t h a t i s ,
(x «-»• y ) A (x A z)
an easy matter.
= (x •<-»• !/) A (1/ A a )
For x = y = * x V 2 = y v z
,
w e m u s t sho w
(x •*-+ y) A (x V z) = (x *-*• y) A (y v z) , which because of distributivity
is again an easy matter.
Finally for
x = y =* (x -<->• 3) = (y -*-»• 3)
it suffices to show
x *->• y < (y *-+ z) *-* (x *-*• z) , that is,
(a; *->• 1/) A
(a: «-»• 2 )
Using the third defining equation of
=
(a; <-^ j/) A
(1/ •<-»• 2 )
.
«-»• , the left-hand side is
(x -<-»• y) A [ (x +•+ y) A x +->• (x *->• y) A 2 ]
and the right-hand side is
(x •«->• !/) A [(x -<-»• j/) A y f-*- (x *-+ y) A 3] , clearly equal since
(x-M-t/)Ax=(x-(-*-j/)Aj/.
D
1.3 Adjoint functors arising from cartesian-closed categories.
PROPOSITION 1.31. Let A be a cartesian-closed category and A' c A
a full reflective subcategory, /? : A -»• A' the reflector. Then R
preserves products iff for all B (. A' , S 4 is in A' .
Proof.
B € A1
Suppose
, that for any
R(A x- C) - RA *• RC , all
C U ,
A, C .
(RC', s") = (C, fl4) .
We wish to show for
(We are invoking the
"Kelly view" of full reflective subcategories, namely that
all those objects
letting
that
Y = X .
X
such that
We obtain from
X •* RX ->• X = 1 .
RX •*• X •*• RX = 1 .
From
(RY, X) ~ (Y, X) .
(RX, X) •++ {X, X)
{RX, X) «• {X, X)
A'
consists of
This can be seen by
a map
RX •* X
such
we obtain
The Ke I I y view is that every full reflective subcategory
I4
Peter Freyd
is definable by stipulating a class of maps
such that
C and then looking at all X
(/, X) is an isomorphism for all / € C .)
To return:
{RC, BA) = (RC*A, B) = (R(RC*A), B) =
(RCxRA, B) = (fl(C*4), s) = (C*A, B) = (c, B 4 ) .
Conversely, if for all B € A' , a € A 1 , then we wish to show that
is, as defined in A' , a product of RA, RC .
)
It suffices, then, to show that
is, for all
B (. A' ,
A1 (RA*RC, -) = A'(R{A*C), -) , that
(RA*RC, B) - [R(A*C) , B) . But
[R(A*C), B) - (.A*C, B) and
, B) == (iJ4, / ^ = (4, / ^ = (4x/?C, B) = (i?C, S 4 )
For any category
objects are
A
A and
B£A ,
A/B denotes the category whose
A-maps of the form A •*• B , and whose maps are triangles
•* A'
\
/
•
B
£ D : A/B •*• A
o
an isomorphism.
denotes the forgetful functor.
Note that
A/1 -»• A is
The naive construction of colimits in A/B works, that
is, given colimits in A . The naive construction of equalizers in A/B
works;
and
and it is the purest of tautologies that the A/B-product of A •*• B
A' •* B
is their
PROPOSITION
A-pullback.
Note that the terminator of A/B is
1.32. Z D : A/B + A preserves and reflects colimits,
D
equalisers, pullbacks and monomorphisms.
If
D
A has finite products, we can define
XB : A -+ A/B by
A t-+ [A*B -£•+ B) .
PROPOSITION
1.33. ZD is the left-adjoint
of
XB .
D
D
The next proposition says for
right-adjoint.
A cartesian-closed, that
XB has a
F i r s t , note the necessity of cartesian-closedness:
if
XB
Aspects of topoi
has a right-adjoint
15
IIg , then
(-xfl, A) = lzB(xBl-)), /) = (XB(-), xB(it)) - [-, nfl(xfl(i4))] ,
that is, given
IIg we can construct
PROPOSITION
1.34.
A
as
II (xB(4)) .
For a artesian-closed
A ,
xB : A -* A/B
has a
right-adjoint.
Proof.
where
Given
1 •* B
[A — £ + fi) f A/S
corresponds to
B
define
>• B .
IIB(4 - ^ * fl) by the pullback
For any
C £ A
we obtain a
pullback in the category of sets
(c, n U -
B))
(c, i)
->- (c, 4B)
[c, sT) .
Three of these sets are naturally equivalent to other things and we obtain
a pullback
(C, IIR(/1 •+ B))
I
1
where the bottom map sends
as a subset of
1 to
•* {C*B,
A)
|(\(C*B, f)
• (C*B, B) ,
C x B —2-> B .
Viewing
(C, U (A •* B)}
(C*B, A) , we note that i t can be described as
{j : C x B + < | C i j -2-»- 4 - £ * B = C * B -&-*• B} , which i s precisely the
description of
A/B(C x B-^-
B, A-£+ B) = A/B{*B{C), A -£+ B) .
D
16
Peter Freyd
We note here, in anticipation of the next chapter's development, that
for any
/
,
B1—^-* B2
E _ : A/B
we can define
•* A/B
f
: K/B^ •* A/B
by composing with
can be viewed as an object in
A/B
/ .
by pulling back along
On the other hand
B —•*-»• B
, and we could consider
: (A/B 2 )/(B 1 -B 2 ) - A/B2 ,
\."]_
But
'2
'"2^
'•I
2^ '
(A/Bp) / (B^ ->• B_) i s isomorphic t o A/B , and the isomorphism reveals
Z C_ B D )i
as
(V 2'
£,, ,
^
xfBX
n •+ BA
2
adjoint, each / , iff
1.4
2^ '
as
f^ .
Therefore
j
has a r i g h t
A/B is cartesian-closed, each B .
Modal operators in Heyting algebras.
In any partially-ordered set, viewed as a category, the full
reflective subcategories are in one-to-one correspondence with the
order-preserving inflationary idempotents, that i s , the functions /
that
i s r
such
fix) < fiy) ,
x s fix) ,
fifix)) = fix) .
Clearly, the reflector of a full reflective subcategory is such.
Conversely, given such
Given
/
then
y (. Im(/) then
Image(/)
is reflective as follows:
x 5 y •» fix) 2 fiy) = y
and
fix) 5 y => x 5 fix) 5 y .
As a corollary of Proposition 1.31 we obtain
PROPOSITION
denoted
x
1.41. Let
H
be a Heyting algebra with a unary operator
such that
x = x ,
x s x
ior x A x = x )j
Aspects of topoi
x 5 y =» x 5 y
V
(x A y = x A y )
x,j/
We shall say that
i//
x
[or
17
i = i A (x v j/) ) .
V
(x = x •=» (j/ + x ) = (y -*• x ) j .
•*•>&
is a closure operator if
x =x,
x 2x
(or x A x = x ),
x A i/ = x A y .
Note that the l a s t equation implies
x 5 y =» x 5 z/ .
By the above remarks, the image of a closure operator i s a reflective
sub-Heyting-algebra with the property that
PROPOSITION
operator.
1.42 for Hey ting algebras.
In particular
Proof.
x = x =* (t/ ->- x) = (y •*• x)
1 ~l x is a closure
1 "1 ( x A j / ) = n i A l l i /
.
We have a l r e a d y noted i n P r o p o s i t i o n 1.23 t h a t
~ l ~ l ~ l ~ l x = ~l~la; .
s
From
x S i / ^ n n x s n i y .
x<y '~\y'S~\x
then
y-*-x = ~\~\(y+x).
3 = ~1 x .
Then
x = ~1 ~1 x
yields
x 2 ~I "I x and
we e a s i l y o b t a i n
By P r o p o s i t i o n l . U l i t s u f f i c e s
x = ~1 ~l x
.
that
t o show t h a t i f
For purposes of c l a r i t y , l e t
x = ~l z .
We u s e only t h i s
last
e q u a t i o n from now on.
x = ~l 3 = (2 -»• 0 ) ,
y -* x = y -* (z -»• 0) = (y A Z) -* 0 = ~l (2/ A 3) ,
1 1 ( j + X) = "1 1 1 ( j A «) = n
(j
A 8
) = i/
+
*
D
(we have proved that the defining equation of a Heyting algebra implies the
equation
~l ~1 (x Aj/) = 1 1 x A ~\ ~\ y , There is, therefore, an entirely
equational proof.
My attempts to find one (essentially by translating the
proof herein) have yielded only the most unbelievably long expressions I've
ever seen.)
The image of double negation is easily seen to be a boolean
algebra.
1.5
Scattered comments.
Besides topoi and Heyting algebras important examples of cartesian-
closed categories are the categories of small categories and various
modifications of categories of topological spaces, which modifications
18
Peter Freyd
exist precisely to gain cartesian-closedness.
The chief such modifications
are fe-spaces and Spanier's quasi-topological spaces.
For any category
as elements and
skeleton of
A
A < B
A' .
P
define the pre-ordered set
iff
A U , B) i- 0 .
Define
A'
with
P(A)
A-objects
to be the
is a reflection from the category of categories to
the category of posets.
P
clearly preserves products and hence the
category of posets is cartesian-closed.
Moreover, if
x
preserves
Let
, +
M
is cartesian-closed then
be a monoid,
Heyting algebra
point,
A
P[^)
P\S )
S
is, and
A •+ P(A)
its category of representations.
changes wildly depending on
= P(S) = {0, 1} . For
M
a group,
be described in terms of sets of subgroups of
numbers
P(A)
and exponentiation.
p(x)
M .
For
p[sP)
The
M
a single
is a set (and can
M ). For
M
the natural
is huge but does satisfy a transfinite descending chain
condition, a fact which requires a long proof and can be found - together
with a description of
P[S )
- in a paper by (of all people) me ([2],
225-229), in a section entitled "When does petty imply lucid?".
For
M
almost anything else,
P(i )
fails the transfinite chain
condition.
2. The fundamentals of topoi
Let
A
be any category with pullbacks.
to be the set of subobjects of
We can make
Sub
A
Given
(so assume that
A f A
A
SubU)
is well-powered).
into a contravariant functor by pulling back.
A topo8 is a cartesian-closed category for which
representable;
define
Sub
is
that is, there is an object ft and a natural equivalence
(-, ft) •*• Sub .
Recall that any
n : (-, A) •*• T
is determined by knowing
D. (l-) € TA , hence there exists fl1 >* ft such that if we define
n : (A, ft) ->• S u b U )
isomorphism;
by
r\(f) = (sub(/))(ft* >* ft) then
that is, for every
A' >* A
A' ->• Q'
is an
there exists a unique
A'
such that there exists
n
such that
A -»• ft
•* A
4-
+
ft' •*• ft
is a pullback.
The
Aspects of topoi
19
representability of Sub is thus revealed as an elementary condition.
we can do better. Note that in particular for any A there exists a
unique
A -*•ftsuch that there exists
A -*•ft'such that +
+
ft' • ft
Given any A •* ft' then for 4-»-ft=i4-»-ftl->-ft
pullback.
But
is a
it is the case
A -±>A
that . +
ft'
t is a pullback.
• ft
The uniqueness condition then says that
(A, ft') has precisely one element.
In other words ft' is a terminator.
Thus we could define a topos as a cartesian-closed category together with
an object ft and a map
unique
A -»• ft such that
A' •*• 1
because
1
1
• ft such that for any A' >* A
A'
+
1
•A
+
-r°
is a pullback.
is a terminator.)
there exists
(We needn't quantify
Of course, this elementary condition
impli.es well-poweredness.
fl° P
PROPOSITION 2.11. For any small A , S
Proof.
is a topos.
Aop
We showed in Proposition l.lU that
S
is cartesian-closed.
For the ft-condition we again assume the result to discover the proof.
ft exists, then
an
U{A) = fa^, ft) = Subfa.). A sub functor of
A-crible, alternatively described as a collection of maps
such that
B->-i4«C=>B'->-B->-/l€C.
Defining
SI (A)
H^
C
If
is called
into
A
as the set of
4-cribles, we make ft into a contravariant functor, again by "pulling
back":
given
B •* A'
such that
Any
A' -*• A
and an 4-crible
C
define
1 •* ft is a choice for each
A
T' c T
as the set of maps
of an element in SiA . Define
t : 1 •+ ft to correspond to the maximal crible, each
4-crible is the set of all maps into
Given
C
B •* A' •* A i C .
define
A .
(The maximal
A ).
T •* Si by sending
x i TA
to the 4-crible of
2"
a l l maps
B -*-* A such that
i s a pullback.
(T/)(x) €T' (B)
and verify that
• T
+
+
1-^ft
The uniqueness of T -*• ft can be directly verified.
•
20
Peter Freyd
PROPOSITION
Proof.
2.12 for topoi. Every monomorphism is an equalizer.
Given
A' »• A
A' + A
+
4- be a pullback.
let
1
Then
4'
is the
•* Q
e q u a l i z e r of A * 1 — ^ A •+ fl a n d 4 * 1 - ^ 1 -»- 0 . B u t 4 X 1 = 4 .
COROLLARY
D
2.13 for topoi. If A •* B is both mono and epi, it is
iso.
Proof.
It can be the equalizer only of
f, f : B •* C .
O
In any finitely bicomplete category we define the "regular image" of
4 —'-*• B
as the equalizer of the cokernel-pair, that is, the equalizer of
B
x, y : B -*• C
where
f\
Ix
is a pushout.
It is a routine exercise to
•• C
B
y
verify that the regular image is the smallest regular subobject through
which
f
factors ("regular" is equal to "appears as an equalizer").
Because all subobjects are regular in a topos, the regular image is the
smallest subobject allowing a factorization of
the image of
/ , denoted
PROPOSITION
4 — £ + B . We shall call it
Im(/)
2.14 for topoi. A ->• lm(f) is epi.
If A -*->• C *- B = / then there exists unique Im(/) = C sueh that
Given
B
+
•D
+
C
there exists unique
Im(/) -»• Jm(g)
such that
9
A •* I m ( / ) •* B
4C
Proof.
By construction,
4-
Im(/) = B
first statement it suffices to show
minimality of
Im(/)
does Just that.
+
iff
/
is epic.
Hence, for the
Im(4 •*• Im(/)) = Im(/) ; but the
Aspects of topoi
21
The other two statements are immediate from the characterization of
Im(/)
as the smallest subobject through which
with Corollary 2.13.
Sub(l)
/
factorizes, together
•
is a Heyting algebra.
It is clearly a lattice (because images
allow us to construct unions of subobjects) and for any
[A, if) - (.A*B, U)
monomorphism.
Uc 1 ,
has at most one element, hence the map
The subobjects of
if •*• 1 - is a
1 , in fact, form a full reflective
subcategory (the reflector constructed by taking images:
X i—> Jm(X -*• l) ) .
Aop
The subobjects of
collections of
1
in
collection and there exists
if
A
may be seen to correspond to
5 •+ A
then
B
A
is in the
is in the collection.
is strongly connected (for example, a monoid) then
two subobjects it always must.
subobjects of
than
S
A-objects with the property that if
1
Aop
S
in
If
A
is
1
•-»••-*••••->••
Hence
has only the
then the
form the linearly ordered set one bit longer
A .
Returning to ft in
5
Aop
, suppose
A
is a monoid
M .
We may view
Aop
S
as the left-representations of
M x X •*• X
such that
left-ideals in
division:
group then
M .
I** = x ,
The action of
ct'A = {&|Ba £ A } .
£2 = 1 + 1 ;
M , that is, sets
a(6*x) = (aS)\r .
M
X
together with.
Then fi is the set of
on ft is not multiplication but
1 — • ft is the unit ideal.
If
M
is a
and, for monoids, conversely.
2.2 The representability of partial maps.
A partial map from
A
Formally, we consider pairs
to
B
is a map from a subobject of
such that
to
< A' ** A, A' •* B > , define
<4' >*• A, A' •*• B > B (A" «• A, A" * B)
A' •*• A"
A
if there exists an isomorphism
B .
22
Peter Freyd
and define
Par(i4, B) as the set of equivalence classes.
Fixing
B,
Par(-, B) is a contravariant functor, by pulling back.
A relation from
of such
A
to B
is a subobject of A x B . Calling the set
Rel(A, B) and fixing
B , Rel(-, S) becomes a contravariant
functor, again by pulling back.
Rel(-, B) is representable, namely by
B
si .
Every map from
a transformation
to B
can be viewed as a partial-map and we obtain
(-, B) -»• Par(-, B) . Every partial-map
yields a relation,
transformation
A
A' >* A
x
B
< A' »• A, A' •* B)
(its "graph") and we obtain a
Par(-, B) •*• Rel(-, B) . Both transformations are monic.
The transformation
(-, B) •* Rel(-, B) must come from a monomorphism
B -*• SI , the singleton map.
B -*• SI
may be computed to correspond to
B —^
B x B -»• £J where
1
4-
*
We shall show that
PROPOSITION
there exists
such that
+
C
Proof.
Par(-, B) is representable.
First:
2.21 for topoi (unique existentiation). Given
Q>* A
4-
is a pullback.
> n
such that
Q-±+ Q
+
+
C—• A
is a pullbaok and for any
is a pullback, there exists
C •* A
X •*• A
X -*• Q •*• A = X •*• A .
»• A
Define
(-, A) * Rel(-, C) by sending
X -*• A
to the pullback
R •* X
4C
4- (H to be viewed as a subobject of X x C ). This transformation is
•* A
C
(
r*A QP C \
i n d u c e d b y a map A •*• Si
Define
( w h i c h c a n b e c o m p u t e d a s A •+• Si
•• SI J .
Q>+ A
(X,Q) * (X,A)
Q by t h e p u l l b a c k 44- . For any X ,
44is a
C - SlC
pullback.
Viewing
X'
iff
in pullback
4-
C
{X, Q)
(X,C) -
as a subset of
{X, A)
(x,SiC)
we see that
/ € (X, Q)
•* X
4- ,
- A
X' c X * C
as a r e l a t i o n from
X •+ C ,
Aspects of topoi
describes a map from
X
to
23
C , that is, X' -*• X
is an isomorphism.
•+• A
C
Given a relation from
A
to
B
described by
D
+
let
Q >*• A
be
B
as described above.
ParU, B) .
We obtain a partial-map,
This operation,
above lemma.
Moreover
Rel(-, B) •+ Par(-, B)
splitting of
g .)
g .
ft
is the identity.
Rel(-, B) -*• Par(-, B) •+ Rel(-, B)
" > ft , and we define ft •* B ,
B-^ft
[B -*• ft can be defined as the equalizer of
(-, ftB) •* (-, B) ,
Clearly
in
is natural by the
Par(-, B) •* Rel(-, B) •* Par(-, B)
The idempotent transformation
from an idempotent
(Q>+A>Q-+C-*-B)
(-, B) "• (-, ftB) splits
must come
as a
1
and
(-, g)
and:
PROPOSITION 2.22 for topoi. Par(-, B) is representable.
ft —**-• ft corresponds to a map
which corresponds to a subobject of
B
' — * B x ft where
B -*• B
s
PROPOSITION
exists unique
B * ft which can be computed to be
2.23. For any partial
A -*• B
n g (l B ) (. Par(B, B) .
fact that
n
is unique
A •*• B
map (A'>+A,A'
A' * A
such that +
+
The transformation
Let
In a telling sense
1 •+ ft , to wit:
B
Proof.
B * ft ->• ft (not the evaluation map)
is the singleton map.
is a generalization of
is a
-*B>
pullback.
r| : (-, B) •*• Par(-, B)
< B' * B, B' •*• B)
represent
is an equivalence says that for any
such that there is
is determined by
1 B (l g ) •
A' •* B'
exists unique
A •* B
is a pullback and
= A->-B.
such that
A-*B'-*-B=A-*-B.
If
A •*• B'
B' -*• B
isomorphism, then we would obtain a contradiction.
Note that
Given a map
Sub (4) = Par (A, l)
B •*• B'
and
we can define
there
->• A
+
In particular, for any
such that there exists
Then the
(A' >* A, A' -*• B >
B'
A'-*-B'+B
there
->• B
A'
pullback and
D
+
is a
•* B
A -*• B
such that
+
B'
there
+
>- B
were other than an
•
1 = ft .
Par(-, B) •* Par(-, B' )
by
24
Peter Freyd
composition and obtain a map B •*• B' .
Alternatively, from the above
B
proposition, there exists a unique map B •* B'
such that
->• B
+
4-
is a
B' •* B'
pullback, and
B is revealed as a covariant functor,
B •* B as a natural
transformation.
2.3.
The fundamental theorem of topoi
A logical-morphism of topoi is a functor that preserves finite limits,
colimits,
£1 , and exponentiation.
THEOREM 2.31
for topoi.
For any topos
T and B e T ,
1 IB
is a
topos.
For
f
back along
bi-continuous
Proof.
: B1 •*• B2
the functor
f , has a left-adjoint
and a logical
f
: T/Bx •* T./B2 defined
I - and a right-adjoint
by
II- j
pulling
/
is
morphism.
We noted at the end of 1.3 that the second sentence follows
from the f i r s t .
The fi condition i s easy:
S u b T / B U ->• B) = Sub T (4) = T[ZB(A •* B), 0.) = T/B(A •* B, U*B ->• B) .
-F
Given
T/B .
A —*-»• B ,
Let B -*• IT
A
that
n
(A-*-B)
a
C - -»- B we wish t o construct
in
T correspond t o
—=-"•—>• J 4 X B
+
B
P -*• B i s
Given
i
"
B
(C •*• B)
,
•*• CT
.
,g
be a pullback.
~A
B -* ET
as follows:
X —»• B ,
T/B(X •* B, P •* B) -* T{X, P)
• T(X, B)
i s a pullback, by definition of
T/B .
in
k : A*B ->• S , t h e unique map such
. P
is a pullback and let
(C -»• B)
By definition of
P ->• B ,
Then
Aspects of Topoi
25
U , P) •+ (x, C4) - Par^x/l, C)
+
4-
+
4
(X, B) •* (X, S ) = Par(Zx4, B )
is a pullback.
Then
T/B(X •+ B, P -+ B) -* ParT(X>^4, C)
1
is a pullback where
X x A -hxl>
• ParT(X*A,
1 •+ Par(J><4, B)
B x A -^+ S .
B)
corresponds to
The element in
Par(,X*A, B)
is therefore the
result of pulling back:
•
->• X*A
+ Vk
B -* B
Q
and that is the same as the pullback
of
X •*• B
and
4 + J
T-partial maps from
X
;
x
that is,
A
to
C
•* X
+
-I- , that is, the product, in
A + B
T/B{X •* B, P •* B)
T/B
is the set of
such that when composed with
C •*• B
yield just what they should.
That
J
preserves exponentiation is reducible, as discussed at the
end of 1.3, to seeing that
wish to compare
XS : T •* T/B
[-, xB(c^)
XBfC 4 ))
* B,
[D - B, XB(C)*B{-A))
and
= \tB(D
preserves exponentiation.
We
(-,
* B),
C4)
= {D, C 4 )
= (D*A,
C)
,
= ((£» + B) x XBU), XB(O) = fsg((£» •* B) * XB(4)), c| .
I t suffices to show D * A - £ g ((0 * B) x xB{A)) .
P
(D -*• B) x xB(i4)
is the pullback
+
D
AW
verified as
p+
•*•
•
+p .
D -»• B
Hence
£ „ ( ( £ •* B)
-\p which can be directly
B
x
xB(4)) = y5 x z) .
26
Peter Freyd
preserves limits and colimits because it has both a left and right
adjoint.
D
COROLLARY 2.32 for topoi. Fullbacks of epimorphisms are
epimorphisms.
Proof.
T/B .
Given
A -£•*• B ,
j (C •*-*• B)
is the pullback.
and terminal objects,
COROLLARY
C ->-»• B , we view
Because
/ (C •* B) •* f (B ->• B)
2.33
for topoi.
Given
C •** B
j
preserves epimorphisms
is an epimorphism.
A -£•+B + C
A, + A =
~<
as an object in
D
there exists
A
I
B+C
Proof.
apply
f
View
B -»• S + C
to obtain
Given a filter
A
F
and
C •+ B + C
->• A , A
->• A
as objects in
in
T/A .
T/B+C
D
(as defined in 1.2) on the Heyting algebra
we obtain a Serre-class of maps in the topos, namely those maps
such that there exists
isomorphism.
U £ F ,
U € F .
(U c l)
so that
T -»• T/£/
T/F(A, B) = lim(/5xU, B) .
T/F
T/F
T/U + T/V
Sub-j-.p(l) = (Sub-j-(l))/F .
to an
which
T/U ,
for
are logical morphlsms and that topoi are essentially algebraic,
easily believed to be a topos.
f
The most insightful
is to take the direct limit of the topoi
Using the fact that all the induced maps
Sub(l)
f : A -*• B
sends
The result of inverting all such maps is a topos
may be also constructed by
way to construct
and
V c V
T/F
is
Finally, one must note that
Strangely enough, we shall not use this
construction.
R
Given relations
define the composition
->- A
+
B
i?2 * B
, +
C
R
° i?2
as
R
let
•* R1
+
-IR2+ B
be a pullback and
Im(if •+ A*C) .
PROPOSITION 2.34 for topoi. Composition of relations
associative.
is
Aspects of topoi
Proof.
Give n
X'
->• B'
Y'
4-
+
, +
X
->- B
Y
*
X •* A*B ,
Y •*• A*B
and
27
B' •* B
let
B'
4-
be pullbacks.
Corollary 2.32 says that if
-*• B
Im(X •* A*B) = Im(y •+A*B) then ImU' ->4><B') = Im(Y' •* A*B' ) . That
fact, together with the associativity of pullbacks provides the proof. Let
R
6 * nh " ffl +
4- +
A
i? 5 •* i? 2 - B
+
+
i? 3 -• C
be such that a l l squares are pullbacks.
= (if ° R ) ° i?3 .
I m ^ -*• A*C) = Ft. ° R~ and hence
Equally
Im(flg •+ AXD) = R o [R ° R ) . D
Given a topos T we obtain a category of relations
Rel(T) and an
embedding T •*• Rel(T) . In particula r Rel i s a bifunctor from T to
s e t s , contravariant on the f i r s t variable, covariant on the second. Given
/ : A •* B the transformation Rel(-, / ) : Rel(-, A) ->• R e l ( - , 5) yields a
transformation
to
il
(-, if) •* (-, il ) which must be induced by a map from
t o be called
3- .
The transformation
( - , - ) -»• R e l ( - , - )
if
is
natural and we obtain
A •+ if
n iv
B •+ i?
Rel(4, B) is of course, a Heyting algebra.
Heyting-algebra-valued functor, but
unions but not intersections.
order.
2.4
Rel(4, -)
Rel(-, B) is a
is not.
3,. preserves
We will use the fact that i t preserves
The propositional calculus of a topos
PROPOSITION 2.41 for topoi. For every B , the subobjeats of B
form a Heyting algebra and the operations of same are preserved by pulling
28
'
Peter Freyd
back.
Proof.
T./B .
The subobjects of
B are the subobjects of the terminator in
D
We can t h e r e f o r e view
Sub a s a H e y t i n g - a l g e b r a - v a l u e d
N e c e s s a r i l y , t h e o b j e c t which r e p r e s e n t s
T .
£2x£2
That i s , t h e r e e x i s t maps
• £2 ,
£2x£2
1
• £2 ,
functor.
Sub must be a Heyting a l g e b r a i n
1 —£+• £2 ,
£2x£2 -^+ £2 ,
• £2 which s a t i s f y t h e e q u a t i o ns of a Heyting a l g e b r a
(t = X, f = 0) , which maps y i e l d t h e Heyting a l g e b r a s t r u c t u r e on each
Sub(S) .
f
1 —-1-*- £2 can be computed as the unique map such that
i s a pullback;
pullback.
£2x£2
>• £2 as the map such that
+
0—
+
1
+/
+n i s a
Let
IX t]
v £2 = Im £2+£2
Then
+
+u
i s a pullback.
unique map such that
+
F i n a l l y , define
++-»• i s a pullback.
ftxfi
>• £2 t o be the
One may directly verify
the equations of the symmetric definition of Heyting algebras for these
maps so defined.
Note that for each
B ,
are two Heyting algebras for
sets,
n
which lives in
f2
B :
becomes a Heyting algebra in
Sub(S)
T .
which lives in the category of
T .
A boolean topos i s one in which
There
Q is boolean.
£2
• £2 i s the
Aspects of topoi
fl
+~1 is a pullback.
unique map such that
+
fl
Note that every
• n — • fl = 1
.
boolean, and given
B' >* B
0
+
-• B'
t
is a pullback,
B"
•+ B
that
B'+B" •* B
Sub(S)
29
Hence
is epi.
is boolean if
is thus forced to be
there exists a complement
B'+B" -*• B
T
B" >*• B
such that
In any topos, such implies
is an iso, just by showing that
B'+B" -*• B
is mono, an
easy matter using Corollary 2.33. Products of boolean topoi are easily
seen to be boolean.
PROPOSITION
2.42 for topoi.
isomorphism.
(1
T ia boolean iff
iff
1 + 1 -±LU fl ie an
1 + 1 -iiX* fl -Lt
Proof. Clearly i f 1 + 1 works as fl then fl
other than the twist map on 1 + 1 , and ~1 ~I = 1 .
Conversely for T boolean, the complement of
and the remarks above yield 1 + 1 = fl .
Q
PROPOSITION
Proof.
If
c
the form S ,
1 +1 .
Aop
2.43. S
A i s a groupoid then
G a group.
Conversely, suppose
is, there is no
is boolean iff
B •*• A
•+ fl can be none
1 —• SI is
1 —*-• fl
A is a groupoid.
Aop
S
i s a product of categories of
c
We observed earlier that fl in S is
A •*• B
such that
in
A
does not have a left inverse, that
B •* A •*• B = 1_ . A •*• B
generates a
D
5-crible, neither empty nor everything; that is fl(fi) has more than 2
elements. But i f 1 + 1 = fl then fl(S) = (l+l)(B) = 1 + 1 .
D
By a closure operation on B we mean a closure operation as defined
in 1.1* on Sub(S) , that is an intersection-preserving inflationary
idempotent Sub(S) •*• Sub(B) . A global closure operation is a choice for
each B of a closure operation that makes Sub •+ Sub natural.
Necessarily such must be induced by a map j : fl •* fl , and j itself is a
30
Peter Freyd
closure operation in the internal sense, that is, j2 = j ,
£2
<1,.7>
luJ
_
—*• £2*£2
n
• £2 = 1
£2x£2-^£2
and JXJ+
ftxft
ftx
+j .
Rather mysteriously:
1- ft
n
PROPOSITION
Proof.
Given
applying
2.44 for topoi.
First a less algebraic proof:
0 :fi•*ftconsider
(B,ft)v
ft = 1 -^+ ft then for
1+
-r
n+
>;/
'•» (B,ft). If
a pullback, the result of
(B, j ) yields
B 1 ->- B
ft' * ft
and t h e r e e x i s t s
i s a pullback and
1 ->• ft' such t h a t
(B, j)
More directly, l e t
< 1 , j > i s monic,
exists
1 •* y such t h a t
is.
that is
+
+ (1, j > be a pullback.
If
Because
1 —>- ft - ^ * ft = 1 — • fi then t h e r e
1 •*• U •* £2 = 1
1—1_
isomorphism;
B' -+~B'
+
1
is inflationary.
+
U •*• 1
1 •* ft1 ->• ft = 1 —*• ft , hence
»• £2 and tf * 1
i s an
i s a pullback.
Hence
ft
+<1, j>
+
•*• ft1
Aspects of topoi
is a pullback and the uniqueness condition on
As shown in Proposition 1.1+2,
Another example arises as follows:
Sub(B) -* Sub(B)
by
is a monomorphism.)
SI
let
• SI
V c 1
(B1 •* B) >-+ (B'u(B*U)} .
31
SI implies the result.
D
is a closure operation.
and define for each
(Notice that
B ,
B*U •* B*l •* B
Clearly this operation is inflationary and idempotent.
To see that it is intersection-preserving consider
B', B" c B .
Then
11
(B'u(B*£/)) n (B"u(Bx£/)) = (B'nB ) u [fl' n{B*V)) u (CBxt/)ng") u (Bxtf) ,
by distributivity.
The middle two terms are contained in
B
x
t/ , hence we
obtain
(B'u(Bxy)) n (B"U(BX£/)) = (s'nB") u (flxy) .
Bx[/ -* B
+
+ i s a pullback.
U +1
The operation is natural because
2.5. Injective objects
2.51 for topoi. a is injective.
PROPOSITION
Proof.
A'
let
Given
and
A + 8
let
A'
•
• A
4- be a pullback.
Then
<• B
+
4- b e a p u l l b a c k .
COROLLARY
Proof.
epic.
A >* B
Then verify that
A->-B + n = A + Q .
O
2.52 for topoi. ftC ie injeotive.
Given
A »• B
we wish to show that
But
(B, QC) - [A, Sf)
I?
I?
, n)
[B, SI ) + (At SI )
is
32
and
Peter Freyd
A*C -*• B*C is monic.
D
Using the singleton map B -*• ft we obtain:
PROPOSITION
injective.
2.52 for topoi.
Every object may be embedded in an
D
COROLLARY
2.53 for topoi.
Pushouts of monomorphisms are
monomorphisms.
Proof.
Given a pushout
A >* B
+
+
choose
C *+ E , E
injective.
There
C •* D
exists
A yB
+ + , h e n c e t h e r e exists
C
•*•
monic.
C-<-D->-E
=C>*E
a n d C •*• D i s
E
•
PROPOSITION
2.54 for topoi.
Snb(D) •* Sub(S)
4+
Snb(C) •* Sub(A)
Proof.
Just use the distributivity,
COROLLARY
intersections
show that
A H- B
$
¥ is a pullback, then
C>+ D
is a pushout.
for Heyting algebras.
Proof.
If
X A («/ v z ) = (x A y) v (x A z)
D
2.55 for topoi.
If
E
is injeative then
(-, E)
carries
(that is, monomorphic pullbaaks) into pushouts.
If E
(-, Cl )
is injective, then
-p
E **• il
splits and it suffices to
carries monomorphic pullbacks into pushouts. But
(-. fi ) - (-*£, ft) and -*£ preserves (as in any category, any E )
monomorphic pullbacks, and Proposition 2.51* says that
desired.
(-, ft) is as
•
PROPOSITION
2.56 for topoi.
Given
A H- 8
\
1 , it is a pullback iff for
C >* D
{D, E) •* (C, E)
injective
objects
Proof.
E3
The family
+
t
(S, E) -* (A, E)
{(-, E)} , E
is a pushout.
injective, is collectively faithful
Aspects of topoi
33
by Proposition 2.52 and hence reflects isomorphisms.
Any family that
reflects isomorphisms reflects the limits it preserves (any category).
•
2.6. Sheaves
Let
that i s
j : ft •* ft be a closure operation as defined in the l a s t section,
j
=j
and j
i s a homomorphism with regard t o ftxft -°->- ft and
1 —• ft . Given B' » B , we will write
(B, j) to Sub(B) .
We will say that
B' = B .
B' H- B i s
B'
j-closed
We say that A i s j-separated
(B, A) •* (B' , A) i s monic.
for the result of applying
if
B' = B'
if for a l l j-dense
We say that A i s a j-eheaf if for a l l ./-dense
(B, i4) •+ (B' , -4) i s an isomorphism.
and j-dense
if
B' >+• B ,
B' H- B ,
A functor i s exact i f i t preserves a l l f i n i t e limits and colimits.
THEOREM 2.61. The fundamental theorem of sheaves. The full
Bubaategoviea of j-eeparated objeote and j-sheaves are reflective and
eaah ie cartesian-closed.
The full subcategory of
exact.
Proof.
We fix
j
j-sheaves is a topos and its reflector
and drop the prefix
B'
then so is
B'*C -*• B*C
is equivalent to
because if
"j-" .
If
is
B' >* B i s dense
• B
+
+3
is a pullback, then density
B -&+ft-2-+ft= B •* 1 -^* ft , and
B' *C •* BXC
+p
B
1
is a pullback.
Hence if A
- ft
is separated (sheaf) and
B' >* B
dense then
34
Peter Freyd
( S , A°) - (B 1 , AC)
\i
I?
(5XC, / I ) •+ ( S ' x C , 4 ) ,
and t h e h o r i z o n t a l maps a r e mono ( i s o ) , and A
i s separated
(sheaf).
Let
Sep. be t h e f u l l subcategory of separated objects and
3
the full subcategory of sheaves.
Sh. be
3
We have just seen that
Sep. and Sh . are cartesian-closed (except
3
3
possibly for finite completeness) and hence by Proposition 1.31, when we
know that
Sep. and Sh . are reflective, we will know that the reflector
3
3
preserves products.
Define
Q . •*ftas the equalizer of 1
3
and j . Because
j is
i d e m p o t e n t , there exists ft •*ft. such that ft •*ft. •*ft= j ,
3
3
ft . -<• ft ->•ft. = 1 . ft. i s i n j e c t i v e .
3
3
3
LEMMA
2.611.
Proof.
Let
of ft . yields
0
B>
ft.
3
is a
B' >* B be dense, and
and we have
Then
B
2.612.
ft..
and B
5 1 n B' = B
COROLLARY
B' •+ Q.
3
given.
The i n j e c t i v i t y
B'—• B
\ J
. We need only the uniqueness condition.
ft.
3
-»- B -£-+ft. = B' •* B -2-+
pullbacks.
sheaf.
Let
4-
4-/ and 4-
are both closed.
n B1 = B1 ,
For any
A,
But B
B^ n B' = B2
tfi
3
is a sheaf.
Suppose
4-0 b e
n B' = B^ n B'
and f = g •
Q
D
The definition of Sep. and S h . easily says that both are closed
3
3
under limits, Sep . under subobjects.
3
The idempotence of j
says that for B' C B , B' is dense in B'
Note that the closed subobjects of B
are in natural correspondence
.
Aspects of topoi
with
35
(B,fi.) .
3
LEMMA
2.613.
Proof.
If A
A
is separated iff A — • A*A is a closed subobject.
is separated, let A c A*A be the closure of A .
A
-A
There is at most one
I •* Ax-A
A •-- J .
hence
Thus
=J
—i*A
Conversely, suppose
/I
B' •+• fl -i-»- A = B' •* B -2-+ A .
S
as the pullback
•
+
S"1 C E C B ; thus
LEMMA 2.614.
_
A
p.
_
po
•* A
A , and hence
A c Eq(p , P2J = A .
•• A
• .4*4 is closed.
Let B' >* B
The equalizer of
/, ^
be dense and
can be constructed
B
^f>9^ . hence
B = B and / = g .
A is separated iff
E1 is closed.
But B' c E and
D
A •+ $/ = A -* fi4 -J2—•• fi!4 .
Proof, j
yields a transformation j : Rel(-, A) -*• Rel(-, A)
alternatively described as taking closures of subobjects of -X/l . The
equation of the lemma is equivalent t o :
(-, A) ->- Rel(-, A) = (-, A) - Rel(-, A) -i+ Rel(-, A)
which may be tested on 1. € (A, A) . The left side yields
the right side yields the closure of A
proof.
O
LEMMA 2.615. For any A the image of
reflection
of A into Sep • .
3
Let
and
A
A •* A be the image of A -*• U •* U .
Sep.
3
A
A •* fir -*—+ fir is the
if •*+fi4.+ j / and A i s a subobject of ft4. .
J
3
u. € Sep.
3
3
• A*A ;
and the last lemma yields the
A
Proof.
A
j
factors as
Lemma 2 . 6 l l said that
i s clearly closed under subobjects, hence A t Sep.
3
36
Peter Freyd
If
B i. Sep • then by the last lemma B •+ B i s an isomorphism.
0
f : A -*• B , S € Sep • we need only show that the outer rectangle of
Given
A •* fT4 -2—*- if
A
commutes to obtain
-*• A
\ ^ 4- .
Equivalently we consider
B
( - , A) •* Rel(-, A)
( - , B) -+ Rel(-, B)
The right-hand square does not commute.
clockwise we obtain
«U Rel(-, A)
T* Rel(-, B)
Given
show t h a t when
R c X*A
graph of
Im(R •
But the fact for inverse
Im(i? -»• X*B) c Im(i? -»• X*B) and because direct images
preserve order we also have
But i f
and chasing
Im(i? -*• X*B) and in the other direction
Inverse, not d i r e c t , images preserve closures.
images yields that
R c: X*A
Im(R •* X*B) c Im(/f -»• X*B) .
R i s the graph of a map then
Im(i? -»• X*B) = Im(R •*• X*B) .
i s the graph of g : X •* A then
X " > A —'-> S
I t suffices t o
Im(i? •* X*B) i s the
and the last lemma says precisely that graphs of
maps i n t o separated objects are closed.
For the r e f l e c t i v i t y of
•
Sh . i t suffices to show i t reflective in
J
SeV. .
LEMMA
2.616.
Proof.
Given
If A' » A is closed, A € Sh. then A' € Sh. .
0
0
B'
B >* B dense and B' •+ A , l e t
•*• B
+
A'
+ commute and
•+ A
B" * B
+
A'
+. a pullback.
Then
B" is closed, B' c B" and hence
B" = B .
D
•* A '
We'll say that a separated object is absolutely closed if whenever i t
appears as a subobject i t appears as a closed subobject.
Aspects of topoi
LEMMA
2.617.
37
Sheaves are absolutely closed and separated.
Absolutely closed objects are sheaves.
Proof.
exists
Given
A •*• A
A >* B , A € Sh. , let A >* B be the closure.
J
A
_
such that + yB , that is, A c A c A .
Conversely, if
4
fl €. Sh .
3
3
3
a sheaf.
B
and if
A
A
is separated then it appears as a subobject of
is absolutely closed, then by the last lemma,
LEMMA
2.618. fi . satisfies the
LEMMA
2.619.
G-iuew
is the reflection of
Proof.
For any
Let
A >* B ,
A
A •*• B
C € Sh. ,
3
There
in
0.-condition for
B
(A, C) = {A, C)
A
D
a sheaf, then the closure of
A
in
because
Lemma 2.6l6 says that
A -*• A
is dense.
A € Sh. .
3
•
can be embedded in a sheaf (for example,
u. ) we obtain that every separated
3
A
has a reflection in
Composing the two reflections, the reflection of an arbitrary
closure of the image of
is
Sh. .
3
be the closure.
Because every separated
Sh. .
A
Sh. .
3
A
is the
A •+• U. .
3
We saw at the beginning of the proof that
Sh . is closed under
3
exponentiation. Lemma 2.618 says that Sh . has an fi , and we have just
3
seen that it is reflective, hence finitely cocomplete. Thus Sh . is a
3
w
topos. Note that for E injective in Sh . , that E >* il. retracts and
3
3
is injective in the ambient category
Since the reflection
T .
H : T -* Sh . preserves products it suffices for
3
exactness to show that it preserves equalizers.
any category, the equalizer of
/, g
Given
/, g : A •+ B
in
may be constructed as the pullback of
38
Peter Freyd
hence i t suffices
t o show t h a t
R : 7" •* Sh .
3
preserves monomorphic
pullbacks.
A y- B
?
I in I , it suffices according to Proposition
C w- D
Given a pullback
2.56 to show that for any injective
{RV, E) ->• (RC, E)
+
+
is a pushout.
(RB, E) •* {RA, E)
{D, E) + (C, E)
+
+
and since
(S, E) •* (A, E)
the finish.
(A'
But that diagram is isomorphic to
is injective in T , Corollary 2.55 provides
D
Consider
3
E
E £ Sh . , it is the case that
3
S* . Si = (3 •*• 2) .
Let i / c i
t h e c l o s u r e o p e r a t o r t h a t sends
•* B) c (A -*• B) .
Then t h e
(A •* 1) . fl . = (2 •*• 1) .
3
reflection of
ft
Recall that
is
ft
•Q
(0 -»• l ) c ( l -»• 1) and
(A' + B' ) c {A ->• B)
to
j - s h e a v e s a r e of t h e form
The r e f l e c t i o n
(3 -»• l )
be
not
of
(A •* B)
is
U -»• l ) . The
Q. .
3
is a closure operator.
PROPOSITION 2.62 fortopoi. Sh-,-, is boolean.
Proof.
1 + 1 —*—>• fl is easily seen to be dense, hence becomes an
isomorphism in Sh-, -. .
COROLLARY
co-continuous
Proof.
•
2.63. Every non-degenerate topos has an exact
functor to a non-degenerate boolean topos.
0
is a 1 1 -sheaf.
D
Following the remarks at the end of Section 2.k, let U c 1 and
consider
B'
Q - ^ + ft t h e closure operator so t h a t for
= B' u (B*U) .
B' c B ,
T •+ Sh. sends 0 •+ V tc an isomorphism,
3
that i s , U is sent to the zero-object. Sh. is degenerate iff U = 1 .
3
COROLLARY
exists
The r e f l e c t o r
2.64 f o r t o p o i .
a boolean topos
Given
x, y : 1 + B ,
6 , exact co-continuous
F •* 8
x f y ,
there
that separates
Aspects of topoi
Proof.
Let
U c 1
39
be the equalizer of
x, y .
described above,
T •*• Sh . sends x, J/ : 1 -»• B
3
Now apply Corollary 2.63.
O
as equalizer.
THEOREM
2.65
f,g:A-*B3
Then for
(the plentitude of boolean topoi).
f ? g , there exists a boolean topos
ao-continuous
T -*• 8
that separates
j
as
.to a pair of maps with • 0
For every
8
exact
3
f, g .
Every small topos can be exactly embedded in a boolean topos.
BM
Proof.
T - T/A
sends f,
g
to
\
/
A*A - 2 ^ 1 * BxA
\ ,
^/
A
,
A
is the terminator for
T/A
5
as follows:
define
J OO
t e a partially ordered set.
for
+
A
and
T/A •* 8
Hence we can apply Corollary 2.6k to obtain
Let
A
.
We define
as desired.
j : ft + ft in
^ P
5
x (. B , Q(x) = {A c H \ u < v £ A "* u £ A ** u < xi ;
to be the set of all
with the property that
then this simply says
V' eH\y
eA>
u € H
such that there exists
(v 2 s) •» (v 2 u)J .
u 5 IM1 .)
j
(If
U4'
A' c A
exists,
is easily seen to be idempotent and
inflationary.
It is natural and it preserves intersections iff
distributive, for example, a Heyting algebra.
(l, ft.) is its completion.
3
(l, ft.) = H
3
For
B
Hence for
H
For
B
B
a Heyting algebra
a complete Heyting algebra
and ft .(a:) = {u i. B \ u < *} .
3
That is, 3 (A) = UA .
x
(l,ft.)
3
Sh . is the classically defined category
3
the lattice of non-empty open sets in a space
is the lattice of all open sets,
is very
X ,
of sheaves.
The definition of a Grothendieck topology on
definition of a closure operator in
more general.
S
A
A
is almost the
op
.
Closure operators are a bit
40
Peter Freyd
2.7.
Insoluble topoi, or how topoi aren't as complete as you'd like
In a powerful way, topoi are "internally complete".
example, define a map
given any
Sub(fT)
stated
B c!T
= fl,
fl
•+ ti
which acts as a union operator.
we obtain a subobject
->- ( l , ?iA) = Sub (4) I
fl
One may, for
UB c A
Hence
[via the functions
which has a l l of the
properly
properties of a union.
There i s a catch.
subobjects of
A
UB i s not necessarily the least upper bound of the
described by
( l , B) c ( l , QA) = SubU) .
completeness we often need i s just t h a t ;
upper bound for
singleton map
( l , B) c Sub(4) .
A •+ u
(1,
A) c Sub (A)
(l,
A) = ( l , A) .
that
1
,
UA = A .
that i s , given
UB f a i l s miserably.
B c tf4 , a least
For the
But the least upper bound of
would be the least subobject
To say t h a t
The type of
A = A , all
A
of
A
such that
A , is equivalent to saying
is a generator (that i s , a well-pointed topos as defined in the
next s e c t i o n ) .
If for every
upper bound of
A
there were such
A c A , then
UB is the least
(l, B) c Sub(j4) . (We shall not use this construction and
hence will not prove it. )
The existence of A c A
independent of the axioms for topoi.
is elementary but
Indeed, as we shall show, it is not
equivalent to any essentially algebraic axioms.
We will call
that
T
solvable if for every
(1, 2) - (1, A)
is the case that
PROPOSITION
Proof.
and for all
Be A
such that
A c A
such
(l, B) - (l, A) it
2.71. Not all topoi are solvable.
We saw at the end of the last section that every complete
(l,fi). Let T-. be such that
order-type of the unit interval and let
exists
there exists
A c B.
Heyting algebra appears as
submodel.
A
TV, c J
(l,fl)is the
be a countable elementary
In Tp , (l,fl)is a dense ordering and countable, hence there
iu c l} , u c \j
such that
n
n
n
n+1
be such that
u
n
c v
al
l
n
•
Uu does not exist.
n
Let V C 1
f
Aspects of topoi
41
/ n
Consider the direct system
T^j ] \ (l+^-)
induced by the obvious
n+l
n
1 T{l+U-) •* 1 [(l+U.) and let
projection maps
T
be the direct limit.
The essentially algebraic nature of topoi insures that
can give a more elementary description of
j
T
is a topos.
We
as follows:
objects: {n, A -+ TT(l-*J / ,
maps:
<n, A
A ^
—>• "TTM^))
(l+^-j) t
to
(m,
(n,
o (m,
from
are equivalence classes of pairs
< k, f)
B
where
k
k
f : A x J~f (!-«/ ) ^S x ] i
such that
T M) X
X
| 7 (1+y) _^U T T M J - TT d^J
i=m+l
i=l
^^f
T
The equivalence relation is generated by
The notation is eased by replacing
in
T . Hence
For
- rr
<k, f> = (k+1, fxl> .
Tg/ ] T(l+^-)
with its image
T/n
T/0 = T 2 , UT/n = T , T(A, B) = lim T/n (/I, S) .
A i T/n , B i T/0 ,
TU,
k
/"
1
S) = lim T/0U x 7 7 (l+y ) , B
—•
(an easy verification).
that
i=m+l
k
] [(l+U.)
If
Thus
"•
i=«+l
J
T(l, 1+7) = lim T/O(T~T(l+i/-) » 1+^
l
—•
W=l
>
•
Note
k
is a coproduct of
2
subobjects of
1 , one of which is
42
42
Peter
Peter Freyd
Freyd
one,
one,one
oneof
ofwhich
which is
is U.
U. ,,the
theother
other smaller
smaller than
than U,
U, .. Hence
Hence the
theunion
union of
of
((k k
))
the
images
of
T/0
the images of T/0MM [{l+U.),
[{l+U.), 1+V\
1+V\ is
is 11 ++ U.
U...
Suppose
Suppose WW>+
>+1+1/
1+1/€€TT were
were such
such that
that (l,
(l,U)
U) -- (l+V)
(l+V) .. We
We shall
shall show
show
that
•»•l+V
that there
there exists
exists W'
W'-»l+V with
with the
thesame
same property,
property, and
andsuch
such that
that
kk
3
V V<£
<£WW .. WW•*
•* l+V
l+V in
in TT appears
appears as
as AA xxYJ(l+U.)
YJ(l+U.) --3-- l+V
l+V some
some kk .. For
For
all
// cc BB all
n ncc Im{g)
all nn ,, 1+U
1+U
Im{g)cc 1+7
1+7.. Let
Let BBcc VV be
be such
such that
that ££
all «« ,,
but
)) .. Then
•»•l+V
ffff
but 1+5
1+5££ Im(
Im(
Then 1+5
1+5-»l+V as
asaa subobject
subobject in
in TT is
is contained
contained in
in
W W -*•
-*• l+V
l+V..
DD
3.3. Well-pointed
Well-pointed topoi
topoi
AAtopos
topos isis well-pointed
well-pointed ifif itit isis non-degenerate
non-degenerate and
and ifif 11 is
is aa
generator.
generator.
PROPOSITION
PROPOSITION 3.11
3.11 for
forwell-pointed
well-pointed topoi.
topoi.
( 1( 1, , fl)
fl) ==22 ; ;
1 1 ++11 ==ftft; ;
( l( ,l , - -) ) preserves
preserves ooproduot8,
ooproduot8, epimorphiemSj
epimorphiemSj epimorphia
epimorphia families,
families, and
and
pushouts
of
monomorphisms;
pushouts of monomorphisms;
AAf f 00=*=*AA isis
injeotive;
injeotive;
44^ ^00, ,l l= => >. .44 isis aa aogeneratOT.
aogeneratOT.
Proof.
Proof. For
For 44^^00 there
there are
areaat t least
least two
twomaps
maps from
from AA ttoo Q
Q ,, hence
hence
t ht he er er e e exxi si st st s 11-»•-»•44 . .
Let
Let UUcc l l . . IfIf UU$$00 then
then there
there exists
exists ll ++U
U and
and 11 •+
•+ UU ++ 11 =
= 1^
1^
forces
forces UU•+
•+ 11 t o
t obe
beepic.
epic. Hence
Hence ((l l, , fi)
fi) ==22 . .
a
11 ++11-*-*nn i si s always
always monic.
monic. Because
Because ((l l, , 1+1)
1+1) a ((ll,, ^^)) and
and 11
r er ef lfel ec ct st s isomorphisms,
isomorphisms, 11++11--flfl. .
For
For ( (11, ,4)
4) ++( (11, ,5)
5) ==((11, ,4+5)
4+5) use
useCorollary
Corollary 2.332.33C C •*•*11
Given
Given AA•*-*
•*-*BB and
and 11-+-+SS l leet t ++ 4-4- be
be aa pullback.
pullback. By
By Corollary
Corollary
AA-*-* BB
Aspects of topoi
2.32,
C •* 1
i s epic;
thus
Cf 0
43
and there exists
1 •+ C yielding
1
4*+ B
Given any family
can view
{A • •* B]
{A. -*• B]
collectively epimorphic, and 1—£-*• B , we
as a collection of objects in T/B which collectively
if
cover the terminal object.
that do the same.
Applying
Hence for some
f^ we obtain a collection of objects
i , / (A . •* s) ^ 0 and
{fl, 4 .) -*- (l, B)} is collectively epimorphic.
A x- B
+
4- we use the booleanness to write
Given a pushout
A •* A+A'
+
+
C * C+A
and we see that
Given
Then
B = A + A'
•*• D
C
is a pushout, clearly preserved by
A >+ B , A \ 0 , again write
B = A+A' •+ A+l •*• A
B = A + A'
(l, - ) .
and choose
is a right-inverse for A -*• B .
1 •+ A .
0
PROPOSITION 3.12. If 8 is a boolean topos then it is well-pointed
iff for all A (. 8 , A f 0 there exists 1 + A .
Proof.
f>g>
Let f,g:
Ac B
f t g . Let E e B be the equalizer of
the complement of E
then there exists
3.2.
B + C,
1 •+ B
in B . Hence if there exists
that distinguishes
f,g.
1 -»• A
•
The plentitude of well-pointed topoi
A logical morphiem of topoi is a functor that preserves all the
structure.
THEOREM 3.21.
there exists
preserves
a well-pointed
epimorphic
Proof.
a topos
& and logical
8
and A € 8 ,
r : B+8 ,
TA % 0 .
A \Q
T
show:
For every boolean topos
8' , logical
epimorphic families;
boolean topos
families.
We f i r s t
LEMMA 3.211.
exists
For every snail
T : 8 •*• 8' ,
and for all
B $. 8
8
and A i 8 ,
T(A) f 0 .
either
T
TB - 0
A J= 0
there
preserves
or there
exists
44
Peter Freyd
1 •*• TB .
Proof of lemma.
object.
Well-order the objects of 8 , taking
A
as first
We construct an ordinal sequence of topoi and logical morphisms as
follows :
If
T : 8 •*• 8 a
sequence at
and
property,
hand, there exist
5*8
terminate
the
a .
on the other
If,
has the described
( 1 , TB) = 0
then take
such that
B to be the first
TB \ 0
such and define
8
cc+l = V ™ •
If
3 is a limit ordinal, and 8 a is defined for all
a < 6,
then Bg is the colimit of the 8 's .
The essentially algebraic nature of topoi insures that
The functor
8 •*• 8
carries
C i8
such that
Moreover for every
Bg is a topos.
TB to an object with a map from
1.
8 (l, TC) * 0 , 8 ( 1 , T'C) * 0 and
eventually the sequence must terminate.
•
Now, for the theorem, define a sequence on the finite ordinals by
8
(as defined in the lemma) and
8
That
8 .
8 = lim B
8 •*• a i s logical and
is boolean because
implies the same in
n+l = K '
Proposition 3.12 says that
8
8 is well-pointed.
T preserves epimorphic families follows from the fact that
colimits of such functors are such functors.
COROLLARY
in a product
families.
1 + 1 - fl in
3.22.
D
Every small boolean topos can be logically
of well-pointed
topoi,
and the embedding preserves
embedded
epimorphic
Aspects of topoi
Proof.
iff
For exact
TA - 0 =» A - 0 .
45
T : 8 •* B' between boolean topoi, T
is faithful
D
Composing with Theorem 2.65 we obtain:
THEOREM
3.23.
Every small topos can be exactly embedded in a product
of well-pointed topoi and the embedding preserves epimorphic families.
Composing with
THEOREM
(l, -) and using Proposition 3.11 we obtain:
3.24. For every small topos T
there exists faithful
T : T ->• IIS .
T
preserves all finite limits, coproducts, epimorphisms, epimorphic
families, and pushouts of monomorphisms.
3.3.
Me ta theorems
By the universal theory of exactness of a category we mean all true
universally quantified sentences using the predicates of composition,
finite limits and colimits.
By the universal Horn theory of exactness we
mean all the universally quantified Horn sentences in the predicates of
exactness, that is, sentences of the form A
A A^ A ... A A
=» A
where
each A . says either that something commutes, or is a limit, or is a
colimit.
By theories of near exactness we mean those using the predicates
of composition, finite limits, coproducts, epimorphisms, and pushouts of
monomorphi sms.
As easy corollaries of Theorem 3.21 through Theorem 3.2U we obtain:
METATHEOREM
3.31. The universal Horn theory of near exactness true
for the category of sets is true for any topos.
The universal Horn theory of exactness true for all well-pointed topoi
is true for all topoi.
The universal Horn theory of topoi true for all well-pointed topoi is
true for all boolean topoi.
The universal theory of topoi true for all well-pointed topoi is true
for all boolean topoi in which (l,fi)= 2 .
We will show later that there do exist universal Horn sentences in
46
Peter Freyd
exactness predicates true for S but not true for all well-pointed topoi.
An equivalence relation on A
is a relation
E c AM
satisfying the
usual axioms.
COROLLARY
3.32 for topoi.
Every equivalence relation is effective;
E •* A
that is, given
E a A*A
there exists
A •* B
such that
+
A
4- is a
•+ B
pullbaok.
Proof.
We get rid of the existential quantifier by defining
to be the coequalizer of the two maps from
E
A •* B
to A . We note that the
statement is in the universal Horn theory of exactness and it suffices to
prove it in well-pointed topoi.
E'
->• A
Accordingly, let +
A
1 -*• E'
E .
be a pullback.
that can not be factored through
x, y : 1 •* A
in
+
such that
Let A'
If E % E' there exists
->• B
E •*• E' . Hence there exists
1 — ^ A •* B = 1 - ^ A + B
be the complement of
but 1
Im(x)' u Im(t/)
X
^
' A*A no 4
and define
1+1+1 by
A' •* A -£+ 1+1+1 = A' •+ 1
x
U
a
1 -^* A -iL* 1+1+1 =
^
E •* A -&-+ 1+1+1
contradiction.
• 1+1+1 ,
1 — ^ * 1+1+1 .
A
equalizes and there must be
• B
\^
/ ,a
1+1+1
•
COROLLARY 3.33 for topoi.
as:
2
1
A -2->- 1+1+1 =
Then
• 1+1+1 ,
Every epimorphism is a
coequalizer.
Proof. We again get rid of the existential quantifier by stating it
every epimorphism is the coequalizer of i t s kernel-pair. (The
E •* A
kernel-pair of A •* B is the pullback
+
+ .)
It suffices to prove i t
A •* B
in well-pointed topoi.
Given
A •*-*• B
let A •*• B'
be the coequalizer of
Aspects of topoi
E -*• A .
x'
1
I t s u f f i c e s t o show t h a t
B' •* B
u'
»• B' •+ B = 1 -a—> B' ->• B and find
1
1 —^+ 4 •+• B = x
E c 4x4
1
,
1 -&+ i4 -• B' = j /
x ' = z/1 .
and
1
47
i s monic.
Let
x , z/ : 1 •* 4
.
Then
a>y
1
so t h a t
» /5M
lies in
D
A «• B
COROLLARY
3.34 for topoi. If *
1 is a pullback and C+B •* D
C y- D
is epi, then it is a pushout.
Proof.
The statement lies in the universal Horn theory of near
exactness and it suffices to prove it in
S , an easy matter.
•
3.4. Solvable topoi
Given a topos
T
and
object if the maps from
1
A i T
we'll say that
to
are jointly epimorphic.
A
A
to be the full subcategory of well-pointed objects.
is a well-pointed
Define
T
c T
In Section 2.7 we
defined a solvable topos , which definition is equivalent to the
coreflectivity of 7"
PROPOSITION
Proof.
is
that
0 .
3.41
Suppose
Let
for topoi.
A, B € J
.
T
is closed under finite products.
We may assume that neither
/, g : A*B •+• C , f * g .
1 •+ XXB—•-• C + 1 •+ i4*B -^*" C .
/, g , and let
1 -2+ A
Because
there exists
B f 0
be such that
We wish to find
Let
f,g
A
1 •+ A*B
nor
B
such
: A •* CT
correspond to
1 — • 4 - ^ if * 1
• 4 - ^ C^ .
1 -• B , and hence
B •* 1
is epi, which
yields:
By following with the evaluation
Cpx-B •* C
we obtain
B •* A*B -£-*• C * B •* A*B -2-- C .
Finally, l e t
C .
D
1 -*• B be such as to separate these two maps from B t o
By a two-valued topos we mean
( l , fi) = 2 .
48
Peter Freyd
PROPOSITION 3.42
well-pointed
topos.
f o r solvable two-valued topoi.
Subobjects
of
T
objects
are in
P
T
is a
T
. hence
T ->• T
P
V
is exact.
Proof.
Because
T
is closed under products and coreflective,
exponentiation is effected by exponentiating in
Given
f : A •* B ,
collection of
B (. T
T/B-objects ,
, consider
{l •* B}
s ubt ermi nat or s to
A
only two subobjects,
Given
in
T .
A «• B
in
T
there exists either
3.5.
T/A .
let
A' •* B
Hence
j
The
applied to that
(Thus in any topos, the maps from
be the negation of
A
1
has
as defined
A u A' = B , for such implies that
T
.
1 + 1
It suffices to show that for any
l*^->-5 = l + S
(i4'ulm(l -»• B)) n A = 0
contradiction.
: T/B •* T/A .
A € T
satisfies the ft condition in
1 -* B
f
form a jointly epimorphic family. ) Because
We wish to show that
neither, then
and then coreflecting.
are such that their maps to the
terminator form a jointly epimorphic family.
family yields a similar family in
T
or
1 + A'-+B=1-+B.
and the maximality of
A'
If
yields a
•
Topoi exactly embeddable in well-pointed topoi
We have shown t h a t every topos i s exactly embeddable in a product of
well-pointed t o p o i , and a residual question presents i t s e l f : which are
exactly embeddable in a single well-pointed topos? Because of the
elementary nature of the l a t t e r (they are closed under ultra-products) , we
are asking which topoi have the universal exactness theory of well-pointed
t o p o i . For example: (U H- l ) =» (0 •** U) v {U ->->• l ) . That i s , such topoi
must be such that ( l , ft) = 2 .
I t suffices to show for each n
f
X'Bl.
B
„
f
2'g2.
B
and
A
f
n*
i = 1, 2, . . . , n
t h a t there i s a well-pointed topos
T'
and exact
T : T •* T'
such that
Aspects of topoi
(/"„•) * T(3J) » i = 1, 2, . . . , n .
and
E. c A
E. = 4. x ... x A . . x E\ x A.
f.,g.
•
Let
E = E
I
T : 1 •* T ,
T
T[fi) * T{9i)
.
(Ultra-products again).
the equalizer of
... * A
u ... u E
.
If
where
E + A
Let
A * A.
E.
is the equalizer of
we know that there exists
71
J
1
49
well-pointed, such that
T{E) * T(A) and hence
It is a statement in the universal exactness theory of well-pointed
topoi that if each E'. t A . then E # A . Hence such is a necessary
Lr
If
condition for exact embeddability into a well-pointed topos.
it elementary by noticing that the case for arbitrary
case
follows from the
n = 2 :
If
(A'*B) u (AXB')
= A x B
then either
A' = A
Further reductions can occur by replacing
A'
of
n
We can make
A
or
with
B' = B .
A/A' , the pushout
•* A
+
1
If
(l*B) u (Axi) = A x B
Together with
(l,fi)= 2
then either
A = 1
or
B = 1 .
this can be seen to be sufficient.
Finally, if one considers
l/T , that is, the category whose objects
^A
are of the form
1 •* A
and whose maps are of the form
1
+ , define
^
(1 •+ A) v (l •* B)
cokernel of
as the coproduct in
l/T ,
(l •* A) v (l •* B) •* (l •*• A*B) ,
B
(l •+ A) A (l •+ B)
(-A-
as the
has a right-adjoint,
namely exponentiation), then the condition for exact embeddability into a
well-pointed topos is that the half-ring of isomorphism types with
addition and
A
Let
L
n-ary
The first order calculus of a topos
be a vocabulary of predicates and operators;
L
as
as multiplication is without zero-divisors.
4.
objects of
v
are either pairs
predicate,
/
an
w-ary
<P, n)
or
operator.
<f, n)
that i s , the
where we call
P
an
50
Peter Freyd
An interpretation
P a Bn
subobject
of
L
for each
in a topos
T
is an object
(P, n> i I , a map
B € j , a
J : B™ ->• B
for each
< / , n> 6 L .
Given any derived
L
n-ary predicate or operator using the vocabulary of
and the classical logical connections and quantifiers, we wish to
stipulate a subobject of
Bn
.
The definition is recursive.
defining maps from expressions in the
two
n-ary predicates
easily define
PTQ
Given an
each
P, Q
,
L-operators are well known.
P v Q , P -* Q
m-ary, we define
Bn .
as a subobject in
P[x-., •••, x )
and operators
is the obvious.
Given an
3
where
Bn
P+
tT -»• B
f , ..., f
• + if
as the pullback +
•)-
P(f , . . . , f )
Given
Sub(s n ) , we can
already assigned values in
n-ary predicate
The rules for
n-ary
P[x., ... , x )
modeled as
P[x , . . . , x )
as the image of
P c B
P ->• B™ •+ Bn~
, we define
.
For
V
we need:
n
PROPOSITION
4.11
exists a maximal
Proof.
S'+B
We define
for topoi.
B' c B
V
is
Given
such that
g
l(i)'-»j().
9
P{x , ... , x j
n
inverse image is contained in
g : A •* B
(B' ) c A' .
A' c A , there
B'
is a model of
T
whose
P .
B
S
(that is, no
, that is, an element of
We shall call such the truth value of the sentence,
T
B1
as the maximal subobject in
unquantified variables) a subobject of
Given any set
is called
•
In this manner we obtain for each sentence
(1, fl) .
and
Such
t(5) .
of sentences, we say that an interpretation of
if every sentence in
T
has truth value
1
• £2 .
L
Aspects of topoi
(Yes, that's why it's called
T
we say that
S^
implies
of
5 2 , and denote same by
S-
in
T
same by
S3. K
S~
S
Given sentences
in
T
^
.
Note that
S^
L
S
1= Sp .
^-r S
We say that
if for all topoi
L ,
S., S^
T , S
H=y S
S.
is a model
t(s,) - £ (<S2) » and denote
1 Hj- (s
we say that
.
and a topos
S-. strongly implies
is equivalent to
and sentences
S,, Sp
if every model of
if for every interpretation of
Given any language
implies
"t" .)
51
S
•* S ) .
semantiaally
We denote same by
52 •
PROPOSITION
4.12. If S 1 !=„ S 2
Proof.
B (. J
Let
t[s ) ^ t(s 2 ) .
model of
S
Then in
.
then for all T,
be an interpretation of
T/i(S )
L
we obtain a model of
SX^TS2.
in
T
and suppose
S
that is not a
G
The definition of semantic implication reduces a host of assertions in
intuitionistic logic to exercises in classical logic:
PROPOSITION
1=%
T
4.13.
For fixed
L
h*
3
obeys Craig's interpolation theorem.
If every finite subset of a given
has a model in some topos, then so does
For each monoid
is recursively enumerable.
T .
D
M , we obtain an intuitionistic logic
H „ .
We
suspect that the connection between such logics and classes of monoids will
be a fruitful pursuit.
Problem:
For every topos
H" ,. for some monoid
4.2.
T
is
\=j the same partial ordering as
M 1
The boolean case
If we restrict our interpretation of a language to boolean topoi, we
can replace
that
v, A, 1
P •*• Q
and
with
3
~1 P v Q
and
V
with
1 3 1
.
The advantage is
are all definable using only the predicates of
52
Peter Freyd
near-exactness.
(p = "I Q i f f
P + Q •* EF i s an isomorphism.)
More p r e c i s e l y , l e t
L = {<P 1 , n>, . . . , <P a , n a >, < / 1 , jn1>
Let
Jg
be the set of interpretations of
elements of
Jc
o
L
</ & , m^ >} .
in a boolean topos
are of the form
{B, P 1 <= B X, . . . , Pa c B a, f± : B X - S
^ : Bb + B) .
PROPOSITION 4 . 2 1 . Given any elementary sentence
universally
quantified
Horn formula
auah that for each boolean
F{B,P~X, ...,Jb)
B and (B, P , ....
iff
there is an exietentially
predicates
G{B, P , ...,
<B,P±,
S
F in the predicates
D
there exists
of
f, ) i J_
1
that
8 . The
a
near-exactness
it is the case
D
.. . , Jb > is a model of
S ,
Also,
quantified conjunction of
near-exactness
f^) with the same property.
Proof. There is a tree. Its root is 5 , each branch a wff (well
formed formula), each leaf a single variable, each branch point (we'll
allow degenerate branch points) marked with either
P , . . . , P , / - , . . . , / " , , A, "1 , = or < 3, n > ; and, if a branch point is
marked
P.
expressions
then
g,,
n.
branches lead into i t , they are all operator
... , g
and the branch leading out is.
i
P.\g , . .., g
I ;
if the branch point is marked
lead in, they are all operator expressions
/.
then
g.s ..., g
m.
branches
and the branch
i
leading out is
f.\g,, ..., g
I ;
if the branch point is marked
two branches lead in, both are predicates
out is
P A Q ;
P, Q
if the branch point is marked
in, it^ is a predicate
P
and the branch out is
g = h ;
~l then one branch leads
"IP;
if it is marked
g, h , and the branch
finally if the branch point is marked
then one branch leads in, it is a predicate
then
and the branch leading
= , two branches lead in, both operator expressions
leading out is
A
P{x , ..., x^)
< 3 , n>
and the branch
Aspects of topoi
leading out is
Let
of
F
JC
a
P
K
and
3
P[x. , . . . , x ) .
n
be the number of variables in
G
53
are defined as follows:
S .
The quantified variables
for each variable in
5 , a map
i
•S ;
•* B
g xB
for each branch other than a leaf, we introduce a variable
if the branch is an operator expression, a variable
the branch is a predicate.
new variable
To each leaf marked
x.
P •+ B
if
we make correspond the
p. .
For each branch point we define a near-exactness predicate as follows:
If a branch point is marked
P.
then let
g , ..., g
variables corresponding to the incoming branches;
and let
A
be the
P •*• b
to the outgoing,
say that
n
_
•
P. •* B v
is a pullback.
If a branch point is marked
/. , then let
g^ , ..., g
be the
1
variables corresponding to the incoming branches;
outgoing and let
A
g :B
•*• B
say that
^- BZ
g = ET
If a branch point is marked
A , then let
^* B .
P •* a
,
variables corresponding to the incoming branches : R •* a
and let
A
to the
say that
JC
R + a
is a pullback of
_
^
+ .
Q •* B^
be the
to the outgoing
54
Peter Freyd
If a branch point is marked
incoming branch;
R •* B
~1
then let P •* IT correspond to the
is the outgoing, and let
A
say that
_
4-
is
a coproduct.
If a branch point i s marked
= , then l e t
correspond to the incoming branches,
say t h a t
P •*• B
P •* a
i s an equalizer of
If a branch point i s marked
t h e incoming branch,
Q •*• a
g : a
-*• B ,
~h : ET •*• B
t o the outgoing, and l e t
A
g, H .
< 3 , n ) , then l e t
P •*• B
t o the outgoing, and l e t
A
correspond to
say t h a t
is a pullback.
Finally, let
S •*• a
be the variable assigned to the root.
F = V [ A / •• (S = / ) ] . G = 2{/\A A (S = / ) ] .
COROLLARY
boolean
4.22. If a theory has a model in any non-degenerate
topos, it has a model in sets.
Proof.
?y Theorem 3.23 there always exists a near-exact functor into
sets, which functor must preserve
COROLLARY
Proof.
G .
4.23. For any boolean
Suppose that
an interpretation of
5
D
L
D
B
if S^ l=s S2
then
S^ £ g 5 2 .
S h o S^ but not S1 ^ g S 2 . Then there exists
in B/t (s )
that is a model for S^ but not
. Let t(S 2 ) be the truth value for Sp , and reflect the
interpretation into
becomes an iso.
Corollary U.22.
Sh . for j
3
the closure operator such that
We then have a model in Sh . of S
•
A ~1 S
0 -»• t{S^\
. Then apply
55
Aspects of topoi
Of course
H ^ is classical logic.
In a later section we find
necessary and sufficient conditions on 8 so that
5.
A pair
1 —^* N ,
short, if for every
NNO's
1
Arithmetic i n topoi
N -^-»- K i s a natural
x
•A
N=g coincides with
t
•A
numbers object,
t h e r e e x i s t s unique
or
N -*• A
NNO for
such t h a t
a r e c l e a r l y unique up t o unique isomorphisms.
PROPOSITION
5 . 1 1 . If
1 -2+ N - ^ N is an
NNO then
1°
: 1+N ^ N
is an isomorphism.
Proof.
Define
1
L e t w . : 1 •+ 1+N a n d w_ : N -*• 1+N b e t h e c o p r o j e c t i o n s .
e ' : 1+N •> 1+N b y us'
•N
og = u
= CM
,
u s ' = ew
*• N i s an NNO , t h e r e i s a u n i q u e
a n d sg = gs'
.
We c l a i m t h a t
g
.
Since
g : N •* 1+N such t h a t
i s t h e inverse of
: 1+N - N .
That
of
g\A
V j
=1
follows from the uniqueness clause in the definition
NNO applied t o the commutative diagram
8
N
3
1+N
1+N
o
8
Peter Freyd
56
That
\ \9 = 1
is
equivalent to og = u
and eg = u . The first we
have, and the second comes by applying the uniqueness clause to the
commutative diagrams
N
N
1+N
-T+
N
1+N
1+N -jr- 1+N .
O
8
PROPOSITION
aoequalizer of
Proof.
if
l
N
is an UNO
then N •* 1 is a
s and !„ .
N •*• 1
to show that if
that
5.12.
is epic (because there exists
N -^-* N -*-+• A = N -^-* A
N -£+ A = N •* 1 -»• A .
Define
1 •*• N ) and it suffices
then there exists
1 ~^* A = 1 - ^ N -£+ A .
N
1 -*• A
such
Then both
N
and
N -2- N
A -r* A
and the uniqueness condition on
KNO's
yields
N -"*-»• <4 = TV ->- 1 -^
A .
We shall show that the exactness conditions of the last two
propositions characterize
NHO's
in topoi.
In the last section we
observed that near-exactness conditions are equivalent to stating that
something is a model for a finite theory.
Here we will see that the
addition of a single coequalizer condition can yield a categorical (in the
old sense) definition.
C
Aspects of topoi
57
We consider first the category of sets:
PROPOSITION
coequalizer of
5.13
s
for sets.
and
A -s* A
If
1. , then either
rl
A - N ,
s = s .
is monio and
A - Z
,
A •*• 1
s{a) = a + 1
is a
or
Yt
D
Hence, if we add the requirement that
s
is not epic, we can
characterize the natural numbers.
Let
is an
T
be a well-pointed topos,
NNO
in
T , then
Ts
1 * 2W
T
*S
• TN
exact.
If
is the standard
1
NNO
•N
in
•N
S .
(Any exact functor from a well-pointed topos to a non-degenerate topos is
faithful.)
Suppose then that in
no natural
n
does
can't exist.
there exists
1 —°->- N -^* N —^+
We obtain such
ultra-power of
T
T
1 —^* N
. . . -^* N = 1 -^+ A? .
pointed topoi.)
Then
T
T'
and
is faithful.
T : T -+T'
(l + 1
suppose
preserves monomorphisms if it is faithful.
1 - ^ N-^
N
is an
C o e q m i J , 2"(s)
Proposition 5.13.
NNO
T
and
= TlCoeqfl^, s)
Thus for
T : T •*• S .
PROPOSITION
in
5.14
T
T(l) = 1 ,
7(1) = 1 ,
, then
is as described in
(the scarcity of right-exact functors).
S , then for every
example, right exact
T ), it is the case that
T : T -*• S
If
T
is
such that
C o e q m i f f ) , T(s)\ - r(Coeq(l, s)) , (for
Suppose for some
View
5 , there
We can go one further step:
and
and define
r(l) + T(l) = T(l+l) ,
1 •+ TN + TN
T(l) + T(l) - 2"(1+1)
r(l) k 0 .
Hence if
a non-principal ultra-power of
a non-principal ultra-power of
1 * T(l)
For
Because non-zero objects in well-pointed topoi are
T
Proof.
T .
is a cogenerator for well-
injective,
1 * 4 ,
3? just
simply by taking a non-principal
T , non-degenerate
17(1+1) = 1 + 1 .
is no such
Then
S .
In fact, we are using very little of the exactness of
well-pointed
such that for
T
4 € T ,
T = 0 .
TA \ 0 .
Then because there exists
as a functor with values in
T' = T •* S/T(l) * S/l .
T'
S/T(l) , choose
may be alternatively
58
Peter Freyd
described by the pullbacks
colimits as T , and
exist. Nor can T .
T' (B) •*• T{B)
+
+
.
1
•* r(l)
T' (l) = 1 .
•
T'
preserves at least the same
Hence by our remarks above,
T'
can not
In particular, Theorem 3- 2U can not be improved to make T exact.
Moreover, we have laid to rest the idea that the exact embedding theorem
for abelian categories has a nice generalization to "base" categories other
than abelian. For:
COROLLARY 5.15. No set of elementary conditions true for the
category of sets implies exact (even right-exact) embeddability into the
category of sets.
(Even if you add countability.)
Proof. We can take the complete elementary theory of S , and let T
be an elementary submodel of a non-principal ultra-power of S and apply
Proposition 5.11*O
The embedding theorem for abelian categories was motivated by the
consequent metatheorem for the universal theory of exactness. The latter
can be true without the former. But, alas, not for topoi. We need, f i r s t ,
a b i t more about NNO's .
5.2.
Primitive recursive functions in topoi
PROPOSITION
for every
5.21
for topoi.
If
1 - ^ N -2->- N is an
A —^->- B and B —• B there exists
unique
A
Proof.
Transfer the problem to solving for
NNO ,
then
A*N •+• B such that
59
Aspects of topoi
N ——• N
D
For the standard natural numbers in
g : A •*• B ,
h : Axflx-B -*• B
S
we know that given
there exists unique
/ : A*N •* B
such that
f(a, 0) = g(a) ,
f(a, i/+l) = h{a, y, f(a, y)) .
(Usually
A
is a power of
PROPOSITION
g : A •* B ,
5.22
N , B = N .)
for topoi. Let 1 -^* N -2+ N
h : A*-N*B -*• B
given.
be a
There exists unique
NNO ,
f : A*N •* B
such
that
A •+ A * l
A*N -J-* B = g ,
A*N
Proof.
k
(Notation:
B =
for any
P±,P2,f
X , X ~^+ N
B .
means
X •*• 1 -2-»- N .)
be such that
k .
(5-231)
as insured by Proposition 5.22.
lt
p28, h>px = (lxe)kp1
We will show that
and
kp~
works.
< 1, o ) ^ = 1 , hence
First:
Let
60
Peter Freyd
AxN
p
works as well as kp. and Proposition 5-22 says, therefore, that
Second:
kp2s = k(p±,
p2s;
h>p2 = {l*s)kp2
and
< 1 , o)kp2 = o
hence
lxs .
N
•- N
s
and
kp2
p2
vrorks as well as
fcp
and again Proposition 5.22 says t h a t
= p2 .
F i n a l l y , for the existence of
k = < px,
p2,
/>
/
, define
/ - kp. .
Then
and
< 1, o >/ = < 1, o >kp3 = < 1, o, g >p3 = g ,
(1*8 )f = (l><s)fcp3 = k<pv
For the uniqueness, suppose
Define
5.22.
h)p3 = < P l , p2,
f)h
.
is as described in the proposition.
k = <p , p , /> , verify that (5.231) commutes and use Proposition
D
Note that
given
f
p2s,
h need not "depend" on A
or N , or B . That is,
h : A*B •* B we could define fc1 : <4XWXB •+ B = AxN*B •* A*B
>• B
and apply the proposition.
Thus we can define on any NNO in a topos
a, m, e : N'X-N -*• N by
Aspects of topoi
61
N = N*N - ^ N - ^ N ,
=
o
= o s , tf xtf - ^ + ffxtf -^+ N = N*N
In the category of sets
exponentiation;
a, m, e
—
•- N*N -^*
N .
are addition, multiplication and
that is,
x + 0 = x , x + (t/+l) = (*+i/) + 1 ,
x • 0 = 0 , x • (j/+l) = (x'jy) + x ,
x° = 1 ,
Take any elementary sentence
x(i/+l) = ( * » ) • * .
5
in the operators
Add to it the six equations above which define
o, s, a,tfi,e .
a, m, e
on
N .
We saw in
Proposition It.21 that there is a universal Horn exactness predicate that
says that a given
<N, o, 3, a, m, e ) is a model of
O
formula to include
[ : 1 + N - N
and
5 .
Coeq(l, s) = 1 .
Enlarge that
The universal
quantification of that formula then says that the arithmetic of the
satisfies
S .
THEOREM
conditions
T
NKO
Hence,
5.23.
FOP any recursively enumerable set of elementary
T , true for the category of sets, there exists a model
T
of
and a universal Horn sentence in the predicates of exactness, true for
sets but false for
Proof.
T .
Add to
T
the axioms of a well pointed topos with an
NKO .
By Godel's Incompleteness Theorem we know that there is an elementary
sentence
5
true for standard arithmetic, whose translation,
S' , into a
universal Horn sentence in the predicates of exactness is not a consequence
of
T
(else number theory would be decidable).
Godel's Completeness Theorem implies that there is a model of
T u D S1 } .
5.3.
D
The exact characterization of the natural numbers
PROPOSITION
5.31
for topoi. If a topos has NNO then
62
Peter Freyd
1 -2+ N -£+ N
is an
NNO
Coeq(l, s) = 1 .
Proof.
iff
: 1 + N •* N
P
is iso and
D
Let
1 -^-* A -£•*• 4
be such that
I*
is iso i
V.*J
Coeq(l, t) = 1 .
1 - ^ ff - 1 * if
Let
be
NNO
and
let
N -^* N
Q,
• A
A
t
commute.
We wish to show
First, f
is epi.
theory, namely that
imply
/
is epi.
Accordingly let
to show that
We have a universal Horn sentence in exactness
isomorphs and
Coeq.(l, a) = Coeq(l, t) = 1
It suffices to prove it in well pointed topoi.
A' = Im(/) and A"
the complement of
t{A") c A" , for such allows us to "split"
A' * A , then
A' . It suffices
t as
t\A' + t\A"
Coeq(l, t) = Coeq(l.,, t\A') + Coeq(l „, t\A") .
A" # 0
suffices to show
Let
an isomorphism.
[° , I*
and obtain a splitting
If
f
and
Coeq.(l, t)
is bigger than
1 . Hence it
t(A") c A" .
t' = t\A' , 1 -±—+ A' = 1 - ^ N •* A' . Then
universal Horn sentence:
" I f A' + A" - A
r,
I5 ;
is iso. The
a n d I , i I, I.I i s o t h e n
V- ) \t)
t(A") c A" " is in the predicates of near-exactness and it suffices to
prove it in
S , an easy matter.
Second, /
is mono.
1
can show t h a t
o+
N
Using just that
•1
A
~T
+ and
and
I
are isomorphs we
N -£+ A
S+
N
+t
A
are pullbacks since such are
y*
sentences in the universal Horn theory of near-exactness and it suffices to
prove them in
S .
Aspects of topoi
Let
Q >* A
63
be as described in Proposition 2.21, that is
A
i s a pullback, and for a l l
X -*• A such that
X + Q+A=X
+A .
there exists
1 •+• Q •* A = 1 — ^ + A .
+
-A
A.
+
+ i s a pullback,
//
there exists
+
N
• A
Then b e c a u s e
o+
+x
N -* A
i s a pullback,
Because
Q -^ Q
4+
is a pullback, there exists
a
N
>• A
It
*• A
Q -&•*• Q
t
Q -**-•• Q + A = Q •* A
1
Q
+ .
+
"~~~~> N
that
such that
Q^+
•A .
Because
Because
1
• It
Q
+
N
+ i s a p u l l b a c k we o b t a i n
• A
>• If i s a
NNO we o b t a i n
It •*• Q such
•• N
N -*• Q •*• N = 1
Hence so is
/ .
.
That i s
Q •*• N
i s e p i and
Q •*• A
i s an i s o m o r p h .
D
Thus, exact functors between topoi with
NNO's
preserve
NNO's .
Hence,
THEOREM
with
5 . 3 2 . If
NNO'e , then
Proof.
1*1'
the elementary
ie exact
for
1,1'
well-pointed
of
1, 1'
Combine Propositions it.21 and 5.31.
D
We can push a bit further.
arithmetics
topoi
coincide.
By the existential second-order arithmetic
of a topos , we mean the second-order sentences in arithmetic in which a l l
second-order quantifiers are existential.
that the truth of such a sentence in
sentence in the exactness theory of
PROPOSITION
well-pointed
5.33.
Propositions 1+.21 and 5-31 say
T is equivalent to an existential
T .
The existential
topos is determined by its
Hence,
second-order arithmetic of a
universal Horn theory of
64
Peter Freyd
exactness.
5.4.
O
Inferring the axiom of infinity
PROPOSITION
A'
5.41
for topoi.
Given
1 -*-A
-^+A
there
exists
V- A and
xy
A' -£-+ A1
i
A
t. A
A -J-"A
,[ : 1 + A' -*• A'
is epi.
Before proving Proposition 5.1*1 we show its consequences:
PROPOSITION
and
5.42 (the Peano property).
Coeq(a;, t) = 1
t(A' ) c A'
Proof.
i t is
then for every
the case that
A' c A
If
[x. : l + A •* A
Xt)
such, that
is iso
Im(a;) c A' ,
A' = A .
The sentence i s in the universal theory of exactness and i t
suffices to prove i t in well-pointed topoi. By applying Proposition 5-^1
x'
t'
(x1)
to
1
• A'
• A' we can assume t h a t
I , , I is epi.
(x)
Let A" a A be the complement of A' . Using just that
is i sio
\v)
and
K , I is epi we can show that t{A") c A" in S , hence everywhere.
I* )
Thus
t splits as t' + t"
(t" = t\A") and
Coeq(l, t) = Coeq(l^ ( , t'} + Coeq(l^,,s t") . If A" f 0 then
is bigger than
THEOREM
1 .
Proof.
5.11, 5.12.
•
5.43 for topoi.
: 1 +N + N
Coeq(l, t)
1 -s- N-$->• N
is an NNO
iff
and Coeq(l, s ) = 1 .
The necessity of the exactness condition was Propositions
For the sufficiency, note first that the Peano property above
yields the uniqueness conditions, that is, if both
/, g : N •* A
were such
65
Aspects of topoi
that
A
•A
t
then the equalizer of
f, g
would satisfy the hypotheses of Proposition
5.k2 and hence would be all of
N .
For the existential condition, let
epi , maps
A •* N , A •* B .
Again, the Peano property says
1 —**-»• B
•B
It suffices to show that
A -*• N
is epi.
A -*• N
is iso.
We need:
5.431. If [tXJ : 1 + A — • A is epi,
\ .)
LEMMA
be given, and apply
[° | : 1 + N ->• N iso,
Xs)
Coeq(l, s) = 1 j and
f
f,
N —• N
s
then
A
^s mono.
Proof.
The sentence is in the universal Horn theory of exactness and
it suffices to prove it in well pointed topoi.
kernel-pair of
that is,
N' .
We need
Note that
to
/ ,
E' = E - A ,
E' - 0 , equivalently
Let
E c A*A
N' = Im(E' * A •+ N) .
N' = 0 .
Let
N"
be the
We need
E = A ,
be the complement of
N" = N .
1 -^+ N
factors through
iff
g
!i factors through
A .
We may verify that
1
N
does, then so does
1 ~^->- N .
hypotheses of Proposition 5.U2 and
N"
That is,
N" = N .
•
N"
has a unique lifting
//' and that if
satisfies the
66
Peter Freyd
THEOREM
5.44 for topoi. The following are equivalent:
(a) there exists an NNO ;
(b) there exists- a monomorphism A >*• A and a map 1 -• 4 such that
0 -• 1
+
A
+ is a pullback;
•* A
(a) there exists an isomorphism 1 + A -A .
Proof.
to obtain
Clearly (a) =» (c) =» (b).' For (b) =» (a) apply Proposition 5.1*1
A'^-A'
1
( ,)
0
f , : 1 + A' -* A'
hence
,
c
l;••}
)
Let
epic.
It is clear that
A' -*• C be a coequalizer of
i
,/
• 4 ' -»• C .
and t h a t
C
along
+i' is a pullback,
is monic, and an isomorphism.
1 -»• C = 1
N,
• 1
+
I .i
l t
1 •+ C t o obtain
:
iV
and define
i4' ->- C1 as an object in
View
J
1 . , , t'
1 + 4 ' = A'
»• N i n
remains t r u e in
T/l .
If
Note t h a t
T/C .
Pullback
We maintain t h e coproduct
conditions and gain the coequalizer condition.
THEOREM 5.45 for topoi.
T/C .
•
( l , n) = 2 then either
there exists
an
A
A
: A •* A
is
NNO or every mono-endo is auto and every epi-endo is auto.
Proof.
Suppose t h a t
U
mono and if
+
A
A
"
f : A >+ A
i s not e p i .
X
+
A
•*• A
is a pullback where
A
1 -»• A
Then
A
j
corresponds to
Aspects of topoi
A — • A , then
11 •* 1
can not be iso.
67
Hence
11=0 and we can apply the
last theorem.
If
g : A •**• A
A* : / - /
.
is not mono, we can repeat the argument for
D
The proof of Theorem 5.1*1 is fairly easy in solvable topoi (see 2.7).
We can there take
such that
A' as the smallest subobject such that
t(A' ) c A' and
lm{x) c A' .
Remarkable enough, even without solvability we can construct
A' , not
as an intersection but as a union.
Proof of Proposition 5.41.
We define
B e !T to correspond to the
family of subobjects of A
such that
US
as the equalizer of the identity map and
works.
where
Define
B c ST
Px"l : #4->-tf4 =
c h a r a c t e r i s t i c map of
For any
B c fi^ i f f
tf4->-l-*-£?4
1
A' c Jm(x) u t(A' )
and
and show that
1 -»• fl4 corresponds t o
A •* fi t h e
•4 .
A' c A , the corresponding map
4 ' c Im(ar) u t ( 4 ' ) .
such remains the ease after
1 •*• u
factors through
Moreover:
application
by any logical morphism.
Let
C — •
B*A
be a pullback.
C"
For any 1 •+• B
C
c Im(x) u
let
•* C
+
+ be a pullback.
1 •+ B
Then
C •*• C •*• A
i s mono and
t(C')
and euch remains the case after application by any logical morphism.
68
Peter Freyd
C' •* 1*4
The proof of this is obtained by noticing that
+
C
+
is a pullback,
•* B*A
hence
C'
is a pullback, where
• 1M
1 —£»• Sr = 1 •+ B •* £r . But (/"xl)e = A -&-* Q the
C ->• 4
map which corresponds to / . +
Ij is a pullback.
1 -0
characteristic map of C' corresponds to a map
through
B
proposition.
Im(C -*• A)
works as A' as demanded by the
The reversal of the above paragraph shows that given any
such that
C' c Im(cc) u t{C' ) we can find
C
+
•+ C
I i s a pullback and
1
-»• B
that
which factors
and C' c Im(x) u t(C) .
We wish to show that
C' >+ A
1 •+ u
Thus the
Im(C ->- /I) c Im (c •+ A
C" •*• C -*• A = C' «• A .
1 •*• B
such that
We wish t o show,
first,
>• A) , equivalently t h a t in the pullback
P •*• 1 + C
1+A
P -*• C
is epic.
(This last "equivalently" is a pair of sentences in
the universal Horn theory of near-exactness and may be verified in sets.)
Suppose
P •*• C
is not epic.
We may transport the entire situation to T/C
P
and obtain a map
1 -*• TC such that in the pullback
1
not epic.
•* TP
+
+
,
P
-* 1
is
•* TC
Because all our constructions are preserved by logical
morphisms , we may drop the "T" and show that for any 1 -»• C
and pullback
Aspects of topoi
69
I+C
1+A
1 + C+ A ,
+ 1
is epi.
Let
+
1
4- be a pullback.
•* C +
Then
B
P~
• 1+C
1+A
1
is a pullback.
Let
4-
4-
-+ C' + A
be a pullback.. It suffices to show that
P2 + 1+C
P
+ P
+ 1
Let
is epi.
4C +
4A
be a pullback.
P
pullback.
P- + 1
is epic.
But
pullbacks and we have shown that
Call
that
is epic.
+ 1+C
P
4-
and
Let
C' c lm(x) u t(C')
A'
and
4-
it is the case that
C
4-
Cc
-*• A) .
C' c A
such
hence
But note that for
Im(x) u t(C ) . Hence
t(A' ) c A' , yielding
we have
be a
are both
+ A = C' + C + A
is the maximal such subobject.
A' -£-* A'
4-
AA
C' = Im(x) u A'
41+C'
Im(C + A) c Im(x) u Im(C + i4
= A' u t(A' ) it is the case that
A u t(A' ) c A'
4-
+ 1+C'
A' = Im(C + A) . We remarked above, that for any
C' c A' . Thus
C
4-
P, •+ C'
C' c Im(x) u t(C ) and
4- . For
> A
t. A
I m U ) <= A' , yielding
70
Peter Freyd
A' -•£-+ A'
s
1
X
I—
Finally,
5.5.
A' c Im(z) u t{A' )
K
: 1 + A' ->•+ A' .
/ , g : A •*• B
let Rf
= Im(A
^ ? f ? > B*B) C flxg and
c S X B the kernel-pair of the coequalizer of f, g . $y Corollary
3.32 s fi'r. is the smallest equivalence relation containing
Suppose
induced by
B —*• C
is epi.
+
B
+C
a pullback.
f, g
iff
E.
Then by Corollary 3-33,
= E. .
Hote that
Rf
and
are defined using only near-exactness.
B
is
B^
Let E-, c B*B be the equivalence relation
h , that i s , +
B - ^ - C i s a coequalizer of
ff,
Q
One c o e q u a l i z e r f o r a l l
Given
£•„
directly yields
• C ie a coequalizer
the smallest
equivalence
of
f , g : A + B iff
relation
containing
relation containing
preserves the
i? c B'x-B define
R .
is epic and EV
/?„
j
In general, given
h
is
Hi? to be the smallest equivalence
Then a near-exact functor is exact iff
it
= operator on binary relations.
Given any
R c A*C we can define a transformation
P •* Q
R e l ( - , A)
• R e l ( - , C)
as follows:
for 8 c W l e t +
I
be a
R •* A
pullback and send
oR : rf4 •+ if
.
R c B*B define
(if
Q to
Im(P •+ X*-C) .
°i? i s natural and there exists
R i s the graph of a map / , then
H c fixfl as
°R = 3
A u R u Im (i? •+• B*B —1-+ B*B) where
.)
Given
T i s the
Aspects of topoi
twist map, and define
k : N -*• H
U
B*B
°R
B*B
• si
be the map such that
71
as above.
1
In a topos with
•N — •
ft
NKO let
corresponds to
B - ^ + Bx-B a n d
N
IV
k
I I
r
If
°R
We o b t a i n a r e l a t i o n from
We may look a t
N
to
B*B ,
Q c tfxgxs ,
Q backwards as a map from
=/?
to
Q
In w e l l - p o i n t e d t o p o i , a t l e a s t , we can d e f i n e
t o t h e non-empty s u b o b j e c t s of
l e a s t e l e m e n t s , and o b t a i n
N
and d e f i n e
=R -*-•• N .
A —>• =R
+
+
i s a pullback
and for
1- N
1
f
(fl )
S'
+
+
+ ,
—>• N
o
.
[U )
•*• N
to
to
properties:
>• =ff
+
1
a;
correspond
t o correspond
has t h e f o l l o w i n g
S —»• 3?
1
Im(Q •+ B*B) = 3? .
pullbaaks,
• A?
xs
S' = (5o?) - S .
These two properties, entirely in the language of near-exactness
except for
N
itself, characterize
PROPOSITION
5.51. If
=R . Hence:
j1 : T + T 1
i8 a near-exact functor of
well-pointed topoi that preserves epimorphic families, then
iff it preserves the coequalizer of
1^ and
s .
T
is exact
•
Moreover given any diagram in a well-pointed topos, we can enlarge it
and add
1 -^+ N
•N
and translate any coequalizer condition into
near-exactness conditions on the enlarged diagram.
We can know the
universal Horn theory of exactness of
T
elementary theories have models in
in which one of the stipulated sorts
T
if we know which two-sorted
is the natural numbers.
If
T
is a well-pointed topos with the axiom of choice, that is every
object is projective, then by a Lowenheim-Skolem argument we can reduce the
two-sorts to one and obtain the converse of Proposition 5-33:
72
Peter Freyd
THEOREM
5.52.
The universal Horn theory of exactness of a
well-pointed topos with
NNO
and axiom of choice is determined by its
existential second order arithmetic.
5.6.
D
A standard recovery
PROPOSITION
5.61
for topoi. If T
is such that for every R c B*B ,
=/? is the union of the sequence A, R, R , ..., through the standard
natural numbers, then T
Proof.
may be exactly embedded in a power of S .
By Theorem 3.2U we know that there is a collectively faithful
family of non-exact functors into S , each of which preserves epimorphic
families, hence unions.
Hence the operation
R i—• 3?
is preserved and by
our remarks in the last section, such functors are exact.
COROLLARY
5.62.
•
Countably complete topoi may be exactly embedded in
a power of S .
Proof.
Whenever
A, R, R , ..., W1, ... has a union it is
=R .
D
From Corollary 5-15 we thus obtain:
PROPOSITION
5.63.
No set of elementary properties true for the
category of sets implies exact (even right-exact) embeddability into a
product of countably complete topoi.
By the standard maps from
1 to
D
N we mean those of the form
1 -*+ N -*+ N -*+ 11 -*+... -*+ N .
We say that
N
is of standard generation if the standard maps form a
collectively epimorphic family.
equivalent to
THEOREM
5.64.
If T
embedded in a power of S
Proof.
In a well-pointed topos, such is
(l, N) having only standard maps.
is a topos with NNO
if
then it may be exactly
N is of standard generation.
Theorem 5.23 says that there is a collectively faithful family
of exact functors into well-pointed topoi each of which preserves
epimorphic families, hence in each of which
has
NNO
(l, N)
is standard.
(Each
by Proposition 5.U2.)
By our remarks in the last section, Hfl is the union of the values of
Aspects of topoi
73
x
a map
N •*• U
, thus a union of a sequence over the standard natural
numbers, and Proposition 5-6l applies.
If
T
is solvable then
N
D
is well-pointed and standard generation is
necessary.
COROLLARY
topos with
WNO
5.65.
If
and every
order arithmetic of
T
T
is a solvable (for example, well-pointed)
1 -»• N
is standard then the existential second
is standard.
D
It seems to me that Corollary 5-65 provides something of a semantics
for existential second-order arithmetic.
6.
Problems
Which categories can be exactly embedded in topoi?
Which well-pointed topoi can be exactly embedded in well-pointed topoi
with
NNO ?
Which of the latter can be exactly embedded in well-pointed topoi with
the axiom of choice?
Which of the latter can be exactly embedded in well-pointed topoi with
AC
and an axiom of replacement?
M i tcheI I and Co Ie have independently shown that the latter are
isomorphic to categories arising from models of Zermelo-Fraenkel.
Hence
we are asking for a metatheorem in which not the category of sets (what's
that?) but a category of sets is the model.
The answers to the last two questions would tell us which existential
second order theories of arithmetic are compatible with
Z-F .
Indeed a
good question is whether each first-order arithmetic compatible with the
axioms of well-pointed topoi is compatible with
Z-F .
Using standard techniques transferred to well-pointed topoi we can see
that each existential second order sentence in arithmetic is equivalent to
one which asserts the existence of a single unary operator
satisfies an equation involving
x
g, + , , - .
g
that
If we know that each such
sentence implies (using the axioms of topoi) that
g
is bounded by a
first-order definable operator (for example, recursive) then the
existential second order theory is determined by the first order theory.
74
Peter Freyd
We should remark, in regard to the second question, that Theorem 5.23
remains true even if we replace the category of sets with the category of
finite sets.
The idea of the proof is to find a finite elementary theory
such that the sentences true for all finite models are not recursively
enumerable.
An example of such is the theory of ordered partial rings, with enough
axioms to insure that the finite models of such are finite intervals of the
integers.
The theory of diophantine equations is thus a subset of the
theory of its finite models.
Editor's note (l May 1972).
The Editor is very grateful to Mr T.G.
Brook and especially also to Professor G.M. Kelly for reading the proofs
with care and amending several mathematical errors.
Some last-minute (and
later) changes are due to the author.
Index of definitions
Paragraph
Page
36
Absolutely closed
2.6
Axiom of choice
5-5
71
Boolean algebra
1.2
10
Boolean topos
2.1*
28
Cartesian-closed
1.1
2
Closed subobjects
2.6
33
Closure operation
2.1*
Closure operator
l.U
29
17
Complement
1.2
10
Crible
2.1
19
Degenerate category
1.1
7
Dense subobject
2.6
Equivalence relation
3-3
33
1+6
Essentially algebraic
1.1
h
Exact functor
2.6
33
Filter
1.2
12
Global closure operation
2.1*
Heyting algebra
1.2
29
9
Aspects of topoi
75
Image
2.1
20
l*.l
51
k.l
50
Logical morphism
2.3
2k
Model in
k.l
50
Implies in
T
Interpretation
T
Natural numbers object
5.1
55
Near-exact
3.3
1*5
Negation
1.2
10
NNO
•
5.1_
55
2.2
21
Relation
2.2
22
Semantically implies
l*.l
51
Separated object
2.6
33
2.6
33
P a r t i a l map
•
Sheaf
.
Singleton map
2.2
22
Solvable topos
2.7
1*0
Standard generation
5-6
72
Strongly implies
It. 1
51
1.2
11
Topos
Symmetric d e f i n i t i o n
of Heyting algebras
2.1
18
Truth value
lt.l
50
Two-valued topos
3.1*
1*7
Universal Horn theory
3.3
^5
Well-pointed object
3. h
1*7
Well-pointed topos
3.1
1*2
References
[/] J.C. Cole, "Categories of sets and models of set theory", Aarhus
Universitet, Matematisk Institut, Preprint series, 52, 1971[2]
Peter Freyd, "Several new concepts:
lucid and concordant functors,
pre-limits, pre-completeness, the continuous and concordant
completions of categories", Category theory, homology theory and
their applications III (Battelle Institute Conference, Seattle,
Washington, 19-68.
Lecture Notes in Mathematics, 99, 196-21*1.
Springer-Verlag, Berlin, Heidelberg, New York, 1969).
76
Peter Freyd
[3]
John W. Gray, "The meeting of the Midwest Category Seminar in Zurich
August 2U-30, 1970, Reports of the Midwest Category Seminar V
(Lecture Notes in Mathematics, 195, 2U8-255.
Springer-Verlag,
Berlin, Heidelberg, New York, 1971).
L41 F.W. Lawvere, "Quantifiers and sheaves", Aates du Congres
International
329-33**.
des Mathematiaiens,
T.I
N i c e , September 1970,
(Gauthier-Villars, P a r i s , 1971).
[5]
F.W. Lawvere and M. Tierney, summarized in [3] above, Reports of the
Midwest Category Seminar V (Lectures Notes in Mathematics, 195,
251-251*. Springer-Verlag, Berlin, Heidelberg, New York, 1971-
[6]
J.-L. Verdier, "Seminaire de geometrie algebrique", fascicule 1,
Mimeographed Notes Inst. Bautes Etudes Sai. (1963/6I1).
To be
published in Theorie des topos et cohomologie Stale des schSmas
(Lecture Notes in Mathematics. Springer-Verlag, Berlin,
Heidelberg, New York),
Department of Mathematics,
University of Pennsylvania,
Phi ladeiphia,
Pennsy I vani a,
USA.