THE L J J . B R O W E R CENTENARY SYMPOSIOM A.S. TroehRa and D. van Dalen (editors) 0 North-Holhnd PubiiShing C O m p l ~ ~ y , 1982 165

THE EFFECTIVE TOPOS

J .M.E. Hyland Department of Pure Mathematics, Cambridge, England.

10 Introduct ion.

The s u b j e c t of t h i s paper is t h e most access ib l e of a series

of toposes which can be constructed from not ions of r e a l i z a b i l i t y :

it i s t h a t based on the o r i g i n a l not ion of r ecu r s ive r e a l i z a b i l i t y

i n Kleene C19451. O f course t h e r e a r e many o t h e r kinds of r e a l i z a -

b i l i t y (see Kleene-Vesley C19651, Kreisel C19591, T a i t C19751). A l l

these (and even the Dia l ec t i ca I n t e r p r e t a t i o n ) f i t i n t o a very ab-

s t r a c t framework descr ibed i n Hyland-Johnstone-Pitts C19801. (Since

w e w i l l r e f e r t o t h i s paper f r equen t ly , w e shorten the reference t o

HJP C19801.) I n t h i s a b s t r a c t framework one passes e a s i l y ( a s i s

becoming customary, see Fourman C19771, Makkai-Reyes C19771, Boileau-

Joyal C198ll) between l o g i c a l and category t h e o r e t i c formulations,

using whichever i s most appropriate . One good example i s worth a

host of g e n e r a l i t i e s , so it i s t h e aim of t h i s paper t o p re sen t t h i s

a b s t r a c t approach t o r ecu r s ive r e a l i z a b i l i t y i n some d e t a i l . The

bas i c s t r a t e g y r e a d i l y extends t o o the r cases .

Many people, most notably Beeson (see f o r example Beeson C19771,

have considered r e a l i z a b i l i t y extended t o give i n t e r p r e t a t i o n s of

complicated formal systems. The f lavour of t he more category theo-

re t ic t reatment is t o have one think i n t e r m s of models. Thus the

approach looks l i k e sheaf models f o r i n t u i t i o n i s t i c l o g i c (see

Fourman-Scott C 1 9 79 1 , where one only has n a t u r a l access t o the models.

(This p a r a l l e l between r e a l i z a b i l i t y and sheaf models was f i r s t made

e x p l i c i t , f o r set theory, i n an u n t i t l e d manuscript, by Powell.)

166 J.M.E. HYLAND

A s i n t he case of sheaves, w e w i l l f i n d ou r se lves looking a t genuine

mathematical s t r u c t u r e s (wi th t h e i r non-standard l o g i c ) when w e

i n v e s t i g a t e t r u t h i n t h e e f f e c t i v e topos . W e w i l l be p re sen t ing

" the world of e f f e c t i v e mathematics" a s it appears t o t h e c l a s s i c a l

mathematician. (Of cour se , it is p o s s i b l e t o p r e s e n t t h e i d e a s i n

t h e contex t of more o r less any mathematical ideology. )

While t h e l o g i c a l approach t o c a t e g o r i e s enables us t o work

wi th concre te s t r u c t u r e s and apply our exper ience of elementary

l o g i c , t h e ca tegory t h e o r e t i c approach t o l o g i c enab le s us t o do

away wi th much l o g i c a l c a l c u l a t i o n and t o use i n s t e a d simple f a c t s

about c a t e g o r i e s ( i n p a r t i c u l a r f a c t s about toposes and geometric

morphisms). I t has become c l e a r i n r e c e n t yea r s t h a t much of con-

s t r u c t i v e l o g i c can be t r e a t e d very e l e g a n t l y i n the con tex t of

topos theory . This is i n harmony wi th work i n t h e i n t u i t i o n i s t

t r a d i t i o n on Beth and Kripke models (see van Dalen C19781, and t h e r e

w e r e many c o n t r i b u t i o n s t o t h e Brouwer Centenary Conference i n t h i s

a r e a . This paper simply does t h e same k ind of t h i n g f o r r e a l i z a -

b i l i t y . Of course t h e r e & a s u r p r i s e he re : t h e topos of t h i s paper

i s most un l ike a Grothendieck topos , and it i s n o t i n i t i a l l y

p l a u s i b l e t h a t theory a b s t r a c t e d from no t ions of c o n t i n u i t y should

have any a p p l i c a t i o n i n t h i s most non-topological s e t t i n g .

The f i r s t t h r e e s e c t i o n s of t h e paper s e rve t o in t roduce t h e

e f f e c t i v e topos a s a world b u i l t ou t of t h e l o g i c of r e c u r s i v e

r e a l i z a b i l i t y . Much d e t a i l is omi t ted i n t h e hope of g iv ing a f e e l

f o r t he s u b j e c t . The main ca t egory - theo re t i c i d e a s axe expla ined

and i n t e r p r e t e d i n 5 84-6. I n p a r t i c u l a r we show why t h e no t ion of

a nega t ive formula a r i s e s n a t u r a l l y i n the theory of sheaves. I n

887-13, w e apply t h i s work t o a s tudy of a n a l y s i s i n t h e e f f e c t i v e

topos. W e show t h a t i n essence it i s c o n s t r u c t i v e r e a l a n a l y s i s

( i n t h e sense of Markov). I am g r a t e f u l t o P ro fes so r T r o e l s t r a f o r

some advice on t h i s t o p i c (I f i n d t h e publ i shed mater ia l unreadable:

The effective topos 167

and i n p a r t i c u l a r f o r d e t e c t i n g an e r r o r i n an e a r l y d r a f t o f t h i s

paper. 5514-17 a r e concerned wi th f e a t u r e s of t h e e f f e c t i v e topos

where t h e power se t ma t t e r s : un i formi ty p r i n c i p l e s and p r o p e r t i e s of

j -ope ra to r s . The paper c l o s e s wi th some gene ra l remarks on the

mathematical s i g n i f i c a n c e o f t h e e f f e c t i v e topos .

F i n a l l y I would l i k e t o thank t h e o rgan ize r s o f t h e Brouwer

Centenary Conference f o r t h e oppor tun i ty t o p r e s e n t t h i s paper ( i n

such p l e a s a n t sur roundings : ) and t o apologize t o everyone f o r be ing

so long i n w r i t i n g it.

51 Recursive r e a l i z a b i l i t y .

Recursive r e a l i z a b i l i t y is based on t h e p a r t i a l a p p l i c a t i v e

s t r u c t u r e (IN ,.) where a s i n H J P 119803 w e w r i t e n.m = n(m) f o r t h e

r e s u l t of apply ing t h e n ' t h p a r t i a l r e c u r s i v e func t ion t o m. (This

saves on b racke t s compared wi th t h e n o t a t i o n { n l n . ) One can de f ine

a no t ion of A-abstraction i n ( I N ,.) i n t h e usua l way from t h e com-

b i n a t o r s , and w e w i l l use it f r e e l y i n what fo l lows , so t h a t ( f o r

example) Ax.x w i l l denote an index f o r t h e i d e n t i t y func t ion . We

a l s o t ake f o r convenience a r e c u r s i v e p a i r i n g func t ion

< , >: I N x I N - > IN; (n,m) --> <n,m>,

and l e t n1,n2 be ( r e c u r s i v e i n d i c e s f o r ) t h e corresponding unpai r ing

€unc t ions .

Recursive r e a l i z a b i l i t y i s usua l ly formula ted i n terms of t h e

no t ion

e r e a l i z e s 0

where e i s a n a t u r a l number and @ i s a sen tence o f (Heyt ing ' s )

a r i t h m e t i c . The c r i t i c a l c l a u s e s i n t h e i n d u c t i v e d e f i n i t i o n a r e

imp l i ca t ion

e r e a l i z e s 4 -> $ i f f f o r a l l n , i f n real izes @ t hen e ( n ) i s

de f ined and realizes $,

168 JM.E. HYLAND

un ive r sa l q u a n t i f i c a t i o n

e r e a l i z e s Vn.$(n) i f f f o r a l l n , e ( n ) is def ined and r e a l i z e s

@ ( g ) Cg t h e numeral f o r n l .

The o t h e r i nduc t ive c l a u s e s a r e

and

e r e a l i z e s $ A $ i f f n (e) r e a l i z e s $ and n 2 ( e ) r e a l i z e s $,

o r

e r e a l i z e s @vJ, i f f e i t h e r a,(e) = 0 and n 2 ( e ) r e a l i z e s $

- 1

-

o r IT^ (e) = 1 and n 2 (e) rea l izes $,

falsity

no numbers rea l ize 1,

e x i s t e n t i a l q u a n t i f i c a t i o n

e rea l izes h . $ (n) i f f n 2 (e) r e a l i z e s @ ( r l (e) ) [ n l (e) t h e - - numerical f o r I T ~ ( ~ ) I .

F i n a l l y w e g i v e t h e i n i t i a l c l a u s e f o r e q u a l i t i e s between c losed

t e r m s

e r e a l i z e s s = t i f f both s and t denote e.

For a c a r e f u l t r ea tmen t of t h e r e a l i z a b i l i t y i n t e r p r e t a t i o n of

a r i t h m e t i c t h e r eade r may c o n s u l t T r o e l s t r a C19731. W e w i l l see i n

53, t h a t t h i s is t h e i n t e r p r e t a t i o n of a r i t h m e t i c w i t h i n t h e

e f f e c t i v e topos . For an account of t h e o r i g i n a l mot iva t ion see

Kleene 119731; it i s i n t e r e s t i n g t o t r y t o understand it i n t e r m s o f

t h e p re sen t paper.

Apparently Dana S c o t t f i r s t no t i ced t h a t r e a l i z a b i l i t y could be

understood "model-theoretically ' ' i n terms of t h e t ru th -va lues

{ e ( e real izes @ I .

t ru th -va lues , and so f o r each set X , a se t Zx of non-standard pre-

d i c a t e s on X.

This g ives us a set C = P ( l N ) of non-standard

W e w r i t e $ = ($,lx E X) and J, = (QX\x E X) f o r

elements of C" and can r e fo rmula t e our ear l ie r d e f i n i t i o n f o r t h e

p ropor t iona l connec t ives by d e f i n i n g ope ra t ions poin twise on Zx as

fo l lows :

The effective topos 169

( $ A $ ) ~ = $ x ~ $ x = {<n,m>ln E $x and m E $,I, ( $ v $ l X = $ x v $ x = t < ~ , n > l n E $ x I u t < l , n > j n E

( $ + I ) ) ~ = $x+$x = {el i f n E $,, then e ( n ) i s def ined and

e ( n ) E

= t h e empty set . IX

The r eade r may a l s o l i k e t o have

TX = IN.

There is a r e l a t i o n lk

Cx by

of en ta i lmen t ( a pre-order ) def ined on each

I-,$ i f f n { ( $ -> $),lx E XI i s non-empty.

The soundness of t h e r e a l i z a b i l i t y i n t e r p r e t a t i o n of i n t u i - t i o n i s t i c p r o p o s i t i o n a l l o g i c i s t h e fo l lowing p ropos i t i on .

Propos i t ion 1.1. (Z i s a Heyting pre-algebra: as a category

t h e preorder has f i n i t e l i m i t s ( m e e t s ) , f i n i t e c o l i m i t s ( j o i n s ) and

i s Car t e s i an c losed (Heyting i m p l i c a t i o n ) .

Proof: The s t r u c t u r e is g iven e x p l i c i t l y i n t h e d e f i n i t i o n s above.

X

W e now in t roduce t h e a b s t r a c t no t ion of q u a n t i f i c a t i o n from

c a t e g o r i c a l l o g i c . For any map f : X -> Y of sets w e de f ine

s u b s t i t u t i o n a long f , f* : C y -> CX a s composition wi th f :

( f*$) , = $ f (x) f * i s a func to r ( i n fact a map of Heyting pre-a lgebras) from

( Z , I - y ) t o ( C , I-x ) and q u a n t i f i c a t i o n a long f i s given by t h e

a d j o i n t s t o f * . As shown i n HJP C19801 t h e s e are de f ined by

r i g h t a d j o i n t

Y X

( v f . + ) y = n t f ( x ) = y -> $ X I X E x l ,

l e f t a d j o i n t

( 3 f . $ ) y = U { f ( X ) = y A $ x l X E XI,

where T , i f f ( x ) = y ,

f f ( x ) = Y]l = ' - J I T l f ( X ) = yj = t L , otherwise,

i s t h e n a t u r a l i n t e r p r e t a t i o n a s a non-standard p r e d i c a t e of

170 J.M.E. HYLAND

f ( x ) = y .

n a t i v e d e f i n i t i o n of t h e l e f t a d j o i n t , r l {$x l f (x ) = y} i s = a

d e f i n i t i o n of t h e r i g h t a d j o i n t un le s s f : X --> Y i s s u r j e c t i v e .

However usua l q u a n t i f i c a t i o n i s q u a n t i f i c a t i o n a long t h e obvious

pro jec t ion , and almost a l l p r o j e c t i o n s are s u r j e c t i v e , so t h i s

nuance w i l l cause t h e r eade r (and au tho r ) no f u r t h e r t r o u b l e .

Note t h a t whi le U { ~ $ ~ l f ( x ) = y} i s a s a t i s f a c t o r y a l te r -

The r eade r w i l l see t h a t what w e have j u s t descr ibed i s an

i n t e r p r e t a t i o n of i n t u i t i o n i s t i c p r e d i c a t e l o g i c : w e have s t anda rd

func t ions and sets, a (non-standard r e p r e s e n t a t i o n o f ) s t anda rd

e q u a l i t y and a c o l l e c t i o n of non-standard p r e d i c a t e s . W e a l s o have

a "gener ic p red ica t e" namely t h e i d e n t i t y i n E L .

a l l t h i s s t r u c t u r e i n t h e fo l lowing p ropos i t i on .

P ropos i t i on 1 . 2 . The toge the r wi th t h e f * and t h e i r a d j o i n t s

3f and Vf and t h e "gene r i c p r e d i c a t e " , form a t r i p o s on t h e ca tegory

of Sets ( i n t h e sense of H J P C19801.

Proof: See H J P C19801.

W e can encapsu la t e

In what w e have s a i d , w e have n o t needed t o d i s t i n g u i s h f o r -

mulae from t h e i r i n t e r p r e t a t i o n s , and w e w i l l cont inue t o b l u r t h i s

d i s t i n c t i o n as f a r as p o s s i b l e . ( W e w i l l use open f a c e b racke t s t o

i n d i c a t e an i n t e r p r e t a t i o n when necessary t o p reven t confus ion . )

W e say t h a t

$ E .Ex i s v a l i d i f f T I - x $ . By a d j o i n t n e s s w e have

9 E Zx i s v a l i d iff T l-lVX.$,

where X: X --> 1 is a unique map from X t o a one element se t . That

is, I$ i s v a l i d i f f Vx.I$(x) , t he un ive r sa l g e n e r a l i z a t i o n of $ is

v a l i d o r r e a l i z a b l e . W e w i l l use t h i s no t ion both t o desc r ibe and

s tudy t h e topos which w e can c o n s t r u c t on t h e b a s i s of ( 1 . 2 ) .

12 Desc r ip t ion of t h e e f f e c t i v e topos .

When cons t ruc t ing a topos from a t r i p o s a s i n H J P C19801, one

The effective topos 171

must

(i) add new subobjects of t h e sets one has s t a r t e d with t o

represent t h e non-standard p red ica t e s , and

(ii) take quo t i en t s of t hese by the non-standard equivalence

r e l a t i o n s . This l eads t o the desc r ip t ion of t h e o b j e c t s of e f f ec -

t i v e topos. An ob jec t of t he e f f e c t i v e topos i s a set X with a

non-standard p red ica t e = on X x X such t h a t

symmetry x = y --> y = x

t r a n s i t i v i t y x = Y A Y = z -> x = z

are v a l i d . Note t h a t w e do no t have r e f l e x i v i t y : ( a s i s t h e case

for Heyting a r i t hme t i c ) t h e r e need be no uniform r e a l i z a t i o n of

(reason why) x = x. W e regard and w i l l w r i t e t he p red ica t e x = x

as an ex i s t ence p red ica t e , Ex, and a s a membership p red ica t e , x E X.

There is a use fu l discussion of the l o g i c of ex i s t ence p red ica t e s

i n Sco t t C19791.

Of course w e need t o consider a l l non-standard maps t o obtain

the e f f e c t i v e topos, and t o do t h a t w e a r e reduced t o considering

funct ional r e l a t i o n s . The m- from (X, =) t o ( Y , =) i n the

e f f e c t i v e topos a r e equivalence c l a s s e s of func t iona l r e l a t i o n s

where

( a ) G 6 cxXy i s a func t iona l r e l a t i o n i f f

r e l a t i o n a l G(x,y)AX = X ' A Y = y ' -> G(X' ,y ' )

s t r ic t G(x,y) -> EXAEY

single-valued G ( x , y ) ~ G ( x , y ' ) --> y = y '

t o t a l EX --> +y.G(x,y)

a re a l l v a l i d ,

(b) 6 i s equ iva len t t o H i f f

G(x,y) <--> H(xrY)

i s v a l i d . W e w i l l say t h a t G r ep resen t s t he map [GI: ( X , = ) - - > ( Y , = ) .

I t i s useful t o note t h a t i f G and H a r e both func t iona l r e l a t i o n s

112 J.M.E. HYLAND

(from ( X I = ) t o ( Y , = ) , t hen t o show G and H e q u i v a l e n t , it s u f f i c e s

t o show t h a t an impl i ca t ion i n one d i r e c t i o n i s v a l i d .

Funct iona l r e l a t i o n s can be composed: i f G E Xxxy and H E C y x z

Also are func t iona l r e l a t i o n s , t hen s o is 3y.G(xly)AH(y,z) E C x x z .

= is a func t iona l r e l a t i o n from ( X , = ) t o i t s e l f . These g ive t h e

composition and i d e n t i t i e s , and so w e have a ca tegory . I n view of

( 2 . 1 ) , w e c a l l t h i s ca tegory t h e e f f e c t i v e topos and denote it by

Eff h e r e a f t e r .

Theorem 2 . 1 . Eff is a topos .

Proof: See H J P C19801 f o r d e t a i l s .

W e can ex tend t h e non-standard i n t e r p r e t a t i o n of 91 t o g ive an

account of t h e i n t e r n a l l o g i c of t h e ca tegory Eff. This goes a s f o r

t h e l o g i c of sheaves except f o r obvious modi f ica t ions t o d e a l wi th

t h e f a c t t h a t func t ions are (only) r ep resen ted by func t iona l rela-

t i o n s . A gene ra l account of t h e i n t e r n a l f i r s t - o r d e r l o g i c of

c a t e g o r i e s i s given i n Makkai and Reyes C19771, and accounts of t h e

h ighe r o rde r l o g i c of toposes can be found i n Fourman C19771 and

Boileau-Joyal C19811. A s t h e s e accounts make c l e a r , c a t e g o r i c a l

cons t ruc t ions can be de f ined by means of t h e i n t e r n a l l o g i c . Thus,

n o t on ly can (an ex tens ion o f ) v a l i d i t y i n t h e sense of 9 1 , be used

t o determine what i s t r u e i n Eff, b u t it can a l s o be used t o d e f i n e

c a t e g o r i c a l c o n s t r u c t s . (Now cont inuing t h e i n t e r p l a y , t h e s e c a t e g o r i -

c a l c o n s t r u c t s can then be used t o e s t a b l i s h f u r t h e r f a c t s about what

i s t r u e i n Eff.) W e now g ive some simple examples of t h e l o g i c a l

d e s c r i p t i o n of t h e s t r u c t u r e of Eff. (On a few occasions w e w i l l

need t o quote some more complicated f a c t s of t h e same kind . )

1) A map [GI: (X,=) --> (Y,=) i s monic i f f

G ( x , y ) ~ G ( x ' , y ) --> x = X'

i s v a l i d .

A subob jec t of (X,=) can always be r ep resen ted (though n o t

uniquely) by a canonica l monic of t h e form

The effective topos

[ = ' I : (X,=') --> (X,=)

where

I[ x = ' x q = A ( X ) A ux = x l n

f o r some A E Zx s t r i c t and r e l a t i o n a l f o r (X,=). Thus subobjec ts

173

always arise by r e s t r i c t i n g t h e membership p r e d i c a t e whi le (as f a r

as p o s s i b l e ) l eav ing t h e e q u a l i t y a lone .

2 ) Given two maps CG1,CHI: (X,=) --> (Y , =) , t h e i r e q u a l i z e r is

represented by t h e canonica l monic obta ined from t h e s t r ic t and

r e l a t i o n a l

3y .G(x ,y)hH(xfy) E E x .

The cons t ruc t ion of o t h e r f i n i t e l i m i t s i s analogous.

The diagram

W , = ) - [ G " > (Z,=)

[H'I 1 1 [HI V

( X r = ) - > ( Y r = ) [GI

i s a pullback i f f CHIo[G'I = CGIo[H'I, (CG'1,CH'I):W -> Z X X i s a

monic and

G ( x , y ) ~ H ( z , y ) --> 3W.G' ( W , Z ) A H ' (w,x)

is v a l i d . The cond i t ion t h a t o t h e r diagrams g ive f i n i t e l i m i t s can

be expressed s i m i l a r l y i n the l o g i c .

3 ) A map [GI: (X,=) --> (Y,=) is s u r j e c t i v e i f f

Ey -> 3x.G(xry)

i s v a l i d .

A q u o t i e n t can always be r ep resen ted as

[ - I : ( X r = ) --> ( X v - )

where - i s s t r i c t r e l a t i o n a l f o r (X,=) and such t h a t

"- is an equiva lence r e l a t i o n on (X,=)"

i s v a l i d . Thus q u o t i e n t s are a ma t t e r of ex tending t h e e q u a l i t y

r e l a t i o n and l eav ing t h e membership p r e d i c a t e a lone .

174 J.M.E. HYLAND

W e can now show t h a t any ob jec t ( X I = ) of Eff i s a quo t i en t of a

subobject of an "ordinary set" , j u s t i f y i n g the explanat ion a t t h e

s t a r t of t h i s s e c t i o n . For a s e t X w e l e t AX ( a s i n 5 4 ) be t h e

ob jec t of Eff with underlying set X and (non-standard r ep resen ta t ion

o f ) standard equa l i ty .

Proposit ion 2 . 2 . Any ob jec t ( X , = ) of Eff i s a quo t i en t by = of the

subobject EX of AX obtained from t h e ex i s t ence p red ica t e of ( X , = ) . Proof: Obvious i n view of 1) and 3) above.

- Note. W e have s t a r t e d using open f ace brackets t o ensure r e a d a b i l i t y

( e spec ia l ly i n connection with e q u a l i t y ) , a s promised i n §I. W e

a l s o abuse no ta t ion and w r i t e X f o r ( X , = ) where context makes the

meaning obvious.

5 3 . Some o b j e c t s and maps i n Eff. - W e can e a s i l y descr ibe a terminal ob jec t 1 i n Eff. I n view of

5 2 , 1 is ( I * } , = ) where { * I i s a s i n g l e t o n , and

[ I* = * I = T

Of course any p equ iva len t t o T i n Z { * ' = Z, t h a t i s , any non-empty

p would do a s the value

sec t ions of an a r b i t r a r y o b j e c t (Y,=) of Eff , t h a t is t h e maps from

1 t o (Y,=) . Since { * I i s a s i n g l e t o n , such maps a r e represented by

degenerate func t iona l r e l a t i o n s G E Z y , such t h a t

1 * = * I . W e now c a l c u l a t e t h e g loba l

G ( Y ) A Y = y ' --> G(y')

G(y) --> Ey

G(y)AG(y') --> y = Y '

3y.G(y)

a r e a l l v a l i d .

G(yo) i s non-empty. The r e l a t i o n a l and single-valued condi t ions

imply t h a t

The t o t a l condi t ion t e l l s us t h a t f o r some y , yo say,

G ( Y o ) --> (G(y) <--> yo = y)

and hence ( s ince G(yo) i s non-empty)

The effective topos 175

G(y) <--> yo = y

a r e v a l i d . C lea r ly i f [I yo = y l l is non-empty, then

Yo = Y <-> Y 1 = Y

i s v a l i d . W e deduce a t once t h e fo l lowing c h a r a c t e r i z a t i o n .

Propos i t ion 3.1. Each map [ G I : 1 -> (Y,=) determines and i s com-

p l e t e l y determined by [y I G(y) non-empty}, which is an equiva lence

c l a s s f o r t h e ( p a r t i a l ) equiva lence r e l a t i o n

" U y = y ' l i s non-empty".

Conversely any such equiva lence class de termines a map from 1 t o

(Y,=).

F i n i t e c o l i m i t s i n Eff a r e ha rd t o g e t used t o because f o r a

s t a r t coproducts a r e odd: t he r e a l i z a b i l i t y i n t e r p r e t a t i o n of d i s -

junc t ion is very r e s t r i c t i v e . I n p a r t i c u l a r , t h e coproduct 2 of 1

with i t s e l f i s n o t t h e obvious o b j e c t A2 wi th s t anda rd e q u a l i t y

(see 54). L e t us look a t maps from A2 t o an a r b i t r a r y o b j e c t IY,=)

of Eff. Suppose G ( i , y ) r e p r e s e n t s such a map (where 2 = { O , l } ) .

Then s i n c e EO = E l = T , t h e t o t a l cond i t ion t e l l s us t h a t t h e r e a r e

yo,yl such t h a t G(O,yo)nG(l,yl) i s non-empty. Arguing a s f o r t h e

te rmina l o b j e c t w e f i n d t h a t

G ( i , y ) <--> yi = y

i s v a l i d . However [GI does n o t correspond simply t o a p a i r o f

equivalence c l a s s e s i n IylEy non-empty?:the union of t h e ex i s t ence of

t h e two equiva lence c l a s s e s must i n t e r s e c t n o n - t r i v i a l l y , and t h i s is

a r e a l r e s t r i c t i o n .

I n f a c t t h e o b j e c t 2 i n Eff can be r ep resen ted as ( 2 , = ) where

EO = 101, E l = {l}, [li = j]l = E i n E j .

(Of course any p,,pl wi th p npl empty would do as t h e va lues EO, E l . )

An argument a s above shows t h a t maps from 2 a r e p a i r s of maps from 1.

Note a l s o t h a t t h e only maps from A2 t o 2 are cons t an t ( t h a t i s ,

f a c t o r through 1). There is an obvious monic from 2 t o A2. I n 516

0

176 J.M.E. HYLAND

we w i l l show t h a t t h e whole s t r u c t u r e of Eff depends on 2 --> A2

n o t be ing i s o , i n t he sense t h a t t h e topology i n v e r t i n g 2 --> A2

c o l l a p s e s Eff back t o Sets. Since 2 is n o t A2, w e would ha rd ly expec t t h e n a t u r a l number

o b j e c t N i n Eff t o be A m . I n f a c t it is t h e o b j e c t ( I N , =) where

En = i n ] , [ n = m ] = EnnEm.

There are maps 0: 1 --> IN and s : IN-> IN i n Eff r ep resen ted res-

p e c t i v e l y by Go and Gs where

Go(*,n) = IOjnIn) and G,(n,m) = In+l)n{m).

P ropos i t i on 3 . 2 . IN t o g e t h e r w i th 0: 1 --> IN and s: IN-> IN i s a

n a t u r a l number o b j e c t i n Eff . Proof: Suppose t h a t w e are g iven maps a : 1 --> ( X , = ) and

g: ( X , = ) --> ( X , = ) r ep resen ted r e s p e c t i v e l y by Ga E C x and

G E 1'''. W e can d e f i n e r e p r e s e n t a t i v e s Gn f o r gn i n d u c t i v e l y by 4 9

( x , x l ) = ~ X ~ ~ . G ~ ( X , X ~ ~ ) A G ( X ~ * , X ' ) . Gn+l G O ( X , X I ) = U X = x 8 n ,

g 9 9 9 Now w e can d e f i n e a func t ion f : IN-> ( X , = ) r ep resen ted by

Gf (n ,x ) = EnA3x'.Ga(x')AGn(x' ,x). 9

W e c la im t h a t

f i fl (X,=)->(X,=) a

9

commutes. This amounts t o showing t h a t

G (x ) <--> 3n.Go(*,n)AGf(n,x)

and

3x ' .Gf ( n , x ' ) A G ( x ' , x ) <--> 3m.Gs (n,m) A G f (m,x) 9

a r e both v a l i d . These can both be e s t a b l i s h e d by use of e lementary

l o g i c .

It remains t o show t h a t f is unique such t h a t ( * ) commutes. SO

suppose t h a t f ' r ep resen ted by Gf I i s ano the r such map.

l o g i c w e see r e a d i l y t h a t

By use of

The effective topos 177

G (0 ,x ) <-> Gf I (0 ,x ) f i s v a l i d and t h a t

G f ( n + l , x ) <-> 3x ' .Gf(n ,x ' )AG ( x ' , x )

G f , (n+l,x)<-> 3 x ' . G f , ( n , x ' ) AG (x ' , x ) g

g a r e both v a l i d . But i n t e r m s o f t h i s d a t a w e can d e f i n e , by primi-

t i v e r e c u r s i o n , a p a r t i a l r e c u r s i v e f u n c t i o n uniformly mapping

Gf ( n ,x ) t o G f , ( n , x ) , and t h i s i s enough t o show t h a t f = f ' ,

Remark. S ince q u a n t i f i c a t i o n i n ou r l o g i c (see Fourman-Scott C 1 9 7 9 1

and S c o t t C19791) i nvo lves t h e e x i s t e n c e p r e d i c a t e , w e see a t once

on the b a s i s of ( 3 . 2 ) t h a t t h e r e a l i z a b i l i t y i n t e r p r e t a t i o n co r re s -

ponds t o the l o g i c of t h e n a t u r a l number o b j e c t i n Eff. Corol la ry 3.3. A s en tence of Heyting a r i t h m e t i c is r e c u r s i v e l y

r e a l i z e d i f f it i s t r u e of t h e n a t u r a l number o b j e c t i n Eff. A l l t he s p e c i f i c o b j e c t s w e have looked a t so f a r have been

o b j e c t s (X,=) where [ x = x ' J non-empty impl i e s x = x ' ( i n X).

Indeed a l l t h e o b j e c t s w e cons ide r u n t i l 1 1 4 w i l l be (isomorphic t o )

ones o f t h i s s o r t (see 5 6 f o r a d i scuss ion of what t h e cond i t ion

means). I t is a s w e l l t o have an example of an o b j e c t n o t o f t h i s

form. The most obvious example i s t h e subob jec t c l a s s i f i e r i n Eff ( t h a t i s , t h e o b j e c t of t r u t h v a l u e s ) . A s i n d i c a t e d i n HJP C19801

t h i s is t h e o b j e c t (X ,<->), t h a t i s , t h e se t X = P(JN ) wi th

e q u a l i t y given by t h e non-standard b i - impl i ca t ion . W e leave it a s

an easy e x e r c i s e t o show t h a t (I,<-->) i s n o t isomorphic t o any

o b j e c t (X,=) where I[x = x ' l non-empty impl i e s x = x ' . (Show

f i r s t t h a t X would have t o have j u s t two e lements . ) As f u r t h e r

examples it i s n a t u r a l t o cons ide r (EA,<->) where now e q u a l i t y is

the poin twise b i - imp l i ca t ion . These are e s s e n t i a l l y t h e o b j e c t s ou t

of which w e cons t ruc t ed Eff i n t h e f i r s t p l a c e ; they a r e i n f a c t t h e

power sets of t h e o b j e c t s AX (see 9 4 ) .

178 J.M.E. HYLAND

5 4 . The inc lus ion of t h e category of sets i n the e f f e c t i v e topos.

In t h e l a s t s e c t i o n w e s a w glimpses of a func to r A from t h e

category Sets of sets t o Eff, t h e e f f e c t i v e topos.

Def in i t i on . For a se t X, de f ine AX t o be ( X , = A x ) l where

is t h e n a t u r a l i n t e r p r e t a t i o n of t he e q u a l i t y i n Sets. For a map

f : X -> Y i n Sets, def ine A f : AX -> AY t o be the map represented

by the func t iona l r e l a t i o n ,

This d e f i n i t i o n can be made whenever w e cons t ruc t a topos from a

t r i p o s (see P i t t s [ 1 9 8 l l ) , and w e always have o w f i r s t r e s u l t .

Proposi t ion 4 . 1 . A : Sets -> Eff is a Cartesian func to r ( t h a t i s ,

func to r preserving f i n i t e l i m i t s ) . Proof: F u n c t o r i a l i t y i s obvious. That A i s Cartesian follows

e a s i l y from t h e way f i n i t e l i m i t s a r e def ined l o g i c a l l y i n 53. For

d e t a i l s see H J P C19801 o r P i t t s C19811.

The next r e s u l t i s a general f e a t u r e of r e a l i z a b i l i t y toposes .

- Propcsi t ion 4 . 2 . A : Sets -> Eff i s f u l l and f a i t h f u l .

Proof: Suppose t h a t f , g : X --> Y i n Sets, and t h a t

= y n <--> u g ( x ) = y~

i s va1i.d. Then I [ f (x ) = y l = T i f f f g ( x ) = yl] = T whence

f ( x ) = y i f f g ( x ) = y: t hus f = g. This shows t h a t A i s f a i t h f u l .

To show t h a t A i s f u l l , l e t G E Z x x y be a func t iona l r e l a t i o n from

AX t o A Y . The r e l a t i o n a l and s t r i c t cond i t ions axe au tomat i ca l ly

s a t i s f i e d , t h e s ingle-valued condi t ion implies t h a t f o r given x

t h e r e is a t most one y with G(x,y) non-empty ( c o n s t r u c t i v e l y , in-

h a b i t e d ) , and t h e t o t a l condi t ion imples t h a t t h e r e is a t least one

such y f o r given x. Thus w e have g: X -> Y such t h a t G(x,y) i s

non-empty i f f g(x) = y . Then c l e a r l y

~ ( x , y ) --> u g w = yn

The effective topos

i s v a l i d : an index f o r t h e i d e n t i t y realizes it. But t h e t o t a l

condi t ion becomes

E x -> G ( x , g ( x ) )

i s v a l i d , whence

I g ( x ) = y n -> G

i s v a l i d . Thus G r e p r e s e n t s t h e map Ag.

Remark. ndr6 Joya l has po in t ed o u t t h a

179

X I Y )

This shows t h a t A is f u l l .

A i s analogous t o t h e

Yoneda embedding: it i s Car t e s i an , f u l l and f a i t h f u l , and (so) pre-

s e rves exponen t i a t ion . But I do n o t understand t h e f o r c e of t h i s

analogy.

The main r e s u l t o f t h i s s e c t i o n i s ano the r gene ra l f e a t u r e of

r e a l i z a b i l i t y toposes . Reca l l t h a t i n (3.1) w e showed i n e f f e c t

t h a t t h e g l o b a l s e c t i o n f u n c t o r on Eff i s n a t u r a l l y isomorphic t o

r : Eff --> Sets def ined by

(i) r ( X , = ) = {XIEX i s non-empty]/_ where x - x ' i f [Ix = x i ] '

i s non-empty;

(ii) i f G i s a f u n c t i o n a l r e l a t i o n from ( X , = ) t o (Y,=)

r e p r e s e n t i n g g , t hen

T(g) ([XI) = {ylG(x ,y) i s non-empty} where [XI denotes

t h e equiva lence c l a s s o f x .

r i s a conc re t e ve r s ion of t h e g loba l s e c t i o n f u n c t o r , w i th which

w e can work, even c o n s t r u c t i v e l y : s t a r t i n g from an a r b i t r a r y base

topos E , T ( X , = ) s t i l l makes sense a s t h e i n t e r p r e t a t i o n i n E of

" the set o f maps from 1 t o ( X I = ) " . (Of course , "non-empty" must be

r ep laced by " inhab i t ed" ) . Theorem 4.3. A is t h e d i r e c t image f u n c t o r of a geometric morphism,

whose i n v e r s e image f u n c t o r i s r . Proof: The g l o b a l s e c t i o n f u n c t o r i s always Car t e s i an : a l t e r n a t i v e l y ,

r a s de f ined is C a r t e s i a n by t h e l o g i c a l c o n s t r u c t i o n of f i n i t e

l i m i t s de sc r ibed i n 53. So w e concen t r a t e on the a d j o i n t n e s s .

180 J.M.E. HYLAND

W e d e f i n e t h e u n i t o f t h e ad junc t ion ny : Y --> ATY f o r (Y,=) i n Eff by t h e f u n c t i o n a l r e l a t i o n

(y , [y ' l ) - -> U{Eyly E Cy ' l}={yy ' i f y - y ' , o the rwise .

Now l e t G be a f u n c t i o n a l r e l a t i o n from (Y,=) t o AX. The t o t a l and

s ingle-va lued c o n d i t i o n s imply t h a t i f Ey i s non-empty, t hen t h e r e

i s a unique x E X wi th G(y ,x ) non-empty. The r e l a t i o n a l cond i t ion

impl i e s t h a t i f y = y ' ] is non-empty, then w e g e t t h e same x

f o r y ' a s f o r y . Thus w e have a wel l -def ined map g: rY --> X . By

l o g i c , t h e composite A ( 9 ) o r l y i s r e p r e s e n t e d by

~ ( y , x ) = U I E ~ A ug(cy11) = x n ~y E iy11 E ~ Y I .

By t h e s t r i c t c o n d i t i o n

G(y,X) --> EY

i s v a l i d ; s o s i n c e G(y ,x) i s non-empty i f f x = q(Cy1) and s i n c e

c l e a r l y

Ey --> H ( y , g ( c y l ) )

is v a l i d , w e deduce t h a t

G(y ,x) --> H(y,x)

i s v a l i d . S ince both G and H a r e f u n c t i o n a l r e l a t i o n s , t h i s shows

( a s remarked i n 5 2 ) t h a t t h e y r e p r e s e n t t h e same f u n c t i o n , and w e

have our f a c t o r i z a t i o n

[GI = A(q) o n y .

I t remains t o show t h a t q i s unique wi th t h i s p r o p e r t y . But i f

9 ' : TY --> X is such t h a t

G ( Y , X ) <--> U{EYA u g i ( c y i i ) = x n Iy E c y ' i E TY}

i s v a l i d , t h e n G ( y , g ' ( C y l ) ) i s non-empty, so t h a t g ' = q. This

completes t h e p roof .

Remark. I t is an easy c o r o l l a r y of t h e proof of ( 4 . 3 ) t h a t maps

( Y , = ) --> A X i n Eff have a s imple canon ica l r e p r e s e n t a t i v e . L e t

g: TY --> X cor respond under t h e ad junc t ion t o a map IY,=) -> AX.

Then t h i s l a t t e r map i s r e p r e s e n t e d by t h e f u n c t i o n a l r e l a t i o n

T h e effective topos 181

(y ,x) --> UiEylg ( [y l ) = XI = { y : W e can now i n d i c a t e how category theo ry may be appl ied t o s tudy

r e a l i z a b i l i t y . (4 - .2 ) and ( 4 . 3 ) t oge the r say t h a t A : Sets --> Eff

i s an inc lus ion of toposes (see Johnstone C19771) so t h a t Sets i s

j-sheaves on Eff f o r a s u i t a b l e topology j . W e g ive an i d e n t i f i -

cat ion of j , which depends on the f a c t t h a t Sets has c l a s s i c a l l o g i c ,

Proposi t ion 4 . 4 . The topology j such t h a t E f f . Sets i s t h e double

negation topology

Proof: In t h e f i r s t p l ace , Sets is dense i n Eff s ince A

preserves t h e i n i t i a l o b j e c t (see (8.1) ) ; so j is a t most 71 ( t h e

g r e a t e s t dense topology) . But Sets i s boolean, and from t h i s

it follows t h a t j must be T-, .

-J

W e can now desc r ibe what the use of "classical o b j e c t s " i n

in tu i t i on i sm amounts t o i n our context : s i n c e they are defined by

l i b e r a l use of 7-,, they are when i n t e r p r e t e d i n Eff, t h e o b j e c t s i n

the image of A . Thus A ( = ) should be regarded a s t h e world of

c l a s s i c a l mathematics w i th in Eff.

15. Basic f a c t s from t h e l o g i c of sheaves.

While the material presented i n this s e c t i o n is i m p l i c i t i n t h e

topos t h e o r e t i c l i t e r a t u r e , it can n o t be found i n the form w e r e -

quire . With Grothendieck toposes one has t y p i c a l l y a subtopos E

of a topos E which one understands ( E i s usua l ly a func to r category)

and one r e q u i r e s r e s u l t s which enable one t o d i scuss E i n terms of

E and t h e topology j . For us however t h e s i t u a t i o n i s d i f f e r e n t .

It is the topos E ( t h a t is Sets) which w e understand and w e wish t o

obtain information about E ( t h a t i s Eff) i n terms of E and j .

j

j

j

j W e p r e sen t t h e material i n the following general context . E is

a topos with a topology j , E . i s t h e f u l l subcategory of E cons i s t ing

of j-sheaves and L: E -> E j i s t h e s h e a f i f i c a t i o n func to r l e f t J

182 J.M.E. HYLAND

a d j o i n t t o the inc lus ion E -> E . W e g ive t h e b a s i c d e f i n i t i o n s

i n a number of u se fu l equ iva len t forms which a r e i m p l i c i t e i t h e r i n

Johnstone C19771 o r i n Fourman-Scott 119791.

Def in i t i on . An ob jec t F of E i s j -separated i f f any of t he following

equ iva len t cond i t ions is s a t i s f i e d :

(i) f o r any j-dense monic m: Y' >--> Y and maps f , g : Y --> F

with fm = gm, w e have f = g;

(ii) t h e u n i t nF : F --> L(F) of t h e adjunct ion i s monic;

(iii) E I= V f , f ' E ~ . j ( f = f ' ) --> ( f = f ' ) .

A subobject (monic) A >--Z E of an o b j e c t E of E i s j - c losed

i f f any one of t h e following equ iva len t cond i t ions i s s a t i s f i e d :

(i) i f a: E --> R c l a s s i f i e s A >--> E , then j a = a ;

(ii) the commutative square A --> L ( A ) i s a pul lback;

E --> L ( E j

(iii) E I= V e . j ( e E A) --> e E A .

Of t hese d i f f e r e n t formulat ions, (i) i s the t r a d i t i o n a l category

t h e o r e t i c one, (ii) i s p a r t i c u l a r l y use fu l f o r understanding Eff and (iii) i s t h e l o g i c a l formulation ( t r e a t i n g j a s a p ropos i t i ona l

ope ra to r ) . I t i s obvious from t h e d e f i n i t i o n s t h a t F is j -separated i f f

t h e e q u a l i t y on F i s j -c losed, and t h a t a subobject of a j -separated

o b j e c t i s i t s e l f j -separated. W e c o l l e c t some f u r t h e r f o l k l o r i c

f a c t s about t hese not ions i n the nex t theorem.

Theorem 5 . l . ( a ) I f E and F are j -separated, then SO i s EXF. A l s o

n E x F : ExF --> L(ExF) = L(E)xL(F) i s n E X n F .

( b ) I f F i s j -separated, then so i s FE f o r any E . A l s o

t he composite of nFE: FE --5 L(FE) with t h e n a t u r a l map

L ( F ~ ) %> L ( F ) L ( E ) followed by t h e isomorphism L(F)nE: L ( F ) L ( E )

-> L ( F ) E is the monic q F : FE --> L ( F ) E , and t h e eva lua t ion map E

The effective topos 183

E F X E --> F i s obtained by f a c t o r i n g evo(aon Xn ) through qF . FE

( c ) If c >--> F i s j -c losed and a: E --> F then Cr*(C) >--> E

i s j-closed. A l s o L ( a * ( C ) ) = L ( a ) * ( L ( C ) ) .

( d ) I f A >--> E and B >--> E are j-closed then SO is

AAB >--> E. A l s o L(AAB) = L ( A ) A L ( B ) .

(e) I f B 2-> E is j -c losed then so is ( A --> B ) >--> E f o r

any A >--> E. Also L ( A --> B) = L ( A ) --> L ( B ) .

( f ) I f A >--> E i s j-closed and a: E --> F then Va .A >--> F i s

j-closed. A l s o L ( V a . A ) = t l L ( a ) . L ( A ) .

(9) I f R >--z ExE i s a j-closed equivalence r e l a t i o n on E l

then the q u o t i e n t E/R i s j -separated. Also the image ( o r s u r j e c t i v e

monic) f a c t o r i z a t i o n of

E- > L ( E ) - > L(E) /L (R) nE

i s

E- > E/R --> L ( E / R ) = L ( E ) / ~ ( ~ ) . n E/R

Proof: A l l t r i v i a l by t h e l o g i c of j -operators (sketched a t t h e end

of Fourman-Scott C19791) . Category t h e o r e t i c proofs are ( impl i c i t )

i n Johnstone C19771.

L e t us now exp la in why w e a r e i n t e r e s t e d i n c losed subobjects .

Our understanding of Grothendieck toposes rests on the f a c t t h a t

inverse image func to r s preserve coherent l o g i c ( t h a t is A , v , ~ ) . But

t h e inc lus ion of Sets i n Eff i s i n the wrong d i r e c t i o n i f W e wish t o

see some of t h e l o g i c of Sets preserved i n Eff. In general a d i r e c t

image func to r preserves l i t t l e , bu t w e can g e t r a t h e r s t rong r e s u l t s ,

when dea l ing with inc lus ions E --> E l by r e s t r i c t i n g a t t e n t i o n t o

j -c losed subob jec t s . This is s i g n i f i c a n t because a j -c losed subobject

A >--> E "agrees with i t s meaning i n E " i n t h e sense t h a t

nE*(LA) = A .

j

j

(This i s vers ion (ii) of the d e f i n i t i o n . )

184 J.M.E. HYLAND

Given an i n t e r p r e t a t i o n of t h e atomic formulae of a f i r s t

order language i n E w e g e t

(i) an i n t e r p r e t a t i o n U $ 1 of an a r b i t r a r y formula i n E , and (ii)

by applying L an i n t e r p r e t a t i o n of t h e atomic formula i n E and

hence an i n t e r p r e t a t i o n [ $ I of an a r b i t r a r y formula i n E .

Clear ly i f II $ 1 i s a subobject of E , then B $ i s a subobject of

L ( E ) . W e a r e i n t e r e s t e d i n when I[ $ ]I "agrees with t h e in t e rp re -

t a t i o n II $ ]I i n E " i n t h e sense t h a t

j

3 '

j

j j n E * ( 1 4 n j) = u $ n .

The r e l e v a n t d e f i n i t i o n i s of a form fami l i a r from T r o e l s t r a C19731.

Def in i t i on . I n a f i r s t o rde r language, t h e negative formulae ( o r

formulae i n the negat ive fragment) a r e those b u i l t up from atomic

formulae using A , -> ,V.

Theorem 5 . 2 . I f an i n t e r p r e t a t i o n of a f i r s t order language i n E

i n t e r p r e t s t h e atomic formulae a s j-closed subobjects and $ i s a

negativeformula with I $ 1 >-> E , then

r l E * ( v ~ n j ) = ~ $ 1 .

Proof: Induction on the complexity of $ using (5.1) (c) (d) (e) and

( f ) .

Remarks 1) W e can only have e q u a l i t y f o r j-separated o b j e c t s .

2) As $ n i s j -c losed, nE*( I$ 1 j ) = f $ ]I i s equ iva len t

to n + n j = L ( u $ D ) .

3) The r e s u l t i s j u s t a consequence of the " j - in t e rp re t a t iod '

of t he l o g i c of E. For negative formulae w e a r e reading it no t a s a

p re sc r ip t ion f o r de r iv ing the l o g i c of E from t h a t of E , bu t a s t h e

statement t h a t the l o g i c of E agrees with t h a t of E j

j '

16 Separated o b j e c t s and closed subobjects i n Eff. In t h i s s e c t i o n , we desc r ibe what (5.1) means f o r t he p a r t i c u l a r

case when E i s Eff and E

topology. ( I n f a c t w e do n o t use t h i s l a s t f a c t , so t h a t t he

i s Sets so t h a t j i s t h e double negation j

The effective topos 185

mater ia l r e l a t i v i z e s t o an a r b i t r a r y base topos E . ) . W e w i l l say

t h a t an ob jec t of Eff is separated when it is j -separated and t h a t

a subobject of an ob jec t is closed when it i s j -c losed.

Proposit ion 6 . 1 . An ob jec t of Eff is separated i f f it is isomorph

t o one of t he form ( X , = ) where fi x = x ' 1 non-empty implies x = x '

Proof: By version (ii) of t h e d e f i n i t i o n of j -separated, w e see

t h a t i f an ob jec t i s sepa ra t ed , it is a subobject of some A X . But

J

C

any canonical monic i n t o A X i s of t he r equ i r ed form. Conversely any

ob jec t of t he required form is a subobject of a AX Cthe obvious map

is monic), and subobjects of separated o b j e c t s a r e separated.

Defini t ion. An ob jec t ( X , = ) of Eff, where 1[ x = x ' i non-empty

implies x = x', i s a canonical ly separated ob jec t of Eff.

(Such an ob jec t i s completely determined by t h e values [Ix E X i f o r

each x i n X, and i s e s s e n t i a l l y ( t h a t i s , modulo t r i v i a l coding)

given a s a canonical monic i n t o A X . )

Proposit ion 6 . 2 . I f ( X , = ) and (Y,=) a r e canonical ly separated, then

so i s t h e usual product ( X x Y , = ) where

u ( x ,y ) = ( x l , y ~ ) n = IIX = x l n A uy = y l n . Proof: Immediate from (5.1) ( a ) and the d e f i n i t i o n of t he product of

maps i n the l o g i c .

The case of funct ion spaces i s more complex than t h a t of products.

Since t h e general desc r ip t ion of a funct ion space (see HJP C19801 i s

too clumsy, w e must use (5.1) (b) t o cons t ruc t a s u i t a b l e represen-

t a t i o n .

Proposit ion 6 . 3 . L e t ( Y , = ) and (Z,=) be o b j e c t s of Eff with (Z,=)

canonically separated.

taken t o be t h e canonical ly separated o b j e c t (rZrY,=) where ( taking

r ( z , = ) 5 2)

Then the funct ion space (Z,=) ("=) may be

I n = II V ~ ~ Y . ~ ( C ~ I ) = ~ I ( c Y n n = n~ II Ey-->a(Cyl)=a' ( c y i ) i l y c ~ i

and where t h e evaluat ion map is represented by t h e func t iona l r e l a t i o n

186 J.M.E. HYLAND

I [ E ~ A EY A a ( c y 1 ) = z n . Proof:

i n t he l o g i c by

(5.1) (b) g ives us a monic from (Z,=) ("=) t o rZrY def ined

uvY Y . ~ C C ~ I ) z n which i s equ iva len t t o t h e formulae given a s CZ,=) i s canon ica l ly

sepa ra t ed . The r e p r e s e n t a t i o n of t h e eva lua t ion map fo l lows from

t h e d e f i n i t i o n i n t h e l o g i c of t h e map descr ibed i n (5 .1) (b ) by

elementary l o g i c .

Remark. I f f o r every y i n Y , Ey i s non-empty (and w e may a s w e l l

d i s r e g a r d t h e o t h e r s ) , then t h e fo l lowing a l t e r n a t i v e r e p r e s e n t a t i o n

of t h e func t ion space is canon ica l ly sepa ra t ed : IZy ,=) where

a = a l n = n{ I I ~ = ~ ~ - - > ~ ( ~ ) = ~ ~ ( ~ ~ ) n l y , y l E ~ ~

and where t h e eva lua t ion map i s r ep resen ted a s above. (We g e t t h i s

a l t e r n a t i v e r e p r e s e n t a t i o n by cons ide r ing t h e obvious map from

r Z ( y l = ) t o Z E Y l where EY i s t h e canonica l subobjec t of AY of which

( Y , = ) is a q u o t i e n t , )

Ea is non-empty, w e can cont inue t h i s process and o b t a i n a simple

d e s c r i p t i o n of i t e r a t e d func t ion spaces of s epa ra t ed o b j e c t s . W e

cons ider t h i s f u r t h e r i n 5 5 7 and 11.

Then i f w e d i s r ega rd those a i n Z y such t h a t

W e nex t cons ide r c losed subob jec t s i n Eff. Propos i t i on 6 . 4 . A subob jec t of an o b j e c t (X,=) of Eff i s c losed

i f f it i s r ep resen ted by a canonica l monic determined by A E Cn of

f o r s o m e A 5 I ' ( X , = ) .

( I t does no harm t o l e t A denote t h e subse t of r ( X , = ) , t h e canonica l

monic as de f ined and t h e c losed subob jec t which it r e p r e s e n t s ) .

Proof: By ve r s ion (ii) of t h e d e f i n i t i o n of j - c losed , a c losed sub-

o b j e c t of (X,=) must be of t h e form rlx (AA) f o r some A TIXI=) . -1

The effective topos 187

But what w e have desc r ibed i s e a s i l y seen t o be e q u i v a l e n t t o t h e

d e f i n i t i o n of nx (AA) i n t h e l o g i c .

Def in i t i on .

-1

A monic of form (X,=') - [="> (X,=) where

[x=xD, i f [XI E A , I x = ' x ' ] = U { I [x=x ' JJ IlxleA} = { I , otherwise,

f o r some A 5 T ( X , = ) i s a canon ica l c losed monic. ( ( 6 . 4 ) shows

e s s e n t i a l l y t h a t t h e c losed subob jec t s a r e j u s t those r ep resen ted

by canon ica l c losed monics) .

Remark. On many occas ions it i s more n a t u r a l t o d i s r e g a r d i n (X,=')

the x which a r e n o t i n A. W e s h a l l s u i t terminology t o need and

r e f e r t o t h i s mod i f i ca t ion a l s o a s a canon ica l c losed monic. Note

t h a t t h e no t ion becomes p a r t i c u l a r l y simple i n case (X,=) i s

canon ica l ly s e p a r a t e d , a s t hen w e may t a k e A 5 X ( t ak ing

r ( X , = ) 5 X a g a i n ) .

We now say what (5 .1) ( c ) , ( d ) , ( f ) mean f o r t he e f f e c t i v e topos .

P ropos i t i on 6 . 5 . L e t A >-> (X,=) and B >-> (X,=) be subob jec t s

of (x,=), C >--> (Y,=) a subob jec t of (Y,=) and [GI: (X,=) --> (Y,=)

a map i n Eff. I f C i s a canon ica l c losed monic (de f ined from C 5 r ( Y , = ) ) , then

[ G l - l ( C ) i s t h e canon ica l c l o s e d monic de f ined from

( r ( G ) - l ( C ) = {Cxl l{ylG(x ,y) non-empty) E C).

I f A , B are canon ica l c l o s e d monics (de f ined from A , B 5 r ( X , = ) ) , then

AAB i s t h e canonica l c losed monic de f ined from AnB. I f B i s a

canonica l c losed monic (de f ined from B 5 r ( X , = ) ) , then A -> B i s

t h e canon ica l c l o s e d monic de f ined from

TA -> B = {Cxll i f [ X I E rA then [XI E B},

and VCG1.B is t h e canon ica l c losed monic de f ined from

V r ( C G 1 ) .B = {Cyll i f G(x,y) non-empty then [-XI E B}.

Proof: (5 .1) t e l l s us t h a t t h e r e l e v a n t subob jec t s are c losed and

t h a t w e g e t a r e p r e s e n t a t i o n by apply ing T, doing t h e r equ i r ed

c o n s t r u c t i o n i n Sets, and t a k i n g t h e cor responding canon ica l c losed

monic.

188 J.M.E. HYLAND

Remark. The cons t ruc t ions desc r ibed i n 6 . 5 are p a r t i c u l a r l y simple

i n case t h e o b j e c t s ( X , = ) and (Y,=) are canon ica l ly sepa ra t ed .

F i n a l l y w e cons ide r t h e meaning of (5 .1) (9) f o r t h e e f f e c t i v e

topos. I t g ives a converse t o t h e obvious remark t h a t i f ( X I = ) i s

canon ica l ly sepa ra t ed ] t hen t h e e q u a l i t y ( o r d iagonal ) i n

( X l = ) x ( X l = ) i s t h e canon ica l c losed monic de f ined by t h e d iagonal i n

xxx. Propos i t ion 6 . 6 .

l ence r e l a t i o n on ( X , = ) i n Eff . Then t h e q u o t i e n t ( X , - ) i s i s o -

morphic ( i n t h e obvious way) t o t h e canon ica l ly sepa ra t ed o b j e c t

( r ( X I - ) , z ) where

1 ~ x l ~ C x l l J

Suppose t h a t - E C x x x r e p r e s e n t s a c losed equiva-

= U { I[ x ' -x l ' J Ix' E [ X I and xl' E [ x l l } .

Proof: The composite ( X , = ) - > ( X , - ) -> A r ( X l - ) i s represen-

t ed by [ - I rl ( X I - )

H(x,Cxll) = U { x-xl') Ixl' E I x l l ) .

By (5 .1) (9) w e r e q u i r e t h e image f a c t o r i z a t i o n of [HI, and what w e

have is a s t anda rd d e f i n i t i o n of t h i s f a c t o r i z a t i o n i n the l o g i c .

57. The e f f e c t i v e o b j e c t s .

Since Sets i s inc luded i n Eff, Eff con ta ins c l a s s i c a l mathema-

t i c s so much of it i s n o t p a r t i c u l a r l y " e f f e c t i v e " . I n t h i s s e c t i o n

w e cons ider o b j e c t s whose c l o s e r e l a t i o n t o t h e a p p l i c a t i v e s t r u c -

ture (IN , .) ensu res t h a t ope ra t ions on them a r e genuinely "ef fec t ive" .

I n l a t e r s e c t i o n s w e w i l l show t h a t t h e o b j e c t s of a n a l y s i s i n Eff a r e ( q u i t e familiar) o b j e c t s of t h i s k ind .

Def in i t i on . An o b j e c t ( X , = ) i s ( s t r i c t l y ) e f f e c t i v e i f f

(i) 1 x E X J is non-empty each x E X ,

(ii) U x E X J n f x ' E X J non-empty impl i e s x = x ' ,

and (iii) U x = x ' l = II x E X I n Ux' E X I . (Occasionally w e may d e s c r i b e an o b j e c t as e f f e c t i v e when it i s

isomorphic t o one of t h e above form. It w i l l be obvious when t h i s

l oose sense is meant.)

The effective topos 189

Clearly e f f e c t i v e o b j e c t s a r e (canonical ly) separated, and w e

can e a s i l y show t h a t they sha re the c losu re p rope r t i e s of separated

objects . Proposition 7 .1 . ( a ) I f ( X , = ) and (Y,=) a r e e f f e c t i v e , then so i s

t h e i r product.

(b) I f (Z,=) i s effect ' ive , then so i s t h e funct ion

space (z,=) (',=) f o r any (Y,=) i n ~ f f .

(c) A subobject of an e f f e c t i v e ob jec t is

e f f ec t ive .

(d) A q u o t i e n t of an e f f e c t i v e o b j e c t by a c losed

equivalence r e l a t i o n i s e f f e c t i v e .

Proof: ( a ) i s t r i v i a l : look a t ( 6 . 2 ) .

(b) follows by in spec t ion of (6.3). I f w e res t r ic t t o those

a E r Z r Y with Ea

(i) , (ii) and (iii) above.

non-empty, then w e g e t an ob jec t s a t i s f y i n g

(c) r equ i r e s more work. L e t ( X I = ) be s t r i c t l y e f f e c t i v e and

l e t (XI=') --> ( X , = ) be a canonical monic with

[ x ='XI II = R ( x ) A Il x=x'n for some s t r i c t r e l a t i o n a l R E c . Write x E' X f o r x ='x and put X ' = Ix E XI Ux E ' X 1 i s non-empty}.

Since [ x E ' x n n [x' € ' X I non-empty implies Ilx E x D n [ x u E x n non-empty which implies x = x', w e g e t a s t r i c t l y e f f e c t i v e ob jec t

(XI,=) with [ x E X ' l = I[ x E ' X 1 . I t is isomorphic t o (XI=')

because

X

n ( ( u x E x'I n Ux' E xi] 1 <--> I [ x ='x'II )

is non-empty . (d) follows from ( 6 . 6 ) . I f - i s a c losed equivalence

r e l a t i o n on (X,=) which is s t r i c t l y e f f e c t i v e , then [x-xi n [xl-x'D

non-empty imples x E X I n I[ X'E X 1 non-empty which implies x = x'.

I t follows t h a t ( r ( X I = ) ,%) i s s t r i c t l y e f f e c t i v e .

The f u l l subcategory of Eff whose o b j e c t s a r e t h e e f f e c t i v e

ones has a concrete r ep resen ta t ion f a m i l i a r t o log ic i ans i n

190 J.M.E. HYLAND

connection with t h e e f f e c t i v e ope ra t ions . Take p a r t i a l equivalence

r e l a t i o n s on IN ( t h a t i s equivalence r e l a t i o n s on t h e i r f i e l d s )

R,S, ... and w r i t e m / R = {CnlR(n E F i e l d ( R ) } f o r t h e set of

equivalence c l a s s e s of R . L e t a m x F: R -> S be a map F: N/R->lN/S

such t h a t t h e r e is f E IN with

F(CnlR) = [ f ( n ) l S

f o r a l l n E F i e l d ( R ) . C lea r ly w e have a category.

Each p a r t i a l equivalence r e l a t i o n R gives r ise t o a s t r i c t l y

e f f e c t i v e o b j e c t ( B / R , = ) of Eff where E ( l n l R ) = Cnl,.

F: R --> S gives r ise t o a map (IN/R ,=) -> (B/S ,=) represented

A map

by

F(CnlR,[mlS) = U{CnlRh[mlSIF([nlR) = [mlsl ,

and so w e have a func to r i n t o Eff which i s c l e a r l y f a i t h f u l and i s

f u l l by applying g loba l s e c t i o n s t o ( 7 . 1 ) ( b ) . Clear ly any s t r i c t l y

e f f e c t i v e o b j e c t i s isomorphic t o one obtained from a p a r t i a l

equivalence r e l a t i o n .

category of p a r t i a l equivalence r e l a t i o n . I t i s given by

L e t us desc r ibe t h e funct ion space SR i n the

R eS f i f f nRm implies e ( n ) S f ( m ) . A moment's thought shows t h a t t h i s corresponds t o the p r e s c r i p t i o n

f o r f ind ing t h e space of func t ions from ( B / R ,=) t o (N/S ,=) given

by ( 7 . 1 ) (b) . This i s a use fu l way t o think of t h e m a t e r i a l i n

§ § l o and 11. ( I n f a c t t h e embedding of t h e p a r t i a l equivalence

r e l a t i o n s i n Eff prese rves t h e l o c a l Cartesian closed s t r u c t u r e of

t h e former category. )

One p a r t i c u l a r e f f e c t i v e o b j e c t i s c ry ing o u t f o r a t t e n t i o n :

t h a t corresponding t o t h e e q u a l i t y r e l a t i o n on N . This i s t h e

object . I N = (IN ,=) where

[In = m l l = inlnIm}.

As w e noted i n 13, t h i s i s t h e n a t u r a l number o b j e c t ; w e consider some

of i t s p r o p e r t i e s i n l a te r sec t ions . F i r s t however, w e w i l l use it

t o give a c h a r a c t e r i z a t i o n of e f f e c t i v e o b j e c t s . Reca l l t h a t any

The effective topos 191

ob jec t ( X , = ) i s a q u o t i e n t of a subobject of AX. For e f f e c t i v e

ob jec t s w e can r ep lace AX by ( I N ,=) . Proposi t ion 7 . 2 . Every e f f e c t i v e o b j e c t i s a q u o t i e n t by a closed

equivalence r e l a t i o n of a c losed subobject of (IN ,=) . Proof: I f ( X , = ) corresponds a s above t o the p a r t i a l equivalence

r e l a t i o n R on I N , then the closed subobject of LIN ,=) i s t h a t

determined by Fld(R) 5 I N and t h e closed equivalence r e l a t i o n - i s

given by

That t h e r e s u l t i n g q u o t i e n t of a subobject of ( IN ,=) gives rise t o

the same R is immediate i n view of ( 6 . 4 ) and ( 6 . 6 ) .

Now w e can s t a t e our c h a r a c t e r i z a t i o n theorem.

Theorem 7.3. The following cond i t ions on a o b j e c t X of Eff a r e

equivalent :

( i) X i s isomorphic t o a s t r i c t l y e f f e c t i v e o b j e c t ;

(ii) X i s a c losed q u o t i e n t of a c losed subobject of (IN ,=) :

(iii) X i s a c losed q u o t i e n t of a subobject of ( I N , = I .

Proof: (i) implies (ii) i s (7 .2 ) , (ii) implies (iii) is t r i v i a l

and (iii) implies (i) follows from ( 7 . 1 ) ( c ) and ( a ) .

Remark. Since (m,=) is t h e n a t u r a l number o b j e c t , w e have shown

t h a t t h e e f f e c t i v e o b j e c t s are those subnumerable i n a c e r t a i n way.

However t h e e q u a l i t y on an e f f e c t i v e o b j e c t must be c losed (as it

is a sepa ra t ed o b j e c t ) and t h e r e a r e quo t i en t s o f (m ,=) by

equivalence r e l a t i o n s which are emphat ical ly no t c losed. (The

reader w i l l know where t o look a f t e r reading t h e next s ec t ion ! )

So the e f f e c t i v e o b j e c t s a r e a proper subc la s s of t h e quo t i en t s of

decidable o b j e c t s r e c e n t l y s tud ied by Peter Johnstone i n a general

context.

192 J.M.E. HYLAND

18. Markov's p r i n c i p l e and almost negat ive formulae.

In t h i s s ec t ion w e see how the general r e s u l t of (5.2) can be

extended i n the case of t h e topos Eff and (double negation) topology

j with Effj = Sets. Lemma 8.1. A : Sets -> Eff preserves t h e i n i t i a l ob jec t . Thus

lI is always a c losed subobject , and hence decidable subobjects

a r e closed.

Proof: T r i v i a l category theory.

Lemma 8.2. Markov's p r i n c i p l e

VR L P ( N ) . ( V n . R ( n ) v ~ R ( n ) h ~ l 3 n ( R ( n ) -> 3nRCn))

holds i n Eff. Proof: A s t h e a r i t h m e t i c a l s ta tements holding i n Eff a r e those

r e a l i z e d i n the o r i g i n a l sense of Kleene (see 13) t h i s is t h e

s tandard argument ( T r o e l s t r a C19731). Note t h a t w e do n o t need t o

know about P ( N ) 1

Lemma 8 . 3 . I f R >-> N x X is a decidable subobject i n E X , then

3n.R(n,x) >-> X i s closed and r ( 3 n . R ( n f x ) ) = 3 n . r ( R ( n f x ) ) .

Proof: This amounts t o --3n.R(n,x) 5 3n.R(n,x) which follows by

( 8 . 2 ) .

Remark. Though ( 8 . 2 ) depends on Markov's p r i n c i p l e i n w, and so

does n o t r e l a t i v i z e t o an a r b i t r a r y topos, ( 8 . 3 ) does r e l a t i v i z e :

we w i l l always have j (3n.R(n,x)) 5 3n.R(n,x) .

(8.1) and ( 8 . 3 ) suggest t h a t w e extend the c l a s s of negat ive

formulae.

Defini t ion. A formula is c a l l e d almost negat ive i f f it i s b u i l t up

from atomic formulae using A , ->,V,i, and sequences of 3n appl ied

t o decidable formulae ( t y p i c a l l y equat ions between numerical-valued

terms) . W e now give our extension of (5 .2) .

Theorem 8 . 4 . I f t he atomic formulae of a f i r s t o rde r language a r e

The effective topos 193

i n t e r p r e t e d as c losed subobjects i n Eff and @ i s almost negat ive

with [ @ 1 >-> E, then

n E * ( u + n j ) = t @ n . Proof: As f o r (5.2) using (8.1) and (8.3) as w e l l .

The fo rce of (8.4) is t h a t , f o r @ almost negat ive @ i s t r u e

i n Eff i f f t h e corresponding i n t e r p r e t a t i o n of @ i n Sets i s t r u e :

t h a t i s , t h e meaning of @ i n Eff "agrees with" i t s c l a s s i c a l meaning.

(8.4) i s a vers ion of 3 .2 .11 ( i ) and (ii) of T r o e l s t r a 119731;

w e could o b t a i n a more p roof - theo re t i c ve r s ion by s e l a t i v i z i n g t o

the f r e e topos (with n a t u r a l number o b j e c t ) . For a language which

can "express i t s own r e a l i z a b i l i t y " we could obviously ob ta in

vers ions of 3.2.12 and 3.2.13 of Tsoe l s t r a 119731. For t h e sake of

completeness w e give a ve r s ion of 3.6.5 of T r o e l s t r a C19731.

Defini t ion ( c f . Hyland C19771) PR(a.n.1 is t h e l e a s t class C of

formulae such t h a t

(i) C con ta ins a l l atomic formulae;

(ii) C i s closed under ~ , v , V , 3 ;

(iii) i f @ is almost negat ive (more gene ra l ly almost negat ive

preceded by e x i s t e n t i a l q u a n t i f i e r s ) and $ is i n C, then ( @ -> $1

i s i n C.

Proposi t ion 8.5. In the s i t u a t i o n of ( 8 . 4 1 , i f @ is i n PR(a.n)

with I@ 1 >-> E , then

n + n 5 nE*( I@ n j ) .

Proof: By induct ion on the complexity of @. (Note t h a t v and + 'are

ca l cu la t ed d i f f e r e n t l y i n E from t h e way they are i n E).

Remark. For a general sheaf subtopos E of E w e have

5 nE*( fi @ 1 . ) f o r a l l @ i n PR(j-closed) .

j

j [ @ I So i f atomics are

3 i n t e r p r e t e d as j -c losed, t hen w e g e t t h e r e s u l t f o r a l l @ i n

PR(negative) . The f o r c e of (8.5) i s t h a t , f o r @ i n PR(a.n) , i f @ is t r u e i n

194 J.M.E. HYLAND

- E f f , then t h e cor responding i n t e r p r e t a t i o n of @ i n Sets i s t r u e .

This g ives r ise t o a conse rva t ive ex tens ion r e s u l t (when r e l a t i v i z e d )

a s i n T r o e l s t r a C19731 53.6.

§ 9 Choice p r i n c i p l e s and t h e r e a l numbers.

I n t h i s s e c t i o n w e make a s t a r t towards showing t h a t a n a l y s i s

i n Eff is j u s t c o n s t r u c t i v e r e c u r s i v e a n a l y s i s . (We a l r eady have

Markov's p r i n c i p l e ( 8 . 2 ) . ) W e do t h i s i n two s t e p s . F i r s t w e show

t h a t w e have t h e choice p r i n c i p l e s t o ensu re t h a t t h e Dedekind

r e a l s ( t h e r i g h t r e a l s i n a topos) a r e Cauchy (see Fourman-Hyland

C19791 and a l s o Fourman-Grayson t h i s volume). Then w e use t h e re-

s u l t s of 57 t o show t h a t t h e Cauchy r e a l s i n Eff can be i d e n t i f i e d

wi th a f a m i l i a r s t r i c t l y e f f e c t i v e o b j e c t used i n c o n s t r u c t i v e

r e c u r s i v e a n a l y s i s .

F i r s t we need t o know what t h e space of func t ions from IN t o

an a r b i t r a r y ( X , = ) looks l i k e i n Eff. A s s t a t e d i n HJP C19801, by

l o g i c a l c o n s i d e r a t i o n s it is

(ZIN xx,=)

where

[ G = H I = EGAn{G(n,x) <-> H(n,x) In E TN , x d X I ,

wi th EG t h e non-standard va lue of "G i s a f u n c t i o n a l r e l a t i o n " .

Suppose now t h a t e realizes Vn E IN ,3x E ( X I = ) . @ ( n , x ) . Then

f o r every n , e ( n ) E UIEXA [ @ ( n , x ) ] Ix E X I . For each n p i ck xn

such t h a t e ( n ) E E X ~ A [ @ ( n , x n ) l . S e t G(n ,x) = EnA [ x = x n 1 . Now (uniformly i n e) w e can f i n d numbers r e a l i z i n g EG and

Vn.3x .G(nIx)h$(n ,x) : G i s r e l a t i o n a l , s t r i c t and s ing leva lued i n a

s t anda rd way from i t s d e f i n i t i o n : A n . < n , n l ( e ( n ) ) > r e a l i z e s G i s

t o t a l ; Xn.<a ( e ( n ) ) , < < n , ~ l ( e , n ) > , a 2 ( e , n ) > > r e a l i z e s

Vn.3x .G(x ,n)h@(xln) . Thus (uni formly i n e ) w e have a number

r e a l i z i n g

1

The effective topos 195

3g: Ri-> ( X , = ) .Vn .$ (n ,g (n ) )

and so we have proved t h e fol lowing r e s u l t .

Proposit ion 9 . 1 . A C ( I N , X ) , t he axiom of choice from t h e n a t u r a l

numbers t o an a r b i t r a r y type X , holds i n Eff. Remark. W e used AC ( IN , X ) i n sets i n t h e above proof . But i f ( X , = )

i s e f f e c t i v e , then no use of a choice p r i n c i p l e i n the base topos

i s needed (compare ( 7 . 1 ) (b) : i n t h i s case t h e argument i s contained

within T r o e l s t r a 119731 3.2 15.

By a s i m i l a r proof ( l e f t t o t he r eade r ) w e a l s o have t h e s t ronge r

r e s u l t .

Proposit ion 9 . 2 . D C ( X ) , t h e axiom of dependent choices on an

a r b i t r a r y type X , holds i n Eff. Remark. Again D C ( X ) i s used i n the proof , but i s no t needed f o r

e f f e c t i v e o b j e c t s X.

A C ( W , I N ) is enough t o show t h a t t h e Cauchy and Dedekind r e a l s

a r e the s a m e . To g e t an e x p l i c i t r ep resen ta t ion of R as a

s t r i c t l y e f f e c t i v e o b j e c t , w e use t h e Cauchy sequence d e f i n i t i o n .

Lemma 9.3. The i n t e g e r s Z and r a t i o n a l s Q i n Eff can be taken a s

s t r i c t l y e f f e c t i v e o b j e c t s a,=) and (Q,=) where f o r x i n Z o r Q ,

Ex = # x) where # x is an elementary code f o r X.

Proof: They are obtained successively from (IN ,=) by taking closed

(decidable) q u o t i e n t s of c losed (decidable) subobjects o f products:

so the r e s u l t fol lows from the p r e s c r i p t i o n s involved i n (7.1) ( a ) ,

( c ) [easy case of c losed subob jec t s l and ( a ) . Lemma 9 . 4 . The space of maps from IN t o Q i n Eff i s t h e s t r i c t l y

e f f e c t i v e o b j e c t (Qm ,=) where

Qm = t h e r e c u r s i v e func t ions from IN t o Q

and [[ c1 E QIN! = { e j e ( n ) = # a ( n ) 1 , t h e set of ind ices f o r a .

196 J.M.E. HYLAND

Proof: This i s t h e p re sc r ip t ion i m p l i c i t i n ( 7 . 1 ) (b) . Since w e

have enough choice t o show t h a t any reasonable not ions of Cauchy

sequence give the same r e a l s i n Eff w e de f ine C S , t h e c o l l e c t i o n of

( r e s t r i c t e d ) Cauchy sequences by

Thus d e f i n i t i o n i s i n the negat ive fragment and so s ince < i s

decidable on the r a t i o n a l s and hence by (8.1) c losed, de f ines a

closed subobject of Qm i n Eff. it.

Lemma 9.5. The space of Cauchy sequences i n Eff is t h e s t r i c t l y

e f f e c t i v e ob jec t ( C S , = ) where CS i s t h e set of r ecu r s ive Cauchy

sequences and Ur E C S ] I i s t h e set of i nd ices f o r r .

Proof: By the discussion above.

To ob ta in the r e a l s IR, w e t ake the quo t i en t of CS by t h e equiva-

lence r e l a t i o n

I n view of (8.4) w e can i d e n t i f y

1 r - s i f f vn.lrn-snl < - p - 3 ' (This choice of d e f i n i t i o n gives one p l en ty of "elbow room".)

Proposit ion 9 . 6 . The space IR of r e a l s i n Eff i s t h e s t r i c t l y

e f f e c t i v e ob jec t (IR ,=) where

IR = t he r ecu r s ive r e a l s ( t h a t i s r e a l s with r ecu r s ive

Cauchy sequences converging t o them)

and Ux E IR ]I = t h e set of i nd ices f o r Cauchy sequences converging

t o x.

Proof: As before - def ines a c losed equivalence r e l a t i o n so this

i s by the p re sc r ip t ion of ( 7 . 1 ) ( a ) . W e have shown t h a t t h e r e a l s i n Eff a r e represented j u s t a s

they are i n ( cons t ruc t ive ) r ecu r s ive ana lys i s . Of course, as t hey

t o o a r e def ined i n the negative fragment, t h e operat ions of

add i t ion , m u l t i p l i c a t i o n and so f o r t h a r e what they should be. To

do s e r i o u s ana lys i s however w e need t o consider funct ions which w e

do i n the next few sec t ions .

The effective topos 197

Remark. I t is seldom e f f i c i e n t t o g r ind t h i n g s o u t i n models f o r

cons t ruc t ive ana lys i s : where possible one should use the axiomatic

point of view. Consider f o r example t h e quest ion of t he fundamental

theorem of a lgebra i n Eff . This theorem is proved i n Bishop C19671.

One way of reading Bishop's cons t ruc t ive mathematics (though no t

the intended one!) is t o regard it a s formalized i n an i n t u i t i o n i s t i c

type theory with extensional e q u a l i t y and using ( D C ) . Hence i n view

of ( 9 . 2 ) the fundamental theorem of a lgebra i s t r u e i n Eff ( a s it i s

i n o the r r e a l i z a b i l i t y t o p o s e s ) . I n view of t he obvious represen-

t a t i o n of C i n Eff derived from ( 9 . 6 ) , and t h e f a c t t h a t

"al, ... ,an a re the r o o t s of zn+an-lzn-l+. . .+ao = 0"

de f ines a c losed subobject ( i n C2n) w e can i n t e r p r e t t h i s f a c t a s

follows. There i s an e f f e c t i v e process t ak ing ind ices f o r the

r ecu r s ive complex c o e f f i c i e n t s of a monic polynomial of degree n

over t he r ecu r s ive complex numbers t o i n d i c e s f o r the r ecu r s ive

roots . I t is n o t t r i v i a l t h a t a r ecu r s ive polynomial has recursive

roo t s and any n a t u r a l proof would s e e m t o e s t a b l i s h the s t ronger

r e s u l t and a s such would have t h e form of an a b s t r a c t proof using

(DC) - 510. E f f e c t i v i t y and Church's Thesis.

I t i s t i m e t o give substance t o the claim made i n 17 t h a t

operat ions on e f f e c t i v e o b j e c t s a r e " e f f e c t i v e " . W e f i r s t consider

t he s p e c i a l case of Church's Thesis.

Lemma 10.1. The space of maps from IN t o IN i n Eff i s t h e s t r i c t l y

e f f e c t i v e o b j e c t ( I N r n ,=) where

ININ = t h e r ecu r s ive funct ions from IN t o IN , and 6 a E I N I N ] = { e l e ( n ) = a ( n ) 1 , t h e set of i nd ices f o r a

Proof: This i s the p r e s c r i p t i o n i m p l i c i t i n (7.1) (b) . Proposit ion 1 0 . 2 . "Church's Thesis" t h a t a l l funct ions a r e r ecu r s ive ,

198 J.M.E. HYLAND

~ c i E mIN .3e .Vn .3y(T(e ,n ,y )~U(y) = c i (n ) )

ho lds i n Eff. CT i s Xleene ' s T-predicate and U h i s ou tpu t f u n c t i o n . ]

Proof: I n view of (3.3) elementary r ecu r s ion theory can be developed

i n Eff as i n T r o e l s t r a E19731. So by ( 8 . 4 ) Vn. 3y . (T(e ,n ,y )AU(y)=

a ( n ) )

real izes "Church's Thes is" .

Remark. Church's Thes is a s t r a d i t i o n a l l y formulated i n Heyt ing ' s

Ar i thmet ic (see T r o e l s t r a C19731) is a n amalgam of ou r "Church's

Thes is" and AC(IN ,IN).

i n Eff agrees wi th i t s meaning i n Sets. Then Xe.<e,<e,e>>

W e can hope t o g e n e r a l i z e ( 1 0 . 2 ) t o a l l e f f e c t i v e o b j e c t s i n

view of (7.3) which states t h a t t hey can i n a c e r t a i n way be sub-

numerated (by t h e codes f o r t h e i r e l emen t s ) .

Lemma 10.3. I f (Z,=) i s s t r i c t l y e f f e c t i v e and ( Y , = ) is a r b i t r a r y

i n Eff , t hen t h e space of maps from (Y,=) t o (Z ,= ) i n Eff i s t h e

s t r i c t l y e f f e c t i v e o b j e c t ( Z r Y , = ) where

Z r y = t h e " r ecu r s ive" maps from T Y t o Z ( t h a t i s , t h e maps

wi th i n d i c e s ) ,

and ci E Z Ty = { e l e ( n ) E Eci(y) f o r a l l n E Ey}, t h e set of

i n d i c e s f o r a.

Proof: This i s t h e p r e s c r i p t i o n i m p l i c i t i n (7 .1 ) ( b ) .

Here then is a g e n e r a l i z a t i o n of (10 .2 ) .

Propos i t ion 1 0 . 4 . L e t (Y,=) <- B >-> IN r e p r e s e n t t h e e f f e c t i v e

o b j e c t (Y,=) as a q u o t i e n t of a c losed subob jec t of I N , and l e t

( X , = ) <s A >--> IN r e p r e s e n t (X,=) as a q u o t i e n t of a c losed

subobjec t of I N . Then a "genera l ized Church's Thes is"

sY

X

Yci E YX.3e.Va E A . 3 z ( T ( e , a , a ) h a ( S X ( a ) ) = Sy(U(z)))

holds i n Eff . (One can u s e f u l l y compare t h i s r e s u l t wi th t h e t r ea tmen t of t h e

extended Church's Thes is i n T r o e l s t r a C19731.)

The effective topos 199

Proof: The cond i t ions given ensu re t h a t a ( S x ( a ) ) = Sy(U(z)) i n t e r -

p r e t s as a c losed subob jec t . (Note t h a t U(y) E B i s i m p l i c i t , s o w e

need B c losed . ) S ince

3 z ( T ( e , a , z ) A a ( S X ( a ) ) = Sy(U(z)))

is e q u i v a l e n t t o

3~.T(e,a,z)AVz.(T(e,a,z) --> a ( S , ( a ) ) = Sy(U(z)),

it a l s o i n t e r p r e t s a s a c losed subob jec t . It remains t o determine

e from an index f o r a .

t ak ing any a E A t o an e lement of ESX(a ) :

t o EaSX(a ) ; t he cond i t ion t h a t Sy i s on to p rov ides a map from t h i s

t o some b E B wi th Sy(b) = a S x ( a ) .

which can c l e a r l y be chosen e f f e c t i v e l y i n the index f o r a .

The t o t a l cond i t ion f o r Sx g ives a map

an index f o r a maps t h i s

e i s an index f o r t h i s composite

In p a r t i c u l a r , w e can see t h a t when e f f e c t i v e o b j e c t s a r e

p re sen ted ( v i a p a r t i a l equiva lence r e l a t i o n s ) a s c losed q u o t i e n t s

of c losed subob jec t s of I N , t h e n maps between them a r e e f f e c t i v e i n

the i n d i c e s (and t h i s ho lds i n Eff). This i s t y p i c a l l y t h e s i t u a t i o n

i n c o n s t r u c t i v e r e c u r s i v e a n a l y s i s .

111. The e f f e c t i v e ope ra t ions .

In t h i s s e c t i o n w e use ( 1 0 . 4 ) a s t h e induc t ion s t e p t o show

t h a t t h e s t a t emen t t h a t t h e f i n i t e t ypes ove r IN a r e t h e he red i -

t a r i l y ( e x t e n s i o n a l ) e f f e c t i v e o p e r a t i o n s ho lds i n Eff . A s s u m e f o r n o t a t i o n a l purposes a c o l l e c t i o n of type symbols

genera ted from 0 by x ( f o r p roduc t s ) and --> ( f o r func t ion s p a c e s ) .

The f i n i t e t ypes over t h e n a t u r a l numbers ( m u la a t ype symbol) a r e

de f ined i n d u c t i v e l y by INo = I N ,

= m ' x m IN U X T u 7'

INu+= = (rnT ) 0 . El

The h e r e d i t a r i l y e f f e c t i v e ope ra t ions ( H E O u l u a t ype symbol)

(see Kreisel 119591 and T s o e l s t r a C19731) may be de f ined by f i r s t

d e f i n i n g a c o l l e c t i o n ( R u I a a type symbol) of p a r t i a l equiva lence

200 JM.E. HYLAND

r e l a t i o n s induc t ive ly by

nROm i f f n = m,

nRaxTm i f f r1 (n) Raml ( m ) and r 2 (n )RTr2 (m) ,

eRa+.rf iff i f nRam then e ( n ) , f (m) are def ined and

e ( n ) R T f ( m ) .

W e can then regard HEOa as t h e equivalence classes IN/Ra .

of t h e d i scuss ion i n 9 7 , w e can equa l ly regard HEOa a s b u i l t up

( toge the r with ind ices f o r i t s elements) from t h e n a t u r a l number

o b j e c t i n Eff, by t ak ing t h e usual products and funct ion spaces as

i n (10.3) . Thus i n Eff No = (HEOaI=) where

In view

Ex = t he i n d i c e s f o r x,

and where nRam i f f n,m a r e i n d i c e s f o r t h e same x E HEOg. Then w e

can regard HEOa as t h e g loba l s e c t i o n s of t h e f i n i t e t ypes over JN

i n Eff .

These d e f i n i t i o n s a l l r e l a t i v i z e and our next r e s u l t s tates

t h a t Eff "knows t h a t i t s f i n i t e t ypes a r e t h e e f f e c t i v e ope ra t ions" .

Theorem -11.1.

t h a t t h e products and func t ion spaces correspond.

Proof: The R a t s are def ined by negat ive formulae, and so by 18.4)

i n t e r p r e t i n Eff as c losed p a r t i a l equivalence r e l a t i o n s agreeing

with t h e i r meaning i n Sets. I f we c a l c u l a t e t he equivalence

c l a s s e s i n the obvious way using (7.1) (d) w e j u s t g e t (HEOg,=)

t h a t i s INa i n Eff.

as it ought. ( I f t h i s i s t o o a b s t r a c t , t h e reader can use a

l abor ious induct ion, with ( 1 0 . 4 ) dea l ing with t h e main induct ion

s t e p . )

Remark. Something q u i t e deep i s going on behind (11.1) which is

connected with i t e r a t i o n s of t h e e f f e c t i v e topos cons t ruc t ion a:

s tud ied i n B i t t s C19811. I t i s i n connection with t h e e f f e c t i v e

ob jec t s t h a t w e can g e t a general expression of t h e idempotency o f

For each a , INa = HEOa holds i n Eff, i n such a way

Clear ly t h e rest of t h e s t r u c t u r e corresponds

The effective topos 20 1

of r e a l i z a b i l i t y (see T r o e l s t r a (1973) 3.2.16).

512 . Sequent ia l con t inu i ty .

From ( 8 . 2 ) , ( 9 . 2 ) , ( 1 0 . 4 ) and t h e discussion i n 19, it should be

c l e a r t h a t a n a l y s i s i n Eff i s j u s t cons t ruc t ive r e c u r s i v e a n a l y s i s .

So w e have t h e usual con t inu i ty r e s u l t s which are ve r s ions of t h e

Kreisel-Lacombe-Shoenfield theorem.

Theorem 1 2 . 1 . "Brouwer's Theorem" t h a t every map from IR t o IR i s

continuous holds i n Eff.

Proof: The r eade r w i l l have t o do t h i s himself (along t h e l i n e s of

( 1 2 . 4 ) below) o r else f i n d (as I have f a i l e d t o do) a readable

account from t h e Russian school .

( 1 2 . 1 ) is only moderately spec tacu la r . Recursive maps on the

r ecu r s ive r e a l s , while n o t t h e r e s t r i c t i o n of continuous funct ions

on the ( c l a s s i c a l ) reals (see ( 1 3 . 4 ) ) , are continuous on t h e i r

domain. (This i s s t a t e d as Exercise 15.35 i n Rogers 119671.) So

we j u s t need e f f e c t i v i t y t o g e t ( 1 2 . 1 ) . By passing t o higher types

we g e t a more i n t e r e s t i n g phenomenon: we g e t e f f e c t i v e maps, which

are n o t continuous on t h e i r e f f e c t i v e domain, b u t which a r e s t i l l

continuous from t h e p o i n t of view of Eff. W e consider t h e h e r e d i t a r i l y e f f e c t i v e operat ions. By (11.1) ,

i n Eff t hese are j u s t t h e f i n i t e types.

much of t h e material can be developed f o r an a r b i t r a r y "type

s t r u c t u r e " over IN .) W e de f ine a not ion of sequence cwvergence on

each HEOu i nduc t ive ly a s follows:

(The r eade r w i l l see t h a t

on HEOO = I N , xn -> x i f f 3k,Vn 2 k.xn = x;

on HEOuxT = HEOuxHEOT, (xn,yn) -> (x ,y) i f f xn ->xand yn-->y;

on H E O ~ + ~ = ( H E O ~ ) HE', , f ->f i f f xn->x implies f n (xn) ->f (x) . n W e say t h a t a funct ion f E HEOO+T i s continuous i f f f preserves

sequence convergence.

202 JM.E. HYLAND

Remark. The meaning of t h e s e d e f i n i t i o n s i n Eff does not a g r e e wi th

t h e meaning i n Sets. L e t us i n i t i a l l y res t r ic t a t t e n t i o n t o t h e h e r e d i t a r i l y e f f e c -

t i v e o p e r a t i o n s of pure t y p e (HEOklk a pure type symbol) where each

k+ l denotes (k -> 0 ) . For f n , f i n HEOk+l, w e s a y t h a t p E HEOk+l

i s a modulus f o r f n --> f i f f

Vx E HEOk.Vn 2 p ( x ) . f n ( x ) = f ( x ) .

( W e do n o t assume h e r e t h a t f n --> f i n HEOk+l: t h i s i s f a l s e i n

S e t s , though t r u e i n Eff . )

Lemma 1 2 . 2 . ( I n Eff.) A s s u m e f u n c t i o n s i n HEOk+l axe cont inuous .

I f 1.1 i s a modulus f o r f n -> f i n HEOk+l, t h e n f n -> f .

Proof:

f o r a l l n t k , p ( x n ) = ~ ( x ) = k ' s a y . A s f i s cont inuous t h e r e is

k" such t h a t f o r a l l n L k ' , f (x,) = f ( x ) . n 2 m a x ( k , k ' , k " ) ,

L e t xn --> x. S ince 1.1 i s cont inuous , t h e r e i s a k such t h a t

Then f o r a l l

f n (xn) = f ( x n ) = f ( x ) .

Remark. Th i s argument i s e n t i r e l y e lementary and h a s u s e f u l

a p p l i c a t i o n t o a v a r i e t y of type s t r u c t u r e s i n a v a r i e t y o f t oposes .

Lemma 12.3. A s s u m e a l l f u n c t i o n s i n HEOk a r e cont inuous .

I f f n -> f i n HEOk+l, t hen t h e r e i s a modulus 1.1 fox f n --> f .

Proof:

so w e can deduce f n ( x ) -> f ( x ) , t h a t i s

Vx3kVm 2 k f m ( x ) = f (x ) .

( I n Eff.)

The sequence wi th c o n s t a n t va lue x converges t o x i n HEOk,

By b a s i c a r i t h m e t i c choose k minimal f o r each x . This g i v e s us a

func t ion p : HEOk -> IN which by (11.1) o r ( 1 0 . 4 ) i s i n HEOk+l.

Remark. This argument depends on e f f e c t i v i t y i n Eff. Again t h e r e

a r e many u s e f u l v e r s i o n s of it.

Lemma 1 2 . 4 .

F E HEOk+2, t h e n t h e r e i s an r such t h a t

Vn t r F ( f n ) = F ( f ) .

( I n Eff.) I f p is a modulus f o r f n --> f i n HEOk+l and

The effective top- 203

Proof:

t o f i n d an r such t h a t Vn t r. F ( f n ) = F ( f ) . Following Gandy, use

t h e second r ecu r s ion theorem t o d e f i n e an index b ' by,

From i n d i c e s bn,m,b,c f o r f n l p r f , F r e s p e c t i v e l y w e wish

b ( a ) i f m ( a ) < l e a s t y.y shows c ( b ) = c ( b ' ) C = yo s a y ] , b ' ( a ) = { b n ( a ) f o r n l e a s t t yo wi th c ( b n ) # c ( b ) , o therwise .

l y shows c ( b ) = c ( b ' ) i f f T(c,b,~l(y))AT(c,b',~~(y))~U(~l(y)) =

U ( T 2 ( Y ) ) .I

W e see e a s i l y t h a t yo e x i s t s and t h a t Vn t yo.c(bn)

Markov's p r i n c i p l e ) .

Remark. This e s s e n t i a l l y i s Gandy's proof of t h e Kreisel-lacombe-

Shoenf ie ld theorem. S ince both Vn t k F ( f n ) = F ( f ) and p is a

modulus f o r f n -> f a r e i n t e r p r e t e d a s c losed subob jec t s i n Eff, it makes no d i f f e r e n c e whether w e do ( 1 2 . 4 ) e x t e r n a l l y i n Sets o r

i n t e r n a l l y i n Eff.

Theorem 12 .5 . (i) I n Eff it holds f o r a l l pure types r+l t h a t

f n -> f i n HEOr+l i f f t h e r e i s a modulus p f o r f n -> f i n HEOr+l,

and t h a t a l l m e m b e r s of HEOr+l are cont inuous .

= c ( b ) (us ing

Thus yo is c l e a r l y what w e want.

(ii) I n Eff it holds f o r any types a , ~ , t h a t a l l

members of HEOa+T a r e cont inuous .

Proof: (i) fol lows by induc t ion us ing ( 1 2 . 2 ) , (12.3) and ( 1 2 . 4 ) .

(ii) fo l lows by ex tens ion us ing Car t e s i an c losedness of t h e here-

d i t a r i l y e f f e c t i v e o p e r a t i o n s and o f t h e continuous f u n c t i o n a l s ( i n

the s e q u e n t i a l v e r s i o n , see Hyland C1979 1 ) i

The r e a d e r should compare (12.5) w i th t h e example of Gandy

(see Gandy-Hyland C19771 of a type 3 e f f e c t i v e o p e r a t i o n n o t con-

t inuous on t h e type 2 e f f e c t i v e o p e r a t i o n s ) . Con t inu i ty has a q u i t e

d i f f e r e n t meaning i n t e r n a l l y i n Eff . Remark. The f i n i t e t ypes over R i n Eff co inc ide n o t on ly wi th t h e

h e r e d i t a r i l y e f f e c t i v e o p e r a t i o n s i n Eff , b u t a l s o wi th t h e sequen-

t i a l l y continuous f u n c t i o n a l s i n Eff. The use of t h e modulus was

in t roduced o r i g i n a l l y i n t h e con tex t of r e c u r s i o n theo ry on the

204 J.M.E. HYLAND

( sequen t i a l ly ) continuous f u n c t i o n a l s by Stan Wainer.

513. F a i l u r e of compactness.

A s i s w e l l known, t h e r e are decidable subse t s R of 2 'B, t he

set of f i n i t e b ina ry sequences such t h a t

(i) any r ecu r s ive a E 2m extends some u E R ,

(ii) t h e r e are a E 2 B which extend no u E R (so t h a t no f i n i t e

S 2 R w i l l s a t i s f y (i)). This has an immediate consequence f o r

E f f .

Proposi t ion 13.1. I n Eff t h e r e is a decidable subobject R of 2

such t h a t

- B

(i) any a extends some u E R ,

(ii) f o r any k, t h e r e is an a which extends no u E R of

length 6 k.

Thus i n Eff t h e r e i s a decidable cover of 2 N , Cantor space, by

b a s i c clopen sets, with no f i n i t e subcover.

Proof: EITHER (i) and (ii) a r e almost negat ive and hold i n sets of

t h e r ecu r s ive r e a l s ,

OR immediate from Church's Thesis .

Corol lary 13.2.

F: 2m-> I N .

Proof: S e t F ( a ) = least l eng th of u E R with a extending u.

I n Eff t h e r e i s a continuous but unbounded func t ion

(13.1) shows t h a t t h e Fan Theorem f a i l s a s badly as p o s s i b l e i n

- Eff . This is why w e g e t (13.2) . There a r e Grothendieck toposes i n

which (13.1) holds without t he s t i p u l a t i o n t h a t R is decidable . I n

t h e known exanples, a l l continuous func t ions from 2 m t o JN are

uniformly continuous and so bounded. It is no t known whether t h e r e

are Grothendieck toposes i n which (13.1) ho lds .

The t r a d i t i o n a l way t o ob ta in r e s u l t s analogous t o (13.1) and

(13.2) f o r t h e reals i s t o use " s ingu la r coverings" as s t u d i e d i n

Zaslavski i -Cei t in C19621. (Of course one can set up (13.1) and

(13.2) i n an analogous fashion.)

The effective topos 205

Proposi t ion 13.3. I n Eff, t h e r e i s a sequence of r a t i o n a l i n t e r -

v a l s covering JR, bu t of a r b i t r a r i l y small measure.

Proof: E s s e n t i a l l y a diagonal enumeration, see Zaslavski i -Cei t in

C19621. The proof i s a l s o sketched i n Rogers C19671 Exercises

15.36, without cons ide ra t ions of e f f e c t i v i t y . By t h e cond i t ions , t o

be s a t i s f i e d by t h e sequence of r a t i o n a l i n t e r v a l s , can be expressed

as almost negat ive formulae, so by (8.4) t h i s does no t matter.

Corol lary 13.4. In Eff t h e r e i s a continuous funct ion from IR t o IR

which i s unbounded on some c losed bounded i n t e r v a l , and so i n

p a r t i c u l a r i s no t uniformly continuous on some closed bounded i n t e r v a l

Proof: Same re fe rences a s f o r (13 .3 ) .

Remark. The r e s u l t s of t h i s s e c t i o n can a l l be regarded a s proved

i n t e r n a l l y i n Eff , t h a t i s , they fol low from t h e e f f e c t i v i t y w e

e s t a b l i s h e d i n 510.

Though w e know Grothendieck toposes i n which Il? f a i l s t o be

l o c a l l y compact (Fourman-Hyland C19791), i n a l l known examples, t h e

t y p i c a l consequences of l o c a l compactness f o r a n a l y s i s s t i l l hold.

Ce r t a in ly continuous func t ions on bounded closed i n t e r v a l s are

uniformly continuous. So the e f f e c t i v e topos opens up p o s s i b i l i t i e s

unknown amongst Grothendieck toposes. Fu r the r examples can be

found i n Zaslavskiy-Celtin C19621.

814. Quo t i en t s of c l a s s i c a l o b j e c t s , and power o b j e c t s .

It is a f a m i l i a r f e a t u r e of i n t u i t i o n i s t i c mathematics t h a t

c o l l e c t i o n s of sets ( spec ie s ) can appear f a r more amorphous than

c o l l e c t i o n s of func t ions . W e have seen i n Eff t h a t t h e o b j e c t of

funct ions between "well-behaved".objects is i t s e l f "well-behaved''

(16.3) and ( 7 . 1 ) (b) ) . W e have seen t h i s good behaviour i n o t h e r

con tex t s (Moschovakis C19731, S c o t t C19701, are t h e e a r l y r e fe ren -

ces), and it can be made t h e b a s i s f o r n i c e p roof - theo re t i c r e s u l t s .

However, when the subobject c l a s s i f i e r is i t s e l f complicated, t h e

206 J.M.E. HYLAND

power set of however simple a ( t o some e x t e n t i nhab i t ed ) o b j e c t w i l l

be complex. I t i s t i m e t o look a t such o b j e c t s i n Eff.

A s w e mentioned i n 13, t he subobject c l a s s i f i e r R i n Eff can be

taken a s (Z,<-->) where <--> i s t h e r e a l i z a b i l i t y bi- implicat ion on

C = P ( W ) . W e may and so do think of t h e members of C a s e x i s t i n g

"global ly" . C lea r ly then (Z,<-->) i s a q u o t i e n t of A C . This w i l l

mean t h a t w e can ob ta in maps t o R i n Eff from s u i t a b l e maps t o C i n

S e t s .

Lemma 1 4 . 1 . Suppose A Y --> (Y,=) i s a s u r j e c t i o n . Then a map

f : X --> Y induces a map f : ( X , = ) --> (Y,=) i n Eff such t h a t

-

-

4 J ( X , = ) - > (Y,=) commutes

f

i f f x = x ' --> f ( x ) = f ( x ' ) i s v a l i d . (That i s , i f f f preserves t h e

e q u a l i t y r e l a t i o n . ) Under these circumstances f i s represented by

t h e func t iona l r e l a t i o n EXA [ I f ( x ) = y 1 . Proof: By a rou t ine use of l o g i c .

In the case of t he s u r j e c t i o n A C --> R , every map a r i s e s a s i n

( 1 4 . 1 ) .

Proposi t ion 1 4 . 2 . Any map from (X,=) t o R = (Z,<-->) i n Eff is

a s defined i n ( 1 4 . 1 ) f o r an f : X --> Z such t h a t both

-

-

(i) x = x ' -> ( f ( x ) <--> f ( x ' ) )

and (ii) f ( x ) --> Ex

a r e v a l i d .

Remark. Given f : X -> C with (i) v a l i d , one can e a s i l y de f ine

g: X --> Z with both (i) and (ii) v a l i d , and such t h a t z = s. S e t

g ( x ) = ExAf (x) . Proof: Since maps from (X,=) t o R a r e i n b i j e c t i v e correspondence

with maps 1 --> P(X,=)), ( 1 4 . 2 ) i s immediate from t h e d e s c r i p t i o n

of the power set i n ( 2 . 1 2 ) of H J P C19801. A reader who f i n d s t h a t

The effective topos 207

proof unpa la t ab le , can t a k e a r e p r e s e n t a t i v e G(x ,p) f o r a map ( X , = )

t o f2, se t f (x ) = 1 vq. (vp(G(x ,p) hp --> q ) -> q ]I , and check t h a t

G(x,p) <--> Exh ( f (x) <--> p) is v a l i d : s i n c e (i) i s v a l i d f o r f , t h e

remark above a p p l i e s t o g i v e (ii) f o r g ( x ) = Exhf (x) . (Su i t ab ly

r e l a t i v i z e d , t h i s i s a proof of ( 2 . 1 2 ) of HJP L19801.)

From ( 1 4 . 2 ) w e see t h a t i f ( X , = ) i s sepa ra t ed then any map

( X , = ) t o R f a c t o r s through q: A X --> R . Our next r e s u l t g ives t h i s

wi th in Eff. Propos i t ion 14.3.

f o r any sepa ra t ed (X,=) i n Eff.

Proof:

Then q ( x ' ' ) : A ( Z

The map q( ' '=): A C ( x r = ) --> R (',=) i s a s u r j e c t i o n

From (6 .3) we see t h a t AZ"") i s (isomorphic t o ) A ( Z r X ) .

r x ) --> P ( ( X , = ) ) i s r ep resen ted by

H(f,R) = ERhn{R(x) <-> ExAf([xl) Ix E X I .

But w e can t ake I'X 5 X (assuming ( X , = ) c anon ica l ly sepa ra t ed ) and so

by s e t t i n g f t o be t h e r e s t r i c t i o n of R t o I 'X , we see a t once t h a t

ER --> 3f .H(f , R )

is v a l i d , so t h a t [HI i s s u r j e c t i v e .

§15. The Uniformity P r i n c i p l e .

F i r s t a gene ra l un i formi ty p r i n c i p l e f o r Eff. Propos i t ion 15.1. L e t AX -> ( X , = ) be a s u r j e c t i o n and l e t ( Y , = ) be

an e f f e c t i v e o b j e c t . Then

V t J l V X E ( X , = ) .3y E (Y ,=)$J (x ,y ) --> 3y E (Y,=) .vx E (X ,=)$J (x ,y ) l

holds i n Eff.

Proof: Take (Y,=) s t r i c t e f f e c t i v e and cons ide r f i r s t t h e case when

( X , = ) i s AX. L e t e E [Vx.i 'y.$J(x,y) 1 . Then 0 E Ex each x i n X , so

b = v l ( e ( 0 ) ) E Ey f o r some y i n Y , unique as ( Y , = ) i s s t r i c t e f f e c -

t i v e ; and c = v 2 ( e ( 0 ) ) i s i n 1 $ ( x , y ) 1 . But t hen i f d = Xn.c, w e

f i n d t h a t he .<b ,d> r e a l i z e s t h e formula i n square b r a c k e t s . (There

i s no dependence on E + . ) The r e s u l t f o r a q u o t i e n t of AX is an

immediate consequence of t h e s p e c i a l case.

208 J.M.E. HYLAND

W e have an immediate c o r o l l a r y .

Corol lary 15.2. The "Uniformity P r i n c i p l e "

\d$"dX E P ( I N ) .3n E IN $ (X,n) --> 3n E IN .VX E P(IN ) $ ( x , n I l

holds i n Eff. Proof: IN is an e f f e c t i v e o b j e c t and by (14.3) P ( R ) i s a q u o t i e n t

of A ( Z m ) .

The uniformity p r i n c i p l e is an extreme form of choice p r inc ip l e :

t h e choice func t ion is cons t an t because t h e domain i s amorphous while

t h e range i s well-behaved. Conditions on both t h e range and the

domain a r e necessary. Obviously t h e r e are non-constant func t ions

from IN t o N. As regards cond i t ions on t h e range, t h e r eade r may

l i k e t o show t h a t t he q u o t i e n t map from A E t o 62 does n o t s p l i t .

916. j -operators : f o r c i n g 2 --> A2 t o be i s o .

I n a topos, j -operators a r e maps j : 62 --> 62 s a t i s f y i n g

P 5 j ( p ) o r equ iva len t ly p --> q i j ( p ) --> j ( q )

j (phq) = j ( p ) h j ( q ) T 5 j ( T )

j ( j (p) ) = j (p) j ( j ( p ) ) 5 j ( p ) .

Of course t h e r e i s a l s o an i n t e r n a l o b j e c t of j -operators , a subobject n

of 62" which w e can desc r ibe i n Eff as fol lows.

Proposi t ion 1 6 . 1 . (i) The o b j e c t 62' i n Eff can be taken as ( E x , = )

where

Bf = n = t v p . f ( p ) <--> gip) 1

(ii) The o b j e c t o f j -operators i n Eff i s t h e subobject

of ( E x , = ) represented by t h e canonical monic de€ined by e i t h e r of t h e

above ways of giving t h e not ion of j -ope ra to r . A l t e r n a t i v e l y i t i s

(J,=) where J is t h e set of j -ope ra to r s and where

t j = k l = E j h [Vp . j (p ) <--> k ( p ) J

with Ej = I[j i s a j -operator ]I . Proof: (i) follows from ( 1 4 . 2 ) i n t he manner of (14.3) and (.ii) i s

then immediate.

The effective topos 209

Remark. A s explained i n Johnstone [ 1 9 7 7 ] j -ope ra to r s correspond t o

topologies and so t o subtoposes. I t i s known t h a t t h e l a t t i c e of

j -operators under pointwise 2 i s a complete Heyting a lgeb ra

( i n t e r n a l l y ) . The r eade r should refer t o Fourman-Scott [ 1 9 7 9 ] f o r

an e x p l i c i t c o n s t r u c t i v e t reatment . I t i s perhaps worth commenting

f u r t h e r on t h e o rde r r e l a t i o n . W e have

B j 6 k j = EjAEkA n Vp.j(p) --> k ( p ) J

def ining t h e appropr i a t e subobject i n Eff. I f w e are looking a t

ex te rna l j -operators , then j 6 k i f f

Vp.j (p) --> k ( p )

i s v a l i d . F i n a l l y note t h a t i f j ( i ) is non-empty then j is t h e

degenerate topology which c o l l a p s e s t h e topos . L e t us look again a t t h e double negation topology. ( W e do no t

bother with a cons t ruc t ive ve r s ion . )

i f p non-empty, ( 7 7 ) p = U I T ~ ~ i s non-empty) = {I: Clearly then w e have the fol lowing lemma.

Lemma 1 6 . 2 . For any j , (-,-,) 5 j i f f n { j ( p ) / p non-empty} i s non-

empty . Proof: T r i v i a l .

W e now consider how t o fo rce monics t o be i s o . L e t a subobject

of (X,=) be given by a canonical monic A and de f ine a map R t o by

@ (p) = 112x (x,=) . A ( X ) --> p n . A

Clea r ly i f j f o rces A >--> (X,=) t o be i s o , t hen

@ A ( j (p) ) --> j (p)

i s v a l i d . This gives us a way t o desc r ibe t h e l e a s t j -operator

fo rc ing A >--> ( X I = ) t o be i s o .

Proposi t ion 16 .3 .

f o rc ing A >-> (X,=) t o be i s o , i s

In the above s i t u a t i o n , jA , t h e l e a s t j -operator

j A ( p ) = Uvq. CC@,q -> q ) h ( p --> q ) --> q ) ll .

210 J.M.E. HYLAND

Proof: Obvious, a s i n t h e l o g i c t h i s s ays

l A ( P ) = A {Sl@,q 5 qAp 5 q l

where A i s t aken i n t e r n a l l y i n a. I t is easy t o check t h a t ( a s

s (9 (p ) -> @,(q)) i s v a l i d ) j, i s a j - o p e r a t o r . A ( -> 9)

W e now show t h a t f o r c i n g 2 >-> A2 t o be is0 c o l l a p s e s Eff t o

S e t s .

P ropos i t i on 1 6 . 4 . The l e a s t j - o p e r a t o r f o r c i n g 2 >-> A2 t o be is0

is ( 17).

Proof: L e t j be t h e l e a s t j -ope ra to r f o r c i n g 2 >-> A2, ob ta ined a s

i n (16.3) from @ : -> a . H e r e

@ ( p ) = {01+p u I l 1 - t ~ = I e l e ( 0 ) E p o r e (1 ) E PI. C l e a r l y it s u f f i c e s t o show ( 7 7 ) s j , t h a t i s by ( 1 6 . 2 )

n { j (p) Ip non-empty] i s non-empty.

nI j I n 1 ) In E 1 i s non-empty: f o r i f a i s i n

I n f a c t it i s enough t o show t h a t

VP,q. ( p -> q ) -> ( j ( p ) --> j (9) and x i s i n n { j { n I ) In E N 1 ,

t hen ( a ( X n . n ) ) x is i n n { j ( p ) lp non-empty].

Now t a k e b i n 1Vp.p -> j ( p ) ] , c i n " d p . j ( j ( p ) ) --> j ( p ) J , and

t a k e as 2 >-> A2 i s j-dense d i n j ( { O } ) n j ( { l l ) . Note t h a t

e = Xx.c( ( a x ) d ) i s i n l[Vp.@ ( j ( p ) ) -> j ( p ) 1 . Define u s i n g t h e

second r e c u r s i o n theorem an index f by

( f k ) ( 0 ) = b ( k )

( f k ) (1) = U ( l e a s t y . T ( e , S l ( f , k + l ) , y ) ) . 1

Now by a s t anda rd k i n d of argument, w e can show t h a t

1 S l ( f , k ) ( 0 ) = ( f k ) ( 0 ) = b ( k ) E j ( I k 1 )

and S l ( f , k ) 1 (1) = ( f k ) (1) = e ( S l ( f , k + l ) ) 1

1 a r e a l l de f ined , and t h e n w e see t h a t S l ( f ,O) is i n @ ( j ( { n I ) ) f o r

a l l n and so e ( S l ( f , O ) ) i s i n n { j ( { n } ) In E m l as r e q u i r e d . 1

5 1 7 . j - o p e r a t o r s and d e c i d a b i l i t y .

( 1 6 . 4 ) appea r s t o restrict t h e j - o p e r a t o r s i n Eff, b u t i n f a c t

w e can show t h a t t hey have a r i c h s t r u c t u r e . Apparently it w a s

The effective topos 21 1

Powell who f i r s t r e a l i z e d t h a t t h e r e i s a connection between not ions

of degree and t h e fo rc ing of d e c i d a b i l i t y i n r e c u r s i v e r e a l i z a b i l i t y .

We con ten t ou r se lves wi th a p r e c i s e s t a t emen t and a ske tch of a

proof.

F i r s t w e g i v e a lemma of Andy P i t t s which s i m p l i f i e s t h e pre-

s e n t a t i o n of t h e proof .

Lemma 1 7 . 1 . I n t h e s i t u a t i o n of (16.3) j, can a l t e r n a t i v e l y be de-

f ined by

@,*(p) = n{q 5 I p ~ { * l 5 q and 0,Cq) 5 qlr

where * is an index f o r t h e empty p a r t i a l func t ion , so long as Ex

non-empty impl ies A(x) non-empty.

Proof: L e t us drop t h e s u b s c r i p t A. Note t h a t $ prese rves inc lu -

s ion . Hence because

P --> (PA{*)) 5 $ ( P ) --> @(PA{*))

i s v a l i d , w e can deduce t h a t

@ ( P ) 5 @ * ( P )

is v a l i d . Also w e have

@ ( @ * ( P ) ) 5 n { @ ( q ) IpAI*l 5 q and @ ( q ) 5 ql

- c n{qjpAt*} 5 q and @ ( q ) 5 q l = @ * ( P I ,

so t h a t

@ ( $ * ( P ) ) 5 @*(P)

is v a l i d , r a t h e r t r i v i a l l y . Thus by t h e d e f i n i t i o n of j i n (16.3)

w e have j s @ * i n Eff, and it remains t o show t h a t $ * s j .

W e can t ake a E [Vp,p -> j (p ) J and s i n c e @ Cj Cp)) S j (p) i n Eff, w e can t ake b E UVx E (X,=). ( A ( x ) -> jCp)) --> jCp) 1 . NOW

de f ine an index e by t h e second r ecu r s ion theorem a s fo l lows .

Cons i de r

t h a t (i)

and (ii)

a ( n ) , i f x = -a,*> e ( x ) = 'b(m)) (Xy.e(z(y))), i f x = <m,z>, z # *. f o r any p , t h e set SCp) = I x l e ( x ) E j ( p ) } . W e see e a s i l y

PAC*) 2 S ( P ) ,

$ ( S ( P ) ) 5 S ( P ) ,

212 J.M.E. HYLAND

so w e can deduce t h a t $ * ( p ) 5 S ( p ) . Thus c l e a r l y e r e a l i z e s

Vp.@*(p) 5 j (p) , and t h i s completes the proof.

Now f o r A 5 I N , l e t D (n) = {<O,n>ln E A}u{<l,n>ln 4 A} so t h a t A

DA r ep resen t s canonical ly t h e subobject

AV i A >-> m , the " d e c i d a b i l i t y of A" . W e w r i t e

$,(PI = U3n E IN .DA(n) --> p n , and kA f o r the l e a s t j -operator generated by $,, t h a t i s , the l e a s t

j -operator fo rc ing A t o be decidable .

$A* i n Eff. Theorem 1 7 . 2 . (External v e r s i o n ) .

reducible t o B.

Proof:

(holds i n E f f ) . Suppose t h a t e ( n ) E JIB (DA(n)) f o r each n. W e wish t o show how t o

compute A from B , t h a t is how t o determine DA(n) from a knowledge

of D ( m ) f o r f i n i t e l y many m.

e i t h e r x is of form <y,*> and w e e a s i l y see t h a t y must be DA(n) so

w e a r e home,

- o r x is of form < m , e l > say i n which case el E D B ( m ) --> $B* (DA(n) )

so w e take x1 t o be e l ( < O , m > ) o r e l ( < l , m > ) a s appropriate ,

x1 E JIB* (DA(n)) , and r epea t t h i s process.

qB*, t h i s terminates i n a f i n i t e number of s t e p s giving DA(n) a s

required.

Suppose conversely t h a t A i s Turing reducible t o B v i a an index f .

Define using t h e second recursion theorem e Cn,y) where y i s (a code

f o r ) a f i n i t e set of numbers of form <m,O> o r < m , l > as follows.

By ( 1 7 . 1 ) kA i s equal t o

kA 5 kB i n Eff i f f A i s Turing

Note f i r s t t h a t kA 5 kg i n Eff i f f Vn.$,*(DA(n)) is v a l i d

*

* L e t x = e ( n ) E QB (DA(n) ; then B

From t h e d e f i n i t i o n of

<k ,*> , i f t h e r e i s a computation {f}Y(n) = k (using only information i n y ) ,

e ( n f y ) = 1 < m , q > , i f t h e computation { f l Y ( n ) asks f o r a value n o t i n y f and g is an index for < m , i > -> e ( n , y u I < m , i > l ) . i

The effective topos 213

I t is easy t o see t h a t f o r a l l n , and f o r y information t r u e of B,

e ( n , y ) is def ined and i n @,*(D,(n)).

set, w e have f o r a l l n ,

In p a r t i c u l a r f o r y t h e empty

e (n,y) i n @,* (DA(n) ) . Thus Vn.QB* (DA(n)) i s v a l i d .

This completes the proof .

Remark. In f a c t t h e r e i s a proof of the implicat ion from r i g h t t o

l e f t along t h e following l i n e s : i f j fo rces B decidable , then the

statement t h a t A i s reducible t o B and t h a t t he computation is always

defined, a r e almost negat ive f o r E f f . and so hold i n E f f . . hence i n

E f f . A i s decidable , t h a t is j fo rces A decidable. -1

-1 I -1 '

W e cannot f i n d a crude i n t e r n a l vers ion of (17 .2 ) i n view of

Goodman C19781. However t h e proof of ( 1 7 . 2 ) is e f f e c t i v e , so w e can

ge t something o u t of it.

with A i s i n t e r n a l l y defined i n Eff. "A Turing reducible t o B": w e mean the n a t u r a l not ion of computabili ty

r e l a t i v e t o ( p a r t i a l ) c h a r a c t e r i s t i c funct ions. W e ob ta in a r e s u l t

by r e s t r i c t i n g a t t e n t i o n t o c losed subsets of I N , t h a t i s t o

P ( I N ) = {A 5 IN IVn. 9 - n B A --> n E A ] .

Proposi t ion 17.3. The s ta tement

Clear ly t h e funct ion which a s soc ia t e s kA

W e must say what w e mean by

VA,B E F ( I N ) . (A Turing reducible t o B) <--> k A ' kg

holds i n Eff. Proof: By the e f f e c t i v i t y of t he proof of ( 7 . 2 ) .

118. General remarks on the e f f e c t i v e topos.

The pleasing f e a t u r e of t h e e f f e c t i v e topos i s t h a t i n i t ,

ideas about e f f e c t i v i t y i n mathematics,seem t o have t h e i r na tu ra l

home. W e mention the two main examples.

1) Constructive real ana lys i s . W e have t r i e d t o i n d i c a t e t h a t t h i s

is what a n a l y s i s i n Eff i s i n essence i n 118-13. It is worth not ing

how t h e r e a l i z a b i l i t y l o g i c makes d i s t i n c t i o n s f o r us. Consider t he

214 J.M.E. HYLAND

examples (Kreisel 119591 t h a t t h e in t e rmed ia t e va lue theorem holds

c l a s s i c a l l y bu t n o t e f f e c t i v e l y f o r r e c u r s i v e (cont inuous) func t ions

on t h e r e c u r s i v e r e a l s . I n Eff ,

vf E lRIR .f (O)<oAf (1)>0 -> 3 X E ( 0 , l ) f ( X ) = 0

i s f a l s e whi le

vf E ~ l R . f ( 0 ) < O A f ( l ) > O --> -1 -13x E ( O , l ) f ( X ) = 0

i s t r u e ( a s it is t r u e i n Sets and e q u i v a l e n t t o a nega t ive formula) .

2 ) E f f e c t i v e a lgeb ra . W e have n o t d i scussed t h i s a t a l l , bu t i t

seems worth po in t ing o u t t h a t t h e d e f i n i t i o n s have a n a t u r a l meaning

i n Eff. A r e c u r s i v e l y p re sen ted f i e l d (see Metakides-Nerode C19791

i s an enumerable (dec idab le ) f i e l d i n Eff. I t has a s p l i t t i n g

a lgor i thm i f f i r r e d u c i b i l i t y of polynomials is dec idable i n Eff. Thus t h e e f f e c t i v e con ten t o f a r e c u r s i v e l y presented s t r u c t u x e

corresponds t o p r o p e r t i e s of it which hold i n Eff. This sugges t s

t h a t p o s i t i v e r e s u l t s i n e f f e c t i v e a lgeb ra should be e s t a b l i s h e d by

proving r e s u l t s i n c o n s t r u c t i v e l o g i c from axioms which ho ld i n Eff, and i n t e r p r e t i n g t h e r e s u l t s i n Eff. That i s , one should use t h e

axiomatic method. Of course , nega t ive r e s u l t s ob ta ined i n e f f e c t i v e

a lgeb ra can be i n t e r p r e t e d i n Eff t o g ive independence r e s u l t s .

What w e l a c k , above a l l , i n our t r ea tmen t of t h e e f f e c t i v e topos,

i s any real informat ion about ax iomat iza t ion analogous t o t h e r e s u l t s

ob ta ined i n T r o e l s t r a 119731 ax iomat iz ing r e a l i z a b i l i t y over both

Heyting and Peano a r i t h m e t i c . Of course , one would expec t t o look a t

t h e e f f e c t i v e topos def ined over a topos o t h e r than Sets (say over

t h e f r e e topos wi th n a t u r a l number o b j e c t ) t o g e t a r e s u l t co r re s -

ponding p r e c i s e l y t o an ax iomat iza t ion . But a l l I wish t o p o i n t ou t

is t h a t ( d e s p i t e t h e sugges t ive work of P i t t s C19811 on i t e r a t i o n )

w e have no good informat ion i n t h i s area. W e can n o t p rope r ly be

s a i d to understand r e a l i z a b i l i t y u n t i l w e do.

The effective topos 215

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