Aspects of Predicative Algebraic Set Theory II: Realizability

Asp ects of Predicativ e Algebraic Set Theory I I: Realizability Benno v an den Berg & Iek e Mo erdijk Jan uary 15, 2008 De dic ate d to Je an- Yves Gir ar d on the o c c asion of his 60th birthda y 1 In tro duc ti on This pap er is the se cond in a se r ies on the r elation betw een alg ebraic set theo ry [19] and pr edicativ e formal systems . The pur pose of the present pa per is to show ho w realizabilit y mo dels of constructive set theories ﬁt into the framew ork of algebr aic set theor y . It can b e read indep enden tly fr o m the ﬁrs t par t [5]; how ever, we r ecommend that readers o f this pap er read the introduction to [5], where the genera l metho ds and goals of algebraic set theory are explained in more detail. T o motiv ate our metho ds, let us recall the co nstruction of Hyland’s eﬀective top os E f f [17]. The ob jects of this ca teg ory are pairs ( X , =), where = is a subset of N × X × X satisfying certain co nditions. If we write n  x = y in ca se the triple ( n, x, y ) belong s to this subset, then these conditions ca n b e for m ulated by r equiring the e xistence of natural num bers s and t such that s  x = x ′ → x ′ = x t  x = x ′ ∧ x ′ = x ′′ → x = x ′′ . These conditions hav e to b e read in the wa y usua l in realiza bilit y [34]. So the ﬁrst says that for any natura l num b er n s atisfying n  x = x ′ , the expression s ( n ) should be deﬁned and b e such that s ( n )  x ′ = x . 1 And the second stipulates that for an y pair of natura l n um ber s n and m with n  x = x ′ and m  x ′ = x ′′ , the expressio n t ( h n, m i ) is deﬁned and is such that t ( h n, m i )  x = x ′′ . 1 F or an y tw o natural n umbers n, m , the Kleene application of n to m will b e written n ( m ), ev en when it i s undeﬁne d. When it is deﬁned, this will b e i ndica ted by n ( m ) ↓ . W e also assume that some r ec ursive pairi ng op eration has b ee n ﬁxed, wi th the asso ciat ed pro j ections being recursive. The pair ing of t w o natural num b ers n and m will be denoted b y h n, m i . Ev ery natural nu mber n w i ll code a pair, wi th its ﬁrs t and s ec ond pro jection denoted b y n 0 and n 1 , resp ec tiv ely . 1 The a rrows [ F ] b et w een tw o such ob jects ( X , =) a nd ( Y , =) are equiv alence classes of subse ts F of N × X × Y satisfying certain conditions. W riting n  F xy for ( n, x, y ) ∈ F , one r equires the existence o f realize rs for statements of the form F xy ∧ x = x ′ ∧ y = y ′ → F x ′ y ′ F xy → x = x ∧ y = y F xy ∧ F xy ′ → y = y ′ x = x → ∃ y F xy. Two s uc h subsets F and G represent the same arr o w [ F ] = [ G ] iﬀ they are extensionally equal in the sense that F xy ↔ Gxy is r ealized. As shown by Hyla nd, the logical prop erties o f this to p os E f f are quite re- mark able. Its ﬁrst-order arithmetic coincides with the r ealizabilit y interpreta- tion of Klee ne (1945 ). The interpretation of the higher types in E f f is g iv en by HEO , the hereditary eﬀective oper ations. Its higher-or der arithmetic is captured by realiza bilit y in the manner of Kreisel and T ro elstra [33], so as to v alidate the uniformity principle: ∀ X ∈ P N ∃ n ∈ N φ ( X , n ) → ∃ n ∈ N ∀ X ∈ P N φ ( X , n ) . The top os E f f is o ne in an entire family o f “realizability to p oses” de ﬁned ov er arbitrary partial combinatory algebras (or more general structure s mo d- eling co mput ation). The rela tion betw een these top oses has b een not b een completely clar iﬁed, although muc h interesting work has already b een done in this dir ection [29, 17, 22, 8, 15, 14] (for an ov erview, see [2 6]) . The constructio n of the top os E f f a nd its v a rian ts can be internalised in an arbitra ry topos. This means in particular that one can construct top oses b y iterating (alternating) constructions of sheaf and r ealizabilit y top oses to obtain interesting mo dels for higher-or der intuitionistic a rithmetic HHA . An exa mple of this phenomenon is the mo diﬁed rea liz abilit y topo s , which o ccurs as a closed subtop os of a realiz- ability top os constr uc ted inside a presheaf top os [28 ]. The purp ose of this series of pap ers is to show that thes e results are not only v alid for top oses as mo dels of HHA , but also for cer ta in types of cate- gories equipp ed with a class of small maps suitable for c onstructing mo dels of constructive set theories like IZF and CZF . In the ﬁrst pa p er of this se r ies [5], we hav e axioma tised this type of categorie s, and refer to them as “pr e dic a - tive catego ry with small maps” (for the conv enience of the rea der their pre c ise deﬁnition is recalled in App endix B). A basic re s ult from [5] is the following: Theorem 1 .1 Every pr e dic ative c ate gory with smal l maps ( E , S ) c ontains a mo del ( V , ǫ ) of set the ory. Mor e over, 2 (i) ( V , ǫ ) is a mo del of IZF , whenever the class S satisﬁes t h e axioms (M) and (PS) . (ii) ( V , ǫ ) is a mo del of CZF , whenever t he class S satisﬁes (F) . 2 T o s ho w that r ealizabilit y models ﬁt in to this pictur e , w e prov e that pr e dica- tive categor ies with small maps a r e close d under internal realiza bilit y , in the same way that top oses are. Mo re pr ecisely , re la tiv e to a given predicative cate- gory with small maps ( E , S ), we constr uc t a “predica tiv e r ealizabilit y category ” ( E f f E , S E ). The main r esult of this pa per will then b e: Theorem 1 .2 If ( E , S ) is a pr e dic ative c ate gory with smal l maps, then so is ( E f f E , S E ) . Mor e over, if ( E , S ) satisﬁes (M), (F) or (PS) , then so do es ( E f f E , S E ) . W e show this for the p ca N together with Klee ne applica tion, but the result is also v alid, whe n this is replaced by an arbitrary smal l p ca A in E . The pro of of the theo rem a bov e is technically ra ther inv olv ed, in particula r in the case of the a dditional prop erties needed to ensur e that the mo del of set theo ry satisﬁes the pr ecise axioms of IZF and CZF . Howev er, once this work is out of the way , one can apply the c onstruction to many diﬀerent predicative categories with small maps, a nd sho w that fa milia r realizability models of s et theory (and some unfamiliar ones) app ear in this wa y . One of the most basic exa mples is that where E is the categor y of sets, and S is the class o f maps b et w een sets whos e ﬁb ers are all b ounded in s ize by some inaccessible c a rdinal. The constr uction underly ing Theo rem 1 .2 then pr oduces Hyland’s eﬀective top os E f f , tog ether with the cla ss of small maps deﬁned in [19], whic h in [21] was shown to lead to the F riedman-McCa rt y mo del of IZF [12, 24] (we will repr o v e this in Section 5 ). An imp ortan t p oin t we wish to emphasise is that one can prov e all the mo del’s salient pr o perties without cons tr ucting it ex plicitly , using its universal prop erties instea d. W e explain this p oin t in more detail. A predicative categ ory with small maps co nsists of a category E and a clas s of maps S in it, the intuition being that the ob jects and morphisms of E ar e classes and class morphisms, and the morphisms in S are those that have small (i.e., set-sized) ﬁbres. F o r such predicative categories with small ma ps , one can prov e that the small sub ob jects functor is re pr esen table. This means that there is a p ower class obje ct P s ( X ) which c la ssiﬁes the s ma ll sub ob jects of X , in the sense that ma ps B / / P s ( X ) corres p ond bijectively to jo intly monic diagra ms B U / / o o X with U / / B small. Under this corr espondence, the identit y id : P s ( X ) / / P s ( X ) corres p onds to a membership relation ∈ X / / / / X × P s X . 2 The precise formulat ions of the axioms ( M) , (PS) and (F) can be found in App e ndix B as well. 3 The mo del of set theor y V that every predicative catego ry with sma ll maps contains (Theorem 1.1) is constructed a s the initial algebr a for the P s -functor. Set-theoretic members hip is in terpreted by a sub ob ject ǫ ⊆ V × V , which one obtains as fo llo ws. By Lambek’s Lemma, the s tructure map for this initial algebra V is a n isomorphism. W e denote it by Int, and its inv erse by Ext: P s V Int + + V . Ext l l The membership relatio n ǫ / / / / V × V is the result of pulling ba c k the usual “external” membership rela tion ∈ V / / / / V × P s ( V ) along id × Ext. Theorem 1 .1 partly ow es its applicability to the fact that the theory of the int ernal mo del ( V , ǫ ) o f IZF or CZF corr esponds precisely to what is true in the categorica l lo gic of E for the ob ject V and its external membership re la - tion ∈ . This, in turn, cor responds to a large extent to what is true in the categoric al logic of E fo r the higher arithmetic t ype s . Indeed, by the isomor - phism Ex t: V / / P s (V) and its inv erse Int, any generalised element a : X / / V corres p onds to a sub ob ject Ext( a ) / / / / X × V with Ext( a ) / / X sma ll, and for tw o such elements a and b , one has that (i) a ∈ b iﬀ a fa c tors through Ext( b ). (ii) a ⊆ b iﬀ the sub ob ject Ex t( a ) of X × V is contained in Ex t( b ). (iii) E x t ( ω ) ∼ = N , the natura l num ber s o b ject of E . (iv) Ext( a b ) ∼ = Ext( a ) Ext( b ) . (v) Ext( P a ) ∼ = P s (Ext( a )). (Prop erties (i) and (ii) hold by deﬁnition; fo r (iii)-(v), see the pro of of P r oposi- tion 7 .2 in [5].) Thus, for example, the sentence “the set of a ll functions from ω to ω is s ubcountable” is tr ue in ( V , ǫ ) iﬀ the cor r esponding statement is true for the natural num bers o b ject N in the categor y E . F or this rea son the r ealizabilit y mo del in the eﬀective top os inherits v a r ious principles from the a m bient categ ory and one immediately concludes : 4 Corollary 1.3 IZF is c onsistent with the c onjun ctio n of t h e fol lowing axioms: the Axiom of Countable Choic e (A C) , the Ax i om of R elativise d Dep en d ent Choic e (RDC) , the Pr esentation Axiom (P A) , Markov’s Principle (MP) , Chur ch’s Thesis (CT) , t h e Uniformity Principle (UP) , Unzerle gb arkeit (UZ) , Indep endc e of Pr emisses for Sets and Nu mb ers (IP) , (IP ω ) . 3 Of co urse, Co rollary 1.3 has also be en pr o v ed directly by realiz a bilit y [12, 24]; how ever, it is a basic example which illustrates the genera l theme, a nd on which there are many v ariations. F or example, o ur pro of of Theorem 1.2 is elementary (in the pro of-theoretic sense), hence ca n b e used to prov e relative consis tency results. If we take for E the syntactic categ o ry of deﬁnable cla sses in the theor y CZF , we obtain Rathjen’s realizability in terpretation of CZF [30], and deduce: Corollary 1.4 [3 0] If CZF is c onsistent, t he n so is CZF c ombine d with the c onjunction of the fol lowing axioms: the Ax io m of Countable Choic e (A C) , t h e Axiom of Re lativise d Dep en d ent Choic e (R DC) , Markov’s Principle (MP) , Chur ch’s Thesis (CT) , t h e U nif ormity Principle (UP ) and U nzerle gb arkeit (UZ) . (W e als o recover the same res ult fo r IZF within our fra mew o rk.) Another po ssibilit y is to mix Theor em 1.2 with the similar co nstruction for sheav es [6]. This shows that mo dels o f s e t theo r y ( IZF or CZF ) a lso ex is t fo r v ario us other notions of r ealizabilit y , such as mo diﬁed realiz a bilit y in the sense of [28, 9] or Kleene-V esley’s function realiz abilit y [20]. W e will discuss this in some more detail in Section 5 below. Inside Hyla nd’s eﬀective to pos, or mo re g enerally , in c a tegories of the fo r m E f f E (cf. Theorem 1 .2 ), other classes of small maps exist, which a re not obtained from an earlier class of small maps in E by Theorem 1.2, but nonetheless satisfy the conditions suﬃcient to a pply o ur theorem from [5] yielding mo dels of set theory (cf. Theo r em 1 .1 ab o v e). F ollowing the work of the ﬁrst author in [4], we will present in some detail one par ticular ca s e of this phenomenon, ba sed o n the notion of mo dest set [16, 1 8]. Alrea dy in [1 9 ] a class T inside the eﬀective top os was considere d, consisting of thos e maps which hav e sub countable ﬁbres (in some suitable sense). This class do es not satisfy the axio ms from [ 1 9] necessa ry to provide a mo del for IZF . How ever, it was shown in [4] that this cla ss T doe s satisfy a s et of axioms suﬃcient to provide a mo del of the predica tiv e set theory CZF . Theorem 1 .5 [19, 4] The eﬀe ctive top os E f f and its class of sub c ountable mor- phisms T form a pr e dic ative c ate gory with smal l m a ps. Mor e over, T satisﬁes the axioms (M) and (F) . 3 A precise formulation of these principles can b e found in App endix A. 5 W e will s ho w that the co rrespo nding model o f s et theor y (Theorem 1.1) ﬁts into the general framework of this ser ies of pap ers, and inv es tigate some of its logica l prop erties, as well a s its relatio n to some earlier mo dels o f F r iedman, Stre icher and L ubarsky [1 3, 32, 23]. In particular, we prove: Corollary 1.6 CZF is c ons i stent with the c onjunction of the fol lowing axioms: F ul l sep ar ation, the sub c ountability of al l sets, as wel l as (A C) , (RDC) , (P A) , (MP) , (CT) , (UP) , (UZ) , (IP) and (IP ω ) . Ac knowledgement s: W e would like to thank Thomas Streicher and Jaap v an Oosten for comments on an ea rlier version of this pap er, and for ma king [26] av a ilable to us. 2 The category of assem blies Recall that our ma in a im (Theorem 1.2) is to co nstruct for a predicative cate- gory with small maps ( E , S ) the re a lizabilit y categor y ( E f f E , S E ), and show it is again a predicative categ ory with small maps. F or this and other purp oses, the description of E f f a s an exact (ex/reg) completion of a category of assemblies [11], ra ther than Hyland’s orig inal desc r iption, is us eful. A similar remark a p- plies to the eﬀective top os E f f [ A ] deﬁned b y an arbitra ry small p ca A . In [5] we show ed tha t the clas s of pre dic a tiv e catego ries with sma ll maps is closed under exact completion. More pr e cisely , we form ulated a weaker version of the axioms (a “categ ory with display maps ”; the notion is also r ecapitulated in App endix B), and show ed that if ( F , T ) is a pair s atisfying the w eaker a x ioms, then in the exact co mpletio n F of F , there is a na tural class o f arrows T , dep ending on T , such that the pa ir ( F , T ) is a predica tive c a tegory with small maps. Therefo re our strateg y in this section will b e to co nstruct a categ o ry of ass em blies r elativ e to the pair ( E , S ) and show it is a categ ory with display maps (strictly sp eaking, we only need to assume that ( E , S ) is itself a ca tegory with display maps for this). Its exact completion will then b e cons ide r ed in the next section. In this section, ( E , S ) is as sumed to b e a predica tiv e ca tegory with s mall maps. Deﬁnition 2.1 An assembly (ov er E ) is a pa ir ( A, α ) consis ting of an ob ject A in E together with a r elation α ⊆ N × A , which is surjective: ∀ a ∈ A ∃ n ∈ N ( n, a ) ∈ α. The natural n um be rs n such that ( n, a ) ∈ α are called the re alizers of a , and we will frequently write n ∈ α ( a ) instead of ( n, a ) ∈ α . A morphism f : B / / A in E is a morphism o f a ssem blies ( B , β ) → ( A, α ) if the statement 6 “There is a natural num ber r such that for all b and n ∈ β ( b ), the expression r ( n ) is deﬁned and r ( n ) ∈ β ( f b ).” is v a lid in the internal logic o f E (note that this makes sense, as the in ternal lo g ic of E is a v ersion of HA , a nd ther efore strong enough to do all basic recursion theory). A num b er r witnessing the ab o ve statement is sa id to t r ack (or re alize ) the morphism f . The resulting category will b e denoted by A sm E , or simply A sm . W e inv estigate the structure of the catego ry A sm E . A sm E has ﬁnite limits. The termina l ob ject is (1 , η ), where 1 = {∗} is a one-p oin t set a nd n ∈ η ( ∗ ) for every n . The pullback ( P, π ) o f f a nd g a s in ( P, π ) / /   ( B , β ) f   ( C, γ ) g / / ( A, α ) can be obtained by putting P = B × A C and n ∈ π ( b, c ) ⇔ n 0 ∈ β ( b ) and n 1 ∈ γ ( c ) . Covers in A sm E . A morphism f : ( B , β ) / / ( A, α ) is a cover if, and only if, the statement “There is a natural num b er s such that for all a ∈ A and n ∈ α ( a ) there exists a b ∈ B with f ( b ) = a and such that the expressio n s ( n ) is deﬁned and s ( n ) ∈ β ( b ).” holds in the internal logic of E . F rom this it follows that covers ar e stable under pullback in A sm . A sm E has images. A morphism f : ( B , β ) / / ( A, α ) is monic in A sm if, a nd only if, the under lying mor phism f : B / / A is monic in E . (This means that if ( R , ρ ) is a sub ob ject o f ( A, α ), then R is a lso a subob ject of A .) Hence the image ( I , ι ) of a map f : ( B , β ) / / ( A, α ) as in ( B , β ) f / / e # # # # G G G G G G G G G ( A, α ) ( I , ι ) ; ; m ; ; w w w w w w w w can be obtained by letting I ⊆ A b e the image of f in E , and n ∈ ι ( a ) ⇔ ∃ b ∈ B p ( b ) = a and n ∈ β ( b ) . 7 One could also write: ι ( a ) = S b ∈ p − 1 ( a ) β ( b ). W e conclude tha t A sm is regula r. A sm E is Heyting. F or any diagr am o f the form ( S, σ )     ( B , β ) f / / ( A, α ) we need to c ompute ( R, ρ ) = ∀ f ( S, σ ). W e ﬁr st put R 0 = ∀ f S ⊆ A , and let ρ ⊆ N × R 0 be deﬁned by n ∈ ρ ( a ) ⇔ n 0 ∈ α ( a ) and ∀ b ∈ f − 1 ( a ) , m ∈ β ( b ) ( n 1 ( m ) ↓ and n 1 ( m ) ∈ σ ( b ) ) . If we now put R = { a ∈ R 0 : ∃ n n ∈ ρ ( a ) } and restrict ρ according ly , the sub ob ject ( R , ρ ) will b e the result of universal quantifying ( S, σ ) along f . A sm E is p ositive. The sum ( A, α ) + ( B , β ) is simply ( S, σ ) with S = A + B and n ∈ σ ( s ) ⇔ n ∈ α ( s ) if s ∈ A , and n ∈ β ( s ) if s ∈ B . W e have proved: Prop osition 2.2 The c ate gory A sm E of assemblies r elative to E is a p ositive Heyting c ate gory. The next s tep is to deﬁne the display ma ps in the ca tegory o f assemblies. The idea is that a displayed as sem bly is an ob ject ( B , β ) in which b oth B and the sub ob ject β ⊆ N × B are s ma ll. When one tries to deﬁne a family of s uch display ed ob jects indexed by an ass em bly ( A, α ) in which neither A nor α needs to be s ma ll, one a rriv es at the c oncept of a standar d displa y map. T o for mulate it, we need a piece o f no tation. Deﬁnition 2.3 Le t ( B , β ) and ( A, α ) b e a ssem blies and f : B / / A b e an arbi- trary map in E . W e construc t a new a ssem bly ( B , β [ f ]) by putting n ∈ β [ f ]( b ) ⇔ n 0 ∈ β ( b ) and n 1 ∈ α ( f x ) . Remark 2.4 Note that we obtain a morphism of assemblies of the form ( B , β [ f ]) → ( A, α ), which, by abuse of notation, we will also denote b y f . Mor eo v er, if f was alrea dy a mor phism of assemblies it can now b e decomp osed as ( B , β ) ∼ = / / ( B , β [ f ]) f / / ( A, α ) . 8 Deﬁnition 2.5 A morphism of a ssem blies of the for m ( B , β [ f ]) → ( A, α ) will be c a lled a standar d display map , if b oth f and the mono β ⊆ N × B ar e s mall in E (the latter is equiv alent to β → B b eing small, o r β ( b ) being a small subob ject of N for every b ∈ B ). A display map is a mo rphism of the for m W ∼ = / / V f / / U, where f is a standar d display ma p. W e will wr ite D E for the class of display maps in A sm E . Lemma 2.6 1. L et f : ( B , β [ f ]) / / ( A, α ) b e a standar d display map, and g : ( C , γ ) / / ( A, α ) b e an arbitr ary morphism of assemblies. Then ther e is a pul lb ack squar e ( P, π [ k ]) h / / k   ( B , β [ f ]) f   ( C, γ ) g / / ( A, α ) in which k is again a st anda r d displ ay map. 2. The c omp osite of two standar d display maps is a displa y map. Pro of. (1) W e set P = B × A C (as usual), a nd n ∈ π ( b, c ) ⇔ n ∈ β ( b ) , turning k into a standa rd display map. Moreov er, this implies n ∈ π [ k ]( b, c ) ⇔ n 0 ∈ β ( b ) and n 1 ∈ γ ( c ) , which is precisely the us ua l deﬁnition. (2) Let ( C, γ ), ( B , β ) and ( A, α ) b e assemblies in which γ ⊆ N × C and β ⊆ N × B are small monos, a nd g : C / / B and f : B / / A be display maps in E . The s e data determine a comp osable pair of standard display maps f : ( B , β [ f ]) / / ( A, α ) and g : ( C, γ [ g ]) / / ( B , β [ f ]), in which n ∈ γ [ g ]( c ) ⇔ n 0 ∈ γ ( c ) and n 1 ∈ β [ f ]( g c ) ⇔ n 0 ∈ γ ( c ) and ( n 1 ) 0 ∈ β ( g c ) a nd ( n 1 ) 1 ∈ γ ( f g c ) . So its comp osite ca n b e written as ( C, γ [ g ]) ∼ = / / ( C, δ [ f g ]) f g / / ( A, α ) , where we hav e deﬁned δ ⊆ N × C by n ∈ δ ( c ) ⇔ n 0 ∈ γ ( c ) and n 1 ∈ β ( g c ) .  9 Corollary 2.7 D i splay maps ar e stable under pul lb ack and close d un d er c om- p osition. Pro of. Stability of display maps under pullback follows immediately from item 1 in the prev io us lemma. T o show that they ar e a lso closed under co mposition, we observe ﬁrst that a mo rphism f which ca n b e wr itten as a comp osite W h / / V g ∼ = / / U, where h is a standa rd display ma p and g is an isomor phism, is a display map. F or it fo llows from the previo us lemma that there exists a pullba c k sq uare Q p ∼ = / / q   W h   U g − 1 ∼ = / / V in which q is a standar d dis play map. Ther e fo re f = q p − 1 is a display map. Now the result follows from the lemma ab o ve.  Prop osition 2.8 The class of display maps in t he c ate gory A sm E of assemblies as deﬁne d ab ove satisﬁes the axioms (A1), (A3-5), (A7-9) , and (A10) for a class of displa y maps, as wel l as (NE ) and (NS) (se e App endix B). Pro of. (A1) W e hav e proved pullback sta bility in the corolla ry ab ov e . (A3) It is easy to see that the sum o f tw o sta ndard display maps can b e chosen to b e a sta ndard dis pla y map a gain. F rom this (A3) follows. (A4) It is als o eas y to see that the ma ps 0 / / 1 , 1 / / 1 and 1 + 1 / / 1 are standard display maps. (A5) Closure of display maps under co mposition we showed in the corollary ab o v e. (A7) W e p ostp one the pr oof of the fact that the display maps satisfy the c ol- lection axio m: one will be given in a lemma b elow. (A8) W e s tart with a diagr am o f the form ( S, σ [ i ])   i   ( B , β [ f ]) f / / ( A, α ) , 10 in which b oth maps ar e sta nda rd display maps (this is s uﬃcien t to establish the general cas e ). W e compute ( R, ρ ) = ∀ f ( S, σ ): ﬁrst we put R 0 = ∀ f S ⊆ A , and let ρ ⊆ N × R 0 be deﬁned by n ∈ ρ ( a ) ⇔ n 0 ∈ α ( a ) and ∀ b ∈ f − 1 ( a ) , m ∈ β [ f ]( b ) ( n 1 ( m ) ↓ and n 1 ( m ) ∈ σ [ i ]( b ) ) . F urthermo re, we set R = { a ∈ R 0 : ∃ n n ∈ ρ ( a ) } and denote by j the inclusion R ⊆ A . By restr ic ting ρ , the s ubob ject ( R, ρ ) will be the r esult of the universal quantifying ( S, σ ) alo ng f . In the par ticular cas e we are in, this ca n be done diﬀerently . W e deﬁne τ ⊆ N × R 0 by n ∈ τ ( a ) ⇔ ∀ a ∈ f − 1 ( a ) , m ∈ β ( b ) ( n 1 ( m ) ↓ and n 1 ( m ) ∈ σ ( b ) ) . Note that we hav e a b ounded for mula on the right (using that N is small). No w one can show that R = { a ∈ R 0 : ∃ n n ∈ τ ( a ) } , from whic h it follo ws that j is a display map (again using that N is small). F urthermo re, one can prove that the ide ntit y is an isomorphis m o f assemblies ( R, ρ ) ∼ = ( R, τ [ j ]) , from which it follows that ( R , ρ ) → ( A, α ) is a display map. (A9) The pr o duct of an assembly ( X, χ ) with itself can b e computed by taking ( X × X , χ × χ ), where n ∈ ( χ × χ )( x, y ) ⇔ n 0 ∈ χ ( x ) and n 1 ∈ χ ( y ) . This means that by writing ∆: X / / X × X f or the diagonal map in E , the diagonal map in assemblies ca n be decomp osed a s follows ( X, χ ) ∼ = / / ( X, µ [∆]) ∆ / / ( X, χ ) × ( X , χ ) , where µ ⊆ N × X is the relation deﬁned by n ∈ µ ( x ) ⇔ Alw ays. (A10) W e need to show that for a display ma p f , if f = me with m a mono and e a cov er, then also m is display . Without loss of genera lity , we may a ssume that f is a s tandard display map f : ( B , β [ f ]) / / ( A, α ). F r om Pr oposition 2.2, we know that we can compute its image ( I , ι ) by putting I = Im ( f ) and n ∈ ι ( a ) ⇔ ∃ b ∈ f − 1 ( a ) n ∈ β ( b ) . 11 As the formula o n the right is b ounded, the map m : ( I , ι ) / / ( A, α ) can b e decomp osed as an iso mo rphism follow ed by a standa rd display map: ( I , ι ) ∼ = / / ( I , ι [ m ]) m / / ( A, α ) . (NE) and (NS) The nno in ass e m blies is the pair consis ting of N together with the diagona l ∆ ⊆ N × N .  W e will use the pro of that the display maps in assemblies satisfy collection to illustrate a technique that do es not really s ave an enormous a moun t of la bour in this particula r ca se, but will b e very useful in more co mplica ted situations. Deﬁnition 2.9 An a ssem bly ( A, α ) will b e called p artitione d , if n ∈ α ( a ) , m ∈ α ( a ) ⇒ n = m. F ro m this it follows that α ca n b e consider ed a s a morphism A → N . Lemma 2.10 1. Every assembly is c over e d by a p artitione d assembly. H enc e every morphism b et we en assemblies is c over e d by a morphism b etwe en p ar- titione d assemblies. 2. A morphism f : ( B , β ) / / ( A, α ) b etwe en p artitione d assemblies is display iﬀ f is sm a l l in E . 3. Every display map b etwe en assemblies is c over e d by a display map b etwe en p artitione d assemblies. The deﬁnitions of the notions of a cov e ring square a nd the cov e r ing relation betw een maps from [5] ar e r ecalled in Appendix B. Pro of. (1) If ( A, α ) is an assembly , α can b e considered as a par titioned as sem- bly with n realizing an element ( m, a ) ∈ α iﬀ n = m . This pa rtitioned a ssem bly cov ers ( A, α ). (2) By deﬁnition it is the case that every display map be t ween pa rtitioned assemblies has an underlying map which is small. Conv ersely , if ( B , β ) is a partitioned assembly , the set β ( b ) is a singleton, and therefore small. So the decomp osition ( B , β ) ∼ = / / ( B , β [ f ]) f / / ( A, α ) . shows tha t f is a display map, if the underlying mor phism is s mall. (3) If f : ( B , β [ f ]) / / ( A, α ) is a standard display map b et w een assemblies, then β [ f ] / / f   ( B , β [ f ]) f   α / / ( A, α ) 12 is a covering squa re with a display map b et w een par tit ioned assemblies on the left.  Lemma 2.11 The class of displa y maps in the c ate gory A sm E of assemblies satisﬁes the c ol le ction axiom (A7) . Pro of. In view of the lemma ab ov e, the general case follows by c o nsidering a display ma p f : ( B , β ) / / ( A, α ) b et ween partitioned as sem blies a nd a cov er q : ( E , η ) / / ( B , β ). The fac t that q is a cov er mea ns that there exists a na tural nu mber t such that “F or all b ∈ B , the expres sion t ( β b ) is deﬁned, and there exists an e ∈ E with q ( e ) = b a nd t ( β b ) ∈ η ( e ).” (1) W e will collect all those natur a l num ber s in an ob ject T = { t : t is a natural num ber with the prop erty deﬁned a bov e } , which can b e turned into a partitioned assembly by putting θ ( t ) = t . F rom (1) it follows that T is an inhabited set, and that for E ′ = { ( e, b, t ) : q ( e ) = b, t ( β b ) ↓ , t ( β b ) ∈ η ( e ) } , the pro jection p : E ′ / / B × T will b e a cover. So we c an apply colle c tion in E to o btain a covering squa re D g   h / / E ′ p / / / / B × T f × T   C k / / / / A × T , where g is a sma ll map. It is no t so hard to see that from this diagr am in E , we obtain tw o covering squares in the categor y o f assemblies ( D , δ ) g   ph / / / / ( B × T , β × τ ) / / / / f × T   ( B , β ) f   ( C, γ ) k / / / / ( A × T , α × τ ) / / / / ( A, α ) , where we hav e set γ ( c ) = ( α × τ )( k c ) and δ ( d ) = ( β × τ )( phd ) . 13 Since g is a display map b et w een partitioned assemblies, we only nee d to verify that the map ( D , δ ) → ( B , β ) along the top of the ab ov e diagra m factors as ( D , δ ) l / / ( E , η ) q / / / / ( B , β ) . W e s et l = π 1 h , b ecause one can s how that this morphism is tracked, as follows. If h ( d ) = ( e , t, b ) for some d ∈ D , then the rea lizer of d consists o f a co de n for the partial recurs ive function t , together with the r e a lizer β b o f b . By deﬁnition of E ′ , the expr ession n ( β b ) is deﬁned and a r ealizer for e = ( π 1 h )( d ) = l ( d ).  3 The predicativ e real izabilit y category Let us reca ll from [1 0] the co nstruction o f the (ordinary) exact completion F ex/r eg of a Heyting category F . O b jects of F ex/r eg are the equiv alence re- lations in F , which we will deno te by X/R when R ⊆ X × X is a n equiv a lence relation. Morphisms fro m X/R to Y / S ar e fun ctio nal r elations , i.e., sub ob jects F ⊆ X × Y satisfying the following statements in the internal lo gic of F : ∀ x ∃ y F ( x, y ) , xRx ′ ∧ y S y ′ ∧ F ( x, y ) → F ( x ′ , y ′ ) , F ( x, y ) ∧ F ( x, y ′ ) → y S y ′ . There is a functor y : F / / F ex / reg sending a n ob ject X to X/ ∆ X , where ∆ X is the dia gonal X → X × X . T his functor is a full em bedding preserving the structure of a Heyting category . When T is a class o f display maps in F , one can ident ify the following cla ss o f maps in F ex / reg : g ∈ T ⇔ g is cov ered by a mo rphism of the form y f with f ∈ T . W e refer to the pair ( F , T ), consis ting of the full sub category F of F ex/r eg containing tho se equiv alence r elations i : R / / X × X for which i b elongs to T , together with T , as the exact c ompletion of the pair ( F , T ). In [5] we proved the following res ult for such exact completions: Theorem 3 .1 [5] I f ( F , T ) is a c ate gory with a r epr esentable class of display maps satisfying ( Π E), (WE ) and (NS) , t he n its exact c ompletion ( F , T ) is a pr e dic ative c ate gory with s mall maps. In the res t of the s ection, we let ( E , S ) be a pr edicativ e ca tegory with small maps. F or such a categor y w e hav e constructed and studied the pa ir ( A sm E , D E ) consisting of the category of assemblies and its display maps . W e now deﬁne ( E f f E , S E ) as the exa c t completion of ( A sm E , D E ) and prove our main theorem (Theorem 1.2) as an a pplication of Theore m 3.1. Much of the work has alr eady bee n done in Section 2. In fact, Prop osition 2.8 shows that the o nly thing that 14 remains to b e shown are the repr esen tabilit y and the v alidit y o f a x ioms ( Π E) and (W E) (see App endix B). Prop osition 3.2 The class of display maps in t he c ate gory A sm E of assemblies is r epr esentable. Pro of. Let π : E / / U be the representation for the small maps in E . W e deﬁne t wo partitioned asse mblies ( T , τ ) and ( D , δ ) by T = { ( u ∈ U, p : E u / / N ) } , τ ( u, p ) = 0 , D = { ( u ∈ U, p : E u / / N , e ∈ E u ) } , δ ( u, p , e ) = pe. Clearly , the pro jection ρ : ( D , δ ) / / ( T , τ ) is a displa y map, which we will now show is a representation. Assume f : ( B , β ) / / ( A, α ) is a display map be t ween par titioned assemblies (in view of Lemma 2.10 it is suﬃcient to co nsider this case). Since f is also a display map in E we ﬁnd a diag ram of the form B f   N s   k / / l o o o o E π   A M g / / h o o o o U, where the left square is c o v ering and the r igh t o ne a pullback. This induces a similar picture ( B , β ) f   ( N , ν ) s   k ′ / / l o o o o ( D , δ ) ρ   ( A, α ) ( M , µ ) g ′ / / h o o o o ( T , τ ) in the catego ry of a ssem blies, where we have set: g ′ ( m ) = ( g m, β l k − 1 : E gm / / N ) , µ ( m ) = αh ( m ), so h is track ed and a cover , k ′ ( n ) = ( g ′ s ( n ) , k n ) , ν ( n ) = h µsn, δ k ′ n i , so the righthand squar e is a pullback. Here g ′ is well-deﬁned, beca use N is a pullba ck and therefore the map k induces for every m ∈ M an isomor phism N m k ∼ = / / E gm . 15 It remains to se e tha t l is track ed, and the tha t left ha nd sq ua re is a quasi- pullback. F or this, one unwinds the deﬁnition of ν : ν ( n ) = h µsn, δ k ′ n i = h µsn, δ ( g ′ s ( n ) , k n ) i = h µsn, δ ( g s ( n ) , β l k − 1 , k n ) i = h µsn, β l k − 1 k n i = h µsn, β l n i . F ro m this description of ν , we see that l is indeed track ed (by the pro jection on the second co ordinate). T o see that the squar e is a quasi-pullback, one uses ﬁrst of all that it is a q ua si-pullbac k in E , and secondly that the r ealizers for an element in N are the sa me as that of its ima ge in the pullback ( M × A B , µ × β ) along the canonical map to this ob ject.  Prop osition 3.3 The display maps in the c ate gory A sm E of assemblies ar e exp onentiable, i.e., satisfy the axiom ( Π E) . Mor e over, if ( Π S) holds in E , then the it holds for the display maps in A sm E as wel l. Pro of. Let f : ( B , β [ f ]) / / ( A, α ) b e a standa r d display map a nd g : ( C , γ ) / / ( A, α ) an ar bitrary map with the same c odomain. F or showing the v alidity of ( Π E) it suﬃces to prove that the exp onen tial g f exists in the slice over ( A, α ). Since f is small, one can for m the exp onential g f in E / A , whose typical elements are pairs ( a ∈ A, φ : B a / / C a ). If we se t n ∈ η ( a, φ ) ⇔ n 0 ∈ α ( a ) a nd ( ∀ b ∈ B a , m ∈ β ( b )) [ n 1 ( m ) ↓ and n 1 ( m ) ∈ γ ( φb )] , E = { ( a, φ ) ∈ f g : ( ∃ n ∈ N ) [ n ∈ η ( a, φ )] } , the assembly ( E , η ) with the obvious pro jection p to ( A, α ) is the ex ponential g f in assemblies. This shows v a lidit y of ( Π E) for the display maps in ass em blies. If g : ( C, ˆ γ [ g ]) / / ( A, α ) is another standard dis pla y map, the exp onential can also b e co nstructed by putting n ∈ ˆ η ( a, φ ) ⇔ ( ∀ b ∈ B a , m ∈ β ( b )) [ n 1 ( m ) ↓ and n 1 ( m ) ∈ ˆ γ ( φb )] , ˆ E = { ( a, φ ) ∈ f g : ( ∃ n ∈ N ) [ n ∈ ˆ η ( a, φ )] } . It is not hard to see tha t ˆ E = E , and the identit y induces an isomorphism o f assemblies ( ˆ E , ˆ η [ p ]) = ( E , η ). This shows the stability of ( Π S) .  Prop osition 3.4 The display maps in t he c ate gory A sm E of assemblies satisfy the axiom (WE) . Mor o ever, if (WS) holds in E , t h en it holds for the display maps in A sm E as wel l. 16 Pro of. Let f : ( B , β [ f ]) / / ( A, α ) b e a standard dis play map. Since (WE) holds in E , w e ca n form W f in E . On it, we wish to deﬁne the relatio n δ ⊆ N × W f given by n ∈ δ (sup a ( t )) ⇔ n 0 ∈ α ( a ) and ( ∀ b ∈ f − 1 ( a ) , m ∈ β ( b )) [ n 1 ( m ) ↓ and n 1 ( m ) ∈ δ ( tb )] (2) (w e will sometimes call the elemen ts n ∈ δ ( w ) the de c or ations of the tr e e w ∈ W ). It is not so obvious that we can, but for that purp ose we in tro duce the notion of an attempt . An a ttemp t is an element σ of P s ( N × W f ) such that ( n, sup a ( t )) ∈ σ ⇒ n 0 ∈ α ( a ) and ( ∀ b ∈ f − 1 ( a ) , m ∈ β ( b )) [ n 1 ( m ) ↓ and ( n 1 ( m ) , tb ) ∈ σ ] . If we now put n ∈ δ ( w ) ⇔ ther e exists a deco ration σ with ( n, w ) ∈ σ , the relation δ will have the desire d prop ert y . (One direction in (2) is trivial, the other is more inv olved and us es the collection axio m.) The W- type in the category of assemblies is now g iv en by ( W, δ ) where W = { w ∈ W f : ( ∃ n ∈ N ) [ n ∈ δ ( w )] } . This shows the v a ilidity of (WE) for the display maps. If A is small and (WS) holds in E , then W f is s ma ll. Moreover, if α ⊆ N × A is s mall, one ca n use the initiality of W f to deﬁne a map d : W f / / P s N by d (sup a ( t )) = { n ∈ N : n 0 ∈ α ( a ) and ( ∀ b ∈ f − 1 ( a ) , m ∈ β ( b )) [ n 1 ( m ) ↓ and n 1 ( m ) ∈ d ( tb )] } . Clearly , n ∈ δ ( w ) iﬀ n ∈ d ( w ), s o δ is a small sub ob ject of N × W f . This shows that ( W, δ ) is display ed, and the s tabilit y of (WS) is prov ed.  T o summarise, we have proved the ﬁrst half of Theor em 1 .2, which we phra se explicitly as: Corollary 3.5 If ( E , S ) is a pr e dic ative c ate gory with smal l maps, then so is ( E f f E , S E ) . 4 Additional axioms T o complete the pro of o f Theorem 1.2, it remains to show the stability of the additional axioms (M) , (PS) and (F) . That is wha t we will do in this (rather techn ical) section. W e assume ag ain that ( E , S ) is a pr edicativ e categ ory with small maps. 17 Prop osition 4.1 Assume the class of sm a l l maps in E satisﬁes (M) . Then (M) is valid for t h e displa y maps in the c ate gory A sm E of assemblies and for the smal l maps in the pr e dic ative r e alizability c ate gory E f f E as wel l. Pro of. Let f : ( B , β ) / / ( A, α ) be a monomo rphism in the ca tegory of as s em- blies. Then the underlying map f in E is a mo no morphism as well. Therefor e it is small, as is the inclus io n β ⊆ N × B . So the morphis m f , which factor s as ( B , β ) ∼ = / / ( B , β [ f ]) / / ( A, α ) , is a display map of a ssem blies. Stabilit y o f the a xiom (M) under exact completion [5 , Pro position 6 .4] shows it holds in E f f E as well.  Prop osition 4.2 Assume t h e class of smal l maps in E satisﬁes (F) . Then (F) is valid for the display maps in the c ate gory A sm E of assemblies and for the smal l maps in the pr e dic ative r e alizability c ate gory E f f E as wel l. Pro of. It is s uﬃcient to show the v alidity of (F) in the ca tegory of a ssem blies, for we show ed the stability of this a xiom under ex act completion in [5 , Prop o- sition 6.25 ]. So we need to ﬁnd a ge neric mvs in the categor y of ass e m blies for any pair of display maps g : ( B , β ) / / ( A, α ) and f : ( A, α ) / / ( X, χ ). In view of Lemma 6.23 from [5] and Lemma 2.10 ab ov e, we may without lo ss of gener alit y assume that g a nd f are display maps b et w een p artitione d assemblies. W e a pply F ullness in E to obtain a diagra m of the form P / / / / # # # # G G G G G G G G G Y × X B / /   B g   Y × X A / /   A f   Y s / / X ′ q / / / / X , where P is a generic display ed mvs for g . This allows us to obta in a similar diagram of partitioned as sem blie s ( ˜ P , ˜ π ) / / / / ' ' ' ' N N N N N N N N N N N ( ˜ Y × X B , ˜ υ × β )   / / ( B , β ) g   ( ˜ Y × X A, ˜ υ × α ) / /   ( A, α ) f   ( ˜ Y , ˜ υ ) ˜ s / / ( X ′ , χ ′ ) q / / / / ( X, χ ) , 18 where we hav e set χ ′ ( x ′ ) = χ ( qx ′ ) for x ′ ∈ X ′ , ˜ Y = { ( y , n ) ∈ Y × N : n rea liz es the s tatemen t that P y → A qsy is a cov er } = { ( y , n ) ∈ Y × N : ( ∀ a ∈ A qsy )( ∃ b ∈ B a ) [( y , b ) ∈ P a nd n ( α ( a )) = β ( b )] } , ˜ υ ( y, n ) = h sq y , n i for ( y , n ) ∈ ˜ Y , ˜ P = ˜ Y × Y P = { ( y , n, b ) ∈ Y × N × B : ( y , n ) ∈ ˜ Y , ( y , b ) ∈ P } , ˜ π ( y , n, b ) = h n, β ( b ) i for ( y , n, b ) ∈ P. The reader should verify that: 1. q is track ed and a cov er. 2. ˜ s is track ed and display , since ˜ Y is deﬁned using a bo unded formula. 3. The inclusion ( ˜ P , ˜ π ) ⊆ ( ˜ Y × X B , ˜ υ × β ) is tracked. 4. It follows from the deﬁnition of ˜ Y that the map ( ˜ P , ˜ π ) → ( ˜ Y × X A, ˜ υ × α ) is a c o v er. W e will now prov e that ( ˜ P , ˜ π ) is the generic mvs for g in a ssem blies. Let R b e a n mvs of g ov er Z , as in: ( R, ρ ) / / i / / ' ' ' ' O O O O O O O O O O O ( Z × X B , ζ × β )   / / ( B , β ) g   ( Z × X A, ζ × α ) / /   ( A, α ) f   ( Z, ζ ) t / / ( X ′ , χ ′ ) q / / / / ( X, χ ) . Since every ob ject is cov ered by a partitioned as sem bly (see Lemma 2.10), w e may assume (without loss of generality) that ( Z , ζ ) is a partitioned assembly . Now we obtain a commuting squa re ( ˜ R, ˜ ρ )   / / ( R, ρ )   ( ˜ Z , ˜ ζ ) d / / / / ( Z, ζ ) , 19 in which we hav e deﬁned ˜ Z = { ( z , m, n ) ∈ Z × N 2 : m tracks i and n r ealizes the statement that R z → A qtz is a cov er } = { ( z , m, n ) : ( ∀ ( z , b ) ∈ R, k ∈ ρ ( z , b )) [ m ( k ) = ( ζ × β )( z , b )] and ( ∀ a ∈ A qtz )( ∃ b ∈ B a ) [( z , b ) ∈ R and n ( α ( a )) ∈ ρ ( z , b ) } ˜ ζ ( z , m, n ) = h ζ z , m, n i for ( z , m, n ) ∈ ˜ Z ˜ R = { ( z , m, n, b ) ∈ ˜ Z × B : ( z , b ) ∈ R and n ( α ( g b )) ∈ ρ ( z , b ) } ˜ ρ ( z , m, n, b ) = h ˜ ζ ( z , m, n ) , β ( b ) i fo r ( z , m, n, b ) ∈ ˜ R It is easy to see that all the ar ro ws in this diagr am ar e track ed, and the pro jection ( ˜ Z , ˜ ζ ) → ( Z , ζ ) is a cover. It is also ea sy to see that ( ˜ R, ˜ ρ ) is still an mvs of g in assemblies. Note also that ( ˜ R, ˜ ρ ) and ( ˜ Z , ˜ ζ ) are partitioned ass em blies . Since the for getful functor to E preser v es mvs s in general, and dis pla y ed ones betw een pa rtitioned assemblies, ˜ R is also a display ed mvs of g in E . Therefore there is a diagr am o f the form ˜ R   l ∗ P   / / o o P   ˜ Z T k o o o o l / / Y in E with tdk = sl . W e turn T into a partitioned a ssem bly by putting τ ( t ) = ˜ ζ ( k t ) for all t ∈ T . Claim: the map l : T / / Y factors thro ug h ˜ Y → Y via a ma p ˜ l : T / / ˜ Y which can be track ed. Pro of: if k ( t ) = ( z , m, n ) and l ( t ) = y for s ome t ∈ T , we set ˜ l ( t ) = ( y , ( m ◦ n ) 1 ) , where m ◦ n is the co de of the par tial recursive function obtained by comp osing the functions co ded b y m with n . W e ﬁrs t hav e to show that this is well-deﬁned, i.e., ˜ l ( t ) ∈ ˜ Y . Since P is an mvs in E , we can ﬁnd for any a ∈ A qsy an element b ∈ B a with ( y , b ) ∈ P . If we take such a b , it follows fro m P y = P lt ⊆ ˜ R kt , that ( z , m, n, b ) ∈ ˜ R , and therefore n ( α ( a )) ∈ ρ ( z , b ). More over, it follo ws fro m the fact that ( z , m, n ) ∈ ˜ Z , that ( m ◦ n ) 1 ( α ( a )) = β ( b ). This shows tha t ˜ l ( t ) ∈ ˜ Y . That ˜ l is tra c ked is ea sy to see. As a result, we obtain a dia gram of the for m ( ˜ R, ˜ ρ )   ˜ l ∗ ( ˜ P , ˜ π )   / / o o ( ˜ P , ˜ π )   ( ˜ Z , ˜ ζ ) ( T , τ ) k o o o o ˜ l / / ( ˜ Y , ˜ υ ) . 20 Given the deﬁnitions of ˜ ρ a nd ˜ π , o ne sees that ˜ l ∗ ( ˜ P , ˜ π ) → ( ˜ R, ˜ ρ ) is tr ac k ed. This completes the pro of.  When it comes to the axiom (PS) concerning p o wer types, there seems to b e no reason to b eliev e that it will be inherited by the assemblies. But, for tuna tely , it will be inher ited by its ex a ct completion, and for our pur p oses that is just as go od. Prop osition 4.3 Assume t h e class of smal l maps in E satisﬁes (PS) . Then (PS) is valid in the r e alizability c ate gory E f f E as wel l. Pro of. F or the purp ose of this pro of, we intro duce the no tion of a weak p o w er class ob ject. Recall that the p ow er cla ss ob ject is deﬁned a s: Deﬁnition 4.4 B y a D - i ndexe d family of sub obje cts o f C , we mea n a subo b ject R ⊆ C × D . A D -indexed family of subo b jects R ⊆ C × D will b e called S - displaye d (or simply displaye d ), whenever the comp osite R ⊆ C × D / / D belo ngs to S . If it exists, the p ower class obje ct P s X is the classifying ob ject for the displayed families of sub ob jects of X . This means that it comes equipped with a display ed P s X -indexed fa mily of sub ob jects of X , denoted by ∈ X ⊆ X × P s X (or s imply ∈ , whenever X is unders too d), with the prop ert y that for any displayed Y - indexed family of sub ob jects of X , R ⊆ X × Y s a y , there ex ists a unique map ρ : Y / / P s X such that the square R     / / ∈ X     X × Y id × ρ / / X × P s X is a pullback. If a classifying map ρ as in the ab o ve diagr am exists, but is not unique, we ca ll the p o wer c lass ob ject we ak . W e will denote a weak pow er class ob ject o f X by P w s X . W e will show that the categor ies o f a ssem blies has weak p o w er class ob jects, which ar e moreov er “ small” (i.e., the unique map to the terminal ob ject is a display map). This will be s uﬃcien t for proving the stability o f (PS) , as we will show in a lemma b elo w that real p ow er ob jects in the exac t completio n are constructed from the weak ones by taking a quotient. Let ( X , χ ) b e an assembly . W e deﬁne a n ass em bly ( P, π ) by P = { ( α ∈ P s X , φ : α / / P s N ) : ( ∀ x ∈ α )( ∃ n ∈ N ) [ n ∈ φ ( x )] and ( ∃ n ∈ N ) ( ∀ x ∈ α, m ∈ φ ( x )) [ n ( m ) ∈ χ ( x )] } , π ( α, φ ) = { n ∈ N : ( ∀ x ∈ α, m ∈ φ ( x )) [ n ( m ) ∈ χ ( x )] } . 21 W e claim that this assembly together with the membership relation ( E , η ) ⊆ ( X, χ ) × ( P , π ) deﬁned by E = { ( x ∈ X , α ∈ P s X , φ : α / / P s N ) : ( α, φ ) ∈ P and x ∈ α } , η ( x, α, φ ) = { n ∈ N : n 0 ∈ φ ( x ) and n 1 ∈ π ( α, φ ) } is a weak p o w er ob ject in ass em blies. F or let ( S, σ ) b e a (standardly) displayed ( Y , υ )-indexed fa mily of sub ob jects of ( X, χ ). This means that the under lying mor phism f : S / / Y is small, and σ = σ [ f ] for a small relation σ ⊆ N × S . Since f is small, we obtain a pullback diagram of the for m S     / / ∈ X     X × Y id × s / / X × P s X in E . W e use this to build a similar diag ram in the catego ry of a ssem blies: ( S, σ )     / / ( E , η )     ( X, χ ) × ( Y , υ ) id × s / / ( X, χ ) × ( P , π ) , where we hav e set s ( y ) = ( sy , λx ∈ sy .σ ( x, y )) . One quickly veriﬁes that with s b eing deﬁned in this wa y , the s quare is a ctually a pullback. This shows that ( P, π ) is indeed a weak p ow er ob ject. If ( X , χ ) is a displayed as s em bly , so bo th X a nd χ ⊆ N × X are s ma ll, and (PS) holds in E , then P and π ar e deﬁned by b ounded separa tion from small ob jects in E . Therefore ( P , π ) is a display ed ob ject. In the exac t co mpletion, the p ow er class o b ject is co ns tructed from this by quotienting this ob ject (se e the le mma below), and is therefore sma ll.  T o complete the pro of of the prop osition a bov e, we ne e d to show the following lemma, which is a v ariation on a result in [5]. Lemma 4.5 L et y : ( F , T ) / / ( F , T ) b e the ex a ct c ompletion of a c ate gory with display m a ps. When P w s X is a we ak p ower obje ct for an T - small obje ct X in F , then the p ower class obje ct in F exists; in fact, it c an b e obtaine d by qu otie nting y P w s X by extensional e quality. Pro of. W e will drop o ccurences of y in the pro of. 22 On P w s X one ca n deﬁne the equiv a lence relation α ∼ β ⇔ ( ∀ x ∈ X )[ x ∈ α ↔ x ∈ β ] . As X is assumed to b e T -small, the mono ∼ ⊆ P w s X × P w s X is sma ll, a nd therefore this equiv alence relation ha s a q uo tien t. W e will wr ite this quotient as P s X a nd pr o v e that it is the p o w er class o b ject o f X in F . The elementhoo d relation of P s X is given by x ∈ [ α ] ↔ x ∈ α, which is clearly well-deﬁned. In par ticular, ∈ X     / / / / ∈ X     X × P w s X X × q / / / / X × P s X is a pullback. Let U ⊆ X × I / / I b e an T -display ed I -indexed family of sub ob jects of X . W e need to show that there is a unique map ρ : I / / P s X s uc h that ( id × ρ ) ∗ ∈ X = U . Since U / / I ∈ T , ther e is a map V / / J ∈ T such that the outer rectangle in V f   / / / / U     X × J / /   X × I   J p / / / / I , is a cov ering square. Now also f : V / / X × J ∈ T , and by replacing f by its image if necessa r y and using the ax io m (A10 ) , w e may assume tha t the top square (and hence the e ntire diagram) is a pullback and f is mo nic. So there is a map σ : J / / P w s X in E with ( id × σ ) ∗ ∈ X = U , b y the “universal” prop ert y of P w s X in E . As pj = pj ′ ⇒ V j = V j ′ ⊆ X ⇒ σ ( j ) ∼ σ ( j ′ ) 23 for all j, j ′ ∈ J , the map q σ co equalises the kernel pair of p . Therefore there is a map ρ : I / / P s X such that ρp = q σ : V   f   / / / / U     / / ∈ X     X × J / / / /   X × I   / / X × P s X   J p / / / / qσ 1 1 I ρ / / P s X . The desired equality ( i d × ρ ) ∗ ∈ X = U now fo llows. The uniqueness o f this map follows from the deﬁnition o f ∼ .  The pr o of of this pro position co mpletes the pro of of our main result, Theorem 1.2. 5 Realizabili t y mo dels for set theory Theorem 1.1 and Theo rem 1.2 together imply that for any pr edicativ e categor y with small maps ( E , S ), t he categ ory ( E f f E , S E ) will contain a mo del of set theory . As a lready mentioned in the int ro duction, man y known co nstructions of rea lizabilit y mo dels of intuitionistic (or constructive) s et theo ry ca n b e viewed as sp ecial cases of this metho d. In addition, our re s ult als o shows that these constructions ca n b e per formed inside weak metatheories s uc h a s CZF , or inside other sheaf or rea lizabilit y mo dels. T o illustr a te this, we will w ork out one sp eciﬁc example, the re a lizabilit y mo del for IZF descr ibed in McCar t y [2 4 ] (we will comment on other exa mples in the remar k clo sing this section). T o this e nd, let us star t with the category S e ts a nd ﬁx an inaccessible ca rdinal κ > ω . The car dinal κ can be used to deﬁne a class o f small ma ps S in S e ts by declaring a morphism to b e sma ll, when all its ﬁbres hav e ca rdinalit y less than κ (these will b e ca lled the κ -sma ll maps). Because the axiom (M ) then holds both in E and the catego r y of assemblies, the exact completion A sm of the assemblies is rea lly the ordinar y exact completion, i.e., the eﬀective top os. This means we hav e deﬁned a class of small maps in the eﬀective top os. W e will now verify that this is the same class of small maps as deﬁned in [19]. Lemma 5.1 The fol lo wing t wo classes of smal l maps in the eﬀe ctive top os c o- incide: (i) Those c over e d by a map f b etwe en p artitione d assemblies for which the underlying map in E is κ -smal l (as in [19 ] ). 24 (ii) Those c over e d by a displa y m a p f b etwe en assemblies (as ab ove). Pro of. Immediate from Lemma 2.10, and the fact that the cov ering relation is transitive.  By the ge neral existence r esult, the eﬀective top os contains a mo del of IZF which we will call V . Prop osition 5.2 In V the fol lowing principles hold: (A C), (RDC), (P A), (MP), (CT) . Mor e over, V is u nif orm, and henc e also (UP) , (UZ) , (IP) and (IP ω ) hold. Pro of. The Axioms of Countable and Relativised Dependent Choic e hold in V , beca us e they hold in the eﬀective top os (recall the r emarks on the re la tion betw een truth in V and truth in the sur rounding categ ory from the intro duction; in par ticula r, that Int( N ) ∼ = ω ). The same applies to Markov’s Principle and Chu rch’s Thesis (for Churc h’s thesis it is also essential that the model V and the eﬀective top os agree on the meaning of the T - and U -predicates). The Presentation Axiom holds, beca use (internally in E f f ) every small ob ject is cov er ed by a sma ll par titio ne d ass em bly (see the Lemma a bov e), and the partitioned assemblies a r e internally pro jective in E f f . The Uniformity Principle, Unzer legbarkeit a nd the Indep endence o f Premiss es principles are immediate conseque nce s of the fact that V is uniform (of cour se, Unzerlegbarkeit follows immediately the Unifor mity Principle; note that for showing that the principles of (IP) a nd (IP ω ) hold, we use classica l logic in the metatheory). T o show that V is uniform, we recall from [5] that the initial P s -algebra is constructed as a quotient of the W-type asso ciated to a represe ntation. In Prop osition 3.2, w e hav e seen that the representation ρ can be c hosen to be a morphism b et w een (partitioned) assemblies ( D , δ ) / / ( T , τ ), where T is uniform (every e le men t in T is r ealized by 0). As the inclus ion of A sm in E f f preser v es W-t yp es, the as sociated W-type might just as well be computed in the catego ry of a ssem blies. Therefo r e it is cons tructed as in Pro position 3.4: fo r building the W-t yp e ass ociated to a ma p f : ( B , β ) / / ( A, α ), one ﬁrst builds W ( f ) in S ets , and deﬁnes (by transﬁnite induction) the r ealizers of an e le ment sup a ( t ) to b e those natural num b ers n co ding a pair h n 0 , n 1 i such that (i) n 0 ∈ α ( a ) and (ii) for all b ∈ f − 1 ( a ) and m ∈ β ( b ), the expressio n n 1 ( m ) is de ﬁned and a rea lizer of tb . Using this description, one sees that a solution of the recursion equa tion f = h 0 , λn.f i rea lizes every tree. Hence W ( ρ ), and its quotient V , are uniform in E f f .  W e will now show that V is in fact McCa rt y’s mo del for IZF . F or this, we will follow a strategy diﬀerent from the one in [21]: we will simply “unwind” 25 the existence pro of fo r V to obtain a co ncrete description. First, we c o mpute W = W ( ρ ) in assemblies (see the pro of ab o ve). Its underlying set consis ts o f well-founded trees, w ith every edge la belled by a natural num ber. Moreov er, at every no de the set of edges into that no de should hav e c a rdinalit y less than κ . One could also descr ibe it as the initial a lgebra of the functor X 7→ P κ ( N × X ), where P κ ( Y ) is the se t of all subsets of Y with cardina lit y less than κ : P κ ( N × W ) I , , W . E m m Again, the re a lizers of a well-founded tr ee w ∈ W are deﬁned inductively: n is a realiz er of w , if for every pair ( m, v ) ∈ E ( w ), the expressio n n ( m ) is deﬁned and a rea liz er o f v . The nex t step is dividing o ut, internally in E f f , b y bisimu lation: w ∼ w ′ ⇔ ( ∀ ( m, v ) ∈ E( w )) ( ∃ ( m ′ , v ′ ) ∈ E( w ′ ))[ v ∼ v ′ ] and vice versa. The internal v alidity of this statement should be tra ns lated in terms o f realizer s . T o make the expression more succinct o ne could introduce the “abbr eviation”: n  w ′ ǫw ⇔ ( ∃ ( m, v ) ∈ E( w )) [ n 0 = m and n 1  w ′ ∼ v ] , so that it b ecomes: n  w ∼ w ′ ⇔ ( ∀ ( m, v ) ∈ E( w ))[ n 0 ( m ) ↓ and n 0 ( m )  v ǫ w ′ ] a nd ( ∀ ( m ′ , v ′ ) ∈ E( w ′ )) [ n 1 ( m ′ ) ↓ and n 1 ( m ′ )  v ′ ǫ w ] . By app ealing to the Recursion Theorem, one ca n c hec k that we hav e deﬁned an equiv alence re lation on W ( ρ ) in the eﬀective top os (altho ug h this is guaranteed by the pro of o f the exis tence theorem for V ). The quotient will b e the set- theoretic model V . So, its underlying set is W and its equality is given by the formula for ∼ . Of co urse, when one unwinds the deﬁnition of the in ternal mem be rship ǫ ⊆ V × V , o ne obtains precisely the fo r m ula ab o ve. Corollary 5.3 The fol lowing clauses r e cursively deﬁne what it me ans that a c ertain statemen t is r e alize d by a natura l num b er n in t he mo del V : n  w ′ ǫw ⇔ ( ∃ ( m, v ) ∈ E( w )) [ n 0 = m and n 1  w ′ = v ] . n  w = w ′ ⇔ ( ∀ ( m, v ) ∈ E( w ))[ n 0 ( m ) ↓ and n 0 ( m )  v ǫ w ′ ] and ( ∀ ( m ′ , v ′ ) ∈ E( w ′ )) [ n 1 ( m ′ ) ↓ and n 1 ( m ′ )  v ′ ǫ w ] . n  φ ∧ ψ ⇔ n 0  φ and n 1  ψ . n  φ ∨ ψ ⇔ n = h 0 , m i and m  φ , or n = h 1 , m i and m  ψ . n  φ → ψ ⇔ F or al l m  φ, we have n · m ↓ and n · m  ψ . n  ¬ φ ⇔ Ther e is n o m s uch that m  φ. n  ∃ x φ ( x ) ⇔ n  φ ( a ) for some a ∈ V . n  ∀ x φ ( x ) ⇔ n  φ ( a ) for al l a ∈ V . 26 Pro of. The internal log ic of E f f is realizability , so the statemen ts for the log ical connectives immediately follow. F or the quantiﬁers one uses the uniformity of V .  W e conclude that the mo de l is iso morphic to tha t of McCarty [24] (based on earlier work by F riedman [12]). Remark 5.4 Ther e are ma n y v a r iations and extensions of the construction just given, s ome of which we already alluded to in the introductio n. Fir st o f all, instead of working with a ina ccessible cardinal κ , we can also work with the category of classes in G¨ odel-Bernays se t theor y , and call a map small if its ﬁbres are sets. (The sligh t disadv antage of this approach is that o ne cannot directly refer to the eﬀective top os, but has to build up a v ersion of that fo r classe s ﬁrst.) More genera lly , one can of course sta r t with any predicative ca tegory with a class of sma ll maps ( E , S ). If ( E , S ) satisﬁes condition (F) , then so will its rea lizabilit y extension, and by Theorem 1.1, this will pro duce mo dels of CZF r ather than IZF . F or example, if we take for ( E , S ) the syntactic categor y with sma ll ma ps asso ciated to the the theory CZF , then one obtains Ra thj en’s syntactic version of McCa rt y ’s mo del [30]. Alternatively (or, in addition), one can als o r eplace num ber realizability by realizability for an arbitr ary small pa rtial combinatory algebr a A internal to E . V ery basic exa mples arise in this w ay , alre ady in the “trivia l” ca se where E is the topo s of s he aves on the Sierpinsk i space, in which case a n internal pc a A can be identiﬁed with a suitable map b et w een p ca’s. The well-known Kleene-V esley realizability [20] is in fact a sp ecial case of this constructio n. More ge nerally , one can s tart with a pr edicativ e category with small maps ( E , S ) and intert wine the construction o f Theorem 1.2 with a similar result for sheaves, announced in [6] and discussed in deta il in Part II I o f this se ries [7]: Theorem 5 .5 [6] L et ( E , S ) b e a pr e dic ative c ate gory with sm a l l maps s a tisfying ( Π S) , and C a smal l site with a b asis in E . Then the c ate gory of she aves Sh E [ C ] c arries a natura l class of maps S E [ C ] , su ch that the p air (Sh E [ C ] , S E [ C ]) is again a pr e dic ative c ate gory with smal l maps satisfy ing ( Π S) . Mor e over, this latter p air satisﬁes (M) , (F) or (PS) , resp e ctively, whenever the p air ( E , S ) do es. Thu s, if C is a small s ite in E , and A is a sheaf o f pca’s on C , o ne o btains a predicative categ o ry with small maps ( E ′ , S ′ ) = ( E f f Sh E [ C ] [ A ] , S Sh E [ C ] [ A ]), as in the case of Kleene-V esley realiza bility [9]. An y op en (resp. clo sed) subtop os deﬁned by a small site in ( E ′ , S ′ ) will now deﬁne another such pair ( E ′′ , S ′′ ), a nd he nc e a mo del of IZF or CZF if the co n- ditions of Theorem 1.1 ar e met by the original pair ( E , S ). O ne migh t r efer to its semantics as “ r elativ e rea lizabilit y” (resp. “mo diﬁed relative realiza bilit y” ). It has been shown by [9] that relative realiz abilit y [3, 31] and mo diﬁed r ealizabilit y 27 [28] ar e sp ecial cases of this, where Sh E [ C ] is again sheav es o n Sie r pinski space (see also [26]). 6 A mo del of CZF in whic h all sets are s ub- coun table In this sec tio n we will show that CZF is consis ten t with the principle saying that all sets are s ubcountable (this was ﬁrst sho wn by Streicher in [32]; the account that now follows is based o n the work o f the ﬁrst author in [4]). F or this purp ose, we consider a g ain the eﬀective top os E f f rela tiv e to the classical metatheory S ets . W e will show it car ries another clas s of sma ll maps. Lemma 6.1 The fol lo wing ar e e quivalent for a m o rphism f : B / / A in E f f . 1. In t h e internal lo gic of E f f it is tru e that al l ﬁ br es of f ar e quotients of sub obje ct s of N (i.e., sub c ountable). 2. In t h e internal lo gic of E f f it is tru e that al l ﬁ br es of f ar e quotients of ¬¬ -close d sub obje cts of N . 3. The morphism f ﬁt s into a diagr am of t he fol lowing shap e X × N " " F F F F F F F F F Y o o o o / / / / g   B f   X / / / / A, wher e the squar e is c overing and Y is a ¬¬ -close d sub obje ct of X × N . Pro of. Items 2 and 3 express the same thing, once in the internal logic and once in diagr a mmatic la ng uage. Tha t 2 implies 1 is triv ia l. 1 ⇒ 2: This is an application of the internal v a lidit y in E f f o f Shanin’s Principle [27, Pr oposition 1.7]: every sub ob ject of N is covered by a ¬¬ -closed o ne . F or let Y be a sub ob ject o f X × N in E f f /X . Since every o b ject in the eﬀective top os is cov ered b y an asse m bly , we ma y just as well a ssume that X is an assembly ( X , χ ). The sub ob ject Y ⊆ X × N ca n b e identiﬁed with a function Y : X × N / / P N for which there exists a natural num b er r with the proper t y that for every m ∈ Y ( x, n ), the v alue r ( m ) is deﬁned and co des a pair h k 0 , k 1 i with k 0 ∈ χ ( x ) and k 1 = n . One ca n then form the ass e m bly ( P, π ) with P = { ( x, n ) ∈ X × N : n co des a pair h n 0 , n 1 i with n 1 ∈ Y ( x, n 0 ) } , π ( x, n ) = { h k 0 , k 1 i : k 0 ∈ χ ( x ) and k 1 = n } , 28 which is actually a ¬¬ -clos ed subo b ject of X × N . P covers Y , clearly . The diagram P / / / / # # # # F F F F F F F F Z { { { { x x x x x x x x X × N do es not co mm ute, but comp osing with the pr o jection X × N / / X it do es.  Let T b e the cla ss of maps having an y of the e quiv a len t prop erties in this lemma. Remark 6.2 The morphisms b elonging to T were called “q ua si-modest” in [19] and “discr ete” in [18]. In the la tter the a uthors prove another characterisatio n of T due to F reyd: the morphisms b elonging to T a re those ﬁbre wise orthog onal to the sub ob ject classiﬁer Ω in E f f (Theor em 6.8 in lo c.ci t. ). Prop osition 6.3 [19, Prop osition 5.4 ] The class T is a re pr esentable class of smal l maps in E f f satisfying (M) and (NS) . Pro of. T o s ho w that T is a class of small maps, it is conv enient to re g ard T as D cov (the class of maps covered by elemen ts of D ), where D co ns ists of those ma ps g : Y / / X for which Y is a ¬¬ -closed s ub ob ject of X × N . It is clea r that D s a tisﬁes axioms (A1, A3-5) for a cla ss of display ma ps, a nd (NS) as well (for (A5) , one uses that ther e is a n iso morphism N × N ∼ = N in E f f ). It also satisﬁes ax iom (A7) , b ecause a ll maps g : Y / / X in D are choice maps, i.e., internally pr o jective as elements of E f f /X . The reason is that in E f f the par titioned assemblies are pro jective, and every ob ject is c o v ered by a partitioned assembly . So if X ′ is some pa r titioned assembly cov ering X , then also X ′ × N is a partitioned as sem bly , since N is a pa rtitioned assembly a nd partitioned a ssem blies ar e clo sed under pro ducts. Moreover, Y × X X ′ as a ¬¬ - closed sub ob ject of X ′ × N is also a partitioned ass em bly . F rom this it follows that g is internally pro jective. A representation π for D is obtained via the pullback ∈ N / / / / π   ∈ N   P ¬¬ ( N ) / / / / P ( N ) . F urthermo re, it is ob vious tha t all monomo rphisms b elong to T , since all the ﬁbres of a monic map a re s ubcountable (internally in E f f ). Now it follows that T is a representable class of small maps sa tisfying (M) and (NS) (along the lines o f Pr oposition 2.14 in [5]).  29 Prop osition 6.4 [4] The class T s a tisﬁes (WS) and (F) . Pro of. (Sk e tch.) W e ﬁrst obs erv e that fo r any tw o mor phisms f : Y / / X and g : Z / / X b elonging to D , the exp onen tial ( f g ) X / / X b elongs to T . Without loss o f generality we may a s sume X is a (partitioned) ass e mbly . If Y ⊆ X × N and Z ⊆ X × N a re ¬¬ -closed sub ob jects, then every function h : Y x / / Z x ov er some ﬁxed x ∈ X is deter mined uniquely b y its realizer, and s o all ﬁbres o f ( f g ) X / / X ar e sub countable. T o show the v a lidit y of (F) , it suﬃces to show the existence o f a generic T - display ed mvs s for maps g : B / / A in D , with f : A / / X als o in D (in view of Lemmas 2.1 5 and 6.23 fro m [5]). Because f is a choice map, one can take the ob ject o f all sections of g ov er X , which is sub coun table by the preceding remark. The a rgumen t for the v alidity of (WS) is s imilar. W e use aga in that every comp osable pair o f maps g : B / / A and f : A / / X belo nging to T ﬁt into cov ering squares of the for m B ′ g ′   / / / / B g   A ′ / / f ′   A f   X ′ p / / / / X , with g ′ and f ′ belo nging to D . W e may a lso assume that X ′ is a (par titioned) assembly . The W-type asso ciated to g ′ in E f f /X ′ is sub coun table, b ecause every element o f W ( g ′ ) X ′ in the slic e ov er some ﬁxed x ∈ X ′ is uniquely deter mined by its realizer. The W-type asso ciated to p ∗ g in the slice o ver X ′ is then a sub q uotien t of W ( g ′ ) X ′ (see the pro of of Prop osition 6.16 in [5]), and ther efore also sub countable. Finally , the W-t ype a ssocia ted to g in the slice ov er X is also sub coun table, by des c e n t for T .  W e w ill obtain a mo del of CZF and F ull separa tio n by considering the initial algebra U for the p o w er class functor as sociated to T , which we w ill denote by P t . P t U Int + + U. Ext l l In the prop osition b elo w, we show that it is not a mo del of IZF , for it r efutes the p o w er set a x iom. Prop osition 6.5 The statement t h at al l sets ar e sub c ountable is valid in the mo del U . Ther efor e it r efutes the p ower set ax i om. 30 Pro of. As we explained in the int ro duction, the s tatemen t that all sets are sub- countable follows fro m the fact that, in the internal log ic of the eﬀective top os, all ﬁbr es of maps b elonging to T are subcountable. But the principle that all sets ar e sub count able immediately implies the non-existence of P ω , using Can- tor’s Diago na l Argument. And neither do es P 1 whe n 1 = {∅} is a set cons isting of o nly one element. F or if it would, so would ( P 1) ω , by Subse t Collection. But it is not hard to see that ( P 1 ) ω can be reworked into the p ow er set o f ω .  Prop osition 6.6 The choi c e principles (CC), (RDC), P A) ar e valid in the mo del U . Mor e over, as an obje ct of the eﬀe ctive top os, U is un if orm, and ther efor e the principles (UP) , (UZ) , (IP) and (IP ω ) hold in U as wel l. Pro of. The pro of is very similar to that of Prop osition 5.2. The Axioms of Countable and Rela tivised Dependent Choice U inherits from the eﬀective top os E f f . T o see that in U every set is the sur jectiv e imag e of a pro jective set, notice that every set is the sur jectiv e image of a ¬¬ -close d subset of ω , a nd these are internally pro jective in E f f . T o s ho w that U is uniform it will suﬃce to p oin t out that the repr esen tation can b e chosen to be of a morphism of assemblies with uniform co domain. Then the ar gumen t will pr oceed a s in Prop osition 5.2. In the present case, the r epre- sentation π ca n be chosen to b e o f the form ∈ N / / / / π   ∈ N   P ¬¬ ( N ) / / / / P ( N ) . So ther efore π is a morphism b et w een ass em blies, where P ¬¬ ( N ) = ∇P N , i.e. the set of all subsets A of the natural num bers , with A be ing rea liz ed by 0, say , and ∈ N = { ( n, A ) : n ∈ A } , with ( n, A ) being r e a lized by n . So π is indeed of the desired for m, and U will be uniform. Therefore it v alidates the pr inciples (UP) , (UZ) , (IP) and (IP ω ) .  Remark 6.7 It fo llo ws from results in [25] that the Regular Ex tens io n Axiom from [2] also holds in U . F or in [25], the a uthors prov e that the v alidit y o f the Regular Extens ion Axio m in V follows fro m the axioms (WS) a nd (AMC) for T . (AMC) is the Axiom of Multiple Choice (see [25]), which holds here bec ause every f ∈ T ﬁts into a cov ering square Y / / / / g   B f   X / / / / A, 31 where g : Y / / X is a small choice map, hence a sma ll collection map over X . The mo del U has a ppeared in diﬀerent forms in the litera ture, its ﬁr st ap- pea rance b eing in F riedman’s pap er [1 3]. W e discuss several of its incar nations. W e hav e seen above that for any s tr ongly inaccessible cardinal κ > ω , the eﬀective top os carrie s another class of s mall maps S . F or this c lass o f small maps, the initial P s -algebra V is precisely McCarty’s r ealizabilit y mo del for IZF . It is not ha rd to see that T ⊆ S , and therefore there exists a po in twise monic natural tr a nsformation P t ⇒ P s . This implies that our present mo del U embeds into McCarty’s mo del. P t U / / Int   P t V     P s V Int   U / / / / V Actually , U consists o f those x ∈ V that V b eliev es to be hereditarily sub- countable (in tuitiv ely s peaking, b ecause V and E f f agree on the mea ning of the word “ subcountable”, see the in tro duction). T o see this, write A = { x ∈ V : V | = x is hereditarily sub countable } . A is a P t -subalgebra of V , and it will b e isomor phic to U , once one prov es that is initial. It is obviously a ﬁxed p oin t, so it suﬃces to show that it is well-founded (see [5, Theore m 7 .3]). So let B ⊆ A b e a P t -subalgebra of A , and deﬁne W = { x ∈ V : x ∈ A ⇒ x ∈ B } . It is not hard to see that this is a P s -subalgebra of V , so W = V and A = B . This a lso shows tha t principles like Ch ur c h’s Thesis (CT) and Ma rk ov’s Principle (MP) ar e v a lid in U , since they are v alid in McCarty’s mo del V . One could also unr avel the c onstruction of the initial a lgebra for the power class functor fro m [5] to obtain an e xplicit descriptio n, as we did in Sectio n 5. Combining the explicit descr iption of a repre sen ta tion π in Pro position 6.6 with the observ ation that its a ssocia ted W-type can be computed as in assemblies, one obtains the following descr iption o f W = W π in E f f . The underlying set consists o f well-founded trees wher e the e dg es are lab elled by natural num b ers, in such a wa y that the edges in to a ﬁxed no de are lab elled by distinct natural nu mbers. So a typical element is of the form sup A ( t ), where A is a subse t of N and t is a function A → W . An alternative would be to regar d W as the initial algebra for the functor X 7→ [ N ⇀ X ], where [ N ⇀ X ] is the s e t of partial functions from N to X . The decorations (realizers) of a n element w ∈ W are 32 deﬁned inductively: n is a r ealizer of sup A ( t ), if for every a ∈ A , the express io n n ( a ) is deﬁned and a r ealizer of t ( a ). W e need to quotient W , internally in E f f , by bisimulation: sup A ( t ) ∼ sup A ′ ( t ′ ) ⇔ ( ∀ a ∈ A ) ( ∃ a ′ ∈ A ′ ) [ ta ∼ t ′ a ′ ] a nd vice versa. T o tra ns late this in terms o f rea lizers, we ag ain use an “abbrevia tion”: n  x ǫ sup A ( t ) ⇔ n 0 ∈ A a nd n 1  x ∼ t ( n 0 ) . Then the equiv a lence relation ∼⊆ W × W is deﬁned by: n  sup A ( t ) ∼ sup A ′ ( t ′ ) ⇔ ( ∀ a ∈ A ) [ n 0 ( a ) ↓ and n 0 ( a )  ta ǫ sup A ′ ( t ′ )] and ( ∀ a ′ ∈ A ′ ) [ n 1 ( a ′ ) ↓ and n 1 ( a ′ )  t ′ a ′ ǫ sup A ( t )] . The quotient in E f f is precisely U , which is therefore the pair co nsisting of the underlying s e t of W together with ∼ a s eq ua lit y . The r eader s ho uld verify that the internal membership is a gain given by the “abbr eviation” ab o ve. Corollary 6.8 The fol lowing clauses r e cursively deﬁne what it me ans that a c ertain statemen t is r e alize d by a natura l num b er n in t he mo del U : n  x ǫ sup A ( t ) ⇔ n 0 ∈ A and n 1  x = t ( n 0 ) . n  sup A ( t ) = sup A ′ ( t ′ ) ⇔ ( ∀ a ∈ A ) [ n 0 ( a ) ↓ and n 0 ( a )  ta ǫ sup A ′ ( t ′ )] and ( ∀ a ′ ∈ A ′ ) [ n 1 ( a ′ ) ↓ and n 1 ( a ′ )  t ′ a ′ ǫ sup A ( t )] . n  φ ∧ ψ ⇔ n 0  φ and n 1  ψ . n  φ ∨ ψ ⇔ n = h 0 , m i and m  φ , or n = h 1 , m i and m  ψ . n  φ → ψ ⇔ F or al l m  φ, n · m ↓ and n · m  ψ . n  ¬ φ ⇔ Ther e is no m su ch that m  φ. n  ∃ x φ ( x ) ⇔ n  φ ( a ) for some a ∈ U. n  ∀ x φ ( x ) ⇔ n  φ ( a ) for al l a ∈ U. F ro m this it follows that the mo del is the elementary equiv alent to the one us ed for pro of-theoretic purp oses by Lubarsky in [23]. Remark 6.9 In an unpublished note [32], Streicher builds a mo del of CZF based an earlier work on rea lizabilit y mo dels for the Calculus of Constructio ns. In our terms, his work can be understo o d as follows. He star ts with the mor - phism τ in the ca tegory A sm of a ssem blies, whose co domain is the set of all mo dest sets, w ith a modest set rea lized by a n y natural n umber , and a ﬁbre of this map o ver a mo dest set being prec is ely that mo dest set (no te that this map ag a in has uniform co domain). He pro ceeds to build the W- type a ssocia ted 33 to τ , takes it as a universe of sets, while interpreting equality as bisim ulation. One cannot literally quotient by bisimulation, for which one co uld pass to the eﬀective top os. When consider ing τ as a morphism in the e ﬀective top os, it is not hard to see that it is in fact another re pr esen tation for the class o f subcountable morphisms T : for all ﬁbres of the repres e n tatio n π also o ccur a s ﬁbres of τ , and all ﬁbres of τ are quotients of ﬁbres of π . Therefor e the mo del is a gain the initial P t -algebra for the class of sub coun table morphisms T in the eﬀective top os. A Set-theoretic axioms Set theory is a ﬁrst-or der theor y with o ne non-logica l binar y relation s y m b ol ǫ . Since we are concerne d with co nstructiv e set theories in this pap er, the underlying logic will b e intuitionistic. As is customary also in classical se t theor ies like ZF , we will use the abbre- viations ∃ xǫa ( . . . ) fo r ∃ x ( xǫa ∧ . . . ), and ∀ xǫa ( . . . ) for ∀ x ( xǫa → . . . ). Recall that a for m ula is called b ounde d , when all the qua n tiﬁer s it contains a re of one of these tw o for ms. A.1 Axioms of IZF The axioms of IZF ar e: Extensionality: ∀ x ( xǫa ↔ xǫb ) → a = b . Empt y s et: ∃ x ∀ y ¬ y ǫx . P airing: ∃ x ∀ y ( y ǫx ↔ y = a ∨ y = b ). Union: ∃ x ∀ y ( y ǫx ↔ ∃ z ǫa y ǫz ). Set induction: ∀ x ( ∀ y ǫx φ ( y ) → φ ( x )) → ∀ x φ ( x ). Inﬁnit y: ∃ a ( ∃ x xǫa ) ∧ ( ∀ xǫa ∃ y ǫa xǫy ). F ull separation: ∃ x ∀ y ( y ǫx ↔ y ǫa ∧ φ ( y ) ), for any formula φ in which a do es not o ccur. P o wer set: ∃ x ∀ y ( y ǫx ↔ y ⊆ a ), where y ⊆ a abbreviates ∀ z ( z ǫy → z ǫa ). Strong collection: ∀ xǫa ∃ y φ ( x, y ) → ∃ b B( xǫa, y ǫb ) φ . In the last axio m, the expressio n B( xǫa, y ǫb ) φ. has b e en used a s an abbrevia tion for ∀ xǫa ∃ y ǫb φ ∧ ∀ y ǫb ∃ xǫa φ . 34 A.2 Axioms of CZF The set theory CZF , intro duced b y Aczel in [1], is obtained b y r eplacing F ull separatio n by Bo unded separ ation and the Po wer set a xiom by Subset collection: Bounded separation: ∃ x ∀ y ( y ǫx ↔ y ǫa ∧ φ ( y ) ), for any b ounded formula φ in which a do es not o ccur. Subset col lection: ∃ c ∀ z ( ∀ xǫa ∃ y ǫb φ ( x, y , z ) → ∃ dǫc B( xǫa, y ǫd ) φ ( x, y , z )). A.3 Constructivist principles In this pap er we will meet the following constructivist principles asso ciated to recursive mathematics a nd realizability . In writing these down, we hav e freely used the symbol ω for the set of natural n um ber s, as it is deﬁnable in b oth CZF and IZF . W e also used 0 for zer o a nd s for the succe s sor o peration. Axiom of Countable Choi ce (CC ) ∀ iǫω ∃ x ψ ( i, x ) → ∃ a, f : ω / / a ∀ iǫω ψ ( i, f ( i )) . Axiom of Re lativised Dep enden t Choi ce (R DC) φ ( x 0 ) ∧ ∀ x ( φ ( x ) → ∃ y ( ψ ( x, y ) ∧ φ ( y ))) → ∃ a ∃ f : ω / / a ( f (0) = x 0 ∧ ∀ i ∈ ω φ ( f ( i ) , f ( si ))) . Presen tation Axiom (P A) E v ery set is the s urjectiv e imag e of a pro jective set (wher e a set a is pr o jective, if every sur jection b → a has a sectio n). Mark ov’s Principle (MP) ∀ nǫω [ φ ( n ) ∨ ¬ φ ( n )] → [ ¬¬∃ n ∈ ω φ ( n ) → ∃ nǫω φ ( n )] . Ch urc h’s Thesis (CT) ∀ nǫω ∃ mǫω φ ( n, m ) → ∃ eǫω ∀ nǫω ∃ m, p ǫω [ T ( e, n, p ) ∧ U ( p, m ) ∧ φ ( n, m )] for every formula φ ( u, v ), where T and U are the set-theo r etic predi- cates which numeralwise repres en t, r espectively , Kleene’s T and result- extraction predicate U . Uniformit y Principle (UP) ∀ x ∃ y ǫω φ ( x, y ) → ∃ y ǫω ∀ x φ ( x, y ) . Unzerlegbark ei t (UZ) ∀ x ( φ ( x ) ∨ ¬ φ ( x )) → ∀ x φ ∨ ∀ x ¬ φ. 35 Indep endence of Premisse s for Sets (IP) ( ¬ θ → ∃ x ψ ) → ∃ x ( ¬ θ → ψ ) , where θ is assumed to b e closed. Indep endence of Premisse s for Numbers (IP ω ) ( ¬ θ → ∃ nǫω ψ ) → ∃ nǫω ( ¬ θ → ψ ) , where θ is assumed to b e closed. B Predicativ e categories with small maps In the pres en t pap er, the a m bient categor y E is always a ssumed to b e a p ositive Heyting c ate gory . Tha t mea ns that E is (i) ca rtesian, i.e., it has ﬁnite limits. (ii) re gular, i.e., morphisms factor in a stable fas hion a s a cover follow ed by a monomorphism. (iii) p ositive, i.e., it has ﬁnite sums , which are disjoint and sta ble. (iv) Heyting, i.e., fo r any mo rphism f : Y / / X the induced pullbac k functor f ∗ : Sub( X ) / / Sub( Y ) has a right a djoin t ∀ f . Deﬁnition B.1 A dia gram in E of the form D f   / / C g   B p / / A is ca lled a quasi-pul lb ack , when the ca nonical ma p D / / B × A C is a cov er. If p is also a cover, the diagram will b e called a c overing squar e . When f and g ﬁt into a cov ering squa re as shown, we s a y that f c overs g , or that g is c over e d by f . A class of maps in E satisfying the following axioms (A1-9) will b e called a class of smal l maps : (A1) (Pullback stability) In any pullback squar e D g   / / B f   C p / / A where f ∈ S , a lso g ∈ S . 36 (A2) (Descent) If in a pullback squa r e as above p is a cov er a nd g ∈ S , then also f ∈ S . (A3) (Sums) Whenever X / / Y and X ′ / / Y ′ belo ng to S , so do es X + X ′ / / Y + Y ′ . (A4) (Finiteness) The ma ps 0 / / 1 , 1 / / 1 and 1 + 1 / / 1 be lo ng to S . (A5) (Comp osition) S is closed under comp osition. (A6) (Quotients) In a co mmuting tria ngle Z h @ @ @ @ @ @ @ f / / / / Y g ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ X , if f is a cov er a nd h b elongs to S , then so do es g . (A7) (Collection) Any tw o arr o ws p : Y / / X a nd f : X / / A where p is a cov er and f belo ngs to S ﬁt into a cov ering squa re Z g   / / Y p / / / / X f   B h / / / / A, where g b elongs to S . (A8) (Heyting) F or any mor phism f : Y / / X b elonging to S , the right a djoin t ∀ f : Sub( Y ) / / Sub( X ) sends small monos to small mo no s. (A9) (Diagona ls) All diag onals ∆ X : X / / X × X be lo ng to S . In ca se S satisﬁes a ll these axioms, the pair ( E , S ) will b e called a c ate gory with smal l maps . Axioms (A4,5,8,9) express tha t the sub categories S X of E /X whose ob jects and arrows a re b oth given by a rrows b elonging to the class S , a re full sub categories of E /X which are closed under all the op erations of a p ositive Heyting category . Moreover, these catego ries together should form a stack on E with resp ect to the ﬁnite cov er top ology accor ding to the Axioms (A1-3) . Finally , the class S sho uld satisfy the Quotient axiom (A6) (saying that if a comp osition C / / / / B / / A 37 belo ngs to S , so do es B / / A ), a nd the Collection Axiom (A7) . This axiom states that, conv ersely , if B / / A b elongs to S and C / / / / B is a cov er (regular e pimo rphism), then lo cally in A this cov er has a small reﬁne- men t. The following weak ening of a class of small maps will play a rˆ ole as well: a class of maps satisfying the a x ioms (A1), (A3-5), (A7-9 ) , and (A10) (Images) If in a commuting triangle Z f @ @ @ @ @ @ @ e / / / / Y ~ ~ m ~ ~ ~ ~ ~ ~ ~ ~ ~ X , e is a cov er, m is monic, a nd f belongs to S , then m also belo ngs to S . will b e a ca lle d a class of display maps . Whenever a clas s o f small maps (res p. a cla ss of display maps) S has been ﬁxed, an ob ject X will b e called small (resp. displa yed), whenever the unique map fr om X to the terminal ob ject is small (res p. a display map). In this pap er, we will see the following additional axio ms for a class of small (or display) maps. (M) All monomorphis ms be long to S . (PE) F o r any ob ject X the p ow er class ob ject P s X exists. (PS) Moreover, for any map f : Y / / X ∈ S , the p o wer class ob ject P X s ( f ) / / X in E /X be lo ngs to S . ( Π E) All morphisms f ∈ S are exp onen tiable. ( Π S) F o r any map f : Y / / X ∈ S , a functor Π f : E / Y / / E /X right a djo int to pullback exists a nd preserves morphisms in S . (WE) F or all f : X / / Y ∈ S , the W-type W f asso ciated to f exists. (WS) Mor e o v er, if Y is small, a lso W f is small. (NE) E has a natural num bers o b ject N . (NS) More o v er, N / / 1 ∈ S . 38 (F) F o r any φ : B / / A ∈ S ov er some X with A / / X ∈ S , there is a co ver q : X ′ / / X and a map y : Y / / X ′ belo nging to S , to g ether with a dis- play e d mvs P of φ over Y , with the following “gener ic” prop erty: if z : Z / / X ′ is any ma p a nd Q any display ed mvs o f φ over Z , then ther e is a ma p k : U / / Y and a cover l : U / / Z with y k = z l , such that k ∗ P ≤ l ∗ Q as (displayed) mvs s o f φ ov er U . More details are to b e found in [5]. A category with small maps ( E , S ) will b e called a pr e dic ative class with smal l maps , if S satisﬁes the axio ms ( Π E), (WE), (NS) and in addition: (Representa bility) The cla ss S is repr esen table, in the sense that there is a small map π : E / / U (a r epr esentation ) of whic h any o ther small map f : Y / / X is lo cally (in X ) a quotient of a pullback. Mor e explicitly: any f : Y / / X ∈ S ﬁts into a diagr a m of the for m Y f   B   / / o o o o E π   X A / / o o o o U, where the left ha nd s quare is co vering and the right hand square is a pullback. (Exactness) F o r any equiv alence r e lation R / / / / X × X given by a small mono, a stable q uotien t X/ R exists in E . References [1] P . Aczel. The t yp e theoretic int erpretation of constructive set theory . In L o gic Co l lo quium ’77 (Pr o c. Conf., Wr o c law, 1977) , volume 96 of Stud. L o gic F oundations Math. , pages 55 –66. No r th-Holland, Amsterdam, 19 7 8. [2] P . Aczel and M. Rathjen. 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Aspects of Predicative Algebraic Set Theory II: Realizability

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