Regular Functors and Relative Realizability Categories

Regular F unctors and Rela tiv e Realizabilit y Categories W outer Pieter Stek ele n burg Departmen t of Mathematics Utrec h t Univ ersit y No v em ber 21, 2018 Abstract The r elative r e alizability top oses that Awodey , Birkedal and Scott intro- duced in [1] satisf y a universal property that i nv olves re gular functors to other categories. W e u se this universal prop erty to define what relative realizability categories are, when based on other categories than of the top os of sets. This pap er ex plains the prop erty and giv es a construction for relative realizabilit y categories that works for arbitrary base H eyting categories. The u n iversa l prop erty shows us some new geometric morphisms to relative realizabilit y top oses t o o. 1 In tro duction This pap er concer ns the re lative r e alizab ility top oses that Awodey , Bir kedal a nd Scott intro duced in [1]. Just like realiza bility topos e s , rela tive rea liz a bility topos e s implicitly assign subsets of a partial com binatory algebra to the prop ositions of their int ernal la nguages. The mem b ers of these subsets are said to r e alize the prop ositio ns they a re assigned to. While realiz a bility topose s s atisfy every prop osition that has an inhabited set o f realizer s, rela tive realizability top os es only satisfy pro po sitions whose se t of realizers intersect a suitable subset of the partial combinatory alge br a. Relative realizability to po ses hav e a universal prop er t y that dictates the b e- havior of regular functors into other ca tegories . Using this universal prop erty we develop relative rea lizability categor ies for or der p artial c ombinatory algebr as (see subsection 2.1) that live in arbitr ary Heyting categor ies. W e also cons ider new rela- tive realizability top os es and ge o metric mor phisms that result fro m the gene r alized construction. 1.1 Realizabilit y top oses This pa p er builds on the following resear ch in realiza bility and category theo ry . The use o f top o ses to study realiz a bility started with Hyla nds effe ctive t op os [1 4]. The cons truction o f this top os is easily gene r alizes to other pa rtial combinatory algebras using trip os the ory (see [21], [15]), a nd this is wher e realiza bility to po ses come from. In his thesis [17] Longle y defined appli c ative morphisms b etw een pa r tial com- binatory alg e bras and prov ed a n equiv alence be t ween these morphisms a nd certain regular functors b etw een r ealizability topo ses. He also show ed that realiza bility top oses s atisfy a universal prop er ty rela tive to one o f their small sub categor ies (the category o f mo dest sets ). 1 Carb oni, F reyd and Scedr ov [8] show ed tha t every r ealizability top os is a n exact c ompletion (definition in subsection 3.3) of its regular subca tegory of ¬¬ -sepa rated ob jects, a lso ca lled assemblies . Menni derived conditions that make an exact com- pletion a top os [19], [2 0]. In [12] v an Oosten a nd Hofstra introduce o rder par tial combinatory algebra s, and the realizability top os es related to them. They gener alize Longle y ’s applicative morphisms and c hara cterize the a pplicative morphisms that corr esp ond to geo metr ic morphisms b etw een top os es. W e see tw o g eneralizatio ns of r ealizability coming together in re la tive re a liz- ability , namely , a more flexible definition of v a lidity , and the idea of developing realizability in a non classical co nt ext. Kleene and V esle y prop ose d an early example of r e la tive realiza bilit y in [16]. The partia l combinatory algebra is Kle ene’s se c ond mo del (example 9 below), whose mem b ers a r e functions N → N ; howev er, only prop o sitions realized by total recurs ive function are v alid. The r elative r ealizability of Kleene and V esley a lready ha d an int uitionist co ntext, i.e., it considered how to prov e the rela tive realiza bility of prop ositions co nstructively . Aw o dey , Birkedal and Sco tt introduced topose s for this v aria tion of rea lizability in [1] and Ba uer and Birkedal s tudied the abs tract prop er ties of relative rea liz ability top oses in their Ph. D. theses [3 ], [2]. In [4] v an Oosten and Birkedal describ ed relative realiz a bility as realiz a bility ov er a internal par tial combinatory alg ebra in another to po s. 1.2 In this pap er Next se ction gener alizes the c o nstruction of the category of a ssemblies so that it works for order partial co mbin ator y algebras that live in a rbitrar y Heyting cate- gories. This is a ca tegorica l wa y to develop rela tive rea lizability in non predicative constructive co ntexts. Ho w ever, we use a universal prop erty to define these cate- gories, a nd introduce the construction as a c o nstructive existence pro of. Section 3 shows that the e x act completion of a ca teg ory of assemblies constr ucted for an order partial combinatory a lg ebra in a to p o s, is a top os . It shows, in other words, that r elative realizability top oses are to p o ses, even if we work with another base top o s than the to p o s of sets. W e use the univ ersal prop erty to find r egular functors form relative realiza bility top oses into other categories in section 4. In particular , we lo ok a t geometric morphisms from lo calic top o ses and fro m other rea lizability top oses into realiza bility top oses. 2 Relativ e Realizabilit y Cat egories This sectio n defines r elative realiza bilit y categ ories b y a universal prop erty , a nd then prov es the existence o f ca tegories that satisfy this pro p er ty . It star ts b y introducing the structure of the ob ject of realizer s, and ends b y considering the adv a ntages of pro jective termina l o b jects. Though we are mainly interested in the developmen t of realizability in top oses , the fact that top o ses hav e power ob jects do es not play an ess e nt ial ro le in real- izability . W e therefor e develop relative r e alizability in the larg er class of H eyting c ate gories . A Heyting categ ory is a categor y tha t has first or der in tuitionistic logic as its internal langua ge. Spec ific a lly , E is a Heyting catego r y if for every o b ject X the cla ss Sub ( X ) of sub ob jects of X is a Heyting algebra a nd for every arr ow f : X → Y the inv erse image map f − 1 : Sub ( Y ) → Sub ( X ) has bo th adjoints 2 ∃ f ⊣ f − 1 ⊣ ∀ f . W e will often use the internal langua ge to define ob jects of Heyting categorie s. 2.1 OPCA pairs In this subsection we will define or der e d p artial c ombinatory algebr as as co mbinatory complete ordered par tial applica tive structures . Definition 1. An or der e d p artial applic ative str u ctur e o r OP AS is an ob ject with a n ordering ≤ and a monotone partia l bina r y op er ator ( x, y ) 7→ xy ca lled applic ation , whose do ma in is downw ard closed. If x , y and z ar e e le ment s o f a n OP AS, w e write xy ↓ z for : ‘the applica tion of x to y is defined and is equa l to z ’. The formula xy ↓ means that there is a z such that xy ↓ z , i.e, that ( x, y ) is in the domain o f the application o p e rator. W e single out certain partial monotone a rrows o f OP ASes. Definition 2. F or each OP AS A , a ∈ A , n ∈ N , U ⊆ A n and f : U → A , we say that a r epr esents or r e alizes f or that f is r epr esentable , if for all ~ x ∈ dom f , ther e is a y ≤ f ( ~ x ) such that (( ax 1 ) . . . ) x n ↓ y . W e call such a r rows p artial r epr esentable arr ows . Remark 3 . W e interpret this definition in the internal lang uage o f the Heyting category . So rela tive to an O P AS A a partial mo rphism f : U ⊆ A n ⇀ A is representable if and only if the following subo b ject of A is inhabited. J f K = { a ∈ A | ∀ ~ x ∈ U. ∃ y ∈ A.y ≤ f ( ~ x ) ∧ (( ax 1 ) . . . ) x n ↓ y } This obje ct of r e alizers of f may not hav e any global sectio n. W e a re interested in OP ASes that repres ent all partial arrows that are co n- structed by r ep eated use of application. Definition 4. The s et o f p artial c ombinatory arr ows is the least set of par tial arbitrar y-ary arrows A n ⇀ A that contains pro jections ~ x 7→ x i and is closed under po int w is e application. So ( x, y ) 7→ x and ( x, y , z ) 7→ xz ( y z ) a re bo th examples of partial co mbinatory arrows. An OP AS is c ombinatory c omplete , if every partial combinatory arr ow is representable. C o mbinatory complete OP ASes are called or- der e d p art ial c ombinatory algebr as or OPCAs [12]. Partial c ombinatory algebr as or PCAs a r e OPCAs that have the discrete or de r ing. Although combinatory co mpletenes s uses universal quantification in its defini- tion, ther e is a way to for ma lize this pr op erty using only regula r logic. Lemma 5. Ther e is a r e gular the ory whose mo dels ar e OPCAs. Pr o of. W e easily transla te our own definitio n of OP ASes in a regular theory . W e use a binary r elation ≤ and one terna r y r e la tion α , but wr ite xy ↓ z instead of α ( x, y , z ). ⊢ a ≤ a a ≤ b, c ≤ a ⊢ c ≤ b a ≤ b, b ≤ a ⊢ a = b ab ↓ c ∧ ab ↓ d ⊢ c = d a ≤ b, c ≤ d, bd ↓ e ⊢ ∃ f .ac ↓ f ∧ f ≤ e W e expr e ss that the OP AS r e presents a partial combinatory function f : A n ⇀ A by extending this theo ry as follows. W e add a predica te F to our language, and an 3 axiom that say it is inhabited: ⊢ ∃ x.F ( x ). W e add a list of a xioms to say that if F ( a ), then (( ax 1 ) . . . ) x n ↓ y for some y ≤ f ( ~ x ): F ( a ) ⊢ ∃ y 1 .ax 1 ↓ y 1 F ( a ) , ax 1 ↓ y 1 ⊢ ∃ y 2 .y 1 x 2 ↓ y 2 . . . F ( a ) , ax 1 ↓ y 1 , y 1 x 2 ↓ y 2 , · · · ⊢ ∃ y n .y n ≤ f ( ~ x ) ∧ y n − 1 x n ↓ y n W e ca n do this for each partial combinatory function a nd get a recursively enum erable theo r y; we c an also use the k , s -basis of combinatory lo gic (see [9 ]) to get an eq uiv alent finitely axiomatized r egular theory . Either wa y , a mo del A for these axioms is a n O P AS that repres ent s all partia l combinatory functions and therefore an OPC A. Remark 6. Every O PCA is a mo del for this theor y , but not alwa ys in a unique wa y . F or each partial combin ator y f we may int erpreted the related pr edicate F as any inhabited sub ob ject o f J f K . Corollary 7. Re gular functors pr eserve OPCAs. OPCAs are mo dels for co mputation. W e can view an OP C A A as the set of co des for progr ams in a functiona l pro gramming language. The applicatio n op er- ator r epresents the execution of one pro gram o n the co de of another. F or rela tive realizability we wan t to apply a limited set of progr ams to a la rger set of co des. This le a d to the following gener alization. Definition 8. An O P CA pair ( A ′ , A ) is a pair of O P ASes, where • A ′ is a sub o b ject of A , a nd application of A ′ is the res tr iction of a pplication in A • A ′ is clo sed under the application in A . So if x, y ∈ A ′ and there is a z ∈ A such tha t xy ↓ z , then z ∈ A ′ . • All partial co mbinatory a rrows of A ar e repr esentable in A ′ . So if f : U ⊆ A n → A is combinatory , then J f K intersects A ′ . Note that if ( A ′ , A ) is an OPCA pair b oth A ′ and A a re OPCAs themselves. Also, the la st c o ndition is equiv a lent to the conditio n that the s ets o f rea lizers fo r the partial co mb inatory arrows ( x, y ) 7→ x and ( x, y , z ) 7→ ( xz )( y z ) int ersect A ′ , for reasons o utlined [9]. Finally , if A is an OPCA, then ( A, A ) is an O PCA pair. F or this r eason ‘abso lute’ realiz a bility is a sp e cial cas e of r elative realizability . Example 9 (Kleene’s second mo del) . There is a universal partial co ntin uous func- tion N N × N N → N N for the pro duct top ology o n N N . With this function N N is a PCA K 2 . The total recurs ive functions fo r m a subPCA K rec 2 and ( K rec 2 , K 2 ) is an OPCA pair . As we will see form the definition in the next par agra ph, this OPCA pair exis ts in every topo s with natura l num b er o b ject. A partial contin uous function φ : N N ⇀ N has an i ∈ N for ea ch x ∈ do m φ such that f ( x ) = f ( x ′ ) whenever x ′ ∈ N N and x j = x ′ j for a ll j < i ; i.e., these c o ntin uo us functionals only us e a initial seg ment to deter mine their o utput. F or m a bijection β betw een N and the set o f finite sequences of natur al num bers N ∗ , we can constr uc t a surjection N N to the set of par tial contin uous functions N N ⇀ N . F or each y ∈ N N we let φ y ( x ) = k if y ( β − 1 ( x 0 , . . . , x j )) = k + 1 for the leas t j ∈ N such that y ( β − 1 ( x 0 , . . . , x j )) > 0, a nd o therwise undefined. Using the bijection N N ≃ N N × N , we a lso get a s urjection N N to N N ⇀ N N , a un iversal p artial c ontinuous function . When the bijection β b etw een N and N ∗ is a recursive function, K rec 2 is close d under application, a nd it represents all pa rtial co mb inatory functions . 4 Example 10 (OPCAs of downsets) . If A is an O PCA pair, let ∂ A b e the set o f downsets , i.e., down w ard closed s ubo b jects, of A . Inclusions o rder ∂ A , and ∂ A has a par tial application oper a tor that s a tisfies U V ↓ W if for all x ∈ U and y ∈ V , xy ↓ and if for all z ∈ W there are x ∈ U and y ∈ V suc h that xy ↓ z . In fact ∂ A is a new OPCA. This construction motiv a tes the generaliza tion from PCAs to OPCAs. Like Kleene’s seco nd mo del, this co nstruction is av ailable in any top os. Example 11. Another constructio n o f OP CAs us e s the reg ular functor s Set /I → Set that come fro m filter quotient c onstructions for filters on the set I . Because regular functors preserve OP CAs, filter quotients o f I -indexed families o f OPCAs are O PCAs. This construction do es not generalize ea sily to other to p o ses. 2.2 Regular Mo dels The co nstruction of the re alizability top o ses solves the following pr oblem. F or each Heyting categor y E and each OPCA pair ( A ′ , A ) in E , we would lik e to cons tr uct a slightly large r categor y E [ ˚ A ], where ˚ A is a subOPCA of A , that is closed under all partial o p erator s A n ⇀ A that are realized by members of A ′ , but not closed under other par tial op erator s. W e a ppr oach this pro blem with tw o dimensional categor y theory . A pseudoinitial obje ct in a 2-categ ory is a n ob ject for which ther e is a n up to isomo rphism unique a rrow to every other ob ject. W e co nstruct a 2- c ategory of suita ble functors F : E → C a nd sub ob jects C ⊆ F A , such that a pseudoinitial ob ject in the 2- categor y should b e like E [ ˚ A ]. Definition 12. Let E b e a Heyting categ ory , ( A ′ , A ) an O P CA pair in E , C a r egular category and F : E → C a r egular functor. An F -fi lt er is a sub ob ject C ≤ F A that satisfies: • If x ∈ C a nd x ≤ y , then y ∈ C . • If x, y ∈ C and xy ↓ z for so me z ∈ F A , then z ∈ C . • If U ⊆ A intersects A ′ , then F U intersects C . Regular functor pr eserve O P AS a nd filters, b ecause their definition inv olves only commutativ e diagr ams, pullbacks a nd ima ges. This also means that for each pair of r egular functor s F : E → C a nd G : C → D and each F -filter C the ob ject GC is a GF - filter. Definition 13. Let a r e gular mo del for ( A ′ , A ) b e a regula r functor F : E → C with a n F -filter. F or ea ch regula r G : E → D , each F - filter C a nd each G -filter D a morphism ( F , C ) → ( G : E → D , D ) is a regular functor H : C → D with a n isomorphism η : H F → G , such that η A : H F A → GA r estricts to an is omorphism betw een F C and D . A r e gu lar r elative r e alizability c ate gory for the pair ( A ′ , A ) is a pseudoinitial regula r mo del i.e.: there is an up to isomor phism unique re gular functor from a regular relative r ealizability category to any regular mo del. Remark 14. F or each regula r mo del ( F, C ), n ∈ N and U ⊆ A n the set of partial arrows F U ∩ C n → C contains the ima g es o f partial A ′ -representable arrows U → A . In that sense it is a mo del of the r egular theor y of a subs e t o f A that is closed under a set of pa rtial o p er ators. Theorem 15. Ther e is a pseudoinitial r e gular m o del for every OPCA p air in every Heyting c ate gory. In the next couple o f subsections, we define a ca tegory , a functor a nd a filter, and prove that these form a pseudoinitial reg ular mo del. 5 2.3 Assem blies This subsection explains the constructio n and some pro p e rties o f the c ate gory of assemblies Asm ( A ′ , A ) for an OP C A pair ( A ′ , A ) in so me Heyting categ ory E , the construction o f a functor ∇ : E → Asm ( A ′ , A ) and a ∇ -filter ˚ A that form a pseu- doinitial reg ular mo del together . Definition 16. An assembly is a pair ( X , Y ) where X ∈ E and where Y is a sub o b ject of A × X , such tha t • for all x ∈ X there is an a ∈ A such that ( a, x ) ∈ Y ; • if ( a, x ) ∈ Y and b ≤ a , then ( b, x ) ∈ Y . F or ea ch pair of a s semblies ( X , Y ) and ( X ′ , Y ′ ), a nd each f : X → X ′ , a V ⊆ A tr acks or r e alizes f if for all a ∈ V and b ∈ A , if ( b, x ) ∈ Y then ab ↓ and ( ab, f ( x )) ∈ Y ′ . A morphism ( X , Y ) → ( X ′ , Y ′ ) is an arrow f : X → X ′ for which there exists a sub ob ject V of A that intersects A ′ and tha t tra cks f . W e summarize this by saying the following diagra m m ust commute. V × Y ( v, y ) 7→ y   ( v, a,x ) 7→ ( v a,f ( x )) # # ● ● ● ● ● ● ● ● ● Y ( a,x ) 7→ x   Y ′ ( a,x ) 7→ x   X f / / X ′ Remark 17. When developing rea liz ability in a top o s , we can pres ent assemblies as P A -v alued functions. W e work with binar y relations instead, so that w e can apply to constr uction to Heyting c a tegories where P A do es no t exist. W e will prove that the ca tegory of assemblies is a Heyting categ ory , after we int ro duce some extr a struc tur e that will help us to do so . Remark 18. Note that if V tracks a morphis m, then s o doe s ↓ V = { a ∈ A | ∃ v ∈ V .a ≤ v } . Lemma 19 . Assemblies and morphisms form a c ate gory. Pr o of. Let I = J x 7→ x K . This combinator in tersects A ′ and tr a cks id X : ( X , Y ) → ( X, Y ). Let B = J ( x, y , z ) 7→ x ( y z ) K . If U and V in tersect A ′ , U tracks f : ( X , Y ) → ( X ′ , Y ′ ) and V tracks g : ( X ′ , Y ′ ) → ( X ′′ , Y ′′ ), then B V U tracks g ◦ f . Definition 20 . W e denote the c ate gory of assemblies by Asm ( A ′ , A ). Remark 21. Our definition of the categor y assemblies is c o mplicated, but equiv- alent to the conv en tional definition (see [2 4]) in the in ternal lang uage of a top os. If ( X, Y ) is an a ssembly , then the pro jection π 1 : Y → X is a family of inhabited downsets of X , which ca n b e represented as an inhabited downset v alued morphism X → P A . Our definition of morphism lets the underlying Heyting category b elieve there is an element of A ′ that tra cks it. Remark 22. Let D( X , Y ) = X . D : Asm ( A ′ , A ) → E is a faithful functor. F or an OPCA pair ( A ′ , A ) in the categor y of sets this functor is not is omorphic to the global sections functor unless A ′ = A . F or that reason we use the D of domain rather than the Γ o f glob al se ction to symbolize this functor . This catego ry has quite a bit more s tructure then just any regular categ ory . 6 Lemma 23 . The c ate gory of assemblies is a Heyting c ate gory. Pr o of. W e start with finite limits. If 1 is termina l, then for ea ch a s sembly ( X , Y ) the unique map ! : X → 1 is a morphism ( X , Y ) → ( 1 , A × 1 ). T o help constr uct pullbacks, let T = J ( x, y ) 7→ x K F = J ( x, y ) 7→ y K P = J ( x, y , z ) 7→ z xy K Given f : ( X, F ) → ( Z , H ) and g : ( Y , G ) → ( Z, H ) let p : W → X and q : W → Y be a pullback cone for f and g in E . Then let K =  ( a, w ) ∈ A × W     ∀ t ∈ T .at ↓ , ( at, pw ) ∈ F , ∀ f ∈ F .af ↓ , ( af , q w ) ∈ G  Let H = J ( x, y ) 7→ y x K , then HT tracks p : ( W, K ) → ( X , F ) and HF tracks q : ( W , K ) → ( Y , F ). Therefore p and q form a commutativ e square with f a nd g in the categor y of assemblies. If L tr acks r : ξ → ( X , F ) and M tra cks s : ξ → ( Y , G ) for an y other assembly ξ , let N = J ( p, x, y , z ) 7→ ( p ( xz )( y z ) K . There ex ists a unique factorization ( r, s ) : D ξ → W thro ugh p and q and N P LM tracks ( r, s ). W e se e bo th that Asm ( A ′ , A ) ha s all finite limits and that D pr eserves them. Next: ima g es. Giv en f : ( X , Y ) → ( X ′ , Y ′ ) let ∃ f ( X, Y ) = ( ∃ f ( X ) , ∃ 1 × f ( Y )). By definition D( ∃ f ( X ) , ∃ 1 × f ( Y )) = ∃ f (D( X, Y )), so I tra cks f : ( X , Y ) → ∃ f ( X, Y ). If p, q : ξ → ( X , Y ) is a kernel pair for f in Asm ( A ′ , A ) then it is a kernel pair for f in E , b ecause D preser ves finite limits. If V tracks so me g : ( X , Y ) → ψ that satisfies g ◦ p = g ◦ q , then it also tracks the factorizatio n of g thro ugh the image of f . Hence ∃ f ( X, Y ) is a co equalizer for the kernel pair. W e need to show that reg ular epimo rphisms are stable. An epimor phism e : ( X, Y ) → ( X ′ , Y ′ ) is r egular, if ∃ e ( X, Y ) ≃ ( X ′ , Y ′ ). Ther efore, we can a ssume that ( X ′ , Y ′ ) = ( ∃ f ( X ) , ∃ id × e ( Y )) w itho ut loss o f generality . Since D(( ∃ f ( X ) , ∃ id × e ( Y )) = ∃ e (D( X, Y )), the functor D preserves reg ular epimorphisms a nd is itself regular. F or any f : ( Z , H ) → ( X ′ , Y ′ ), let p : ( W, K ) → ( X , Y ), q : ( W, K ) → ( Z, H ) be a pullback co ne for e and f like the one we constructed ab ov e. The arr ow q is a r egular epimo r phism in E , b ecaus e D preser ves pullbacks and E is a r egular category . F ur ther more, I tracks e , HT tracks p and HF tracks q . If V tracks f , then J ( p, v , x ) 7→ px ( v x ) K P V tracks id Z : ( Z, H ) → ∃ q ( W , K ), while HF tracks id Z : ∃ q ( W , K ) → ( Z , H ). So ∃ q ( W , K ) ≃ ( Z , H ) and q is a regular epimorphism in Asm ( A ′ , A ). So pullbacks of r egular epimor phism ar e regula r epimorphisms. W e see that Asm ( A ′ , A ) is a regular categor y a nd tha t D : Asm ( A ′ , A ) → E is a regular functor. W e construct joins of sub o b jects to show that all subob ject p osets are lattices. First we s how how to repres ent a sub ob ject of an assembly ( X ′ , Y ′ ) as a sub ob ject F ⊆ A × X . A mor phisms m : X → Y is monic if and o nly if ∃ m ( X ) ≃ X . Therefo r e, given a mono m : ( X , Y ) → ( X ′ , Y ′ ) in Asm ( A ′ , A ) we hav e ( X , Y ) ≃ ( ∃ m ( X ) , ∃ id × m ( Y )). Let Z = { x ∈ X | ∃ a ∈ A. ( a, x ) ∈ Z } fo r all F ⊆ A × X . F or each sub ob ject U of ( X, Y ) there is an F ⊆ A × X s uch that id F : ( F , F ) → ( X , Y ) is tr ack ed a nd such that id F : ( F , F ) → ( X, Y ) repr esents U . F or all X ∈ E and all F, G ≤ A × X , say that U tracks F ≤ G if it tracks id F : ( F , F ) → ( G, G ). On to joins. F or any pair F , G ≤ A × X , let F ∨ G =    ( a, x ) ∈ A × X       ∃ b ∈ A, p ∈ P , t ∈ T . ( b, x ) ∈ F , a ≤ ptb ∨ ∃ b ∈ A, p ∈ P , f ∈ F . ( b, x ) ∈ G, a ≤ pf b    Now PT tracks Y ≤ Y ∨ Y ′ and PF tra cks Y ≤ Y ∨ Y ′ , ther efore Y ∨ Y ′ is an upp er b o und of { Y , Y ′ } . If U trac ks Y ≤ Z a nd U ′ tracks Y ′ ≤ Z , then 7 J ( t, f , u, u ′ , a ) 7→ at [ u ( af )][ u ′ ( af )] K TF U U ′ tracks Y ∨ Y ′ ≤ Z . Therefor e Y ∨ Y ′ is the lea st upp er b ound. There is only one ass embly ( ⊥ , ⊥ ) over the initial ob ject ⊥ of E , and it is embedded in every other assembly . This is the bo ttom element of the p oset of sub o b jects over every assembly , which p o set we can now call a lattic e o f sub o b jects. W e now constr uct r ight a djoints to the inv erse ima ge maps. F or each f : ( X, Y ) → ( X ′ , Y ′ ) and F ≤ Y let: ∀ f ( F ) = { ( a, y ) ∈ A × X ′ | ∀ ( b , x ) ∈ Y .f ( x ) = y → ab ↓ ∧ ( ab, x ) ∈ F } Pullbacks induce the inv erse ima g e map. Therefor e , if G ≤ A × X ′ represents a sub o b ject of ( X ′ , Y ′ ), then the following ob ject represe nt s its inv erse image. f − 1 ( G ) =  ( a, x ) ∈ A × X     ∀ t ∈ T .a t ↓ , ( at, x ) ∈ Y , ∀ f ′ ∈ F .af ↓ , ( af ′ , f ( x )) ∈ G  If U tra cks G ≤ ∀ f ( F ), let h ( t, f , u, x ) = u ( xt )( xf ) then J h K TF U tracks f − 1 ( G ) ≤ F . If V tracks f − 1 ( G ) ≤ F , then J ( p, v , x, y ) 7→ v ( pxy ) K P V trac ks G ≤ ∀ f ( F ). W e see that ∀ f is right adjoint to f − 1 . W e now hav e shown that Asm ( A ′ , A ) is r egular, that sub ob jects form a la t- tice and that inv erse ima g e maps have b oth left a n rig ht adjoints. W e construct Heyting implications form thes e right a djo ints: if m : ( X, Y ) → ( X ′ , Y ′ ) re p- resent a sub ob ject U of ( X ′ , Y ′ ), and V is another sub ob ject of ( X ′ , Y ′ ), then U → V = ∀ m m − 1 ( V ). W e conclude tha t la ttices of sub ob jects a re Heyting algebra s and that the inv erse image maps have b oth left and rig ht adjoints. Ther efore Asm ( A ′ , A ) is a Heyting category . On to the functor. Definition 24. F or each o b ject X in A let ∇ X = ( X, A × X ). F or each ar row f : X → Y , let ∇ f = f . The arr ow ∇ f is a mor phism ∇ X → ∇ Y b ecause D ∇ = id E and id A × f : A × X → A × Y . The functor D is a faithful Asm ( A ′ , A ) → E and ∇ is a right inverse. In fact ∇ is rig ht adjoint to D, b ecause I tra cks the inclusio n id X : ( X , Y ) → ∇ D( X , Y ) for every assembly ( X , Y ). Lemma 25 . The functor ∇ is re gular. Pr o of. In regula r categor ies e : X → Y is a r egular e pimorphism if and only if ∃ e ( X ) ≃ Y . So let e ∈ E b e a reg ular epimorphism. Ima ges lift to Asm ( A ′ , A ), a nd ∃ ∇ e ( ∇ X ) = ( Y , ∃ id A × e ( A × X )) ≃ ∇ Y Therefore ∇ preserves regular epimorphis ms . Since ∇ is right adjoint to D, it als o preserves all limits. That ma kes it a regular functor . W e hav e a ca tegory and w e ha ve a functor. Now we need a filter, which is some sub o b ject of ∇ A . Lemma 26 . L et { ≤ } b e  ( x, y ) ∈ A 2   x ≤ y  and let ˚ A = ( A, { ≤ } ) . The iden- tity map id A : ˚ A → ∇ A is a monomorphism that r epr esent s a fi lter on ˚ A . Pr o of. Tha t id A is a morphism follows fro m the fact that id A × id A : { ≤ } → A 2 is just the inclusion. If f , g : ( X, Y ) → ˚ A sa tisfy id A ◦ f = id A ◦ g then f = g , so id A is a monomo r phism, and monomorphisms r e present sub ob jects. 8 Because ∇ is regula r ∇ { ≤ } is a partial order ing o f ∇ A . Relative to this order ing ˚ A is a n upw ard clo sed subo b ject. The order { ≤ } has tw o pro jections { ≤ } → A . By pulling ˚ A back alo ng the fir st pr o jection we get the ob ject o f pair s of element of ∇ A , where the first is some element ˚ A and the second is a grea ter element of ∇ A . The sub ob ject I tra cks the s e c ond pr o jection of this pullback to ˚ A . This shows ˚ A is upw ard clo sed under the o r dering ∇ { ≤ } . If U is a sub ob ject o f A that intersects A ′ , then ˚ A intersects ∇ U . This mea ns the the supp ort o f the pullba ck of the inclus io ns of ˚ A and ∇ U is a terminal ob ject. Using the co nstructions in the pro of of lemma 23 we find that the assembly ( 1 , ↓ U ) represents this supp or t. Let K = J ( x, y ) 7→ x K . The unique arr ow id 1 : 1 → 1 is a mo rphism ( 1 , A ) → ( 1 , ↓ U ), be c a use U intersects A ′ and K U tracks it. Therefo r e ˚ A ∧ ∇ U is inhabited and ˚ A intersects ∇ U if A ′ int ersects U . Let D ⊆ A 2 be the doma in of the applica tion ope r ator. W e in tersect ˚ A × ˚ A with ∇ D by pulling ba ck alo ng the inclus io n id D : D → A 2 . T o get a simpler representation, we pro ject down a long the inclusion of ( ˚ A × ˚ A ) ∩ ∇ D . This wa y ˚ A 2 ∩ ∇ D ≃ ( D, E ), where E = { ( a, b , c ) ∈ A × D | ∀ t ∈ T , f ∈ F .at = b, af = c } Let G = J ( t, f , x ) 7→ ( xt )( xf ) K . The application op erator α 1 : D → A is a mor- phism ( D , E ) → ˚ A b ecause GTF tra cks it. This means in the internal languag e of Asm ( A ′ , A ) that if x, y ∈ ˚ A and xy ↓ , then xy ∈ ˚ A . The a ssembly ˚ A is a filter b ecause it is down w ard closed, it intersects ∇ U when A ′ int ersects U and it is clos ed under application. W e hav e a category Asm ( A ′ , A ), a reg ular functor ∇ : E → Asm ( A ′ , A ) a nd a ∇ - filter ˚ A ≤ ∇ A , so we have a regula r mo del for ( A ′ , A ). If this mo del is pseudoinitial, the commo n str uc tur e of all regular mo dels g enerates every o b ject and morphism: the base categ o ry , imag es and pr eimages and the filter. W e show this in the next couple of lemmas. Lemma 27. F or e ach assembly ( X , Y ) let a : Y → A b e the first pr oje ction and x : Y → X b e the se c ond pr oje ction. ( X, Y ) ≃ ∃ ∇ x (( ∇ a ) − 1 ( ˚ A )) W e can compute this using the co nstructions for pullbacks an images given in the pro of of lemma 23. A more traditional wa y to state this lemma is as follows. Definition 28. An assembly ( X , Y ) is p artitione d if there is an a rrow f : X → A in E such that ( X, Y ) ≃ ( ∇ f ) − 1 ( ˚ A ) Lemma 29 . Partitione d assemblies c over al l assemblies. W e will r efer to r egular epimor phis ms from partitioned assemblies to o ther as- semblies as p artitione d c overs . Remark 30. While par titioned assemblies are pro jective o b jects in r ealizability categorie s o ver the catego r y of sets and other c a tegories where epimor phisms split, this do e s not genera lize to all top ose s , let alo ne all Heyting categor ie s. The s tr ucture of r egular mo dels als o gener ates the cla ss of mo rphisms of Asm ( A ′ , A ). The pr o of of the following lemma reveals how our definition of mor phism works. 9 Lemma 31. Each morphism f : ( X , Y ) → ( X ′ , Y ′ ) is the u n ique factorization of ∇ D f c omp ose d with id X : ( X , Y ) → ∇ D( X , Y ) t hr ough id X ′ : ( X ′ , Y ′ ) → ∇ D( X ′ , Y ′ ) . Pr o of. Let U track f : ( X , Y ) → ( X ′ , Y ′ ). According to the definition of mor phisms the following diagra m co mmut es and the vertical a rrows a re regular epimorphisms. U × Y ( u,a,x ) 7→ ( ux,f ( x )) # # ● ● ● ● ● ● ● ● ● ( u,y ) 7→ y ❴   Y ( a,x ) 7→ x ❴   Y ′ ( a,x ) 7→ x ❴   X f / / X ′ W e will use the internal language her e to define so me pullbacks. Let P = n ( u, ( a, x )) ∈ ∇ ( U × Y )    ( u, a ) ∈ ˚ A 2 o P ′ = n ( a, x ) ∈ Y ′    a ∈ ˚ A o The a ssembly P covers ( X , Y ). The assemblies ∃ ∇ (( u, ( a,x )) 7→ x ) ( P ) and ∃ ( a,x ) 7→ x ( Y ) are the same subo b ject of ∇ X , beca use ∇ U in tersects ˚ A . The r estriction of ( u, a, x ) 7→ ( ux, f ( x )) to P lands in P ′ , b eca use ˚ A in clos ed under applica tion. And ( X ′ , Y ′ ) = ∃ ∇ (( a,x ) 7→ x ) by lemma 27. Consider the following diagra m. P ( u, ( a,x )) 7→ x ❴   ( u,a,x ) 7→ ( ux,f ( x )) / / P ′ ( a,x ) 7→ x ❴   ( X, Y ) id X   f / / ( X ′ , Y ′ ) id X ′   ∇ D( X, Y ) ∇ D f / / ∇ D( X, Y ) W e just prov ed that the outer square co mm utes and the low er squa re co mm utes b y the definition of morphism. The upp er square commutes beca use id X ′ is mo nic. Conclusion: each morphism of as semblies f : ( X , Y ) → ( X ′ , Y ′ ) equa ls the unique fac torization of ∇ D f ◦ id X ov er id X ′ . With these lemmas in hand, we can prov e that Asm ( A ′ , A ) is a pseudoinitial regular mo de l. 2.4 Existence Theorem W e this section we will show that ( ∇ , ˚ A ) is a pseudo initial regular mo del. Thus w e prov e theorem 1 5. Theorem 32. Ther e is a pseudoinitial mo del for every OPCA p air in every Heyting c ate gory. Pr o of. Given a regula r mo del ( F , C ) for an OPCA pair ( A ′ , A ), we choose an ob- ject map F C . F or each assembly ( X , Y ), let a : Y → A and x : Y → X b e the pro jections. Let F C ( X, Y ) be isomorphic to ∃ F x ( F a − 1 ( C )). By definition 10 U ( X , Y ) = ∃ x ( Y ), therefore F U ( X , Y ) = ∃ F x ( Y ) and F C ( X, Y ) is a sub ob ject of F U ( X , Y ). While the ob ject map requires a strong form of choice or a s mall categor y E , once we hav e this map, there is a unique wa y to extend it to a functor, thanks to lemma 3 1. If U tra cks f : ( X , Y ) → ( X ′ , Y ′ ), then the following squar e commutes, and the vertical arrows are epic b ecause F is a regular functor. F ( U × Y ) F (( u,a,x ) 7→ ( ua,f ( x ))) / / ( u, ( a,x )) 7→ x ❴   F Y ′ ( a,x ) 7→ x ❴   F X F f / / F X ′ Because C is a filter, the subo b ject  ( u, ( a, x )) ∈ F ( U × Y )   ( u, a ) ∈ C 2  cov ers F C ( X, Y ) and the restr iction of F ( g × f ) factors through { ( a, x ) ∈ F Y | a ∈ C } , the sub o b ject of F Y ′ that cov ers F C ( X, Y ). Therefore there is a unique factorization through ( X ′ , Y ′ ) of F D f re stricted to ( X , Y ). W e define F C f to be that mor phism. This functor preser ves ima ges a nd preimage s by definition a nd ther efore is reg- ular. Also F C ∇ X ≃ F X and F C ˚ A ≃ C , so this regular functor is a morphism of regular mo de ls . Every regular G : Asm ( A ′ , A ) → co d F such that G ˚ A ≃ C and G ∇ ≃ F is isomorphic to F C . P ullbacks preser ve the isomor phism F C ( ˚ A ) ≃ G ˚ A , s o that the functors have to agree o n all partitioned a ssemblies. The isomorphism F C ∇ ≃ G ∇ , and the re lation of each mor phism f to ∇ D f now for c es the functors to agr ee on all ass emblies. W e co nclude that the functor ∇ : E → Asm ( A ′ , A ) a nd the filter ˚ A ⊆ ∇ A together form a pseudoinitial regular mo del for every OPCA pair ( A ′ , A ) in every Heyting c ategory E . W e take this result one step further to prov e that certain catego ries of reg ular functors ar e equiv alent to cer tain categ ories of sub ob jects. Definition 33. F or every Heyting ca tegory E , O PCA pair ( A ′ , A ), reg ula r c a tegory C and regular functor F : E → C , a r e gular extension of F is a reg ular functor G : Asm ( A ′ , A ) → C with a n isomorphism φ : G ∇ → F . A morphism of reg ular extensions ( G, φ ) → ( H , ψ ) is a natural tra nsformation η : G → H that commutes with the isomor phisms, i.e., η ∇ ◦ φ = ψ . Corollary 34. F or a fixe d r e gular F : E → C ther e is an e quivalenc e of c ate gories b et we en the p oset of F -filters, whose or dering is inclusion, and the c ate gory of r e g- ular ext ensions of F . Pr o of. W e first show how natural trans formations induce inclusions of filter s. Let G, H : Asm ( A ′ , A ) → C b e regular functors, let η : G → H and let η ∆ : G ∆ → H ∆ be an isomor phis m of functors . Cons ide r the following natur ality square. G ˚ A η ˚ A / / id A   H ˚ A id A   G ∇ A η ∇ A / / H ∇ A Since G and H ar e b oth r egular, the v ertical arrows are monic and the low er arrow is an is o morphism. Therefore η ˚ A m ust be monic to o. If there are iso morphisms φ : G ∇ → F and ψ : H ∇ → F , and if η ∇ comm utes with these isomorphis ms , then η ∇ is a n isomo r phism. Hence G ˚ A ⊆ H ˚ A . 11 Next we construct a natura l transfor ma tion from an inclusion o f filters. Let C ⊆ C ′ be F -filter s. Pullbacks preserve the inclusion C ⊆ C ′ and since par titioned assemblies are pullbacks, we ca n define η P : F C P → F C ′ P to b e this pulled back inclusion. E a ch ass embly X has a partitioned cover e : P → X , which we use the construct this diag r am. F C P η P / / F C e ❴   F C ′ P F C ′ e ❴   F C ( X, Y ) F C id X   F C ′ ( X, Y ) F C ′ id X   F C ∇ D( X, Y ) ∼ / / F C ′ ∇ D( X, Y ) There is a unique arrow F C ( X, Y ) → F C ( X ′ , Y ′ ) that commutes with a ll the a rrows in the diag ram, a nd we define η X,Y to equal this arrows. Th us we get a natural transformatio n η for which η ∇ is an isomo r phism. The natural transfor mation η we constructed in the last paragr aph induces the inclusion C ⊆ C ′ . Also, the diag ram above s hows tha t any transfor mation that induces this inclusion m ust eq ual η . Therefor e, there is an equiv alence of ca tegories betw een the po set of F - filters and the r egular extens io ns of F . 2.5 Pro jective T erminals If the terminal ob ject of the underlying Heyting category E is pro jective, e.g ., in the categor y o f sets, then every inhabited s et has a glo ba l section. This s implifies the c o nstruction of the categ ory of assemblies. Since each inhabited ob ject has a section, global sections realize e ach r epresentable a rrow o f each OP AS and therefo r e each partial combinatory function of each OPCA. Definition 35. Let A b e an OP AS. An a rrow f : A n → A is glob al ly repr e sentable if J f K (see r emark 3) ha s a glo bal section. Lemma 36 . In every H eyting c ate gory E every glob al ly r epr esentable morphism is r epr esentable. If t he terminal obje ct is pr oje ctive, any r epr esentable morphism is glob al ly r epr esentable. Pr o of. Any ar row f : U ⊆ A n → A is repr esentable if the following o b ject is inhabited. J f K = { a ∈ A | ∀ ~ x ∈ U. ∃ y ∈ A.y ≤ f ( ~ x ) ∧ (( ax 1 ) . . . ) x n ↓ y } If f is globally repr esentable, then J f K has a globa l s ection. This makes J f K inhabited and therefor e f represent able. If g is representable and the terminal ob ject is pro jective, then J f K has a global section, a nd this sectio n g lobally represents f . W e ca n use g lobal r epresentabilit y to co ns truct categorie s of asse m blies for cer- tain pairs of or der ed pa rtial applicative structure s in categories that hav e finite limits, but a re not nece s sarily reg ula r or Heyting. In the following lemma we fo r- m ulate o ne pr op erty global repr esentabilit y that lets us do this. Lemma 37. F or every fin ite limit c ate gory C and let Γ : C → Set b e the glob al se ct ions functor. F or every or der e d p artial applic ative structu r e A ∈ C , a p artial arr ow f : A n ⇀ A is glob al ly r epr esentable in A if and only if Γ f is r epr esentable in Γ A . 12 Pr o of. The set o f realizer s J Γ f K ⊆ Γ A is inhabited precisely when J f K ha s a g lobal section. One p oss ible definition o f a category o f assemblies for a global OP CA pair is now the following. Definition 38. F or any finite limit catego ry C let Γ : E → Set b e the global sectio ns functor. A pair of OP ASes A ′ ⊆ A in C is a glob al OPCA p air , if (Γ A, Γ A ′ ) is an OPCA pair. The catego r y o f assemblies for the globa l OP CA pair ( A ′ , A ) is the fibred pro duct of U : Asm (Γ A ′ , Γ A ) → Set and Γ : C → Set . W e r eturn to our o wn definition of a category of assemblies ov er arbitr ary Heyt- ing ca tegories. Assuming a pr o jectiv e terminal ob ject, we can simplify the definition of a morphism of ass emblies. Lemma 39. F or e ach morphism f : ( X , Y ) → ( X ′ , Y ′ ) ther e is a glob al c ombinator r : 1 → A such that ( r a, f ( x )) ∈ Y ′ for al l ( a, x ) ∈ Y . F or every f ′ : X → X ′ and every p air of assemblies ( X , Y ) and ( X ′ , Y ′ ) and e ach glob al se ction r ′ : 1 → A ′ , if r ′ a ↓ and ( r ′ a, x ) ∈ Y ′ for al l ( a, x ) ∈ Y , then f ′ is a morphism. Pr o of. F or any tracking U of f that intersects A ′ , ther e is a glo bal sectio n r : 1 → U ∩ A ′ that satisfies our requir ements. The sub ob ject { r ′ } tra cks f ′ : ( X , Y ) → ( X ′ , Y ′ ), so f ′ is a morphism. This is the definition of morphism o f assemblies o ne finds in other s ources, like [24]. So known categories of assemblies are sp ecia l cases of o ur cons truction. The categ ory of assemblies is a pseudoinitial regula r mo del of an O PCA pair . In the next section we will show a simila r definition of relative realiza bilit y to p o ses. 3 Relativ e Realizabilit y T op oses In this s ection we assume that the underlying categ ory E is a top os. Under that condition, we can construct a top os out of the categor y of assemblies. Definition 40. F or every top os E and every O P CA pair ( A ′ , A ) in E a n exact mo del is a regula r functor F from E to a n ex a ct c ategory C , together with an F - filter. A r elative r e alizability top os RT ( A ′ , A ) is an pseudoinitial exact mo del. Theorem 41. R elative r e alizabilty top oses ex ist for every O PCA p air in every top os. Pr o of. W e start with a co nstruction that turns regular categor ies int o exa ct one s . The 2- c ategory of exact categ ories is a r eflective subc a tegory of the 2-catego ry of regular categ ories (see [5]). This means that fo r every reg ular ca tegory C there is an exact category C ex / r e g and a regular functor I : C → C ex / r e g such that every regular functor from C to a n e x act ca teg ory D factor s thro ugh I up to isomorphism. Categorie s w ith this prop er ty of C e x/reg are ca lled exact c ompletio ns of C . Let E be a to p o s, D a n e xact categor y and F : E → D a r e gular functor. If ( F, C ) is an exact model, then there is a n up to iso morphism unique regu- lar functor F C : Asm ( A ′ , A ) → D such that F C ∇ ≃ F and F ˚ A ≃ C , b ecause exact mo dels are r egular mo dels. F C factors up to isomor phism through e xact completions o f Asm ( A ′ , A ) bec ause its co domain is exa ct. The re gular functor I : Asm ( A ′ , A ) → Asm ( A ′ , A ) ex/r eg creates an exact mo del ( I ∇ , I ˚ A ), and we see now that it is pseudoinitia l. 13 W e give a construction for a n exact completio n of Asm ( A ′ , A ) in sec tio n 3.3. Before that we w ant to prov e that r elative r ealizability topo s es ar e indeed to p o ses. W e will use a res ult from Mat ´ ıas Menni’s thes is [19] fo r this: if a regula r categor y is lo cally Ca rtesian closed and has a generic mono, then its ex act c o mpletion is a top os. 3.1 Lo cal Cartesian Closure Lo cal Cartesian closure means Car tesian closure of all slice categ o ries. W e prov e that the ca tegory of ass emblies is lo cally Cartesia n clo s ed in tw o steps. Fir stly we prov e that if a Heyting category has a Cartesia n closed reflective sub categor y , then it is Cartesian clo sed under so me conditions on the reflector. Secondly w e prove that for each assembly ( X , Y ), the slice categ ory E /X is a reflective sub ca tegory of Asm ( A ′ , A ) / ( X , Y ). F or each Z ∈ E the slice E / Z is C a rtesian c losed b ecause E is a top os, therefore Asm ( A ′ , A ) is lo c a lly Car tesian closed. Lemma 42. L et E b e a Heyting c ate gory, let D b e a Cartesian close d ful l sub c ate gory and let L : E → D b e a finite limit pr eserving left adjoint to the inclusion of D into E , such that the unit η : L → 1 is a natur al monomorphism. Then E is Cartesian close d. Pr o of. F or simplicity , we will use the v alidit y of fir st order logic and s imply typed λ -calculus in the in ternal languages o f r esp ectively Heyting and Cartesian closed categorie s. W e define for all Y , Z ∈ E Z Y =  f ∈ LZ LY   ∀ y ∈ Y . ∃ z ∈ Z .f ( η Y y ) = η Z z  F or all f : X → Y Z , x ∈ X and y ∈ Y , there exists a z ∈ Z such that f ( x )( η Y y ) = η Z z and b e cause η Z is a mo nomorphism, this z is unique. So let f t ( x, y ) = z if f ( x )( η Y y ) = η Z z for a ll x ∈ X , y ∈ Y and z ∈ Z . F or all g : X × Y → Z , x ∈ X a nd y ∈ Y we hav e η Z ◦ g ( x, y ) = L g ( η X x, η Y y ). Note that we use L ( X × Y ) ≃ LX × LY by the way . Because the sub categor y is Cartesian clos ed, we can let g t ( x ) = λy .Lg ( η X x, η Y y ). F or each f : X → Y Z , x ∈ X a nd y ∈ Y we hav e ( f t ) t ( x )( η Y y ) = η Z z if and only if f ( x )( η Y y ) = η Z z . Therefo r e ( f t ) t = f . F or each g : X × Y → Z , x ∈ X and y ∈ Y we have ( g t ) t ( x, y ) = z if and only if g t ( x )( η Y y ) = η Z z while g t ( x )( η Y y ) = η Z ◦ g ( x, y ). Since η Z is mono we hav e ( g t ) t = g . This mea ns that Z 7→ Z Y is right adjoint to X 7→ X × Y and that E is Cartesian clos ed. Lemma 43. F or e ach ( X, Y ) ∈ Asm ( A ′ , A ) , ther e is a fu l l and faithful fun ctor E / X → Asm ( A ′ , A ) / ( X , Y ) with finite limit pr eserving left adjoint. Pr o of. The functor ∇ is right adjoint to D and the unit of this adjunction ( X , Y ) → ∇ D( X, Y ) is a monomorphism. F or each ( X , Y ) ∈ Asm ( A ′ , A ), we let ∇ ( X , Y ) : E / X → Asm ( A ′ , A ) / ( X , Y ) be the functor that maps f : Z → D( X , Y ) to ( ∇ f ) − 1 ( Y ): the sub ob ject of ∇ D( X , Y ) represented b y Y . This functor is faithful and D acts as reflec to r Asm ( A ′ , A ) / ( X , Y ) → E /X that preserves finite limits, and the unit is still a monomor phism. Theorem 44. F or e ach lo c al ly Cartesian close d Heyting c ate gory E and an OPCA p air ( A ′ , A ) in E , the c ate gory of assemblies is a lo c al ly Cartesian close d Heyting c ate gory. Pr o of. Lemma 23 tells us Asm ( A ′ , A ) is Heyting. F or each assembly ( X , Y ) lemma 43 embeds the Cartesia n closed Heyting catego ry E / D( X , Y ) into the Heyting cate- gory Asm ( A ′ , A ) / ( X , Y ) in such way that the inclusion has a finite limit preserving 14 left adjoint. Therefor e every slice of Asm ( A ′ , A ) is Cartesia n closed a ccording to lemma 42, and that means Asm ( A ′ , A ) is a lo ca lly Cartesian closed Heyting cate- gory . 3.2 Generic Monomorphisms W e construct a gener ic monomo rphism for the categ ory of assemblies. Lemma 45. L et E b e a top os and ( A ′ , A ) an OPCA p air in E . L et D ∗ A ≤ Ω A b e the obje ct of inhabite d downwar d close d su b obje ct s of A , and let { ∈ } b e the element-of r elatio n { ( a, U ) ∈ A × D ∗ A | a ∈ U } . The inclusion id D( D ∗ A, { ∈ } ) : ( D ∗ A, { ∈ } ) → ∇ D ∗ A is a generic monomorphism. Pr o of. If m : X → Y is monic, then ∃ m ( X ) ≃ X . Therefore we c a n fo cus o n monomorphisms o f the form id D( X,Y ) : ( X , Y ) → ( X , Y ′ ). T o Y ≤ A × X b elong s a characteris tic ma p y : X → D ∗ A ≤ Ω A : y ( x ) = { a ∈ A | ( a, x ) ∈ Y } , which by the definition of a ssemblies is a down ward closed set. I f we pull back ( D ∗ A, { ∈ } ) a long y using the cons tr uctions from lemma 23, we g et the as s embly ( X , Y ∧ Y ′ ), wher e Y ∧ Y ′ = { ( a, x ) ∈ A × x | ∀ t ∈ T .at ↓ ∧ ( at, x ) ∈ Y , ∀ f ∈ F .af ↓ ∧ ( af , x ) ∈ Y ′ } Since Y ≤ Y ′ we have Y ∧ Y ′ ≃ Y . Theorem 46. L et E b e a top os and ( A ′ , A ) an OPCA p air in E . The re lative r e alizability top os RT ( A ′ , A ) = Asm ( A ′ , A ) ex/r eg is a top os. Pr o of. The categ o ry o f assemblies is lo ca lly Cartesia n clo sed an has a gener ic mo- nomorphism. This implies that its exac t completion is a top os, according to Matias Menni [1 9]. Remark 47. Given any assembly ( X , Y ) let a : Y → A and x : Y → X b e the pro jections. Let { ∈ } = { ( a, ξ ) ∈ A × D ∗ A | a ∈ ξ } , and let b : { ∈ } → A and d : { ∈ } → D ∗ A b e the pro jections. Ther e is a y : X → D ∗ A such that the square in the following commut ative diagra m is a pullback: Y ( a,y ) / / a + + x   { ∈ } d   b / / A X y / / D ∗ A Because ∇ and Asm ( A ′ , A ) a re r e gular and be c ause o f lemma 27, this means: ( X, Y ) ≃ ∇ y − 1 ∃ ∇ d ∇ b − 1 ( ˚ A ) So the gener ic mono mo rphism is the inclusio n of ∃ ∇ d ∇ b − 1 ( ˚ A ) into ∇ D ∗ A . Note the reg ular epimorphism d : ∇ b − 1 ( ˚ A ) → ∃ ∇ d ∇ b − 1 ( ˚ A ). It is a generic partitioned cov er. If C is reg ular, F : Asm ( A ′ , A ) → C preserves finite limits, F ∇ is reg ular and F d is a regular epimorphism, then F is a reg ular functor. Example 48. F or the O PCA pair ( K rec 2 , K 2 ) from exa mple 9, a version of which exists in e very to p o s with a natural num ber ob ject, we now c a n construct the Kle en e- V esley top os RT ( K rec 2 , K 2 ) (see [2 4]). This is a top os theor etic version of Kleene and V esleys int uitionism in [1 6]. The lattice of subterminal ob ject is dual to the Medvedev la ttice [18] and has be e n studied as a mo del fo r constructive prop ositiona l logic, e.g ., in [23]. 15 3.3 Exact Completions In this section we r e call the co ns truction of the ex act completion of a reg ular cat- egory . Using this construction we give a concre te description of the rela tive realiz- ability topos . Definition 49. Given a regula r c a tegory C let a sub quotient b e a pair ( X , E ) wher e X ∈ C , E ⊆ X 2 and E satisfies: ( x, y ) ∈ E → ( y , x ) ∈ E ( x, y ) , ( y , z ) ∈ E → ( x, z ) ∈ E Given any tw o sub quotients ( X , E ) and ( X ′ , E ′ ) a nd tw o sub ob jects F , G ⊆ X × X ′ let F ≃ E → E ′ G if b oth ( x, y ) ∈ E → ∃ z ∈ X ′ . ( z , z ) ∈ E ′ ∧ ( x, z ) ∈ F ∧ ( y , z ) ∈ G, ( x, x ) ∈ E ∧ ( x, y ) ∈ F ∧ ( x, z ) ∈ G → ( y , z ) ∈ E ′ If F ⊆ X × X ′ satisfies F ≃ E → E ′ F , then it is ca lled a functional r elation . A morphism of sub quotients ( X , E ) → ( X ′ , E ′ ) is an equiv a lence clas s for ≃ E → E ′ . W e explain how this definition works. F o r every s ubq uotient ( X , E ), the r e la tion E is s y mmetric and transitive in the in ternal la nguage o f C . It defines an equiv alence relation on { x ∈ X | ( x, x ) ∈ E } . W e use this pair to represent that quotient. The relations ≃ E → E ′ are symmetric and tr ansitive re la tion o n the p oset of sub ob jects of X × X ′ . This defines an equiv alence relatio n of an subset to o , but this relatio n is external to C . If F ⊆ X × X ′ and F ≃ E ,E ′ F , then F induces a function for m equiv a le nce classes of E to equiv alence classes for E ′ . Ther efore F represents a morphism b etw een quotients. If G ⊆ X × X ′ , G ≃ E ,E ′ G and G ≃ E ,E ′ F , then G induces the same function a s F . That is why morphisms ( X , E ) → ( X ′ , E ′ ) are equiv a le nce class es for ∼ E → E ′ . Lemma 50. Su b quotients and m orphisms for a r e gular c ate gory C to gether form a c ate gory C ex / reg . This c ate gory C ex / reg is an exact c ompletion of C . Pr o of. W e comp ose relations F ⊆ X × Y and G ⊆ Y × Z by letting G ◦ F = { ( x, z ) ∈ X × Z | ∃ y ∈ Y . ( y , z ) ∈ G, ( x, y ) ∈ F } . If F ≃ F ′ and G ≃ G ′ relative to some s ubq uotients, then F ◦ G ≃ F ′ ◦ G ′ . F o r e very sub quotient ( X , E ) we hav e E ≃ E ,E E and its equiv alence class is an ident ity morphism. The functor that sends each o b ject X to the pair ( X , ∆ X ), where ∆ X is the diagonal, and each arrow f : X → Y to the equiv alence cla ss of its graph, is an embedding o f C . Finally , if F : C → D is a regular functor to an exact category , and ( X , E ) is a sub quotient, then F E is an equiv alenc e relation o n a sub ob ject of X . The sub q uotient F X/ F E e xists here be cause of exactness , and that is where we map ( X, E ) to o. If G ≃ G b etw een ( X , E ) and ( X ′ , E ′ ), then comp osition with F G induces a map F X/F E → F X ′ /F E ′ . If G ≃ G ′ then F G a nd F G ′ induce the same map. Th us F fa c to rs thr ough the categor y of sub quotients in a n up to iso morphism unique way . The inclusion id D( X,Y ) : ( X , Y ) → ∇ D( X , Y ) is a monomor phis m in Asm ( A ′ , A ). That mea ns every assembly is a s ubo b ject of an ob ject in the image o f ∇ . In turn every subquotient is a sub q uotient of a n ob ject in the image of ∇ . If m : ( Y , E ) → ∇ X is a monomorphis m that r epresents such a relation, then so do es the isomor phic assembly ∃ m ( Y , E ) ≃ ( X 2 , ∃ id A × m ( E )). Therefor e assemblies ( X 2 , E ) that define a sub q uotients of ∇ X represent all ob jects of the relative r e a lizability top os. W e use these fa cts to get a simpler construction of r elative r ealizability topos es. 16 Definition 51 . Let E b e a top os a nd let ( A ′ , A ) b e an OP C A pair . The standar d relative realiza bility top os is defined a s follows. The o b jects ar e pair s ( X , E ⊆ A × X 2 ) such the the assembly ( X 2 , E ) is a s ymmetric and transitive rela tion on ∇ X . A morphism ( X , E ) → ( X ′ , E ) is a n is omorphism class of ass e mblies ( X × X ′ , Y ), where Y is a functiona l r elation. F or ea ch O PCA pair , the categor y o f as s emblies is the pseudo initia l reg ula r mo del a nd the relative realizability topo s is the pse udo initial exact mo del. That is the main po int of our pap er. In the nex t section we explain some consequences of our definitions . 4 F unctors In this section, we use initial mo dels to find examples of regular functors from relative realizability catego ries into other categories . W e no long e r demand tha t the underly ing categor y is a top os. Howev er, when the underlying categor y is a topo s many of the functor s we construct hav e right adjoints and therefor e are inv erse ima ges parts of geometric morphisms. F or completeness we will also prov e the exis tence of these r ight a djoints. The first tw o s ubs ections deal with g eometric morphisms fr om lo calic top os e s ov er the bas e ca tegory to relative r ealizability top oses. The ex amples w e pro - vide ther e a re mostly new. More is known ab out morphisms betw een realizability top oses, which are the s ub ject of the las t subsectio n. 4.1 P oin ts A p oint of a top os T is a g eometric mo rphism Set → T , where Set is the top os of se ts . The inv erse image part of a g e o metric mor phism is a reg ular functor, and this allows us to use our universal prop erty . If ( A ′ , A ) is an OPCA pair in E , then regular mo de ls ( F, C ) where F is a se t v alued reg ular functor repr esent eac h p o int of R T ( A ′ , A ). The ana lysis of set-v a lued regular mo dels will hav e to wait fo r another pap er. Here, we fo cus on regula r mo dels of the form (id E , C ) in stead. The r eason that w e hang on the to word ‘p oint’, is that in the cas e that E = Set , these mo dels corres p o nd to the class of all p oints of RT ( A ′ , A ) that satisfy f ∗ ∇ ≃ id Set . As g eometric morphisms, these a r e precisely the s ubmo rphisms o f D ⊣ ∇ : E → R T ( A ′ , A ). F or every Heyting categor y E the iden tity functor id E : E → E is reg ular. If ( A ′ , A ) is an O PCA pair in E , we can construct r egular functors with filters of A . A id E -filter is a subo b ject C of A tha t sa tisfies: • F or a ll x ∈ C and y ∈ A if y ≥ x then y ∈ C . • F or a ll x, y ∈ C and z ∈ A if xy ↓ z then z ∈ C . • F or a ll U ⊆ A if U intersects A ′ then U in tersects C . Remark 52. This la st condition must b e interpreted externa lly , not in the internal language o f E . An internal interpretation is p o ssible if E is a top os, but that condition implies A ′ ⊆ C . F or each filter C ⊆ A , we c o nstruct the C -induced regula r functor a s follows. F or each assembly ( X , Y ) we let D C ( X, Y ) = { x ∈ X | ∃ c ∈ C. ( c, x ) ∈ Y } The functor then maps f : ( X , Y ) → ( X ′ , Y ′ ) to D f restricted to D C ( X, Y ) fa ctored through D C ( X ′ , Y ′ ). 17 Quite surpr isingly , all regula r functors G : R T ( A ′ , A ) → E that satis fy G ∇ ≃ id E are in verse image parts of ge o metric mo rphisms. Ther efore relative realizability top oses c a n hav e many po int s, just like Grothendieck top oses . Theorem 53. L et E b e a top os, let ( A ′ , A ) b e an OPCA of E and let C b e a id E - filter. Then t he induc e d r e gular functor D C : R T ( A ′ , A ) → E has a right adjoint. Pr o of. W e use the construction of the re lative rea lizability top o s for m subsection 3.3 to g et a clear pictur e o f D C : RT ( A ′ , A ) → E . As D C m ust preserve sub quo tient s, we can construct the functor as follows. F or each s ub quo tient ( X , E ), each F : ( X, E ) → ( X ′ , E ′ ) and each ξ ∈ C ( X, E ) we let D C ( X, E ) =  ξ ∈ Ω X   ∃ x ∈ X .ξ = { y ∈ X | ∃ a ∈ C. ( a, x, y ) ∈ E }  D C F ( ξ ) = { y ∈ X ′ | ∃ a ∈ C. ( a, x, y ) ∈ F } W e now co nstruct a functor ∇ C : E → R T ( A ′ , A ). F or each X ∈ E , let E X =  ( a, f , g ) ∈ A × (Ω X ) 2   a ∈ C → ∃ x ∈ X .f = g = { x }  The assembly ( X 2 , E X ) is a par tial equiv alence rela tion on ∇ X . F or any ar r ow f : X → Y the mo rphism ∇ f commutes with the partial equiv alence relation of either side. Therefore w e g et a functor ∇ C by mapping each X to ( X, E X ) a nd each f : X → Y to the morphism of sub quotients it induce s . By computation we find that D C ∇ C X is isomorphic to X for all X ∈ E . C R C X = n ξ ∈ Ω Ω X    ∃ x ∈ X .ξ = { { x } } o Let e X : D C ∇ C X → X b e the inv erse of x 7→ { { x } } . F or each ( X, E ) ∈ RT ( A ′ , A ) define f ( X,E ) : X → Ω D C ( X,E ) by f ( x ) = { { y ∈ X | ∃ a ∈ C. ( a, x, y ) ∈ E } } If ( a, x, y ) ∈ E and a ∈ C , then there is an z ∈ D C ( X, E ) such that f ( x ) = f ( y ) = { z } , na mely z = { y ∈ X | ∃ a ∈ C . ( a, x, y ) ∈ E } . So ( a, f ( x ) , f ( y )) ∈ E D C ( X,E ) , and ther efore f is a morphism of the par tial e quiv ale nce relatio ns. Hence f ( X,E ) : ( X, E ) → ∇ C D C ( X, E ). F or ξ ∈ D ( X , E ) we hav e D C f X,E ( ξ ) = { f ( x ) | x ∈ ξ } = { { ξ } } . Therefor e e D C ( X,E ) ◦ D C f ( X,E ) = id C . F or g ∈ Ω X we have f ∇ C X ( g ) = { { g } } . Therefore ∇ C e ◦ f ∇ C X = id ∇ C . Hence we hav e an adjunction D C ⊣ ∇ C . Remark 54. It is not clear that all ge o metric morphisms f : E → RT ( A ′ , A ) satisfy f ∗ ∇ ∼ id E , even in the case that E = Se t . 4.2 Characters W e gener alize the notio n of p o int from the previous subsection. F or each top os E and e ach OPCA pa ir ( A ′ , A ), we co nsider g eometric mo rphisms E P op → RT ( A ′ , A ) where P is a preorder ed ob ject of E , a nd E P op the topo s o f internal presheaves ov er P . Indirectly , we are lo oking a t how top o s es that a r e lo calic ov er E map into RT ( A ′ , A ), b eca use all lo ca lic top oses e m b ed int o a to po s of the form E P op . The top os of internal pr esheav es E P op is co ns tructed a s follows. Each internal presheaf is a n arrow p : X → P in E , together w ith a r estriction op erator r : { ( x, u ) ∈ P × X | x ≤ p ( u ) } → X that satisfies p ◦ r ( x, u ) = u . Each morphism f : ( p, r ) → ( p ′ , r ′ ) is just an a rrow f : X → X ′ such that p ′ ◦ f = p a nd r ′ ◦ f = f ◦ r . The constant sheaf functor ∆ : E → E P op has b oth adjoints and is therefore a regular functor. Let D P be the ob ject of downsets of P . The ∆-filters of ( A ′ , A ) corres p o nd to arr ows A → D P . 18 Definition 55. Let P b e a preor dered set a nd ( A ′ , A ) a nd OPCA pa ir in E . A character γ is an arrow A → D P that satisfies: • If x ≤ y then γ ( x ) ≤ γ ( y ). • If xy ↓ z then γ ( x ) ∩ γ ( y ) ≤ γ ( z ). • If a ∈ A ′ then γ ( a ) = P . W e derive the next coro llary from theor em 15. Corollary 56. Char acters c orr esp ond t o r e gular functors RT ( A ′ , A ) → E P op . Pr o of. Ther e is a bijection b etw een E ( A, D P ) a nd the sub ob jects o f ∆ A : E ( A, D P ) ≃ E ( A, ΓΩ) ≃ E DP op (∆ A, Ω) ≃ Sub (∆ A ) This bijectio n turns characters in to ∆-filters. Because the functor γ ∗ : RT ( A ′ , A ) → E P op is re g ular, we can give a n explicit definition. Let ( X , E ) be any ob ject and let fo r all x ∈ X J x K u = { y ∈ X | ∃ a ∈ A.u ∈ γ ( a ) , ( a, x, y ) ∈ E } Let γ ∗ ( X, E ) = ( X ′ , p, r ) with X ′ =  ( u, ξ ) ∈ P × Ω X   ∃ x ∈ X .ξ = J x K u  p : γ ∗ ( X, E ) is just the pro jection to the first co ordina te. W e let r ( ξ , u ) = S x ∈ ξ J x K u ; r now satisfies r ( J x K u , v ) = J x K v for v ≤ u . Let f : ( X , E ) → ( X ′ , E ′ ) b e a ny functional relation. Let for all ( u, ξ ) ∈ γ ∗ ( X, E ) γ ∗ f ( u, ξ ) = ( u, { y ∈ X ′ | ∃ a ∈ A, x ∈ ξ .u ∈ γ ( a ) ∧ ( a, x, y ) ∈ f } ) In the cas e that the underlying category is a top os, the functors that characters induce a re not just r egular, how ever. Theorem 57. L et E b e a top os, P a pr e or der e d set and ( A ′ , A ) an OPCA p air. Char acters A → D P induc e ge ometric morphisms E P op → RT ( A ′ , A ) . Pr o of. F or each ob ject ( X , p, r ) ∈ E P op , let x, y ∈ X e ( x, y ) = { u ∈ P | u ≤ p x, u ≤ py , r ( x, u ) = r ( y , u ) } γ ∗ ( X, p, r ) = ( X , { ( a, x, y ) ∈ A × X × Y | γ ( a ) ⊆ e ( x, y ) } F or each morphism f : ( X , p, r ) → ( X ′ , p ′ , r ′ ) we let: γ ∗ f = { ( a, x, y ) ∈ A × X × X ′ | γ ( a ) ⊆ e ( f x, y ) } By writing out the definitions we find that if γ ∗ γ ∗ ( X, p, r ) = ( X ′ , p ′ , r ′ ) then ( u, ξ ) ∈ X ′ ⇐ ⇒ ∃ x ∈ X .ξ = { y ∈ X | u ≤ p ( y ) , r ( y , u ) = r ( x, u ) } This new presheaf is isomorphic to ( X , p, r ), by the following iso morphism: g ( x ) = ( px, { y ∈ X | u ≤ p ( y ) , r ( y , u ) = x } ) ∀ x ∈ ξ . ǫ ( X,p,r ) ( u, ξ ) = r ( x, u ) The s e cond family of mo rphisms a c ts as counit. 19 If γ ∗ γ ∗ ( X, E ) = ( X ′ , E ′ ), then X ′ =  ( u, J x K u ) ∈ P × Ω X   x ∈ X  with J x K u defined as befo re. W e simplify the par tial equiv alence re la tion. E ′ = { ( a, ( u, J x K u ) , ( v , J y K v )) | ( ∀ w ∈ γ ( a ) .w ≤ u, w ≤ v ) ∧ ( a, x, y ) ∈ E } W e define a family of functiona l r elations ( X , E ) → γ ∗ γ ∗ ( X, E ) by η ( X,E ) = { ( a, x, ( u, J x K u )) ∈ A × X × X ′ | u ∈ γ ( a ) } The s ub o b ject I tr a cks a ll o f these, and tog ether they form the unit. W e conclude tha t ǫ γ ∗ ◦ γ ∗ η = id γ ∗ bec ause γ ∗ η ( X,E ) ( u, ξ ) = ( u, { ( v , J x K v ) ∈ X ′ | x ∈ ξ } ) = g ( u, ξ ) By writing out definitions we also find that ( a, ( u, ξ ) , y ) ∈ γ ∗ ǫ ( X,p,r ) if for all v ∈ γ ( a ) and x ∈ ξ , v ≤ u and r ( x, v ) = r ( y , v ), while ( b, x, ( u, J x K u )) ∈ η γ ∗ ( X,E ) if u ∈ γ ( b ). W e hav e γ ∗ ǫ ◦ η γ ∗ = id γ ∗ , b ecause fo r any p ∈ P we hav e γ ( pab ) = γ ( a ) ∩ γ ( b ). So γ ∗ is right a djoint to γ ∗ . Since γ ∗ is regular their combination is a geometric morphism. Remark 5 8. An internal Grothendieck to p o logy J on a preorder ed ob ject P a llows us to define a top os of she aves Sh ( P, J ). This top os is embedded in E P op by a geometric morphism. Therefore, we ca n r elate geometric morphisms Sh ( P, J ) → RT ( A ′ , A ) to characters γ : A → D P of which the v a lues ar e J -closed se ts . Remark 59. F or the trivia l po set that is the terminal ob ject 1 we hav e E 1 op ∼ = E , and characters ar e po ints. T op os es of sheav es are b etter understo od then r e lative realiza bilit y topo ses. By inducing geo metric morphisms betw een these tw o kinds of top os es, characters may clarify the theory of r elative realizability . 4.3 Applicativ e Morphisms In this subse ction we co nsider reg ular functor s b etw een realizability categor ies for different OPCA pa irs. The filters tha t induce these functors ar e the applicative morphisms that were defined by Longley [17], Hofstra and v a n Oosten [12] and Hofstra [1 3]. Definition 60. Let ( A ′ , A ) and ( B ′ , B ) b e tw o O PCA pa ir s in an Heyting ca tegory E . An applic ative morphism γ : ( A ′ , A ) → ( B ′ , B ) is a B -assembly ( A, C ) over A , such tha t the following sub ob jects o f B intersect B ′ . { u ∈ B | ∀ ( x, y ) ∈ C, y ′ ∈ A.y ≤ y ′ → ( u x ↓ ∧ ( ux, y ′ ) ∈ C } { r ∈ B | ∀ ( x ′ , x ) , ( y ′ , y ) ∈ C.xy ↓ → (( rx ′ ) y ′ ↓ ∧ (( r x ′ ) y ′ , xy ) ∈ C ) } ∀ a ∈ A ′ { b ∈ B | ( b, a ) ∈ C } Theorem 61. F or e ach applic ative morphism γ : ( A ′ , A ) → ( B ′ , B ) ther e is an up to isomorphism u nique r e gular functor F : Asm ( A ′ , A ) → Asm ( B ′ , B ) su ch that F ˚ A ≃ ( A, C ) and F ∇ ≃ ∇ . F or e ach r e gular functor F : Asm ( A ′ , A ) → Asm ( B ′ , B ) such that F ∇ ≃ ∇ , t her e is an up t o isomorphi sm unique applic ative m orphism γ : ( A ′ , A ) → ( B ′ , B ) . 20 Pr o of. γ is a filter for ∇ : E → Asm ( B ′ , B ), so ( ∇ , γ ) is a reg ular mo del for ( A ′ , A ). Therefore there is an up to is omorphisms unique regular functor Asm ( A ′ , A ) → Asm ( B ′ , B ) satisfying the c onditions. An y regular functor F suc h that F ∇ ≃ ∇ will map id A : ˚ A → ∇ A to some monomorphism F ˚ A → F ∇ A . The imag e of F ˚ A along the comp osition of F id A with the isomorphism F ∇ A → ∇ A is a n a pplicative morphism becaus e F preserves filters. Unlik e characters, a pplicative morphisms do not genera lly induce geometric mor- phisms if the underlying category is a top os. The ones that do hav e the following prop erty . Definition 62. F or γ : ( A ′ , A ) → ( B ′ , B ) we define the a rrow γ : A → D B by γ ( a ) = { b ∈ B | b ∈ γ ( a ) } . W e define the following rela tion on D B : U V ↓ W if and only if ∀ x ∈ U , y ∈ V . ∃ z ∈ W .xy ↓ z The term U V stands for the lea st W ∈ D B such that U V ↓ W and remains undefined if no such W exists. The applica tive mor phism γ is c omput ational ly dense if there is some µ ⊆ B intersecting B ′ such that for each U ∈ D B that int ersects B ′ the following sub ob ject o f A intersects A ′ . U µ = { a ∈ A | ∀ x ∈ A.U γ ( x ) ↓ → ax ↓ ∧ µγ ( ax ) ↓ U γ ( x ) ↓ ) } Theorem 63. Computational ly dense applic ative morphi sms induc e ge ometric mor- phisms b etwe en r elative r e alizability top oses. Pr o of. W e leave to the reader to chec k that for e a ch relative realizability topo s RT ( A ′ , A ) over a bas e top os E the assignment X → Sub ( ∇− ) is a trip os ov er E and that an adjoint pair of tr ansformatio ns of tr ipo ses induces a geometr ic mor phism [24]. F or clarity , let ( ∇ A , ˚ A ) b e an initial exact model for ( A ′ , A ) a nd ( ∇ B , ˚ B ) fo r ( B ′ , B ). Since the regular functor that γ pr eserves ∇ and sub ob jects, the functor relates to a transfor mation of trip os es Sub ( ∇ A − ) → Su b ( ∇ B − ). So we nee d to find a rig ht a djo int to that tr ansformatio n. Fixing X ∈ E , we may repr e s ent sub ob jects of ∇ A X by sub ob jects o f A × X and sub o b jects of ∇ B by sub ob jects of B × X . W e can repre sent the transformatio n induced by γ = ( A, C ) with the following map. γ ∗ Y = { ( b, x ) ∈ B × X | ∃ a ∈ A. ( b, a ) ∈ C ∧ ( a, x ) ∈ Y } Now we finally star t constructing a r ight adjoin t. γ µ Y = { ( a, x ) ∈ A × X | µγ ( a ) ↓ { b ∈ B | ( b, x ) ∈ Y } } Automatically µ tr acks the inclusio n id D( X,γ ∗ γ µ Y ) : ( X, γ ∗ γ µ Y ) → ( X , Y ). T o find a tracking for the inclusio n ( X , Y ) → ( γ µ γ ∗ Y ) let ι = { b ∈ B | ∀ x ∈ B . ∃ y ≤ x.bx ↓ y } Since the identit y a r row is combinatory , the sub ob ject ι in tersects B ′ and ι µ int er- sects A ′ . The tracking we need is ι µ . T o establish that γ µ is a well defined mapping Sub ( ∇ B X ) → Sub ( ∇ A X ), let ( X, Y ) and ( X ′ , Y ′ ) b e any pair of assemblies for ( B ′ , B ), and let U = { b ∈ B | ∀ ( x, y ) ∈ Y .bx ↓ , ( bx, y ) ∈ Y ′ } If ( a, x ) ∈ γ µ ( Y ) and u ∈ U µ , then ua ↓ and µγ ( ua ) ↓ U γ ( a ). This implies U µ tracks the inclusio n of ( X , γ µ ( Y )) into ( X , γ µ ( Y ′ )). Thu s we g et a right adjoint to γ ∗ , and a ge ometric mor phism of r elative r ealiz- ability topos e s. 21 5 Conclusion The r elative rea lizability top os for an OPCA pair ( A ′ , A ) in Set satisfies a universal prop erty: RT ( A ′ , A ) is the universal exact categor y that adds a new subo b ject to A that is closed under applica tion and that intersects all subsets that intersect A ′ , while pr eserving r egular pr op ositions. There is a construction fo r relative rea liz- ability categor ies for OPCA pairs in other Heyting ca teg ories that sa tisfies a s imilar universal prop er ty . The universal pro p erty allows us to s tudy reg ular functor s by studying filters of order partial combinatory algebr as. I thank the referees for their useful remarks. 5.1 F urt her Though ts W e consider a co uple of topics fo r future publications. Carb oni and Celia Magno [6] describ ed the exact completion o f left ex a ct cat- egories. Robinson and Ros olini showed [2 2] tha t r ealizability top oses constr uc ted ov er the categor y of sets are exa ct c ompletions of subca tegory o f par titioned as- semblies. Car b oni noted [5] that the categor y o f a ssemblies is a n intermediate step, being the r e gular c ompletion of the category of partitio ned a ssemblies. The relatio n betw een the v ar ious completions is explained in [7]. Relative realiza bilit y top oses ov er top oses where e pimo rphisms don’t split no longer a re exact completions of their categor ies of pa rtitioned assemblies. Hofstra developed the alternative no tio n of r elative c ompletio n [11], to de a l with the more general case. Relative co mpletions works for OPCA pairs ( A ′ , A ) where A ′ has enough globa l sections, which means that every inhabited sub o b ject has a global low er b ound in A ′ . It may be interesting to see if there is a na tural co mpletion construction that works for other pa irs. While the limitatio ns of Heyting categor ies requir e the universal prop erty we gav e in this pap er, it is p ossible to character ize relative realizability top oses by an- other, p o s sibly mor e useful pseudo initia lit y prop er ty . In a to p o s E every OPCA pair ( A ′ , A ) ha s a ‘completion’ ( B ′ , B ), where B = D ∗ A (see lemma 45) and B ′ ⊆ D ∗ A is the o b ject of downsets of A that intersect A ′ . W e call this a ‘co mpletion’, b ecause D ∗ A is closed under joins o f inha bited sub ob jects. Because of this completeness prop erty , every r epresentable function is g lobally r epresentable: any repres entable f : B n ⇀ B is represented by the globa l section S J f K : 1 → B , b ecause application in B preser ves joins. W e belie ve that rela tive re alizability top ose s, and so me of their subtop oses, hav e a the following universal pro p er ty inv olving complete OP CA pair and left exact functors. Let a left ex act mo del b e the combination of a left exact F : E → C with a C ⊆ F B that is upw ard clo s ed a nd clo sed under application, and through which every F x : 1 → F B ′ ⊆ F B factors. The functor ∇ : E → Asm ( A ′ , A ) together with ˚ B = ( B , { ( a, b ) ∈ A × B | a ∈ b } ) is a pseudoinitial left exa ct mo del. Using F r e y ’s analysis of the trip os-to-top o s constr uction [10], we may b e able to derive another universal prop e rty of RT ( A ′ , A ). In [13], Hofstra uses b asic c ombinatory obje cts to provide a framework for all kinds of realizability . 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