A Note on "Extensional PERs"

A Note on “Extensional PERs” W.P .Stek elen burg No v em b er 2, 2021 Abstract The pap er “E xt en s ional PERs” by P .F reyd, P .Mulry , G.Rosolini and D.Scott ([2]) i dentifies a reflective sub categ ory of the category of PERs, namely the category C of p oin ted complete ext en s ional PERs, which has the interesting prop ert y of b eing algebr ai c al ly c omp act with resp e ct to r e alizable functors. Unfortunately , th e definition of realizable functors used in [2] is too restric- tive, and this is a problem, b ecause sp ecifically th a t part of the definition that is to o restrictive, is a necessary premise to the given algebraic compactness proof. Here, I present t wo w ays to bypass this problem, and thus to complete the pro of . The pap er “ Extensional P ERs” by P .F reyd, P .Mulry , G.Ro solini and D.Sco tt ([2]) identifies a reflective s u b category of the categor y of PERs, namely the cate- gory C of p oin ted complete extensio na l PE Rs, which has the in teres t ing prop ert y of be ing algebr aic al ly c omp act . Algebr a ic compactness ensur e s the exis t ence of so- lutions to recurs iv e do main e quations (see [1]). In other words: given a functor F : ( C op ) n × C m → C , there is a fixobje ct X with a fix m a p f : F ( ~ X ) → X , which is an iso morphism. Due to this pro perty , C is an in teresting candidate for a catego r ical semantics of progr amming languages with recursively defined types. There is one res t ric t ion though: the functor F has to be r e alizable . The ca teg ory of PERs and this sub category of p ointe d CEPERs , are internal in the effective top o s. An y internal functor b et ween these categor ie s comes with a rea lizer for its functor ial prop erties. Hence the na me ‘realizable functor’. Unfor t unately , the definition given in the pap er se e ms to b e more res t ric t ive: it is not c lear to me that a ll internal functors are r ealizable according to the definition found in the pap er. And this is a problem, b ecause sp ecifically that par t of the definition that is to o restrictive, is a necessary premise to the given algebr aic compactness pro of. In the research for my ma ster thesis I found tw o wa ys to bypass this prob- lem. Firstly , weakly c omplete int er na l c ategories, like the catego ry of PE Rs and the c ategory of p oin ted CEPE Rs, already s a tisfy the weak er pr operty of algebra ic completeness. Secondly , any internal functor is iso morphic to some other internal functor that do es sa tisfy the more r estrictiv e definition, showing that the original pro of can b e used witho ut loss of genera lity . 1 The Category of PERs: Notation In stea d o f using the notation of [2], I will wr it e ab out PE Rs with a mor e usual mathematical symbolism. So: Definition 1.1 A PER is a p artial e quivalenc e r elation o n the natura l num b ers. So a PER R is a subset of N 2 such that: • for all ( n, m ) ∈ N , ( m, n ) ∈ N ( symmetry ) • for all ( n, m ) and ( m, p ) ∈ N , ( n, p ) ∈ N ( tr ansitivity ) 1 An y PER R for ms a total equiv a lence r elation on its domain dom R := { n | ( n, n ) ∈ R } , a n d the quo tien ts dom R/R are used to define mor phisms b et ween PERs. Given n ∈ dom R , I use [ n ] R to denote the equiv a lence class containing n in dom R/ R : Definition 1.2 A morphism of PERs f : R → S , is a function f : dom R /R → dom S/S , which is t r acke d b y a par tial recursive function. This means that there is a partial recursive function φ s u ch that for all n ∈ dom R φn is defined and f [ n ] R = [ φn ] S . These ob jects a nd morphisms for m the ca tegory of P ERs P . This categ o ry is Cartesian closed, basically b ecause we c a n define S R to be the P ER of indices of tracking pa rtial recur siv e functions. So a ny f : R → S can b e identified with the set of thos e natura l num b ers that are indices o f tracking functions of f . Therefo r e I will sometimes use [ n ] R → S to re f er to the function R → S that is track ed by the n -th partia l recursive function. Finally , I write the a pplication of the n -th par t ial recursive function to some num b er m as a simple juxtap osition: nm . 2 Realizable and Monotone F unctors The pro per definition of re alizable functors , bas e d on the idea that they a re internal functors in the effective top os is: Definition 2.1 An endofunctor F of the categor y o f P ERs is r e alizable , if there is a single par t ial r ecursiv e function φ tha t tr ac ks F : ho m ( R, S ) → hom( F R , F S ) for all R and S . This means that φx conv erges whenev er [ x ] : R → S for every pair of PERs R and S , and that F ([ x ] R → S ) = [ φx ] F R → F S (1) W e say that φ tra c ks F is this case. The definition is similar to the definition found in [2]. What is left out, is the requirement that for some index i of the iden tity function on N , φi = i . Because F preserves identities, a nd b ecause i tracks the identit y function on any PE R R , we know that F ([ i ] R → R ) = [ φi ] F R → F R = [ i ] F R → F R . So i ∈ T R F ([ i ] R → R ) do es hold. This still do esn’t guar an tee that φi = i , how ever. Let ψ i = i a nd ψ x = φx if x 6 = i . ψ is a recurs iv e function, and o ne might wonder if it can tak e the place of φ , saving the orig inal definition. Obviously , (1 ) is satisfied for x 6 = i . In the case that S = R , the same equation holds for i . So w e’re left with the case x = i and S 6 = R . Now note that [ i ] : R → S iff R ⊂ S . Therefore if R ⊂ S , and if F ([ i ] R → S ) = [ ψ i ] F R → F S = [ i ] F R → F S , then F R ⊂ F S . This mea n s that all functors which a re track ed b y an i preser ving function are monotone mappings o f PE Rs. On the other hand, if a functor is monotone this way , and has a tra c king function φ , the function ψ defined ab ov e is another tra c king function, and this one is i pr e serving. So the functors defined in [2] are a sp ecial kind of r e a lizable functor: Definition 2.2 A r ealizable endofunctor F of the categ ory of PERs is monotone , if its ob ject map is monotone with r espect to the inclus ion order ing on P ERs. In other words: if R ⊂ S , then F R ⊂ F S . I hav e not b een able to prove (or r efut e, b y the w ay) that all re alizable functors are monotone , or to find a pro of in the literature. Sadly , in [2] the least fixp oin ts that monotone functors ha ve, are used in the algebra ic compactness pro of: for any monotone functor F we hav e a fixp oin t X := T { R | F R ⊂ R } , wher e F X = X and i represents the fixmap. 2 3 Algebraic Completeness I’ll start with the following genera l result: Lemma 3.1 Given any top os E , and any c omplete internal c ate gory C . C is al- gebr aic al ly c omplete, me aning: for any internal endofunctor F , ther e is an initial algebr a. Pro of . E allows the constructio n of the category of algebras o f any endofunctor F of C internally , so b oth the category of F -algebr as F - a lg, and the underlying ob ject functor U : F -alg → C are in ternal to E . Now this underlying functor creates limits, and since C is c omplete (rela tiv e to E ), F -alg must b e complete to o. Therefore it has an initial ob ject, which is a n initial a lg ebra for F .  In this genera l pro of we actually only need the limit over one functor , na m ely the underlying ob ject functor U : F -alg → C . The c ategory of PERs P is a we akly complete full in ternal subca t eg ory of the effective topos. W eak completeness means that although for arbitrar y internal categor ies D and internal functors F : D → C a limiting cone exists, ther e is no int er nal functor C D → C adjoint to the functor K : C → C D that maps ob jects to constant functor s. This, how ever, still suffices to prove a limit exists for the underlying PER functor U : F -alg → P , and that weakly complete catego ries are also algebraic ally c o mplete. When we work out the constructio n of this limit, which is supp osed to be a sub o b ject o f the pro duct of a ll alg ebras, we get something like this: fir s tly , if we fix a P ER R , then [ F R → R ] is a PER of a ll alg e b r a s based on R . Lets say the limit of U is R 0 , then every element f ∈ R 0 restricts to a mapping f R : [ F R → R ] → R . This is a mor phis m of PERs, b ecause the categ ory of PERs is a full subc a tegory of the effective top os. As a co ns equence f R itself is an element o f the P E R [[ F R → R ] → R ]. No w [[ F R → R ] → R ] R ∈P is a family of PE Rs indexed by the class of PERs itself, and we can take R 0 ⊂ Q R ∈P [[ F R → R ] → R ]. Secondly , the ob ject o f PE Rs exists within the effective top os, and is unifor m. This mak es T R ∈P [[ F R → R ] → R ], the intersection o f this family of P ERs, alre ady its pr oduct inside the ca tegory of P ERs (see [4]). So to find R 0 , we only need to select thos e elements of T R ∈P [[ F R → R ] → R ] that commute w ith all the alge br a morphisms. The r esults in the pa p er [6] seem to suggest tha t R 0 = T R ∈P [[ F R → R ] → R ]. But in any case, the underlying P ER of the initial algebra is: R 0 :=    ∀ ( R, a ) , ( S, b ) , ( T , c ) ∈ F -alg 0 , ( f , f ′ ) ∈ N 2 ∀ m : ( R, a ) → ( T , c ) , m ′ : ( S, b ) → ( T , c ) . ( m ( f a ) , m ′ ( f ′ b )) ∈ T    (2) Given an alge br a ( R, a ), we g et the pro jection ma p π a : R 0 → R , simply defined by ev a luation: π a ( f ) = f a . These pr o jections taken together form the limiting cone, which justifies calling R 0 the limit of U . O bviously , any alg e b r a structur e c on R 0 has to make the following diag ram commute for any a lg ebra ( R, a ): F R 0 F π a / / c   F R a   R 0 π a / / R That means that for all ( x, y ) ∈ F R 0 , ( cxa, a ( F π a y )) ∈ R . This can be a c hieved by letting c := λxa.a ( φπ a x ) (for an y φ tracking F ). The construction ab o ve shows that the category P of PE Rs is algebra ically complete. The catego ry C o f p oin ted CEPERs is a reflec t ive s ubcategory of P , as is 3 shown in [2] (the prove of this fact do esn’t use mono to n y , and is sound). Therefor e it inherits w eak completeness from P , and we may use a similar construction to find an initial a lgebra for any endofunctor . With the theory developed in [3] the fact that the ca tegory of pointed CEPERs is a CPO category can b e used to pr o ve that it is algebr a ically compact, b e c ause it is a lgebraically complete. 4 Y o neda Before we get to the Y oneda lemma, we need to know some things a bout natural transformatio ns betw een realiza ble functors: Definition 4.1 A natural transforma tion η b et ween t wo rea liz a ble endo functor s F and G of the ca tegory of PE R s is r ealizable, iff there is a single num b er e such that η R = [ e ] F R → G R for all PERs R . Again, rea liz abilit y is what makes the transfo r mations internal to the effective to p os. In this case the definition given in [2] is cor rect. Because natura l tr ansformations a re r epresen ted by natural n umbers – or b e- cause the categ o ry of PE Rs is weakly complete and internal: it al dep ends on your per spective – we can cons tr uct a PER of natural transfor m atio ns b et ween any pa ir of PER v alue d functors. In fact: categories o f PER v alued functors ar e enriched ov er the ca tegory o f PERs, a s long as the do m a ins ar e int er n al categ ories of the effective top os. Theorem 4. 2 Every endofunctor of P is n a tu r al ly isomorphic t o a monotone end- ofunctor. Pro of . W e know b ecause o f Y oneda’s lemma that F X ≃ nat(hom( X , − ) , F ) nat- urally in b oth F and X , and when F is and endofunctor of P , then the map- ping F ∗ satisfying F ∗ X = na t (hom( X , − ) , F ) can be turned into an endofunctor to o: for a n y morphism f : X → Y , if φ tr ac ks f , then the induced morphism f ∗ : na t( hom( X , − ) , F ) → nat(hom( X , − ) , F ) is track ed b y λxy .x ◦ y ◦ φ . Now F ∗ happ ens to b e mono tone: If X ⊂ Y and [ n ] : Y → Z , then [ n ] : X → Z b ecause ( nx, ny ) ∈ Z whenever ( x, y ) ∈ Y and ( x, y ) ∈ Y whenever ( x, y ) ∈ X . Therefore hom( Y , − ) ⊂ hom( X, − ) po in t wise. F urthermore: if i is a n index of the identit y function, it determines a natural transfor ma t io n [ i ] : hom( Y , − ) ⇒ hom( X , − ). Let [ i ] : F ⇒ G for any t wo functors F and G , a nd let n : G ⇒ H , then n : F ⇒ G , b ecause n ◦ i r epresen ts the sa me p.r.f as n . Therefore nat( G, − ) ⊂ nat( F, − ), and even i : nat( G, − ) ⇒ nat( F, − ) p oin t wise. W e see that if X ⊂ Y , then F ∗ X ⊂ F ∗ Y . Ther efore F is a monotone functor .  Although so m e internal functors may not be mo notone, the a ssumption that realizable functors a re, can b e made without lo s s of genera lit y . With this informa- tion added the original pro of suffices to show that the categor y o f p ointed complete extensional PERs is indeed algebr aically compact. References [1] P . F reyd: ‘Algebraica lly Complete Categor ies’, p. 95- 104 in A. Carb oni, M. C. Pedicc hio, G. Ros olini: ‘Categ ory Theory - P roceeding s of the International Conference held in Co mo , Italy , July 22-2 8, 1990 ’, Springer-V erla g Berlin Heidelber g 1 991 4 [2] P . F reyd, P .Mulry , G.Rosolini, D.Scott: ‘Extensional P ERs’, p. 3 46-354 in ‘Pro ceedings of the fifth annual conference of Lo gic in Computer Science 199 0’ (LICS 90) [3] P . F r eyd: ‘Recursive Types Reduced to Inductiv e Types ’, p. 4 98-507 in ‘Pro- ceedings of the fifth annual conference of Lo g ic in Computer Science 1990 ’ (LICS 90) [4] G. Ros o lini: ‘Ab out Mo dest Sets’, Int. J. F ound. Comp. Sci. 1:341- 353,1990 [5] J. M. E. Hyla nd : ‘The effective topos ’, p. 165 - 216 in A. S. T ro elstra, D. v an Dalen ‘The L. E. J . Br ou wer Centenary Symposium’, No ord Holla nd Publishing Company Amsterdam 198 2 [6] P . F re y d, E. P . Robinson, G.Roso lini: ‘Dinaturalit y for free’, Pro cs. SA CS (M.F our ma n, P .Johnsto n e, A.Pitts, eds.) p. 107 -118, C UP , 1 992 5