A Note on 'Extensional PERs'

📝 Abstract

In the paper “Extensional PERs” by P. Freyd, P. Mulry, G. Rosolini and D. Scott, a category $\mathcal{C}$ of “pointed complete extensional PERs” and computable maps is introduced to provide an instance of an \emph{algebraically compact category} relative to a restricted class of functors. Algebraic compactness is a synthetic condition on a category which ensures solutions of recursive equations involving endofunctors of the category. We extend that result to include all internal functors on $\mathcal{C}$ when $\mathcal{C}$ is viewed as a full internal category of the effective topos. This is done using two general results: one about internal functors in general, and one about internal functors in the effective topos.

💡 Analysis

In the paper “Extensional PERs” by P. Freyd, P. Mulry, G. Rosolini and D. Scott, a category $\mathcal{C}$ of “pointed complete extensional PERs” and computable maps is introduced to provide an instance of an \emph{algebraically compact category} relative to a restricted class of functors. Algebraic compactness is a synthetic condition on a category which ensures solutions of recursive equations involving endofunctors of the category. We extend that result to include all internal functors on $\mathcal{C}$ when $\mathcal{C}$ is viewed as a full internal category of the effective topos. This is done using two general results: one about internal functors in general, and one about internal functors in the effective topos.

📄 Content

arXiv:0901.3967v2 [math.LO] 20 Sep 2010 A Note on “Extensional PERs” W.P.Stekelenburg November 2, 2021 Abstract The paper “Extensional PERs” by P.Freyd, P.Mulry, G.Rosolini and D.Scott ([2]) identifies a reflective subcategory of the category of PERs, namely the category C of pointed complete extensional PERs, which has the interesting property of being algebraically compact with respect to realizable functors. Unfortunately, the definition of realizable functors used in [2] is too restric- tive, and this is a problem, because specifically that part of the definition that is too restrictive, is a necessary premise to the given algebraic compactness proof. Here, I present two ways to bypass this problem, and thus to complete the proof. The paper “Extensional PERs” by P.Freyd, P.Mulry, G.Rosolini and D.Scott ([2]) identifies a reflective subcategory of the category of PERs, namely the cate- gory C of pointed complete extensional PERs, which has the interesting property of being algebraically compact. Algebraic compactness ensures the existence of so- lutions to recursive domain equations (see [1]). In other words: given a functor F : (Cop)n × Cm →C, there is a fixobject X with a fixmap f : F( ⃗X) →X, which is an isomorphism. Due to this property, C is an interesting candidate for a categorical semantics of programming languages with recursively defined types. There is one restriction though: the functor F has to be realizable. The category of PERs and this subcategory of pointed CEPERs, are internal in the effective topos. Any internal functor between these categories comes with a realizer for its functorial properties. Hence the name ‘realizable functor’. Unfortunately, the definition given in the paper seems to be more restrictive: it is not clear to me that all internal functors are realizable according to the definition found in the paper. And this is a problem, because specifically that part of the definition that is too restrictive, is a necessary premise to the given algebraic compactness proof. In the research for my master thesis I found two ways to bypass this prob- lem. Firstly, weakly complete internal categories, like the category of PERs and the category of pointed CEPERs, already satisfy the weaker property of algebraic completeness. Secondly, any internal functor is isomorphic to some other internal functor that does satisfy the more restrictive definition, showing that the original proof can be used without loss of generality. 1 The Category of PERs: Notation In stead of using the notation of [2], I will write about PERs with a more usual mathematical symbolism. So: Definition 1.1 A PER is a partial equivalence relation on the natural numbers. So a PER R is a subset of N2 such that: • for all (n, m) ∈N, (m, n) ∈N (symmetry) • for all (n, m) and (m, p) ∈N, (n, p) ∈N (transitivity) 1 Any PER R forms a total equivalence relation on its domain domR := {n|(n, n) ∈ R}, and the quotients domR/R are used to define morphisms between PERs. Given n ∈domR, I use [n]R to denote the equivalence class containing n in domR/R: Definition 1.2 A morphism of PERs f : R →S, is a function f : domR/R → domS/S, which is tracked by a partial recursive function. This means that there is a partial recursive function φ such that for all n ∈domR φn is defined and f[n]R = [φn]S. These objects and morphisms form the category of PERs P. This category is Cartesian closed, basically because we can define SR to be the PER of indices of tracking partial recursive functions. So any f : R →S can be identified with the set of those natural numbers that are indices of tracking functions of f. Therefore I will sometimes use [n]R→S to refer to the function R →S that is tracked by the n-th partial recursive function. Finally, I write the application of the n-th partial recursive function to some number m as a simple juxtaposition: nm. 2 Realizable and Monotone Functors The proper definition of realizable functors, based on the idea that they are internal functors in the effective topos is: Definition 2.1 An endofunctor F of the category of PERs is realizable, if there is a single partial recursive function φ that tracks F : hom(R, S) →hom(FR, FS) for all R and S. This means that φx converges whenever [x] : R →S for every pair of PERs R and S, and that F([x]R→S) = [φx]F R→F S (1) We say that φ tracks 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 identity function on N, φi = i. Because F preserves identities, and because i tracks the identity function on any PER 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) does hold. This still doesn’t guarantee that φi = i, however. Let ψi = i and ψx = φx if x ̸= i. ψ is a recursive function, and one might wonder if it can take the place of φ, saving the original definition. Obviously, (1) is satisfied for x ̸= i. In the case that S = R, the same equation holds for i. So we’re left with the case x = i and S

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