Representation and duality of the untyped λ-calculus in nominal lattice and topological semantics, with a proof of topological completeness Murdoch J. Gabbaya, Michael J. Gabbayb aHeriot-Watt University, Scotland, UK bUniversity of Cambridge, UK Abstract We give a semantics for the λ-calculus based on a topological duality theorem in nom- inal sets. A novel interpretation of λ is given in terms of adjoints, and λ-terms are interpreted absolutely as sets (no valuation is necessary). Keywords: Nominal algebras, fresh-finite limits, lambda-calculus, spectral spaces, lattices and order, variables, nominal techniques, mathematical foundations, Fraenkel-Mostowski set theory Contents 1 Introduction 5 1.1 A very brief summary of the contributionsof this paper . ...... 5 1.2 Thepointofdualityresults . 6 1.3 Mapofthepaper ............................. 7 1.4 Alistofinterestingtechnicalfeatures. ..... 8 1.5 Whyisthispapersolong? . 9 2 Background on nominal techniques 10 2.1 Basicdefinitions............................. 10 arXiv:1305.5968v3 [cs.LO] 6 Oct 2016 2.2 Examples ................................ 12 2.2.1 Atomsandbooleans . 12 2.2.2 Cartesianproduct. 12 2.2.3 Tensorproduct . .... .... .... ... .... .... 13 2.2.4 Fullfunctionspace . 13 2.2.5 Small-supportedfunctionspace . 13 2.2.6 Fullpowerset .......................... 13 2.3 The principleof equivarianceand the NEW quantifier . ..... 14 2.4 Twolemmas................................ 17 URL: www.gabbay.org.uk (Murdoch J. Gabbay) 2.5 Furtherexamples ............................. 17 2.5.1 Small-supportedpowerset . 18 2.5.2 Strictlysmall-supportedpowerset . 18 2.6 TheNEW-quantifierfornominalsets. 19 I Nominal distributive lattices with quantification 20 3 Nominal algebras over nominal sets 20 3.1 Definition of a sigma-algebra (σ-algebra) ............... 20 3.1.1 A termlike σ-algebra ...................... 20 3.1.2 A σ-algebra............................ 21 3.2 Definitionofanamgis-algebra . 21 3.3 DualityI:sigmatoamgis . 22 3.4 DualityII:amgistosigma . 23 3.4.1 The pointwise sigma-action on subsets of an amgis-algebra . 23 3.4.2 The σ-powerset Powσ(P) ................... 24 3.4.3 Somefurtherremarks. 26 4 Nominal posets 27 4.1 Nominalposetsandfresh-finitelimits . ... 27 4.2 Compatible σ-structure ......................... 28 4.3 Definition of a nominal distributive lattice with ∀ ........... 30 4.4 Impredicativenominaldistributivelattices . ....... 32 5 The σ-powerset as a nominal distributive lattice with ∀ 34 5.1 Basicsetsoperations . 34 5.2 Setsquantification. 35 II Nominal spectral space representation 38 6 Completeness 38 6.1 Filtersandprimefilters . 38 6.1.1 Filters.............................. 39 6.1.2 Growingfilters .......................... 41 6.1.3 Growingideals . .... .... .... ... .... .... 42 6.1.4 Maximalandprimefilters . 42 6.1.5 TheZornargument. 44 6.2 Theamgis-actionon(prime)filters . 47 6.3 Injecting D intothesetofsetsofprimefilters . 48 6.4 Themapfrom x, to prime filters containing x,asamorphism. 50 2 7 Nominal σ-topological spaces 51 7.1 Thebasicdefinition... .... .... .... ... .... .... .. 51 7.2 Themap F from distributive lattices to nominal σ-topological spaces 52 7.3 Technicalinterlude: two importantpropositions . ....... 52 7.4 Compactness .............................. 55 #a 7.5 Coherent spaces: closure under σ, ∩ and T ............. 56 7.6 Completelyprimefiltersinacoherentspace . .... 57 7.7 Impredicativity ............................. 58 7.8 Themap G fromcoherentspacestodistributivelattices . 59 7.9 Soberspaces............................... 60 7.10 Nominalspectralspaces . 62 8 Morphisms of nominal spectral spaces 62 8.1 The definitionof inSpect∀, and F viewedasafunctortoit . 62 8.2 Theactionof G on morphisms in inSpect∀ .............. 66 8.3 Theequivalence.............................. 67 III Adding application and its topological dual the combination operator 69 9 inDi∀• and inSpect∀• 69 9.1 Adding • and ⊸• to inDi∀ to get inDi∀• ................. 69 9.2 The combination operator ◦: a topological dual to • and ⊸• ....... 71 9.2.1 Thebasicdefinition. 72 9.2.2 • and ⊸•,andspectralspaces. 72 9.2.3 Usefultechnicallemmas . 74 9.3 Filters in the presence of • and ⊸• ................... 74 9.3.1 Combiningfilters I (q◦y).................... 74 9.3.2 Combiningfilters II (q•x) ................... 75 9.4 Thesecondrepresentationtheorem . 77 9.4.1 Thebasicactions . 77 9.4.2 Combiningfilters III (p•q).................... 77 9.4.3 Therepresentationtheorem. 78 9.4.4 Propertiesoftherepresentation . 79 9.5 Construction of the topological space F (D), with • and ⊸• ...... 80 9.6 The duality, in the presence of • and ⊸• ................. 81 IV Application to the λ-calculus 84 10 The λ-calculus 84 10.1 Syntax of the λ-calculus ........................ 84 10.2 λ, β, and η usingadjoints........................ 85 10.2.1 λ using ∀ and ⊸• ........................ 85 10.2.2 λ as a sets operation in F (D) ................. 86 3 10.3Idioms.................................. 86 10.4 A sound denotation for the λ-calculus ................. 88 10.5 Interlude: axiomatising the λ-calculusinnominalalgebra . 90 11 Representation of the λ-calculus in inSpect∀• 91 11.1 Π-points and σ-freshness ........................ 92 11.2 Constructing Π-points,andtheiramgis-algebrastructure . 94 11.3 The left adjoint p[a7→u] to the amgis-action p[u←[a] .......... 95 11.3.1 Basicdefinition. 96 11.3.2 Two characterisations of p[a7→u] ................. 97 11.3.3 Additional lemmas about the σ-actionasanadjoint . 98 11.4 The left adjoint p[a7→u] as a σ-actiononpoints . 98 11.4.1 It is indeed a σ-action ..................... 98 11.4.2 The σ-actiondistributesoverunionandsubset . 100 11.5 Somefurtheroperationsonpoints . 100 11.5.1 The operations: ∧, ∀, •, and ⊸• onpoints . 100 11.5.2 ⊸• and ∀ make λ ......................... 101 11.6 How the σ-actiononpointscommutes . 102 11.7 Operationsonsetsofpoints. 104 11.7.1 Basicdefinitions . 104 11.7.2 Commutationproperties . 106 11.8Atopology ............................... 108 11.8.1 Giving Pnt Π atopology .................... 108 11.8.2 The compact open sets of Pnt Π ................ 109 11.8.3 Interaction of • and ⊸• with ∪ and ⊆ ............. 110 #a 11.8.4 Interaction of T withunions .................. 111 11.8.5 Proof that Pnt Π iscoherent .................. 112 11.8.6 Proof that Pnt Π issober,thusspectral . 113 11.9 Logical properties of the topology, and completeness . ........ 114 11.10Interlude: aninterestingdisconnect . ...... 115 12 Conclusions 116 12.1Relatedwork ............................... 117 12.1.1 Algebraicsemantics . 117 12.1.2 Absolutesemantics . 118 12.1.3 η-expansion........................... 119 12.1.4 Surjectivepairing. 119 12.1.5 Previous treatment of λ-calculusbytheauthors . 119 12.1.6 No conflict with topological incompleteness results ...... 120 12.1.7 Gamemodels........................... 121 12.1.8 Sheaves.............................. 121 12.1.9 Prooftheory .......................... 122 12.1.10Inwhatuniversedoesthispapertakeplace? . ... 122 12.1.11Discussionofsomedecisions . 122 12.2Futurework ............................... 123 12.2.1 Finestructureofthecanonicalmodel . 123 4 12.2.2 Weakentheaxioms. 124 12.2.3 Morestructureintheexistingaxioms . 124 12.2.4 Dualityandcardinality . 125 12.2.5 Sizeofpermutations . 125 A More on fresh-finite limits 130 A.1 Nominal algebra axiomatisation of fresh-finite limits . 130 A.2 Moreonfresh-finitelimits................. 131 B Additional properties of the canonical model 134 B.1 λ commuteswithunions. 135 B.2 Anexistentialquantifier ................. 135 1. Introduction In this paper we build a topological duality result for the untyped λ-calculus in nominal sets and prove soundness, completeness, and topological completeness. This means the following: • We give a lattice-style axiomatisation of the untyped λ-calculus and prove it sound and complete. • We define a notion of topological space whose compact open sets have notions of application and λ-abstraction. • We prove that the categories of lattices-with-λ and topological-spaces-with-λ are dual. • We give a complete topological semantics for the λ-calculus. So this paper does what universal algebra and Stone duality do for Boolean algebras [Joh86], but for the λ-calculus. We do this by combining three threads of previous work: • the nominal algebra of [GM09b]; • the study of the logical structure of nominal powersets from [Gab16] (applied in [Gab16] to first-order logic with equality, and in this paper to the λ-calculus); and • the Kripke-style models of the untyped λ-calculus from [GG14, Gab14a]. 1.1. A very brief summary of the contributions of this paper We summarise some contributions of this paper; this list will be fleshed out in the rest of the Introduction: 1. No previous duality result for λ-calculus theories exists. Duality results are interesting in themselves (see next subsection), and it is inter- esting to see how nominal techniques help to manage the technical demands of such a result. We are extremely grateful to the editor Phil Scott and to the anonymous referees for their time and help in improving the mathematics and readability of what follows. 5 2. The lattice semantics of this paper is abstract, yet complete. The well-known domains-based denotational semantics are incomplete (whereas semantics such as term models quotiented by equivalence are complete, but concrete). 3. We prove a topologicalcompletenessresult—but this should be impossible: topo- logical incompleteness results exist in the literature. Thisdependson the topology,so the fact thata notionof