Cubical Categories for Higher-Dimensional Parametricity

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by  Patricia Johann

Cubical Categories for Higher-Dimensional
Parametricity
Patricia Johann and Kristina Sojakova
Appalachian State University, Boone, NC, USA
email: {johannp, sojakovak}@appstate.edu
7
Abstract—Reynolds’ theory of relational parametricity formal- way that the relational actions interpreting its type formers
1
0 izes parametric polymorphism for System F, thus capturing the propagate relatedness up its type hierarchy. For each logical
ideathatpolymorphicallytypedSystemFprogramsalwaysmap
2 relation for a language, a parametricity theorem can then be
relatedinputstorelatedresults.ThispapershowsthatReynolds’
n theory can be seen as the instantiation at dimension 1 of a proved. Such a theorem states that (the basic interpretation
a theory of relational parametricity for System F that holds at all of) each of the languages’ programs is related to itself by
J higher dimensions, including infinite dimension. This theory is therelationalinterpretation,viatheassociatedlogicalrelation,

3 formulatedintermsofthenewnotionofap-dimensionalcubical of that program’s type. When instantiated judiciously, this
2 category, which we use to define a p-dimensional parametric
seemingly simple result can be used to prove, inter alia,
model of System F for any p ∈ N ∪ {∞}. We show that
invariance of polymorphic functions under changes of data
] every p-dimensional parametric model of System F yields a
O split λ2-fibration in which types are interpreted as face map- representation[1],[5],equivalencesofprograms[11], andso-
L anddegeneracy-preserving cubicalfunctorsandtermsareinter- called “free theorems” via which properties of programs can
. preted as face-map and degeneracy-preserving cubical natural be inferred solely from their types [20].
s transformations. We demonstrate that our theory is “good” by
Therecentbifibrationaltreatmentofrelationalparametricity
c
showing that the PER model of Bainbridge et al. is derivable
[ as another 1-dimensional instance, and that all instances at in [8] has put forth a more abstract notion of a parametric
1 all dimensions derive higher-dimensional analogues of expected model of polymorphism. In this treatment every type is still
v results for parametric models, such as a Graph Lemma and the givenainterpretationinasufficientlystructuredbasecategory,
4 existence of initial algebras and final coalgebras. Finally, our together with a relational interpretation in a category of (now
4 technical development resolves a numberof significant technical abstractlyformulated)relationsoverthatbasecategory,butthe
2 issues arising in Ghani et al.’s recent bifibrational treatment twointerpretationsaredefinedsimultaneouslyandarerequired
6 of relational parametricity, which allows us to clarify their
0 approach and strengthen their main result. Once clarified, their to be connected via a sufficiently structured bifibration. The
. bifibrational framework, too, can be seen as a 1-dimensional expressaim of [8] is to provide a very generalframeworkfor
1 instance of our theory. relationalparametricitythatis directlyinstantiablenotonlyto
0
recoverwell-knownrelationallyparametricmodels—suchas
17 I. INTRODUCTION Reynolds’ original model1 and the PER model of Bainbridge
: Strachey [19] distinguishedbetween ad hoc and parametric et al. — but also to deliver entirely new models of relational
v
polymorphic functions in programming languages, defining parametricity for System F.
i
X a polymorphic program to be parametric if it applies the Unfortunately, however, models of relational parametricity
r sametype-uniformalgorithmateachofitstypeinstantiations. often require more careful notions of functor and natural
a
Reynolds[16] introducedthe notion of relationalparametric- transformation than just the standard categorical ones used
ity to model the extensional behavior of parametric programs in the bifibrational framework of [8]. For example, functors
inSystemF[9],theformalcalculusatthecoreofpolymorphic and natural transformations must be internal to the category
functionallanguages.Relationallyparametricmodelscapturea of types and terms in the Calculus of Inductive Constructions
keyfeatureofparametricprograms,namelythattheypreserve with impredicative Set to recover Reynolds’ original model,
allrelationsbetweeninstantiatedtypes.Inotherwords,inrela- and they must be internal to the category of ω-sets to recover
tionally parametric models, parametric polymorphicfunctions Bainbridge et al.’s PER model. As a result, neither of these
always map related arguments to related results. models is a true instance of the bifibrational framework
Implicit in Reynolds’ original formulation of relational of [8]. Said differently, the bifibrational framework of [8] is
parametricity [16] is that a model of System F is relationally not actually an extension of Reynolds’ theory of relational
parametric if equality in the model is induced by a logical parametricity as claimed. In fact, showing that Reynolds’
relation. A logicalrelation assigns to each type of a language original model is parametric in the sense specified by the
not only a basic interpretation as, say, a set or a domain, but
simultaneously an interpretation as a relation on that set or 1Since there are no set-theoretic models of System F, by the phrase
“Reynolds’ original model” we will mean the version of his model that is
domain as well. Logical relations are defined by induction on
internaltotheCalculusofInductiveConstructionswithImpredicativeSet(as
the language’s type structure, and are constructed in such a indicated in[8]).
bifibrationalframeworkrequiresa complete redevelopmentof parametric models for System F are instances of our theory.
the framework internally to the Calculus of Inductive Con- It is also precisely the mechanism by which we restrict the
structions with impredicative Set, and showing that the PER possibleinterpretationsoftypesandtermsinourλ2-fibrations
model is parametric requires a redevelopment internal to the sufficientlytoexcludeadhocpolymorphism.Thisisillustrated
categoryofω-sets.Theneedtoredeveloptheentireframework concretely in Example 29 below.
of [8] internal to a different category for each relationally In addition to remedying all known problems with the
parametricmodelofinterestmakesthebifibrationalframework bifibrational framework of [8], the theory we develop in this
moreofa“blueprint”forconstructingparametricmodelsthan paper also naturally opens the way to a theory of relational
ageneraltheorythatactuallyincludesknownmodelsproperly parametricity at higher dimensions. Indeed, our theory de-
among its instances. The fact that such redevelopments must ploys the two key ingredients identified above not solely
also be carriedouton an adhoc basis, withoutany generally- in a bifibrational setting similar to that in [8], but also in
applicable guidance, only emphasizes the need for a truly combination with ideas inspired by the theory of cubical
instantiable theory of relational parametricity. sets [2], [13], [10]. In this way, it delivers λ2-fibrations that
But even if uniform guidance for instantiating the frame- model more than just the single “level”, or “dimension”, of
work of [8] were to be given, the framework itself would relational parametricity originally identified by Reynolds and
still be problematic. Unless fibred functors are required to consideredin[8].Toenforcerelationalparametricityathigher
preserve equality on the nose, neither composition nor sub- dimensions we introduce the new notion of a p-dimensional
stitution in (what is intended to be) the base category of cubical category, in terms of which we define the equally
the λ2-fibration constructed in the main theorem of [8] can new notion of a p-dimensional parametric model for System
be defined in any standard way. But equality in [8] is only F. Here, the dimension p can be any natural number or ∞.
defined — and therefore can only be preserved — up to Intuitively, cubical categories generalize cubical sets in the
isomorphism. And even if the original bifibration from which obvious way, by considering the codomain Cat instead of
the λ2-fibration in [8] is constructed (U in the terminology Set. Technically, the codomain will be Cat(C), the category
there) were assumed to be split, so that the equality functor of categories internal to some sufficiently structured ambient
were defined uniquely rather than only up to isomorphism, category C. Cubical categories have (essentially) the same
Reynolds’ original model still would not be an instance of algebraic structure as cubical sets, except that the morphisms
the bifibrational framework given there: in that case, neither in their domain category are restricted to just those gener-
productsnorexponentialswouldpreserveequalityonthenose, ated by face maps and degeneracies, i.e., to just those have
as would be needed to properly interpret arrow types. In the naturalinterpretationsas operationson relations. Thisensures
absenceofanyalternativedefinitionsordiscussionoftheexact that morphisms are restricted to those that, intuitively, have
senseinwhichfibredfunctorsarerequiredtopreserveequality, interpretations as operations on relations.
we can only conclude that the standard definitions are the Our main technical result (Theorem 25) shows that every
intended ones. As a result, we regard the entire λ2-fibration p-dimensional parametric model of System F gives rise to a
as being ill-defined, and the beautifulideas explored in [8] as split λ2-fibration. When combined with a suitable variant of
being in need of careful technical reconsideration. Seely’sresultthateverysplitλ2-fibrationgivesrisetoasound
This paper provides precisely such a reconsideration, as modelof System F (Theorem26), thisallows usto provethat
well as a significant extension. We remedy both of the afore- every p-dimensional parametric model of System F gives a
mentioned difficulties by developing a unifying approach to sound model of that calculus in which types are interpreted
relational parametricity that turns the bifibrational “blueprint” asp-dimensionalfacemap-anddegeneracy-preservingcubical
for constructing parametric models for System F given by functorsandtermsare interpretedasp-dimensionalfacemap-
the framework of [8] into a single theory whose instantia- preserving and degeneracy-preserving cubical natural trans-
tion actually delivers such models. Our theory combines two formations. This, our main result, appears as Theorem 27
key technical ingredients to produce λ2-fibrations that not below. It strengthens the analogous result in [8], which states
only are actually well-defined, but do really model relational that natural transformations interpreting terms must be face
parametricity. First, we ensure that the categories necessary map-preserving when p = 1, but does not observe that
to our constructions are well-defined by parameterizing our they can also be proved to be degeneracy-preserving even in
theory over a class of “good” natural isomorphisms, and the 1-dimensional setting. Because they interpret System F
requiringthatfibredfunctorspreserveequalityonlyuptothese terms as face map- and degeneracy-preservingcubicalnatural
isomorphisms and (essentially) that fibred functors transfor- transformations, thereby ensuring that these terms cannot
mations preserve them. Secondly, we ensure that well-known exhibit ad hoc polymorphic behavior, we contend that p-
models of relational parametricity for System F are properly dimensional parametric models of System F are deserving of
instances of our theory by parameterizing it over a suitably theirname.ThatbothReynolds’modelandthatofBainbridge
structured ambient category and working internally to that et al. are both p-dimensional parametric models for System
category. But working internally to an appropriate ambient F when p = 1 further shows our definition is both sensible
category is more than just a technical device ensuring that and good. Additional evidence is provided in Section VI,
all of our constructions are well-defined and that well-known whereitis shownthatallp-dimensionalparametricmodelsof
SystemFvalidatehigher-dimensionalanaloguesofthe“litmus extending the function mapping each object P of E to Σ P
f
test” properties for “good” parametric models. That is, they is the opreindexing functor along f.
validate a higher-dimensional Identity Extension Lemma, a In this paper we will construct a certain kind of fibration,
higher-dimensionalGraphLemma,andthe existenceofinitial called a λ2-fibration [18], that models higher-dimensional
algebras and final coalgebras for face map- and degeneracy- parametricityforSystemF. Fibrationswillalsobeessentialto
preserving cubical functors. defining a higher-dimensional graph functor, and bifibrations
Instantiations of our theory for specific choices for p may will be crucial to formulating an alternative characterization
already be of particular interest. When p = 2, our notion of of the graph functor that allows us to prove both a higher-
a p-dimensional parametric model of System F formalizes a dimensional Graph Lemma, and the existence of initial al-
notion of proof-relevantrelational parametricity that properly gebras and final coalgebras of face map- and degeneracy-
generalizesReynolds’originaltheory.When p=∞, we geta preserving cubical functors.
notionofinfinite-dimensionalrelationalparametricityforSys-
temFthatmayprovideausefulperspectiveonthehomotopy- III. CUBICAL CATEGORIES
canonicityconjectureforhomotopytypetheory,sinceproofof Cubicalcategories,functors,andnaturaltransformationsare
this conjecture involves constructing an infinitely parametric the key structures from which we will construct our models
model of Martin-Lo¨f type theory. Investigation of this matter of higher-dimensional parametricity. To define them we start
is, however, beyond the scope of the present paper. with the following preliminary category.
II. FIBRATIONAL PRELIMINARIES Definition 3. The category (cid:3) is given as follows:
We give a brief introduction to fibrations, mainly to settle • theobjectsare(finite)setsofnaturalnumbersoftheform
notation. More details can be found in, e.g., [12]. {0,...,l−1} for l ∈N
• themorphismsfroml1tol2arefunctionsfroml1tol2+2,
Definition 1. Let U : E → B be a functor. A morphism g : where 2 is the two-element set {⊤,⊥}
Q → P in E is cartesian over f : X → Y in B if Ug = f
• the identity morphism on l is (induced by) the identity
and, for every g′ : Q′ → P in E with Ug′ = f ◦v for some function on l, i.e., is the inclusion map l ֒→l+2
v :UQ′ →X,thereexistsauniqueh:Q′ →Qwith Uh=v
and g′ = g◦h. A morphism g : P → Q in E is opcartesian • lthe→colmpisosthiteiofnunocftitownogm◦ofrp:hlism→s fl :+l12→defiln2eadnbdyg :
over f : X →Y in B if Ug =f and, for every g′ : P →Q′ 2 3 1 3
in E with Ug′ =v◦f for some v :Y →UQ′, there exists a ⋆ if f(i)=⋆,⋆∈2
(g◦f)(i)=
unique h:Q→Q′ with Uh=v and g′ =h◦g. (g(j) if f(i)=j,j ∈N
Wewritef§ forthecartesianmorphismoverf withcodomain
P We will henceforth denote a set {0,...,l−1} of natural
P, and fP for the opcartesian morphism over f with domain
§ numbers by l. (Set-theoretically, these are identical.) We call
P. Such morphisms are unique up to isomorphism. If P is anysuchset,i.e.,anyobjectof(cid:3),alevel.Thecategory(cid:3)can
an object of E then we write f∗P for the domain of f§ and
P also be described as the Kleisli category for the error monad
Σ P forthecodomainoffP.WeomitP fromthesenotations
f § withtwodistincterrorvalues.Thelastbulletpointthendefines
when it is either unimportant or clear from context. composition of morphisms in (cid:3) to be normal composition of
Definition 2. A functor U :E →B is a fibration if for every functions, except that errors are propagated.
objectP ofE andeverymorphismf :X →UP ofB,thereis The category (cid:3) contains all functions from l1 to l2 + 2
acartesianmorphismfP§ :Q→P inE overf.Similarly,U is for all l1 and l2. But to model parametricity, we will want
an opfibration if for every object P of E and every morphism to restrict the set of morphisms to those that, intuitively,
f : UP → Y of B, there is an opcartesian morphism fP : haveinterpretationsasoperationsonrelations.Forthisreason,
§
P →Q in E over f. A functor U is a bifibration if it is both only the face maps and degeneracies defined below, and their
a fibration and an opfibration. compositions, are used to construct our cubical categories.
The interpretations of the face maps and degeneracies in the
If U : E → B is a fibration, opfibration, or bifibration, then
specific setting of [8] are given in Example 6.
E is its total category and B is its base category. An object For any ⋆ ∈ 2 and l,k ∈ N with k ≤ l, we define the
P in E is over its image UP and similarly for morphisms. A function f (l,k):l+1→l by
⋆
morphismisverticalifitisoverid. We writeE forthefiber
X
over an object X in B, i.e., the subcategory of E of objects i if i<k
over X and morphisms over idX. For f : X → Y in B, the f (l,k)(i)= ⋆ if i=k
⋆ 
function mapping each object P of E to f∗P extends to a
i−1 if i>k
functor f∗ : E → E mapping each morphism k : P →
Y X
P′ in EY to the morphism f∗k with kfP§ = fP§′f∗k. The Such a function is called a face map. The terminologycomes
universalpropertyoff§ ensurestheexistenceanduniqueness from regarding each natural number l as defining an l-
P′
of f∗k. We call f∗ the reindexing functor along f. A similar dimensional cube. A face map f (l,k) then can be thought
⋆
situation holds for opfibrations; the functor Σ : E → E of as projecting an (l+1)-cube onto either the “top” or the
f X Y
“bottom” l-dimensional cube in dimension k, according as ⋆ subsumes well-known models as (1-dimensional) relational
is ⊤ or ⊥. Similarly, for any l,k ∈ N with k ≤ l, we define parametricity as instances of our framework. For example,
the function d(l,k):l →l+1 by Reynolds’originalmodelisaninstancewhenC istakento be
the category of types and terms in the Calculus of Inductive
d(l,k)(i)= i if i<k Constructions with impredicative Set3, and the PER model
(i+1 if i≥k of Bainbridge et al. is an instance when C is the category of
ω-sets; see Examples 28 and 29 below for details.
Such a function is called a degeneracy. A degeneracy d(l,k)
Let Cat(C) be the category of categories internal to an
can be thoughtof as constructingan (l+1)-dimensionalcube
ambient category C.
fromanl-dimensionalonebyreplicatingitalongdimensionk.
We also have the following set of categories (cid:3)p: Definition 5. A (p-dimensional) cubical category is a functor
Definition 4. Let p ∈ N ∪ {∞}. The category (cid:3) is the X :(cid:3)p →Cat(C).
p
subcategory of (cid:3) generated by the following data:
To ease the notational and conceptualburden, we will hence-
• levels l≤p forth regard a (p-dimensional) cubical category as a functor
• face maps f⋆(l,k) for l<p X : (cid:3)p → Cat, and similarly identify internal and external
• degeneracies d(l,k) for l <p constructions when convenient. Under this identification, a p-
Alternatively,wecandefine(cid:3) asthefreecategorygenerated dimensional cubical category X becomes the category-level
p
by the the data above and the following relations: equivalentofacovariantpresheafon(cid:3)p,andgivesusa(small)
• f⋆(l,k)◦d(l,k)=1l for l<p category X(l) for each level l ∈ (cid:3)p. The category X(0)
• f⋆(l,j)◦d(l,k)=d(l−1,k)◦f⋆(l−1,j−1) for k <j can be thought of as an abstract category of “0-relations”,
or “types”; X(1) can be thought of as the category of “1-
and l<p
• f⋆(l,j)◦d(l,k)=d(l−1,k−1)◦f⋆(l−1,j) for j <k relations”,orordinaryrelations,ontypes;X(2)canbethought
of as the category of “2-relations”;and so on. Each face map
and l<p
• d(l+1,j)◦d(l,k)=d(l+1,k)◦d(l,j−1) for k <j f :l+1→l,X(f)isthusafunctorprojectinganl-relationout
of a given (l+1)-relation,and each degeneracyd:l→l+1,
and l+1<p
• f⋆2(l,j)◦f⋆1(l+1,k) = f⋆1(l,k)◦f⋆2(l+1,j+1) for aXn(d(l)+is1a)-rfeulnactitoonr.that replicates a given l-relation to obtain
k ≤j and l+1<p
The alternative characterization coincides exactly with Example6. Inthesettingof[8],arelationsfibrationRel(U):
Crans’ combinatorytreatmentof cubes. As provedin [4], any Rel(E) → B ×B induces a 1-dimensional cubical category,
morphismin(cid:3) canbefactoredasacompositionoffacemaps with the action on objects given by 0 7→ B and 1 7→ Rel(E).
p
followed by a composition of degeneracies. Moreover, the The action on morphisms is inducedby mappingthe two face
second arguments to the degeneracies in these compositions maps f⊤(0,0) and f⊥(0,0) to the functors fst◦Rel(U) and
arenon-increasing,andthesecondargumentstothefacemaps snd◦Rel(U)respectively,andmappingthedegeneracyd(0,0)
are strictly decreasing, when read in composition order, i.e., to the equality functor Eq on B.
from right to left. Such a factorization gives a representation
If X is a cubical category we define the discrete cubical
ofeach morphismasa surjectionfollowedbyan inclusion,as
category|X|, andthe productcubicalcategoryXn forn∈N,
well as a canonical form for each morphism in (cid:3) .
p via the usual constructions for functors. Each construction on
The categories(cid:3) will serve as the domainsof our cubical
p cubicalcategories actually requires an analogousconstruction
categories.As such, they are ouranaloguesof the categoryof
on Cat(C), but these are precisely as expected. For example,
names and substitutions, which formsthe commondomain of for cubical categories X,Y : (cid:3) → Cat(C), the cubical
p
all cubical sets in [2]. The differencesbetween our categories category X ×Y : (cid:3) → Cat(C) is defined by (X ×Y)(l) =
(cid:3) and the category of names and substitutions are that (cid:3) p
p p X(l)×Y(l) for all l ≤p. Here, the product on the left-hand
doesnotincludethe “exchangemorphisms”of [2] (and[10]),
side is a product of functors, and the product on the right-
and that membership in (cid:3) does not explicitly require mor-
p hand side is a product of internal categories. The latter exists
phisms to be injective. That all morphisms in each (cid:3) are, in
p because C is finitely complete by assumption.
fact, injective follows from the injectivity of (cid:3) ’s generators.
p
In the remainder of this paper we will always work in- Definition 7. Let X and Y be (p-dimensional) cubical cate-
ternally with respect to a finitely complete locally small gories. A (p-dimensional) cubical functor F from X to Y is
ambient category C, i.e., a category C with pullbacks and a a set of functors {F(l) : X(l) → Y(l) | l ≤ p}. A cubical
terminal object 1C.2 Instantiating C appropriately will impose functor F is face map-preserving if the following diagram
conditions on functors and natural transformations that allow commutes for every face map h:l →l in (cid:3) :
1 2 p
ustoproduceatheoryofhigher-dimensionalparametricitythat
2Therequirement that C hasallpullbacks is actually strongerthan neces- 3Thequitesurprisingfactthatthistheoryhasnotyetactuallybeenproved
sary.Infact,weneedonlyrequire C tohaveall“composable” pullbacks. consistent is,however, worthnoting.
F(l1) namely,therequirementthatcubicalnaturaltransformationsbe
X(l1) Y(l1)
bothfacemap-anddegeneracy-preserving—thusgeneralizes
X(h) Y(h) structure already present in the 1-dimensional setting of [8].
Weexploitthisrequirementoncubicalnaturaltransformations
X(l2) Y(l2) to construct λ2-fibrations in which terms of System F are
F(l2)
interpreted as face map- and degeneracy-preserving cubical
A cubical functor F is degeneracy-preserving if the diagram naturaltransformations.Evenwhenp=1thisgivesastronger
above commutes up to a chosen natural isomorphism ε (h) result than is obtained in [8], where natural transformations
F
for each degeneracy h:l →l in (cid:3) . interpreting terms are not shown to be equality-preserving.
1 2 p
In the setting of [8], a fibred functor from |Rel(U)|n to
IV. FIBRATIONAL MODELSOF HIGHER-DIMENSIONAL
Rel(U) is precisely a face map-preserving cubical functor
PARAMETRICITY
from |Rel(U)|n to Rel(U) (presented as cubical categories).
Similarly,anequality-preservingfibredfunctorfrom|Rel(U)|n In this section we assume a fixed (p-dimensional) cubical
to Rel(U) is a face map-preserving cubical functor that also category Rel. To ensure that composition (and thus substitu-
preservesdegeneraciesupto“good”naturalisomorphisms(for tion) in the base category of the λ2-fibrations we construct is
a suitable notion of “good”; see Section IV below). well-defined, as well as that the cubical functors interpreting
System F types will be not just face map-preserving but
Definition8. LetF andG be(p-dimensional)cubicalfunctors
also degeneracy-preserving, we need to consider equality of
fromX toY.A(p-dimensional)cubicalnaturaltransformation
morphisms up to (certain kinds of) natural isomorphisms.
η from F to G is a set of natural transformations {η(l) :
We therefore assume that our fixed cubical category Rel
F(l) → G(l) | l ≤ p}. A cubical natural transformation η is Σ
comes equipped with a class M = M(l) of “good”
l≤p
face map-preservingifF andG areface map-preservingand,
isomorphisms of the form J → Rel(l) , where J is some
for each face map h : l → l in (cid:3) , the following equality 1
1 2 p object of C and Rel(l) is the object of morphisms in the
1
holds for every object X of X(l ):
1 internal category Rel(l). We require that each M(l) contains
all identity morphisms in Rel(l), is closed under composition
Y(h) η(l )X =η(l ) X(h)X
1 2
and inverses in Rel(l), and is closed under reindexing by
Acubicalnaturaltr(cid:0)ansforma(cid:1)tionη isd(cid:0)egeneracy(cid:1)-preservingif morphismsinC.TheisomorphismsinM willbeusedbelowto
F andG are degeneracy-preservingand,for eachdegeneracy parameterize various constructions on cubical categories over
h:l →l in (cid:3) , the following diagram commutes for every different notions of equivalence of morphisms. For example,
1 2 p
object X of X(l ): taking M to be the class of identity morphisms will ensure
1
that diagrams commute on the nose — as for Bainbridge et
Y(h) F(l1)X εF(h)X F(l2) X(h)X al.’s PER model; see Example 29 below — while taking M
to be the class of all isomorphisms will entail that the same
(cid:0) (cid:1) (cid:0) (cid:1)
diagrams commute up to an arbitrary natural isomorphism.
Y(h) η(l1)X η(l2) X(h)X
LessextremalchoicesforM arepossibleaswell:forexample,
(cid:0) (cid:1) (cid:0) (cid:1)
we can define M by induction on l, letting M(0) consist of
Y(h) G(l1)X εG(h)X G(l2) X(h)X only the identity morphisms, and defining an isomorphism f
(cid:0) (cid:1) (cid:0) (cid:1) to belong to M(l+1) iff Rel(h)◦f belongs to M(l) for any
Here,εF(h):Y(h)◦F(l1)→F(l2)◦X(h)andεG(h):Y(h)◦ face map h : l+1→l, where this composition is in C. This
G(l1)→G(l2)◦X(h)arethenaturalisomorphismswitnessing definitionofM isusedforReynolds’model;seeExample28.
that F and G are themselves degeneracy-preserving. Wewillusethefollowingcubicalcategories—oneforeach
n — to interpret System F types:
By contrast with the diagram in Definition 7, the one in
Definition 8 is required to commute on the nose. Definition 9. The (p-dimensional) cubical category |Rel|n →
Inthesettingof[8],afibrednaturaltransformationbetween Rel is given as follows:
twofibredfunctorsF,G :|Rel(U)|n →Rel(U)inducesaface
map-preserving cubical natural transformation from F to G • the objects are triples (F,εF,υF), where
presented as cubical functors. We note, however, that there is – F is a face map-preserving (p-dimensional) cubical
no notion in [8] that induces cubical natural transformations functor from |Rel|n to Rel
that are both face map- and degeneracy-preserving; indeed, – ε isafamilyofnaturalisomorphismswitnessingthat
F
the frameworkof [8] does not require natural transformations F is also degeneracy-preserving. Moreover, ε (h) is
F
between equality-preserving fibred functors from |Rel(U)|n in M(l2) for each degeneracy h:l1 →l2 in (cid:3)p
to Rel(U) to themselves be equality-preserving.Significantly, – υ is a function associating to each isomorphism
F
the results of [8] can still be obtained even if fibred natural f : Rel(l)m → Rel(l)n with the property that π ◦f
0 1 k
transformations are required to be equality-preserving. The is in M(l) for each k ≤ n an isomorphism υ (f) :
F
analogous requirement for cubical natural transformations — Rel(l)m → Rel(l) in M(l). Moreover, υ respects the
0 1
source and target operations, as well as identities, equality below holds for each face map h:l →l in (cid:3) :
1 2 p
composition, and reindexing of isomorphisms
Rel(h)1 =1
l1 l2
• the morphisms are face map- and degeneracy-preserving
(p-dimensional) cubical natural transformations It is stable under degeneracies if the equality holds up to an
isomorphisminM(l )foreachdegeneracyh:l →l in(cid:3) .
2 1 2 p
Generalizing from the 1-dimensional setting of [8], in
We write 1 rather than 1 below when l is clear from context.
which types with n free variables are interpreted as equality- l
preservingfunctorsfrom|Rel(1)|n to Rel(1), we aim to inter- Lemma 13. If Rel has terminal objects that are stable under
pret System F types as face map- and degeneracy-preserving facemapsanddegeneraciesthenwehaveachoiceofterminal
cubicalfunctorsfrom|Rel|ntoRelforvariousn;therestriction objects 1 in |Rel|n →Rel.
n
to functors with discrete domains here makes it possible to
Definition 14. Rel has products if it comes equipped with a
handle all type expressions in System F, not just the positive
choice of products (× ,fst ,snd ) in Rel(l) for l ≤ p, such
ones. This will require that the total categories of the λ2- l l l
that M(l) for l ≤ p is closed under products. This choice
fibrations we construct have such functors as their objects.
of products is stable under face maps if, for each face map
But for such functors to form a category, they must support
h : l → l in (cid:3) , the equalities below hold for any objects
a well-defined notion of composition. To ensure that this is 1 2 p
A,B of Rel(l ):
the case even though cubical functors are only required to 1
preserve degeneracies up to isomorphism, and even though Rel(h)(A× B)= Rel(h)A × Rel(h)B
l1 l2
thoseisomorphismsare,importantly,inRelratherthanin|Rel|,
Rel(h)(fst [A,B])=fst Rel(h)A,Rel(h)B
we need to arrange that cubical functors from |Rel|n to Rel l1 (cid:0) l2 (cid:1) (cid:0) (cid:1)
Rel(h)(snd [A,B])=snd Rel(h)A,Rel(h)B
for various n preserve enough isomorphisms. The functions l1 (cid:2)l2 (cid:3)
υF accomplish just this: they endow each cubical functor Itisstableunderdegeneraciesif,fo(cid:2)reachdegeneracyh(cid:3):l →
1
F : |Rel|n → Rel with enough structure to preserve all l in (cid:3) , the first equality above holds up to an isomorphism
2 p
“good”isomorphisms,andthis is whatwe need to pushall of
ε(h,A,B) in M(l ) that makes the following two diagrams
2
the constructions we require through. Of course, if F were commute:
a cubical functor with domain Reln rather than |Rel|n we
would get the preservation of (all) isomorphisms in Reln for Rel(h)(A×l1B) ε(h,A,B) Rel(h)A ×l2 Rel(h)B
for free. However, this would make it impossible to handle
(cid:0) (cid:1) (cid:0) (cid:1)
contravariant type expressions.
WhengivingacategoricalinterpretationofSystemF, acat- Rel(h)(fstl1[A,B]) fstl2 Rel(h)A,Rel(h)B
egory for interpreting type contexts is required. We therefore
(cid:2) (cid:3)
associate a category of contexts to each cubical category.
Rel(h)A
Definition 10. The (p-dimensional) category of contexts ε(h,A,B)
Ctx(Rel) is given as follows: Rel(h)(A×l1B) Rel(h)A ×l2 Rel(h)B
(cid:0) (cid:1) (cid:0) (cid:1)
• the objects are natural numbers
• |thReel|mno→rphRiseml s from n to m are m-tuples of objects in Rel(h)(sndl1[A,B]) sndl2 Rel(h)A,Rel(h)B
(cid:2) (cid:3)
Defining the product m×1 in Ctx(Rel) to be the natural Rel(h)B
number sum m + 1, we see that Ctx(Rel) enjoys sufficient
We write (×,fst,snd) rather than (× ,fst ,snd ) when l is
l l l
structuretomodeltheconstructionofSystemFtypecontexts:
clear from context.
Lemma 11. The category Ctx(Rel) has a terminal object 0 Lemma 15. If Rel has products stable under face maps
and a choice of products (−)×1.
and degeneracies then we have a choice of products
(× ,fst ,snd ) in |Rel|n →Rel.
To appropriately interpret arrow types will we need to n n n
know that each cubical category of the form |Rel|n → Rel If,followingthe developmentin [8], we didnotrequirecu-
is cartesian closed. The next three lemmas show that, under bicalnaturaltransformationstopreservedegeneracies,thenwe
reasonable conditions on Rel, this is indeed the case. The would not need to require commutativity of the two diagrams
constructions are variants of familiar ones, except that care above for degeneracies. We would still need Definition 14’s
must be taken to ensure that the isomorphisms in M are requirement on face maps, however.
respected.
Definition 16. Rel has exponentials if it has products and it
Definition 12. Rel has terminal objects if it comes equipped comes equipped with a choice of exponentials (⇒ ,eval ) in
l l
with a choice of terminal objects 1 in Rel(l) for l ≤ p. This Rel(l)forl ≤p with respectto thechosenproducts,suchthat
l
choice of terminal objects is stable under face maps if the M(l) for l ≤ p is closed under exponentials. This choice of
exponentialsisstableunderfacemapsifthechoiceofproducts as the fiber over object n of Ctx(Rel) in the λ2-fibration
isstableunderfacemapsand,foreachfacemaph:l →l in we construct to interpret System F. To interpret ∀-types we
1 2
(cid:3) , theequalitiesbelowholdforanyobjectsA,B ofRel(l ): will require a right adjoint to context weakening that moves
p 1
between such fibers and is appropriate to the cubical setting.
Rel(h)(A⇒ B)= Rel(h)A ⇒ Rel(h)B
l1 l2 Toformalizethisrequirement,wefirstdefinethecategorythat
Rel(h)(evall1[A,B])(cid:0)=evall2 R(cid:1)el(h)A(cid:0),Rel(h)B(cid:1) will be the total category of our λ2-fibration.
It is stable under degeneracies if (cid:2)the choice of produ(cid:3)cts is Definition 19. The (p-dimensional) cubical category
stable under degeneracies and, for each degeneracy h : |Rel|n →Rel is given as follows:
l → l in (cid:3) , the first equality above holds up to an n
is1omorph2ism υ(hp,A,B) in M(l ) that makes the following R • the objects are pairs (n,F), where F is an object in
2 |Rel|n →Rel
diagram commute:
• the morphisms from (n,F) to (m,G) are pairs (F,η),
where F : n → m is a morphism in Ctx(Rel) and η :
Rel(h) (A⇒l1B)×l1A ε(h,A⇒l1B,A) Rel(h)(A⇒l1B) ×l2 Rel(h)A F →G◦F is a morphism in |Rel|n →Rel
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
Since the set of objects of |Rel|n → Rel is, by definition,
υ(h,A,B)×l21 (isomorphicto) the set of morphismsMor(n,1) in Ctx(Rel),
we have not only that |Rel|n → Rel is the total category
n
Rel(h)(evall1[A,B]) (cid:16)(cid:0)Rel(h)A(cid:1)⇒l2(cid:0)Rel(h)B(cid:1)(cid:17)×l2(cid:0)Rel(h)A(cid:1) aofspalifitbfirabtriaotnioonv.er Ctx(RRel), but that this fibration is actually
evall2 Rel(h)A,Rel(h)B Lemma 20. The forgetful functor from |Rel|n → Rel to
(cid:2) (cid:3) n
Ctx(Rel) is a split fibration with split generic object 1.
Rel(h)B R
Moreover, cartesian structure from Rel lifts to this fibration:
Here,ε(h,A⇒ B,A)istheisomorphisminM(l )witness-
l1 2
ing the stability of the product in question under h. Lemma 21. If Rel has terminal objects, products, and expo-
nentials, all stable under face maps and degeneracies, then
Wewrite(⇒,eval)ratherthan(⇒ ,eval )whenlisclearfrom
l l the forgetful functor from |Rel|n → Rel to Ctx(Rel) is a
context. n
split cartesian closed fibration with split generic object 1.
R
Lemma 17. If Rel has exponentials stable under face maps
anddegeneraciesthenwe haveachoiceofexponentials(⇒ , The split cartesian closed structure identified in Lemma 21
n
eval ) in |Rel|n →Rel. will allow us to interpretof function types. To ensure that we
n
canalsointerpret∀-typeswerequiresomeadditionalstructure.
As for products, if we follow the development in [8] and
didnotrequirecubicalnaturaltransformationsto preservede- Definition 22. Let U : E → B be a split fibration with a
generacies, then we would not need to require commutativity distinguishedobjectΩ ofB anda choiceofproducts(−)×Ω
of the above diagram for degeneracies. We would still need in B. We say that U has split simple Ω-products if it comes
Definition 16’s requirement on face maps, however. equipped with a choice of right adjoints ∀ :E →E to
A A×Ω A
Putting Lemmas 13, 15, and 17 together gives: the weakening functors fst[A,Ω]∗ : E → E for objects
A A×Ω
A of B, with the respective unit and counit pairs (η ,ε ),
Proposition18. IfacubicalcategoryRelhasterminalobjects, A A
satisfying the following conditions for every morphism f :
products, andexponentials,all ofwhich are stable underface
A→B in B:
mapsanddegeneracies,then|Rel|n →Reliscartesianclosed.
• the following diagram commutes:
Inthedevelopmentaboveweconsidercubicalcategoriesto
be functors with codomain Cat, as explained in Section III EB×Ω ∀B EB
above. If, however, we more properly view p-dimensional
cubical categories as functors from (cid:3) to Cat(C), then the (f×1)∗ f∗
p
construction of terminal objects, products, and exponentials
mustactuallybecarriedoutinternallytoourambientcategory EA×Ω ∀A EA
C.ThismeansthatinDefinition14,forexample,AandB are
morphisms into C’s object of objects, and their product is an • f∗(ηB(X))=ηA(f∗(X)) for every object X of E(B)
internalproduct.The necessary definitionof internalproducts • (f×1)∗(εB(X))=εA((f×1)∗(X))for everyobjectX
of E
is standard and can be found, for example, in Section 7.2 B×Ω
of [12]. A similar remark applies to Definitions 12 and 16, Fibrations with enough structure to give sound interpreta-
and at several places below, but we will suppress remarks tionsof System F were dubbed“λ2-fibrations”bySeely [18]:
analogous to this one in the remainder of this paper.
Definition 23. A split λ2-fibration is a split cartesian closed
The cubical category |Rel|n → Rel will ultimately emerge fibration U : E → B, that has a terminal object in B, a split
generic object Ω, chosen products (−)×Ω in B, and split ImpredicativeSet.Intheinterestofclarity,wewriteU(rather
simple Ω-products. than Set, as in implementationsof Coq) for the impredicative
universe U. We then define
Definition 24. Rel is a (p-dimensional) parametric model of
SystemFifithasterminalobjects,products,andexponentials, isProp(A):=Π Id(a,b)
a,b:A
all stable underface maps anddegeneracies,andis such that
Prop:=ΣA:UisProp(A)
the forgetful functor from |Rel|n → Rel to Ctx(Rel) has
n isSet(A):=Π isProp(Id(a,b))
split simple 1-products. a,b:A
R Set:=ΣA:UisSet(A)
Our main technical theorem shows that every parametric
model of System F naturally gives rise to a split λ2-fibration. Here, Σ forms dependent sums, Π forms dependent products,
The construction is also a careful variant of familiar ones. and Id is the identity type. Intuitively, Set is the type of types
in U that are “discrete”. We therefore treat the terms of Set
Theorem 25. IfRel is a (p-dimensional)parametricmodelof as if they were types in U. Since U is impredicative, we have
System F, then the forgetful functor from n |Rel|n → Rel to Set:U.
Ctx(Rel) is a split λ2-fibration.
To capture Reynolds’ construction we take our ambient
R
We also have the following variant of Seely’s [18] result category C to be the category whose objects are the types in
that every split λ2-fibration gives rise to a sound model of U,andwhosemorphismsareequivalenceclassesoffunctions.
System F: Here, functions f,g : A → B are considered equal precisely
whenthetypeeq(f,g)isinhabited.Tokeepfromincorporating
Theorem 26. Every split λ2-fibration U : E → B gives a
any particular computational structure into the categorical
model of System F in which:
structure, it is crucial that we use proof-irrelevant propo-
• every type context Γ is interpreted as an object [[Γ]] in B sitional equality types eq(−,−), rather than proof-relevant
• every type Γ ⊢ T is interpreted as an object [[Γ ⊢ T]] in identity types Id(−,−), here; this ensures, for example, that
the fiber E the uniqueness condition for pullbacks is satisfied. With this
[[Γ]]
• every term context Γ;∆ is interpreted as an object [[Γ⊢ definition it is easy to check that C is finitely complete.
∆]] in the fiber E[[Γ]] To see the type Set as a category internal to C we first
• every term Γ;∆ ⊢ t : T is interpreted as a morphism define the type Set(A,B) of morphisms from A to B to be
[[Γ;∆⊢t:T]] from [[Γ;∆]] to [[Γ⊢T]] in the fiber E Set(A,B) := A → B, and then take the object of objects
[[Γ]]
Moreover, if Γ;∆ ⊢ s = t : T, then [[Γ;∆ ⊢ s : T]] = in the internal category to be Set itself and its object of
βη
[[Γ;∆⊢t:T]]. morphisms to be ΣA,B:UA → B. We define a category of
relations by
Theorems25and26togetherimplyourmainresult,namely:
Theorem 27. A (p-dimensional) parametric model Rel of R:=ΣA,B:SetA×B →Prop
R(A ,A ,R )(B ,B ,R ):=Σ Σ
System F gives a sound model of System F in which 1 2 A 1 2 B f:A1→B1 g:A2→B2
• every type Γ ⊢ T is interpreted as a face map- Πa1:A1Πa2:A2RA(a1,a2)→RB(fa1,ga2)
and degeneracy-preserving cubical functor [[Γ ⊢ T]] :
which we can see as a category internal to C whose object of
|Rel||Γ| →Rel objects is R itself and whose object of morphisms is
• every term Γ;∆ ⊢ t : T is interpreted as a face map-
and degeneracy-preserving cubical natural transforma- Σ(A1,A2,RA),(B1,B2,RB):RΣf:A1→B1Σg:A2→B2
tion [[Γ;∆⊢t:T]]:[[Γ⊢∆]]→[[Γ⊢T]] Π Π R (a ,a )→R (fa ,ga )
a1:A1 a2:A2 A 1 2 B 1 2
Taking p = 1 and omitting the requirement that cubical We obviously have two internal functors from R to Set corre-
natural transformationsbe degeneracy-preservingas indicated sponding to the first and second projections, respectively. We
at several places above shows that Theorem 27 naturally also have an equality functor Eq from Set to R defined by
generalizesTheorem4.6of [8] to arbitrary(includinginfinite,
EqA:=(A,A,Id )
whenp=∞)higherdimensions.Inparticular,thefactthatour A
cubical functors interpreting types are degeneracy-preserving Eqf :=(f,f,apf)
gives a higher-dimensionalanalogue of the fibrational formu-
where apf :Id (a ,a )→Id (fa ,fa ) is definedas usual
A 1 2 B 1 2
lation of Reynolds’ Identity Extension Lemma from [8].
by Id-induction.
V. EXAMPLES Weobtaina1-dimensionalcubicalcategoryRelbydefining
Rel(0) = Set and Rel(1) = R, and mapping the two face
InthissectionweshowhowbothReynolds’originalmodel
mapsto the two projections, and the single degeneracyto Eq.
and the PER model of Bainbridge et al. arise as instances of
We can define terminal objects, products, and exponentials
our theory.
for Rel in the obvious ways, relating two pairs iff their
Example 28. We consider Reynolds’ original model, which first and second components are related, and two functions
is internal to the Calculus of Inductive Constructions with iff they map related arguments to related values. It is not
hard to check that all these constructs are preserved on the satisfying theconditionthat,forany k suchthatR (k) holds,
1
nose by the two face maps (projections), and preserved up we have that R hn·fst(k),m·snd(k)i holds as well. The
2
to a natural isomorphism whose first and second projections first two componentsof such a quadruple serve to encode the
(cid:0) (cid:1)
are identities by the single degeneracy (equality functor). All domain and codomain of the morphism. The third component
three constructs are therefore stable under both face maps is a (nonempty) equivalence class under the exponential PER
and degeneracies. As noted in the introduction, the difference A → A . Here the exponential A → B of two PERs
1 2
betweenfibredfunctorspreservingequalityonthenoseoronly is constructed in the standard way, using an encoding of
up to naturalisomorphism is precisely where the construction partial recursive functions as natural numbers and relating
in [8]fails. Compositionandsubstitutionin(whatisintended twofunctionsifftheymaprelatedargumentstorelatedvalues.
to be) the base categoryofthe λ2-fibrationconstructedin the In accordance with [14], we denote the application of the
main theorem there cannot be defined in any standard way nth partial recursive function to a natural number a in its
unlessequalityis preserved onthe nose, butequality in [8] is domain by n·a. To finish the construction of our object of
only defined — and therefore can only be preserved — up to morphisms for Rel(1), we take a quadruple as above to be
isomorphism. realized by a natural number k iff fst(k) ∼ n and
A1→A2
Finally, we define the adjoint ∀ by snd(k)∼ m.
n B1→B2
We obviously have two internal functors from Rel(1) to
∀nF(0)A:=Σf:ΠA:SetF(0)(A,A)ΠR:R Rel(0), corresponding to the first and second projections,
π3 F(1)(EqA,R)(f (π1R))′ (f (π2R))′ respectively. We also have an equality functorEq from Rel(0)
to Rel(1) whose action on objects is given by EqA :=
(cid:0) (cid:1)
∀nF(1)R:= ∀nF(0)π1(R),∀nF(0)π2(R), (A,A,RA), where RA(k) iff fst(k) ∼A snd(k), and whose
action on morphisms is given by Eq(A,B,{n} ) :=
λ (cid:16) λ A→B
f:∀nF(0)π1R g:∀nF(0)π2R (EqA,EqB,{n}A→B,{n}A→B). We therefore have that Rel
π (F(1)(R,R))(f (π R))′ (g (π R))′ is indeed a 1-dimensional cubical category. We can define
3 1 2
terminal objects, products, and exponentials for Rel in the
(cid:17)
Intheabove,theterm(f(π R))′ :π F(1)(EqA,R )stands obviousways,inheritingfromthecorrespondingstandardcon-
1 1
for the term f(π R) : F(0)(A,π R)) transported along the structsonPERs.Itisnothardtocheckthatalltheseconstructs
1 1 (cid:0) (cid:1)
equalitybetweentherespectivetypes,andsimilarlyforπ and arepreserved bothby thetwo face maps(projections),andby
2
g. We emphasize again that these terms all exist because Rel the single degeneracy (equality functor), on the nose.
preserves face maps on the nose. Finally, we define the adjoint ∀ on objects by
n
Example 29. We considerthePERmodelofBainbridgeetal.
∀ F(0)A:= (n,k)| for allA:M′,n∼ k,
internal to the category of ω-sets. We follow the development n A
of [14] for concepts related to this category, In particular, and for allR(cid:8):Rel(1),π3 F(1)(EqA,R) (n,k)
this category is defined in Definition 6.3 of [14], and proved
(cid:0) (cid:1) (cid:9)
in Corollary 8.3 there to be finitely complete.
Weconstructa1-dimensionalcubicalfunctorRelasfollows. ∀nF(1)R:= ∀nF(0)π1(R),∀nF(0)π2(R),
AsourinternalcategoryRel(0)of0-relationswetakethecat- n| for allR:Rel(1),π (F(1)(R,R))n
(cid:0) 3
egory M′ as in Definition 8.4 of [14]. Informally, the objects
M′ are partial equivalence relations, and its morphisms are To define(cid:8)∀ on a morphism η :F →G, we define (cid:9)(cid:1)
n
realizable functions that respect those relations.
To define the internal category Rel(1) of 1-relations, we ∀ η(0)A:= ∀ F(0)A,∀ G(0)A,
n n n
first construct its object of objects. As the carrier of this ω-
{m·0}
set we take the set of triples (A,B,R), where A and B are (cid:0) ∀nF(0)A→∀nG(0)A
partial equivalence relations and R is a saturated predicate (cid:1)
Here, m is any natural number realizing η(0)A¯. It is crucial
onA×B.HeretheproductA×B oftwoPERsisconstructed
that all natural transformations are “uniformly realized”,
in the standard way, using a bijective pairing function h·,·i
in the sense that there is a natural number realizing each
andrelatingtwopairsifftheirrespectiveprojections—which
such transformation and, because all PERs are defined to
we will call fst and snd below — are related. A saturated
be realized by all natural numbers, each is suitably uniform.
predicate on a PER A is a predicate R on natural numbers
In particular, if η were not uniformly realized in the above
that is closed under A, in the sense that m∼ n and R(m)
A sense,then∀wouldnotbewell-defined.Usingthisobservation
implyR(n). To finishthe constructionofourobjectofobjects
it is possible to show that, in the category-theoretic setting
forRel(1)wetakeanytriple(A,B,R)asabovetoberealized
(rather than in the setting of ω-sets), the adjoint ∀ cannot
by any natural number. n
exist precisely becausead hoc naturaltransformations— i.e.,
AsthecarrieroftheobjectofmorphismsforRel(1)wetake
naturaltransformationsthatarenotuniformlyrealizable,even
the set of quadruples of the form
though each of their componentsmay indeed be realizable —
(A ,B ,R ),(A ,B ,R ),{n} ,{m} are not excluded.
1 1 1 2 2 2 A1→A2 B1→B2
(cid:0) (cid:1)
VI. CONSEQUENCES OF PARAMETRICITY • if f : A → B, f′ : A′ → B′, and (g,h) : f →
f′ is a morphism in Rel(l)→, then hg,hi is the
In this section we show that the models constructed in l,k
unique morphism from hfi to hf′i obtained from
Theorem27satisfythepropertiesthat“good”modelsofpara- l,k l,k
metricityforSystemFshouldsatisfy.Inparticular,Lemma34 (Reld(l,k)h)◦(f,idB)§ via (f′,idB′)§
below shows that, under reasonable conditions, our models
support the definition of a graph for each face map- and hfi (f,idB)§//Reld(l,k)B
degeneracy-preservingcubicalfunctor.Moreover,Theorem35 ✤l,k
and its analogue for final coalgebras show that our higher- ∃!hg,hil,k ✤ Reld(l,k)h
dimensional models of relational parametricity for System (cid:15)(cid:15)✤ (cid:15)(cid:15)
F also validate the existence of initial algebras and final hf′il,k //Reld(l,k)B′
coalgebras for such functors. These results serve as a sanity (f′,idB′)§
check for our theory, and show that it is powerful enough Intuitively, one of f (l,k) and f (l,k) acts as the x-
⊥ ⊤
to show that “good” models of relational parametricity for axis, and the other acts as a y-axis, for l-dimensional graphs
System F can be constructed even at higher dimensions. projected onto dimension k. Since reindexing preserves iden-
tities, we have that hid i = (id ,id )∗(Reld(l,k)B) =
A l,k A A
A. A Higher-Dimensional Graph Lemma Reld(l,k)B. This generalizes the observation that hid i =
A
Every function f : A → B between sets A and B defines EqA in the 1-dimensional setting of [8].
a graph relation hfi = {(a,b)|fa = b}. This observation We also have the following alternative characterization of
can be phrased fibrationally by letting U : Rel → Set×Set the graph functor when Rel is a bifibration:
be the standard relations fibration on Set, and noting that Lemma 32. If Rel is a (p-dimensional) cubical bifibration
hfi can be obtained by reindexing the equality relation EqB that has terminal objects, and if f : A → B, then hfi =
l,k
on B. In [8], the notion of a graph was extended to more Σ d(l,k)A.
generalrelationsfibrationsandaGraphLemmawasprovedfor (idA,f)
theirassociatedmodelsof1-dimensionalparametricity.Inthis By contrast with the analogous characterization in Lemma
subsection we give a natural generalization of the definition 5.2 of [8], no Beck-Chevalley condition is required since the
of a graph from [8] to the higher-dimensional setting, and bifibrations f(l,k) are postulated here, rather than derived
prove a Graph Lemma appropriate to this setting. We begin from more primitive bifibrations as is done there.
by introducing the (new) notion of a cubical (bi)fibration. We have the following analogue of Lemma 5.3 of [8]:
Definition 30. A (p-dimensional) cubical category Rel that Lemma 33. h−il,k is full and faithful if Reld(l,k) is.
has products is a (p-dimensional) cubical (bi)fibration if, for
Together, the (fibrational) definition of the graph functor
each l<p, each functor
anditsopfibrationalcharacterizationfromLemma32givethe
f(l,k) = hRel(f (l,k)),Rel(f (l,k))i following Graph Lemma for our higher-dimensionalsetting:
⊥ ⊤
: Rel(l+1)→Rel(l)×Rel(l) Lemma 34. (Graph Lemma) Let Rel be a (p-dimensional)
cubical bifibration that has terminal objects and F : Rel →
for k ≤l is a (bi)fibration.
Relbea(p-dimensional)facemap-anddegeneracy-preserving
As already noted in Example 6, the (bi)fibrations f(l,k) play cubical functor. For any l < p, f : A → B in Rel(l), and
the role of the relations fibrations in [8], while the d(l,k) k ≤l, there exist morphisms
play the role of equality functors. When Rel is a cubical
(bi)fibration,wehavethatRel(d(l,k))Aisindeedover(A,A) φf :hF(l)fil,k →F(l+1)hfil,k
with respect to f(l,k) for every object A in Rel(l), and and
similarly for every morphism in Rel(l).
ψ :F(l+1)hfi →hF(l)fi
f l,k l,k
If C is a category, write C→ for the arrow category of C,
i.e., for the category whose objects are morphisms in C and in Rel(l+1) that are vertical with respect to f(l,k).
whose morphisms from f : A → B to f′ : A′ → B′ in C→
B. Existence of Initial Algebras and Final Coalgebras
are pairsof morphismsg :A→A′ andh:B →B′ such that
f′◦g =h◦f. We define the graph functor for Rel to be the In this subsection we use our Graph Lemma to show that
set of functors {h−i |l < p, k ≤ l}, where each h−i is the models constructed in Theorem 27 validate the existence
l,k l,k
defined as follows: of initial algebras and final coalgebras for face map- and
degeneracy-preserving cubical functors, and thus for all in-
Definition 31. Let Rel be a (p-dimensional) cubical fibration
terpretations of positive type expressions in System F. Our
that has terminal objects. For every l < p and k ≤ l, the
constructions naturally extend those in [8] to the higher-
functor h−i :Rel(l)→ →Rel(l+1) is defined by:
l,k dimensional setting.
• if h : A → B is an object in Rel(l)→, then hhil,k = If Rel is a (p-dimensional) cubical category and F :Rel→
(h,id )∗(Reld(l,k)B) Rel is (p-dimensional) cubical functor, then an F-algebra
B