The Scott model of Linear Logic is the extensional collapse of its relational model

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The Scott model of Linear Logic is the extensional

collapse of its relational model

Thomas Ehrhard

To cite this version:

Thomas Ehrhard.

The Scott model of Linear Logic is the extensional collapse of its relational

model. Theoretical Computer Science, Elsevier, 2012, 424, pp.20-45. �10.1016/j.tcs.2011.11.027�.

�hal-00369831�

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extensional ollapse of its relational model

Thomas Ehrhard

Preuves, Programmes et Systèmes,UMR 7126 CNRS and University ParisDiderot - Paris 7

∗

Mar h 22,2009

Abstra t

Weshowthattheextensional ollapseoftherelationalmodeloflinear logi isthemodelofprime-algebrai latti es,anaturalextensiontolinear logi ofthewellknownS ottsemanti softhelambda- al ulus.

Introdu tion

LinearLogi arosefromdenotationalinvestigationsofse ondorderintuitionisti logi byGirard(systemF[Gir86℄). Heobservedthat thequalitativedomains

1 usedforinterpretingsystemF anbeassumedtobegeneratedbyabinary rela-tiononasetofverti es(theweb): su hastru tureis alleda oheren espa e

2 . The ategory of oheren e spa es, with linear maps (stable maps preserving arbitrary existing unions) asmorphisms, hasremarkablesymmetryproperties that ledhimto thesequent al ulusofLL,and thentoproof-nets[Gir87℄ and totheGeometryofIntera tion.

S ott semanti s ofLL. Inspiteof Barr'sobservation[Bar79℄that the at-egoryof ompletelatti es andlinearmaps is

∗

-autonomous, it wasa ommon beliefin theLinearLogi ommunitythat thestandardS ott semanti softhe lambda- al ulus(S ottdomains and ontinuousmaps) annot provide models of lassi allinearlogi . Huth showedhoweverin [Hut94℄ that prime-algebrai ompletelatti esandlub-preservingmapsprovideamodelof lassi alLLwhose asso iatedCCC (the Kleisli ategoryof the

!

omonad) isafull-CCC of the ategoryofS ott domainsand ontinuousmaps. Huth onsidered howeverhis model as degenerate, as it identies the

⊗

and

`

onne tives of LL

3

. A few yearslater,Winskelredis overedthesamemodel in asemanti alinvestigation

∗This

work as also been partly funded by the ANR proje t CHOCO: http:// ho o.pps.jussieu .fr.

1

Qualitativedomains anbeseenasparti ulardI-domains[Ber78 ℄. 2

Thepurelambda- al ulus,ortheTuring- ompletefun tionallanguagePCF[Plo77℄, an alsobeinterpretedin oheren espa es.

3

Theinterpretationofproofsinthismodelisnon-trivialandinterestingnevertheless.Asin the aseoftherelationalmodel(seebelow),itispossibletoendowthismodelwhi hadditional stru tureswhi hseparate

⊗

and

`

,withoutmodifyingtheinterpretationofproofs.

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aparti ular aseof amoregeneralprofun tor onstru tion, heshowedindeed that the ategorywhoseobje tsarepreorderedsets andwhere themorphisms fromapreorder

S

toapreorder

T

arethefun tionsfromtheset

I(S)

of down-ward losed subsets of

S

to the set

I(T )

whi h preserve arbitraryunions is a modelof lassi alLL.This ategoryisequivalenttoHuth'smodel,butweprefer Winskel's approa h, asitinsistson onsidering preorders(and notlatti es)as obje ts: preordersaresimilartothewebsof oheren espa es,tothesetsofthe relationalmodel,andrepresenttheprimeelementsofthe orrespondinglatti es. Moreover,theLL onstru tionsareeasiertodes ribeintermsofpreordersthan intermsoflatti es. ItisfairtomentionalsothatKrivine[Kri90,Kri93℄usedthe same onstru tion(set

I(S)

ofinitialsegments ofapreorder

S

)fordes ribing models of the pure lambda- al ulusand mentioned that these preorders give risetoamodelofLL,withlinearnegation orrespondingtotakingtheopposite preorder.

Relational semanti s. On the other hand, when one applies the O am's RazorPrin ipletothe oheren espa esemanti s,oneisledtointerpreting for-mulaeassets(thewebs,withoutanystru ture)andproofsasrelationsbetween these sets. Somethingtri kyhappensduringthispro ess: sin e oheren e van-ishes,one annotrestri tthesetinterpretinganof ourseto ontainonlynite liques asGirarddid in [Gir86℄, thebest one an dois takeall nite subsets. Butthen, thedereli tion relation(from

!X

to

X

), whi h isthe setof allpairs

({a}, a)

where

a ∈ X

, isno moreanaturaltransformation. This problem an easilybesolvedbyrepla ingnitesetswithnitemultisets,buttheee tofthis hoi eis thatthe orrespondingKleisli ategoryis nomorewell-pointed. One denesinthatwaytherelationalsemanti soflinearlogi ,whi his ertainlyits simplest (and,maybe,most anoni al)denotationalmodel.

Coe ients. One way of turning the CCC asso iated with the relational modelinto awell-pointed ategoryisby enri hingit with oe ients: instead oftakingsubsetof

X × Y

asmorphismsfrom

X

to

Y

,takeelementsof

C

X×Y

, where

C

isasuitableset (or lass)of oe ients;a anoni al hoi e onsistsin taking

C = Set

, the lass of all sets. An element of

Set

X×Y

should be on-sideredasamatrixwhoserowsareindexedbytheelementsof

Y

,and olumns bytheelementsof

X

: this isbasi allytheideaofGirard'squantitative seman-ti s[Gir88℄,whi hispresentedasamodelofintuitionisti logi ,butisindeeda modelof LL(Girardwrotethispaperbeforehedis overedLL),see[Has02℄. It isalsoaninstan eofthealreadymentionedprofun tor onstru tions[Win99℄.

Finite oe ients belonging to more standard algebrai stru tures (rigs, elds, et .) an also be onsidered, but this requires adding some stru ture to thesesets forguaranteeingthe onvergen eofthesumswhi happearwhen multiplyingthematri es,see[Ehr02,Ehr05,DE08℄: theee tofsu hadditional stru tureisthat obje tsareequippedwithatopologyforwhi h the(generally innite) sumsinvolvedinmultiplyingmatri es onverge.

Extensional ollapse of the relational model. Theotherwayofmaking therelationalmodelwell-pointedisbyperforminganextensional ollapse. This operationiseasilyunderstoodinthetypehierar hyasso iatedwiththe artesian losedKleisli ategoryofthenitemultiset omonadonthe ategoryofsetsand

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relations: ea htype

A

isinterpretedbyitsrelationalinterpretation

[A]

(asimple set),togetherwithapartialequivalen erelation(PER)

∼

A

on

P([A])

. When

A

isthetype

B ⇒ C

,anelementof

P([A])

is amorphismfrom

B

to

C

,andtwo su hmorphisms

f

and

g

are

∼

B⇒C

-equivalentif,forany

x, y

su hthat

x ∼

A

y

, onehas

f (x) ∼

B

g(y)

. Inother words,this PERisalogi alrelation

4

, andthe extensional ollapseof this typehierar hy is obtainedbyquotienting ea hset

P([A])

bythePER

∼

A

(one onsiders onlytheelements

x

of

P([A])

su hthat

x ∼

A

x

,whi hareoften alled invariant elements).

Content of the paper. Weprovethatthis extensional ollapseofthe rela-tional model oin ides pre isely with the S ott model of preorders. The rst problemwehavetofa eistogiveapre isemeaningtothisstatement. Westart fromtheworkofBu iarelli[Bu 97℄,re astingitina ategori alsetting: given aCCC

C

and awell-pointed CCC

E

, wewantto express what it meansfor

E

to be (weshallsayto represent)theextensional ollapseof

C

. Forthis, we introdu etwo ategori al onstru tions.

•

Thehomogeneous ollapse ategory

e(C)

, whose obje tsare pairs

(U, ∼)

where

U

isanobje tof

C

and

∼

isapartialequivalen erelation(PER)on thepointsof

U

(thatison

C(⊤, U )

where

⊤

is theterminalobje tof

C

). Themorphisms are those of

C

whi h preservethis additional stru ture, and it is easy to see that this ategoryis a CCC. The important point inthisdenitionisthat theobje tofmorphismsfrom

(U, ∼)

to

(V, ∼)

is

(W, ∼

W

)where

W

istheobje tofmorphismsfrom

U

to

V

in

C

andthe relation

∼

W

isdenedasalogi alrelation.

•

Theheterogeneous ollapse ategory

e(C, E)

,whoseobje tsaretriples

(U, E, )

where

U

isanobje tof

C

,

E

isanobje tof

E

and

⊆ C(⊤, U ) × E(⊤, E)

shouldbeunderstoodasarealizabilitypredi ate:

x ζ

meansintuitively that

ζ

represents the extensional behavior of

x

. The morphisms are thepairs

(f, ϕ)

of morphisms whi h preservethe relation

, and again, it is easy to he k that this ategory is aCCC. The important point is that,when onstru tingtheobje tofmorphisms,

isdenedasalogi al relation.

These two onstru tions are possible for any CCCs

C

and

E

. We say that

E

representstheextensional ollapseof

E

if

• e(C, E)

ontainsa su ientlylarge (in areasonablesense, to be made pre ise later) sub-CCC

H

whose obje ts

(U, E, )

are modest, meaning that

isapartial surje tion from

C(⊤, U )

to

E(⊤, E)

,and therefore in-du esaPERon

C(⊤, U )

(observethat

E(⊤, E)

anbe onsideredasthe quotientof

C(⊤, U )

bythisPER)

•

andthefun tor

H → e(C)

whi hmaps

(U, E, )

to

(U, ∼)

,where

∼

isthe PERindu edby

(andmapsamorphism

(f, ϕ)

to

f

),isaCCCfun tor (thatis,preservestheCCCstru tureonthenose).

The ni e feature of this denition is that it is ompatible with the standard one(basedontypehierar hies)andthatit aneasilybeextended,forinstan e,

4

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lambda- al ulustorepresenttheextensional ollapseofanotherone.

It would be ni e of ourse to have a similar denition of the extensional ollapseofa ategori almodelofLL,andnotonlyofCCCs,butsin ethe de-nitionofsu hamodelisalreadyquite ompli ated,weprefernottoaddressthis issue. Instead,weperformtheCCC onstru tionsdenedabove on retely,ina ompletelylinearsetting, obtainingbothCCCs

e(C)

and

H

asKleisli onstru -tions of suitable exponential omonads: in thepresent paper,

C

is the Kleisli ategory

Rel

!

asso iatedwiththeLLmodelof setsandrelations,and

E

isthe Kleisli ategory

ScottL

!

asso iatedwith theLLmodelof preordersandlinear mapsbetweentheasso iated ompletelatti es.

Afterhavingintrodu edthene essarypreliminarymaterial,werstbuildin Se tion 2.2 alinearversionof the ategory

e(Rel

!

)

. Morepre isely, we dene a model of LL denoted as

PerL

, whose obje ts are alled PER-obje ts: they are setsequipped withaPER ontheir powersets. TheKleisli ategory

PerL

!

isisomorphi to

e(Rel

!

)

(or,morepre isely,toafull sub-CCCof

e(Rel

!

)

).

Then,inSe tion3,wedes ribetheS ottmodel

ScottL

ofLL.Theobje ts arepreorderedsets,andamorphismfrom

S

to

T

isalinearmap(thatis,amap preservingallunions)from

I(S)

(thesetofalldownward- losedsubsetsof

S

)to

I(T )

. Asfarassetsare on erned,themultipli ativeandadditive onstru tions inthismodel oin idewiththoseofthemodel

Rel

(morethingshavetobesaid abouttheasso iatedpreorders: forinstan e,

S

⊥

istheset

S

equippedwiththe oppositeofthepreorderof

S

). Astotheexponential,thenatural hoi ewould beto dene

!S

astheset of nite subsetsof

S

withasuitable preorder: with that hoi e,theKleisli ategory

ScottL

!

isasub-CCCoftheCCCof omplete latti esand S ott- ontinuousfun tions. Butwe anobtainthesameee t by dening

!S

asthesetofallnitemultisetsofelementsof

S

,andthiswillgreatly simplify our onstru tions,be ausewiththis hoi e, theset interpretinganLL formulain

Rel

oin ideswiththesetinterpretingthesameformulain

ScottL

(rememberthatthisset isequippedwithapreorder).

InSe tion4,weintrodu ethelinearversionoftheheterogeneous ategory

H

of the onstru tiondes ribedabove. An obje tshould beatriple

(X, S, )

where

X

is aset,

S

is apreordered set and

⊆ P(X) × I(S)

(whi h hasto be apartial surje tion). By our hoi e abovefor the denition of

!S

, we an assume

X = S

,soasarstsimpli ation,we anassumeourobje tstobepairs

(S, )

where

S

is apreordered set and

⊆ P(S) × I(S)

hasto be a partial surje tion. A arefulanalysis shows that, when

x u

, wemusthave

u = ↓ x

(the downward losureof

x

in

S

),so that,fordeningthepartialsurje tion

, weonlyneedtoknowitsdomain

D

. Soanobje tofour ategorywillbeapair

(S, D)

where

D ⊆ P(S)

. What onditionshouldsatisfy

D

? Asusual,itshould beequal toits double dual for asuitablenotion ofduality: here, we saythat

x, x

′

_{⊆ S}

areindualityif

x

′

_{∩ ↓ x 6= ∅ ⇒ x}

′

_{∩ x 6= ∅}

,thatis

x

′

annotseparate

x

fromitsdownward losure. Weshowthatthese obje ts( alledpreorderswith proje tions),withsuitablelinearmorphisms,formamodeloflinearlogi

PpL

, whoseasso iatedKleisli ategory

PpL

!

anbe onsideredasafullsub-CCCof

e(Rel

!

, ScottL

!

)

, of whi h allobje tsaremodest. Andweshowthat

ScottL

!

represents the extensional ollapse of

Rel

!

in the sense explained above. We a tuallyexhibit afun torfrom

PpL

to

PerL

whi h preservesthestru ture of LLmodelandwhi hindu estherequiredCCCfun torfrom

PpL

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al ulus,usingnotionsofin lusionsbetweenthevariousstru tureswe onsider, organizing them into omplete partially ordered lasses, and using the fa t thatthelogi al onstru tions(tensorprodu t,orthogonalityet )are ontinuous wrt.these in lusions. Thisprovidesasimplerepresentationof theextensional ollapseofthereexiveobje tin

Rel

!

weintrodu edin[BEM07℄,asareexive obje tintheCCCof ompletelatti esand ontinuousmaps,whi hisprobably isomorphi toS ott'sstandard

D

∞

.

Contents

1 Preliminaries 6

1.1 Notations . . . 6

1.2 Cartesian losed ategoriesandmodelsofthepurelambda- al ulus 6 1.3 Intuitionisti extensional ollapse . . . 7

1.3.1 Representingthe ollapseasaninterpretation. . . 8

1.3.2 Categori alpresentation. . . 8

1.3.3 Conne tionbetweenthetwodenitions. . . 9

1.3.4 Extensional ollapseofareexiveobje t. . . 10

1.4 New-Seely ategoriesandLL-fun tors . . . 10

2 The ollapsepartialequivalen e relation 11 2.1 The ategoryofsetsandrelations. . . 11

2.1.1 Linearstru ture. . . 11

2.1.2 Theasso iatedCCC.. . . 12

2.1.3 In lusions.. . . 12

2.2 The ollapse ategory . . . 12

2.2.1 Pre-PERs,PERobje tsandmorphismsofPERobje ts.. 12

2.2.2 Orthogonalityandstrongisomorphisms. . . 13

2.2.3 Monoidalstru ture. . . 13

2.2.4 Additivestru ture. . . 14

2.2.5 Exponentials. . . 15

2.2.6 Fundamentalisomorphismand artesian loseness. . . 15

2.3 Thepartiallyordered lassofPER-obje ts. . . 16

2.3.1 Completeness. . . 16

2.3.2 VariablePER-obje tsandxpointsthereof. . . 17

2.3.3 AnextensionalreexivePER-obje t. . . 18

3 Alinear S ott semanti s 18 3.1 Star-autonomousstru ture. . . 18

3.1.1 Isomorphisms. . . 19

3.1.2 Monoidalstru ture. . . 19

3.2 Produ tsand oprodu ts. . . 20

3.3 Exponentials . . . 20

3.3.1 Comonadstru tureoftheexponential. . . 21

3.3.2 Weakeningand ontra tion. . . 21

3.4 TheKleisli ategory . . . 21

3.4.1 TheKleisli ategoryofpreorders. . . 22

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4.1 Adualityonpreorders . . . 23

4.2 Thelinear ategory. . . 25

4.2.1 Identityand omposition. . . 25

4.2.2 Tensorprodu t.. . . 25

4.2.3 Strongisomorphisms. . . 26

4.2.4 Asso iativityandsymmetryisomorphisms. . . 26

4.2.5 Linearfun tionspa eandmonoidal loseness.. . . 26

4.2.6 Thepar onne tive. . . 27

4.2.7 Themorphism

mix

isnotanisomorphismin general. . . . 27

4.3 Theadditives . . . 27

4.4 Theexponentials . . . 28

4.4.1 Fundamentalisomorphism. . . 28

4.4.2 Stru turalmaps. . . 29

4.4.3 Cartesian loseness. . . 29

4.5 Thepartiallyordered lassofPPs . . . 29

4.5.1 Completeness. . . 30

4.5.2 VariablePPsandleastxpointsthereof. . . 31

4.5.3 AnextensionalreexivePP. . . 31

4.6 PPsareheterogeneouslogi alrelations . . . 31

4.6.1 Heterogeneousrelationasso iatedwithaPP. . . 31

4.7 Afun torfromPPs toPER-obje ts . . . 33

4.7.1 Continuityof

ε

.. . . 35

4.7.2 Imageofthereexiveobje tof

PpL

!

. . . 36

4.8 Afun torfromPPs topreorders . . . 36

1 Preliminaries 1.1 Notations

A nitemultiset

p

ofelementsof

S

is amap

p : S → N

su hthat

p(a) = 0

for almost all

a ∈ S

. Wewrite

a ∈ p

for

p(a) > 0

,anduse

supp(p)

forthesupport of

p

whi h is the set

{a ∈ S | a ∈ p}

. Weuse

p + q

for thepointwisesum of multisets,and

0

fortheemptymultiset.

Givena ategory

C

andtwomorphisms

f ∈ E(E, F )

and

x ∈ C(⊤, E)

(where

⊤

istheterminalobje tof

C

thatweassumetoexist),wewrite

f (x)

insteadof

f ◦ x

be ausewe onsider

x

asapoint (anelement)of

E

.

1.2 Cartesian losed ategories and models of the pure lambda- al ulus

Webriey re allthat a ategory

C

is artesian losed (isaCCC) ifea h nite family

(E

i

)

i∈I

of obje ts of

C

has a artesian produ t

&

i∈I

E

i

(in parti ular, ithasaterminalobje t

⊤

)together withproje tions

π

j

∈ C(&

i∈I

E

i

, E

j

)

su h that, for any family

(f

i

)

i∈I

with

f

i

∈ C(F, E

i

)

there is an unique morphism

hf

i

i∈I

∈ C(F, &

i∈I

E

i

)

su hthat

π

j

◦ hf

i

i∈I

= f

j

forea h

j

andif,giventwo obje ts

E

and

F

of

C

,thereisapair

(E ⇒ F, Ev)

, alledtheobje tofmorphisms from

E

to

F

, togetherwith anevaluation morphism

Ev

∈ C((E ⇒ F ) & E, F )

(8)

su hthat,forany

f ∈ C(G & E, F )

,there isanunique

Cur(f ) ∈ C(G, E ⇒ F )

su hthat

Ev

◦ (Cur(f ) & Id

E

) = f

.

GiventwoCCCs

C

and

D

,afun tor

F : C → D

willbesaidtobea artesian losed fun tor if it preservesthe artesian losed stru ture on thenose. This meansthat

F(&

i∈I

E

i

) = &

i∈I

F(E

i

)

,

F(π

i

) = π

i

,

F(E ⇒ F ) = F(E) ⇒ F(F )

and

F(Ev) = Ev

.

Areexive obje t in aCCC

C

isatriple

(H, app, lam)

where

H

isanobje t of

C

,

app

∈ C(H, H ⇒ H)

and

lam

∈ C(H ⇒ H, H)

satisfy

app

◦ lam = Id

H⇒H

. One says moreover that

(H, app, lam)

is extensional

5

if

lam

◦ app = Id

H

. If

(H, app, lam)

is a reexive obje t in

C

and if

F : C → D

is a CCC fun tor, then

(F (H), F (app), F (lam))

is areexiveobje tin

D

, whi h is extensionalif

(H, app, lam)

isextensional.

Let

(H, app, lam)

be a reexive obje t in the CCC

C

. Then, given any lambda-term

M

and any repetition-free list of variables

~x = x

1 , . . . , x

n

whi h ontains all the freevariables of

M

(su h a list will be said to be adapted to

M

), onedenes

[M ]

H

~

x

∈ C(H

n

, H)

byindu tion on

M

(

[x

i

]

H

~

x

= π

i

,

[λx N ]

H

~

x

=

lam

◦ Cur([N ]

H

~

x,x

)

and

[(N ) P ]

H

~

x

= Ev ◦ happ ◦ [N ]

H

~

x

, [P ]

H

~

x

i

). If

M

and

M

′

are

β

-equivalent and

~x

is adapted to

M

and

M

′

, wehave

[M ]

H

~

x

= [M

′

]

H

~

x

. If

(H, app, lam)

is extensional, we have

[M ]

H

~

x

= [M

′

]

H

~

x

when

M

and

M

′

are

βη

-equivalent.

If

F : C → D

is a CCC fun tor then, for any lambda-term

M

, we have

F([M ]

H

~

x

) = [M ]

F (H)

~

x

where

[M ]

F (H)

~

x

istheinterpretationof

M

inthereexive obje t

(F (H), F (app), F (lam))

.

1.3 Intuitionisti extensional ollapse

Thepresentanalysisoftheextensional ollapseofamodelofthetyped lambda- al ulusisbasedon[Bu 97℄.

From the usual intuitionisti viewpoint, the extensional ollapse is a log-i al relation. More spe i ally, onsider the hierar hy of simple types based on some type atoms

α

,

β

..., and intuitionisti impli ation

⇒

. Consider a artesian losed ategory

C

(withterminal obje t

⊤

, artesianprodu t

&

and fun tion spa e

⇒

). Given a valuation

I

from type atoms to obje tsof

C

, we haveaninterpretationoftypes

[A]

I

∈ C

. Theextensional ollapse ofthis inter-pretationis atype-indexedfamilyofpartial equivalen erelations

(∼

A

)

, where

∼

A

⊆ C(⊤, [A]

I

)

2

. Thisrelationisdenedbyindu tionontypes.

•

Atea hbasi type

α

,therelation

∼

α

oin ideswithequalityon

C(⊤, I(α))

.

•

Then, given

f, g ∈ C(⊤, [A ⇒ B]

I

) = C(⊤, [A]

I

⇒ [B]

I

) ≃ C([A]

I

, [B]

I

)

, onehas

f ∼

A⇒B

g

if,forall

x, y ∈ C(⊤, [A]

I

)

su h that

x ∼

A

y

,one has

f (x) ∼

B

g(y)

(where were allthat wewrite

f (x)

insteadof

f ◦ x

when thesour eof

x

istheterminalobje t).

By indu tion on types, one proveseasily that

∼

A

is a PER on

C(⊤, [A]

I

)

for ea h type

A

. Sin e the family of PERs

(∼

A

)

is dened asa logi al relation, it is ompatible with the syntax of the simply typed lambda- al ulus, in the sense that, if

M

is a losed term of type

A

, its semanti s

[M ]

I

∈ C(⊤, [A]

I

)

5

Thisnotionofextensionality,whi h orrespondstothe

η

onversionruleofthe lambda- al ulus,shouldnotbe onfusedwiththenotionofextensionalitywearedealingwithinthis paper,whi hisrelatedtothe ategori alnotionofwell-pointedness.

(9)

satises

[M ]

I

∼

A

[M ]

I

. Thisproperty anbeextended tofun tional enri hed versions ofthe simplytyped lambda- al ulus(su h asPCF) under somemild assumptionson

C

and

I

.

1.3.1 Representing the ollapse as an interpretation. Let

E

be an-other artesian losed ategory, that we assume to be well-pointed (mean-ing that, if

ϕ, ψ ∈ E(E, F )

satisfy

ϕ(ζ) = ψ(ζ)

for all

ζ ∈ E(⊤, E)

, then

ϕ = ψ

). Let

J

be a valuation of type atoms in

E

and, for ea h type atom

α

, let

α

⊆ C(⊤, I(α)) × E(⊤, J(α))

be abije tion (to be understood as ex-pressinganequalityrelationbetweentheelementsofthetwomodelsatground types). Thenwedene

A

⊆ C(⊤, [A]

I

) × E(⊤, [A]

J

)

foralltype

A

asalogi al relation( alledtheheterogeneousrelation), thatis

f

A⇒B

ψ ⇔ (∀x, ζ x

A

ζ ⇒ f (x)

B

ϕ(ζ)) .

If

A

issurje tiveforalltype

A

(thatis

∀ζ ∈ E(⊤, [A]

J

) ∃x ∈ C(⊤, [A]

I

) x

A

ζ

), then all the relations

A

are fun tional (in the sense that if

x

A

ζ

and

x

A

ζ

′

, then

ζ = ζ

′

). Thisis easyto he kbyindu tion ontypesand isdue tothewell-pointednessof

E

.

Wesaythat

(

A

)

isarepresentation of the ollapse oftheinterpretation

I

bytheinterpretationof

J

if,foralltype

A

,

A

issurje tive(andbije tivewhen

A = α

isabasi type)andonehas

∀x, y ∈ C(⊤, [A]

I

) x ∼

A

y ⇔ (∃ζ ∈ E(⊤, [A]

J

) x

A

ζ

and

y

A

ζ) .

This meansthat,at ea h type

A

, therelation

A

indu esabije tion between

E(⊤, [A]

J

)

andthequotient 6

C(⊤, [A]

I

)/∼

A

.

Assume that

(

A

)

issu h arepresentation. Sin eit is dened asalogi al relation,wehave

[M ]

I

A

[M ]

J

forea h losedlambda-termoftype

A

,wehave

[M ]

I

∼

A

[N ]

I

i

[M ]

J

= [N ]

J

forall losedterms

M

and

N

oftype

A

.

1.3.2 Categori al presentation. There is another, more on eptual way of des ribing thesituation above. First one denes the ollapse ategory

e(C)

of

C

. Its obje ts are pairs

U = (pU q, ∼

U

)

where

p

U q

is an obje t of

C

and

∼

U

⊆ C(⊤, pU q)

2

is aPER.Giventwoobje ts

U

and

V

of

e(C)

,the elements of

e(C)(U, V )

arethemorphisms

f ∈ C(pU q, pV q)

su hthat

∀x, x

′

∈ C(⊤, pU q) x ∼

U

x

′

⇒ f (x) ∼

V

f (x

′

) .

If the ategory

C

is artesian,thensois

e(C)

(with artesian produ tsdened in the most obvious way). And if

C

is artesian losed, so is

e(C)

. Given two obje ts

U

and

V

of

C

, one denes

U ⇒ V = (pU q ⇒ pV q, ∼

U

⇒V

)

with

f ∼

U⇒V

f

′

i

f (x) ∼

Y

f

′

_(x

′

₎

for all

x, x

′

_{∈ C(⊤, pU q)}

su h that

x ∼

U

x

′

(for

f, f

′

_{∈ C(⊤, pU ⇒ V q) ≃ C(pU q, pV q)}

). The evaluation morphism

Ev

∈

e(C)((U ⇒ V ) & U, V )

is theevaluationmorphism of the ategory

C

,whi h is also a morphism in

e(C)

. We say that an obje t

U

of

e(C)

is dis rete if

∼

U

oin ideswithequality.

Similarly, one denes the heterogeneous ategory

e(C, E)

of

C

and

E

. Its obje ts are triples

X = (pXq, xXy,

X

)

where

p

Xq

is an obje t of

C

,

x

Xy

6

WhenquotientingasetbyaPER,one onsidersonlytheelementsofthesetwhi hare equivalenttothemselves.

(10)

is an obje t of

E

and

X

⊆ C(⊤, pXq) × E(⊤, xXy)

. A morphism

θ

from

X

to

Y

in that ategory is a pair

(pθq, xθy)

where

p

θq ∈ C(pXq, pY q)

and

x

_{θy ∈ E(xXy, xY y)}

satisfy

p

θq(x)

Y

x

θy(ζ)

forall

(x, ζ)

su hthat

x

X

ζ

. Again, if both ategories

C

and

E

are artesian, so is

e(C, E)

, and if they are artesian losed,sois

e(C, E)

, with

X ⇒ Y

dened asfollows:

p

X ⇒ Y q =

p

_{Xq ⇒ pY q}

,

x

X ⇒ Y y = xXy ⇒ xY y

and, given

f ∈ C(⊤, pX ⇒ Y q) ≃

C(pXq, pY q)

and

ϕ ∈ E(⊤, xX ⇒ Y y) ≃ C(xXy, xY y)

, we have

f

X⇒Y

ϕ

if

f (x)

Y

ϕ(ζ)

forall

(x, ζ)

su h that

x

X

ζ

. Letussaythatanobje t

X

of

e(C, E)

ismodest

7

iftherelation

X

isapartial surje tion from

C(⊤, pXq)

to

E(⊤, xXy)

. Let

e

mod

(C, E)

bethefullsub ategory of

e(C, E)

whose obje tsarethemodestobje ts. If

C

and

E

are artesian,then

e

mod

(C, E)

is asub- artesian ategoryof

e(C, E)

. But in general,

e

mod

(C, E)

is not artesian losed. It an benoti ed that, if

X

and

Y

are obje tsof

e(C, E)

whi h are modest (so that, again,

X ⇒ Y

is well dened but notne essarily modest)andif

X⇒Y

issurje tive,then

X⇒Y

isfun tional,andhen e

X ⇒ Y

ismodest.

There is a artesian losed se ond proje tion fun tor

σ : e(C, E) → E

(it maps an obje t

X

to

x

Xy

and a morphism

θ

to

x

θy

). There is also a fun tor

ε : e

mod

(C, E) → e(C)

whi h mapsan obje t

X

to

(pXq, ∼

ε(X)

)

, where

x

1 ∼

ε(X)

x

2

if

x

1 X

ζ

and

x

2 X

ζ

for some(ne essarily unique)

ζ

. Given

θ ∈ e(C, E)(X, Y )

, weset

ε(θ) = pθq

. Indeed,let

x

1 , x

2 ∈ C(⊤, pXq)

su hthat

x

1 ∼

ε(X)

x

2

(with

ζ ∈ E(⊤, xXy)

su h that

x

1 X

ζ

and

x

2 X

ζ

), wehave

p

_θq(x

₁

₎

_Y

x

_θy(ζ)

and

p

θq(x

2 )

Y

x

θy(ζ)

, and hen e

p

θq(x

1 ) ∼

Y

p

θq(x

2 )

,so that

p

θq ∈ e(C)(ε(X), ε(Y ))

.

Wesaythatthe ategory

E

representstheextensional ollapseofthe ategory

C

ifthereexistsasub-CCC

H

of

e(C, E)

su h that

•

ea hobje tof

H

ismodest;

•

thefun tor

ε : H → e(C)

is artesian losed

•

and, for any 8

dis rete obje t

U

of

e(C)

, there is an obje t

X

of

H

su h that

ε(X) = U

(sothat

p

Xq = U

and

X

isabije tion).

1.3.3 Conne tionbetween the twodenitions. Themotivationofthis denition is that, in that situation, if

I

is a type valuation in

C

then, for ea h ground type

α

, we an nd an obje t

J(α)

of

E

su h that

K(α) =

(I(α), J(α),

α

)

is an obje tof

H

, for some bije tion

K(α)

. We anextend

(K(α))

into an interpretation of types

([A]

K

)

in the CCC

H

whi h satises

[A]

K

= ([A]

I

, [A]

J

,

A

)

where

A

oin ides with theheterogeneouslogi al re-lation denedin 1.3.1. Then ourassumption that

E

representstheextensional ollapseof

C

impliesthat

(

A

)

isarepresentationoftheextensional ollapseof

I

by

J

, inthesenseof1.3.1.

Thebenetofthisabstra tionisthat the on eptofaCCC

E

representing the extensional ollapse of a CCC

C

is quite exible and independent of any type hierar hygivena priori. For instan e, it providesa naturaldenition of theextensional ollapseofamodelofthepurelambda- al ulus.

7

Thisis ompatiblewiththestandardterminologyofrealizability,seee.g.[AC98℄. 8

Wea tually don'tneed thispropertyfor alldis rete

U

s,but onlyfor thosewhi h are intended torepresentthe basi typesof the fun tionallanguage wehaveinmind. Forthe sakeofsimpli ity,weadoptthisstrongerhypothesis.

(11)

E

representsthe extensional ollapse of

C

in the sense above, with

H

as het-erogeneous ollapse CCC. Let

(Z, app, lam)

be a reexive obje t in

H

. Then

(ε(Z), pappq, plamq)

isareexiveobje tin

e(C)

,

(pZq, pappq, plamq)

isa reex-iveobje tin

C

and

(xZy, xappy, xlamy)

isareexiveobje tin

E

.

Inthat ase,wesaythatthereexiveobje t

(xZy, xappy, xlamy)

isa repre-sentationoftheextensional ollapseofthereexiveobje t

(pZq, pappq, plamq)

. Remark: Thepre ise synta ti al meaning ofthis denition is not ompletely learyet. Inthispaper,weshallgivearepresentationoftheextensional ollapse of the relationalmodel of thelambda- al ulusintrodu ed in [BEM07℄ (in the senseabove),andthesetwomodelswill learlybequitedierent. However,both modelsindu ethesameequationaltheoryonlambda-terms(namely,thetheory

H

∗

,a ordingtowhi htwoterms

M

and

M

′

areequivalentif,forany ontext

C

,theterm

C[M ]

issolvableitheterm

C[M

′

_]

issolvable). Withthenotations above,thismeansthat,whenrestri tedtotheinterpretationsoflambda-terms, the relation

∼

Z

is just equality. Extending for instan e the lambda- al ulus withaparallel omposition onstru tionbasedonthemix ruleofLinearLogi asin[DK00,BEM08℄,thesituation be omesmoreinterestingandthetheories indu edbythetwomodelsonthelanguagearedistin t.

1.4 New-Seely ategories and LL-fun tors

Following [Bie95℄, we dene a model

L

of LL as a New-Seely ategory. This onsistsof

•

asymmetri monoidal losedstar-autonomous ategory(alsodenotedwith

L

)whi hhasallniteprodu ts(theunitofthetensorprodu tisdenoted with

1

,thedualizingobje twith

⊥

,theterminalobje t

⊤

andthe arte-sianprodu tof

X

and

Y

isdenotedwith

X & Y

),

•

a omonad

! : L → L

(the stru ture morphisms

d

L

X

∈ L(!X, X)

is alled dereli tionand

p

L

X

∈ L(!X, !!X)

is alleddigging),

•

andtwonaturalisomorphisms

!⊤ ≃ 1

and

!(X & Y ) ≃ !X ⊗ !Y

su hthattheadjun tionbetween

L

anditsKleisli ategory

L

!

(whi his artesian losedbythehypothesesabove)isamonoidaladjun tion.

Givenafun tion

I

(valuation)fromthepropositionalatomsofLLtoobje ts of

L

, the interpretation

[A]

L

I

of an LL-formula

A

is dened by indu tion on

A

, using theabove mentionedstru tures of

L

, e.g.

[A ⊗ B]

L

I

= [A]

L

I

⊗

L

[B]

L

I

. Similarly,givenaproof

π

of

A

,onedenes

[π]

L

I

∈ L(1, [A]

L

I

)

byindu tionon

π

(expressedinthestandardsequent al ulusofLL[Gir87℄).

Given two New-Seely ategories

L

and

M

, a fun tor

F : L → M

will be alled an LL-fun tor ifit ommutesonthe nose withall the onstru tions requiredforinterpretingLL,e.g.

F (X ⊗

L

Y ) = F (X) ⊗

M

F (Y )

,

F (d

L

X

) = d

M

X

et . Thenonehas

F ([A]

L

I

) = [A]

M

F◦I

and

F ([π]

L

I

) = [π]

M

F◦I

forallformula

A

and proof

π

ofLL.

Su h an LL-fun tor

F

fun tor indu es a artesian losed fun tor (still de-notedwith

F

)from

L

!

to

M

!

.

(12)

We dene a ategory whose obje ts are sets equipped with a partial equiva-len e relation (PER) on theirpowersets, the intuition beingthat twosubsets are equivalent if they havethe sameextensional behavior. These PERs are dened as logi al relations,in thesense that, when wedenefun tion spa es, twomorphismsareequivalentitheymap equivalentsetstoequivalentsets.

2.1 The ategory of sets and relations

This ategoryunderliesthe ollapse ategorywewanttodene. Morepre isely, the ollapse ategorywedenein Se tion 2.2is anenri hmentof the ategory of sets and relations where ea h obje t is endowed with a partial equivalen e relationexpressingwhentwosetsareextensionallyequivalent,asin1.3.2. 2.1.1 Linear stru ture. The ategoryof sets and relations

Rel

has sets as obje ts, and, given twosets

E

and

F

, the set of morphisms from

E

to

F

is

Rel(E, F ) = P(E × F )

. Composition is dened in the standardrelational way: the ompositionof

s ∈ Rel(E, F )

and

t ∈ Rel(F, G)

is

t · s ∈ Rel(E, G)

. Theidentitymorphismis thediagonalrelation

Id

∈ Rel(E, E)

. This ategory has aquitesimple monoidalstru ture: the tensorprodu tis

E ⊗ F = E × F

and theunit of thetensor is

1 = {∗}

. This tensor produ tis afun tor: given

s

i

∈ Rel(E

i

, F

i

)

for

i = 1, 2

, then

s

1 ⊗ s

2 = {((a

1 , a

2 ), (b

1 , b

2 )) | (a

i

, b

i

) ∈

s

i

for

i = 1, 2}

. Equippedwiththistensorprodu t,

Rel

issymmetri monoidal losed(the asso iativity,neutralityandsymmetryisomorphismsaredenedin the usualobviousway),with anobje tof linearmorphisms

E ⊸ F = E × F

and linear evaluation morphism

ev

∈ Rel((E ⊸ F ) ⊗ E, F )

given by

ev

=

{(((a, b), a), b) | a ∈ E

and

b ∈ F }

.

Thesymmetri monoidal losed ategory

Rel

isastar-autonomous ategory, withdualizingobje t

⊥ = 1

,andthe orrespondingdualityistrivial:

E

⊥

_{= E}

. So

E`F = E ⊸ F = E ⊗ F = E × F

in thismodel.

Remark: Again,this ategoryisadegeneratemodel ofLLin thesensethat itidenties

⊗

and

`

,justas

ScottL

(andevenworse,sin eitequatesaformula withitslinearnegation!). Weshowedin[BE01℄howthismodel anbeenri hed with various stru tures withoutmodifying the interpretation of proofs, making

⊗

and

`

non-isomorphi operations. This anbe onsideredasoneofthemost strikingfeatures ofLL: thislogi alsystemis sorobust thatit survives(in the sense thatproofsarenottrivialized)in su h adegenerateframework.

Given

s ∈ Rel(E, F )

and

x ⊆ E

,onesets

s · x = {b | ∃a ∈ x

and

(a, b) ∈ s}

. The ategory

Rel

is artesian. The artesianprodu tofafamily

(E

i

)

i∈I

of sets is

&

i∈I

E

i

=

S

i∈I

({i} × E

i

)

, withproje tions

π

j

= {((j, a), a) | a ∈ E

j

} ∈

Rel

(&

i∈I

E

i

, E

j

)

. Given a family of morphisms

s

i

∈ Rel(F, E

i

)

, the orre-spondingmorphism

hs

i

i∈I

∈ Rel(F, &

i∈I

E

i

)

is givenby

hs

i

i∈I

= {(b, (i, a)) |

i ∈ I

and

(b, a) ∈ s

i

}

. Theterminalobje tis

⊤ = ∅

.

Theexponential omonad is

!E = M

fin

(E)

, with a tiononmorphisms de-nedasfollows:

!s = {([a

1 , . . . , a

n

], [b

1 , . . . , b

n

]) | (a

i

, b

i

) ∈ s

for

i = 1, . . . , n} ∈

Rel(!E, !F )

for

s ∈ Rel(E, F )

. Dereli tion is given by

d

E

= {([a], a) | a ∈

S} ∈ Rel(!E, E)

and digging by

p

E

= {(m

1 + · · · + m

n

, [m

1 , . . . , m

n

]) | n ∈

N

and

m

1 , . . . , m

n

∈ !E} ∈ Rel(!E, !!E)

. Given

x ⊆ E

, one denes

x

!

₌

M

fin

(x)

. Observethat,asusual,

!s · x

!

_{= (s · x)}

!

,

d

E

· x

!

_{= x}

and

p

(13)

Theisomorphism

!⊤ ≃ 1

identies

[]

and

∗

,andtheisomorphism

!(E & F ) ≃

!E ⊗ !F

maps theelement

[(1, a

1 ), . . . , (1, a

l

), (2, b

1 ), . . . , (2, b

r

)]

of

!(E & F )

to

([a

1 , . . . , a

l

], [b

1 , . . . , b

r

]) ∈ !E ⊗ !F

(thisis alled thefundamental isomorphism in thepresentpaper).

Allthesedatadene anewSeely ategory,see Se tion1.4.

2.1.2 Theasso iatedCCC. TheKleisli ategory

Rel

!

is artesian losed. Givenaset

E

,apointof

E

in

Rel

!

isbydenitionamorphismin

Rel(!⊤, E)

, thatis,asubsetof

E

. Theterminalobje tis

⊤

,the artesianprodu tof

(E

i

)

i∈I

is

E = &

i∈I

E

i

, with proje tions

π

i

◦ d

E

(still denoted as

π

i

). The obje t of morphisms

E ⇒ F

is

!E ⊸ F

, withevaluationmap

Ev

= ev ◦ (d

E⇒F

⊗ Id

!E

)

, that is

Ev

= {(([(m, b)], m), b) | m ∈ !E

and

b ∈ F } .

Applyingamorphism

s ∈ Rel

!

(E, F ) = Rel(!E, F )

toapoint

x ⊆ E

onsists in omposing

s

with

x

( onsidered asa morphismfrom

⊤

to

E

) in

Rel

!

; the resultis

s(x) = s · x

!

= {b | ∃m (m, b) ∈ s

and

supp(m) ⊆ x} .

The ategory

Rel

!

isnotwellpointed,inthesensethattwodistin tmorphisms

s

1 , s

2 ∈ Rel

!

(E, F )

an satisfy

∀x ⊆ E s

1 (x) = s

2 (x)

; take for instan e

s

1 =

{([a], b)}

and

s

2 = {([a, a], b)}

.

Thepurpose ofthe ollapsePER is pre iselyto makeit expli itwhen two su h morphismsshould be identied. This depends of ourse onthe PERs

E

and

F

areequippedwith: the ollapsePERisalogi alrelation. Weshallpresent this onstru tionasanew ategory.

2.1.3 In lusions. Let

E

and

F

be two sets su h that

E ⊆ F

. Then we denoteby

η

E,F

and

ρ

E,F

therelations

η

E,F

= (E × F ) ∩ Id

E

and

ρ

E,F

= (F × E) ∩ Id

E

.

Observethat

ρ

E,F

◦ η

E,F

= Id

E

.

We denote by

RelC

the lass of all sets, ordered by in lusion. This is a partiallyordered lass, whi his ompleteinthesensethatanyfamily

(E

γ

)

γ∈Γ

ofelementsof

RelC

admitsaleastupperbound. Weshall onsidera tuallyonly dire tedfamilies (thatis,where

Γ

isadire ted poset,and

γ ≤ δ ⇒ E

γ

⊆ E

δ

).

2.2 The ollapse ategory

We equip now the obje ts of

Rel

with a partial equivalen e relation whose purposeistoidentifymorphismswhi hyieldequivalentvalueswhenappliedto equivalentarguments.

2.2.1 Pre-PERs,PER obje ts and morphisms ofPER obje ts. Let

E

beaset.Givenabinaryrelation

B

on

P(E)

,wedeneanotherbinaryrelation

B

⊥

on

P(E)

, alledthedual of

B

,asfollows:

x

′

B

⊥

y

′

if

∀x, y ∈ P(E) x B y ⇒ (x ∩ x

′

_{6= ∅ ⇔ y ∩ y}

′

_{6= ∅) .}

As usual, onehas

B ⊆ C ⇒ C

⊥

_{⊆ B}

⊥

and

B ⊆ B

⊥⊥

(as subsetsof

P(E)

2

). We say that the relation

B

is apre-PER ifit is symmetri and satises

x B

(14)

y ⇒ x B x

. Clearly,anyPER isapre-PERandif

B

isapre-PER, then

B

⊥

is aPER.

APER-obje t isapair

U = (|U |, ∼

U

)

,where

|U |

isasetand

∼

U

isabinary relationon

P(|U |)

whi hisapre-PERsu hthat

∼

⊥⊥

U

= ∼

U

. Thissimplymeans that,given

x, y ⊆ |U |

, onehas

x ∼

U

y

assoonas

x ∩ x

′

_{6= ∅ ⇔ y ∩ y}

′

_{6= ∅}

, for all

x

′

_{, y}

′

_{⊆ |U |}

su h that

x

′

_∼

⊥

U

y

′

. Bythis ondition,

∼

U

is automati ally a PER (indeed,

∼

U

ispre-PER, hen e

∼

⊥

U

isaPER, and therefore

∼

U

=∼

⊥⊥

U

is aPER).

Let

PerL

bethe ategorywhoseobje tsarethePER-obje ts,andwherea morphismfrom

U

to

V

isarelation

t ⊆ |U | × |V |

su h, forall

x, y ∈ P(|X|)

,if

x ∼

X

y

then

t · x ∼

Y

t · y

.

Remark: Let

U

beaPER-obje tand

A ⊆ P(|U |)

su hthat

∀x

1 , x

2 ∈ A x

1 ∼

U

x

2

. Then

∀x ∈ A x ∼

C

S A

. Indeed,let

x

′

1 , x

′

2 ⊆ |U |

besu hthat

x

′

1 ∼

U

⊥

x

′

₂

. If

x ∩ x

′

1 6= ∅

, then

x ∩ x

′

2 6= ∅

be ause

x ∼

U

x

, and hen e

S A ∩ x

′

2 6= ∅

. Conversely,if

S A ∩ x

′

2 6= ∅

,there issome

y ∈ A

su hthat

y ∩ x

′

2 6= ∅

and we on ludesin e

x ∼

U

y

. Soea hequivalen e lassof

∼

U

hasamaximalelement, whi histheunionofalltheelementsofthe lass. Theseparti ularelements

x

of

P(|U |)

are hara terizedbythetwofollowingproperties:

• x ∼

U

x

•

and

∀y ∈ P(|U |) y ∼

U

x ⇒ y ⊆ x

.

Lemma1 Let

U

be a PER-obje t and let

(x

i

)

i∈I

and

(y

i

)

i∈I

be families of elements of

P(|U |)

be su h that

x

i

∼

U

y

i

for ea h

i ∈ I

. Then

S

i∈I

x

i

∼

U

S

i∈I

y

i

.

Theproofisstraightforward. Inparti ular

∅ ∼

U

∅

,foranyPER-obje t

U

. 2.2.2 Orthogonality and strong isomorphisms. We dene the PER-obje t

U

⊥

by

|U

⊥

_{| = |U |}

and

∼

U

⊥

= ∼

⊥

_U

,sothat

U

⊥⊥

_{= U}

.

Lemma2 Given two PER-obje ts

U

and

V

, any bije tion

θ : |U | → |V |

su h that,for all

x, y ∈ P(|X|)

,onehas

x ∼

U

y

i

θ(x) ∼

V

θ(y)

is anisomorphism from

U

to

V

. Su h a bije tion will be alled a strong isomorphismfrom

U

to

V

.

Straightforwardveri ation. Of ourse,

θ

−1

isastrongisomorphismfrom

V

to

U

.

Observethatanystrongisomorphism

θ

from

U

to

V

isalsoastrong isomor-phismfrom

U

⊥

to

V

⊥

. Indeed,let

x

′

1 , x

′

2 ⊆ |U |

. Assume rstthat

x

′

1 ∼

U

⊥

x

′

₂

and let us show that

θ(x

′

1 ) ∼

V

⊥

θ(x

′

₂

)

. So let

y

1 , y

2 ⊆ |V |

be su h that

y

1 ∼

V

y

2

. We have

θ(x

′

1 ) ∩ y

1 6= ∅ ⇔ x

′

1 ∩ θ

−1

(y

1 ) 6= ∅

and we on lude sin e

θ

−1

is a strong isomorphism from

V

to

U

. The onverse impli ation

θ(x

′

1 ) ∼

V

⊥

θ(x

′

₂

) ⇒ x

₁

′

∼

_U

⊥

x

′

₂

isprovensimilarly.

2.2.3 Monoidalstru ture. Wedene

U ⊗V

asfollows. Wetake

|U ⊗ V | =

|U | × |V |

,and

∼

U⊗V

= E

⊥⊥

where

E = {(x

1 × y

1 , x

2 × y

2 ) | x

1 ∼

U

x

2

and

y

1 ∼

U

y

2 } ⊆ P(|U ⊗ V |)

(15)

Sin e this relation

E

is apre-PER(but not aPER apriori, sin e one annot re over

x

and

y

from

x × y

whenoneof thesetwosets isempty), therelation

∼

U⊗V

is aPER, and

U ⊗ V

sodened isaPER-obje t. Wedene

U ⊸ V =

(U ⊗ V

⊥

₎

⊥

.

Lemma3 One has

|U ⊸ V | = |U | × |V |

. If

t

1 , t

2 ∈ P(|U ⊸ V |)

, one has

t

1 ∼

U ⊸V

t

2

iforall

x

1 , x

2 ⊆ |U |

su hthat

x

1 ∼

U

x

2

,onehas

t

1 · x

1 ∼

Y

t

2 · x

2

. Moreover,one has

t

1 ∼

U ⊸V

t

2 ⇔

t

_t

1 ∼

V

⊥

⊸U

⊥

t

_t

2

.

Proof. Thisisduetothefa tthat,forany

t ⊆ |U ⊸ V |

,

x ⊆ |U |

and

y

′

_{⊆ |V |}

, onehas

t ∩ (x × y

′

_{) 6= ∅ ⇔ (t · x) ∩ y}

′

_{6= ∅}

2

Sothe morphismsfrom

U

to

V

are exa tlythe

t ∈ P(|U ⊸ V |)

su hthat

t ∼

U ⊸V

t

,andif

t ∈ PerL(U, V )

then

t

_{t ∈ PerL(V}

⊥

_{, U}

⊥

₎

.

Lemma4 Theobviousbije tion

λ

from

|U ⊗ V ⊸ W |

to

|U ⊸ (V ⊸ W )|

de-nes a strong isomorphism between the PER-obje ts

U ⊗ V ⊸ W

and

U ⊸

(V ⊸ W )

. In parti ular, for

s

1 , s

2 ∈ P(|U ⊗ V ⊸ W |)

, onehas

s

1 ∼

U

⊗V ⊸W

s

2

i for any

x

1 , x

2 ∈ P(|U |)

and

y

1 , y

2 ∈ P(|V |)

su h that

x

1 ∼

U

x

2

and

y

1 ∼

U

y

2

,onehas

s

1 · (x

1 × y

1 ) ∼

W

s

2 · (x

2 × y

2 )

.

Proof. Let

t

1 , t

2 ⊆ P(U ⊗ V ⊸ W )

. Assume rst that

t

1 ∼

U

⊗V ⊸W

t

2

, we wantto provethat

λ(t

1 ) ∼

U ⊸(V ⊸W )

λ(t

2 )

. Butthisis learsin e,if

x

1 , x

2 ⊆

|U |

and

y

1 , y

2 ⊆ |V |

satisfy

x

1 ∼

U

x

2

and

y

1 ∼

V

y

2

,thenwehave

x

1 × y

2 ∼

U⊗V

x

2 × y

2

, and therefore

(λ(t

1 ) · x

1 ) · y

1 = t

1 · (x

1 × y

1 ) ∼

W

t

2 · (x

2 × y

2 ) =

(λ(t

2 ) · x

2 ) · y

2

. Assume onverselythat

λ(t

1 ) ∼

U ⊸(V ⊸W )

λ(t

2 )

,weprovethat

t

1 ∼

U⊗V ⊸W

t

2

. Forthis,wepro eedasabove,showingthat

t

_t

1 ∼

W

⊥

⊸(U ⊗V )

⊥

t

_t

2

andapplyingLemma 3.

2

Lemma5 The obvious bije tion

α : |(U ⊗ V ) ⊗ W | → |U ⊗ (V ⊗ W )|

is an isomorphism of PER-obje ts from

(U ⊗ V ) ⊗ W

to

U ⊗ (V ⊗ W )

.

Proof. By2.2.2,itsu estoprovethat

α

isanisomorphismfrom

((U ⊗ V ) ⊗

W )

⊥

to

(U ⊗ (V ⊗ W ))

⊥

,andthisresultsfromLemma4.

2

Given

s ∈ PerL(U

1 , U

2 )

and

t ∈ PerL(V

1 , V

2 )

,onedenes

s⊗t ⊆ |U

1 ⊗ V

1 |×

|U

2 ⊗ V

2 |

as in 4.2.2. ThenoneshowsusingLemma 4that

s ⊗ t ∈ PerL(U

1 ⊗

V

1 , U

2 ⊗V

2 )

,andone he ksthatthe ategory

PerL

equippedwiththis

⊗

binary fun tor,togetherwiththeasso iativityisomorphismofLemma5(aswellasthe symmetryisomorphismet .) isasymmetri monoidal ategory,whi his losed (with

U ⊸ V

asobje tof linearmorphismsfrom

U

to

V

)by Lemma 4. The linearevaluation morphismis

ev

,asdenedin Se tion2.1.

PerL

isstar-autonomous,with

⊥ = ({∗}, =)

asdualizingobje t.

2.2.4 Additivestru ture. Givenafamily

(U

i

)

i∈I

ofPER-obje ts,one de-nes

U = &

i∈I

U

i

bysetting

|U | =

Q

i∈I

({i} × |U

i

|)

,andbysayingthat,forany

x = (x

i

)

i∈I

, y = (y

i

)

i∈I

∈ P(|U |)

(identifying this latter set with a produ t), onehas

x ∼

U

y

if onehas

x

i

∼

U

i

y

i

forall

i ∈ I

. Using thefa t that

∅ ∼

V

∅

in any PER-obje t

V

, one shows that

∼

⊥

U

= ∼

&

i∈I

U

i

⊥

and it followsthat

U

is aPER-obje t. Itisroutineto he kthat

&

i∈I

U

i

sodened isthe artesian

(16)

produ tofthe

U

i

sinthe ategory

PerL

,andthatthis artesianprodu tisalso a oprodu t. Inparti ular,if

U

isaPER-obje tand

I

isaset,wedenotewith

U

I

theprodu t

&

i∈I

U

i

where

U

i

= U

forea h

U

.

Inparti ular,

PerL

hasaterminalobje t

⊤

,givenby

|⊤| = ∅

and

∅ ∼

⊤

∅

. ObservethatthisistheonlyPER-obje twithanemptyweb.

2.2.5 Exponentials. GivenaPER-obje t

U

,wedene

!U

by

|!U | = M

fin

(|U |)

, and

∼

!U

= E

⊥⊥

where

E = {(x

!

1 , x

!

2 ) | x

1 , x

2 ∈ P(|U |) x

1 ∼

U

x

2 }

wherewere allthat

x

!

_{= M}

fin

(x)

. Sin e

E

isapre-PER(anda tuallyaPER, be ause

x

an be re overedfrom

x

!

usingdereli tion:

x = {a | [a] ∈ x

!

_}

), the relation

∼

!U

isa PER. Were all that, if

s ⊆ |!U ⊸ V |

and

x ⊆ |U |

, then we denotewith

s(x)

thesubset

s · x

!

of

|Y |

,seeSe tion2.1.

Lemma6 Let

U

and

V

be PER-obje ts and let

s

1 , s

2 ⊆ |!U ⊸ V |

. One has

s

1 ∼

!U⊸V

s

2

i

∀x

1 , x

2 ⊆ |U |

x

1 ∼

U

x

2 ⇒ s

1 (x

1 ) ∼

V

s

2 (x

2 ) .

Proof. The

⇒

dire tionistrivial.Forthe onverse,oneassumesthatthestated onditionholds, and one he ksthat

t

_s

1 ∼

V

⊥

⊸(!U )

⊥

t

_s

2

, andfor thispurpose,

itsu estoapplyLemma 3.

2

Given

s ∈ PerL(U, V )

, one denes

!s ⊆ |!U | × |!V |

as in the standard relationalmodelbysetting

!s = {([a

1 , . . . , a

n

], [b

1 , . . . , b

n

]) | n ∈ N, (a

i

, b

i

) ∈ s

for

i = 1, . . . , n} .

Then,sin e

!s · x

!

_{= (s · x)}

!

,wehave

!s

1 ∼

!U⊸!V

!s

2

assoonas

s

1 ∼

U ⊸V

s

2

(by Lemma 6); in parti ular, if

s ∈ PerL(U, V )

,onehas

!s ∈ PerL(!U, !V )

andso theoperation

s 7→ !s

isanendofun toron

PerL

.

Onedenes

d

U

⊆ |!U| × |U |

as

d

U

= {([a], a) | a ∈ |U |}

,and sin e

d

U

· x

!

₌

x

for all

x ⊆ |U |

, we get easily

d

U

∈ PerL(!U, U )

. Similarly, one denes

p

_U

⊆ |!U | × |!!U |

as

p

U

= {(m

1 + · · · + m

k

, [m

1 , . . . , m

k

]) | m

1 , . . . , m

k

∈ |!U |}

. Sin e

p

U

· x

!

= x

!!

, we get

p

U

∈ PerL(!U, !!U)

. The naturality in

U

of these morphisms is lear (it holds in the relational model), and

!

_ equipped with these two naturaltransformations is a omonad. Moreover, the fundamental isomorphismalsoholdsinthissetting.

2.2.6 Fundamentalisomorphismand artesian loseness. Let

U

and

V

bePER-obje ts. Let

θ : |!(U & V )| → |!U ⊗ !V |

betheusualbije tiondened by

θ([(1, a

1 ), . . . , (1, a

l

), (2, b

1 ), . . . , (2, b

r

)]) = ([a

1 , . . . , a

l

], [b

1 , . . . , b

r

])

UsingLemma6,oneshowseasilythat

θ ∈ PerL(!(U & V ), !U ⊗ !V )

(asa rela-tion). Forshowingthat

θ

−1

_{∈ PerL(!U ⊗ !V , !(U & V ))}

,oneappliesLemma4 andthen Lemma6,twi e. Thisshowsthat

θ

isastrongisomorphismof PER-obje ts.

(17)

stru tureexplainedabove)isanew-Seely ategory,in thesense of[Bie95℄. Theasso iatedKleisli ategory

PerL

!

is artesian losed. Theobje tof mor-phismsfrom

U

to

V

is

U ⇒ V = !U ⊸ V

andwehaveseenthattheasso iated PER

∼

U⇒V

issu hthat,giventwoelements

s

1

and

s

2

of

PerL

!

(U, V )

,onehas

s

1 ∼

U

⇒V

s

2

i

s

1 (x

1 ) ∼

V

s

2 (x

2 )

forall

x

1 , x

2 ⊆ |U |

su h that

x

1 ∼

U

x

2

. The evaluationmorphismis

Ev

,asdened in2.1.2.

2.3 The partially ordered lass of PER-obje ts

Let

U

and

V

bePERobje ts. Wesaythat

U

isasubobje tof

V

andwrite

U ⊑

V

if

|U | ⊆ |V |

,andmoreover

η

|U|,|V |

∈ PerL(U, V )

and

ρ

|U|,|V |

∈ PerL(V, U )

. Thismeansthatthetwofollowing onditionsaresatised

∀x

1 , x

2 ⊆ |U | x

1 ∼

U

x

2 ⇒ x

1 ∼

V

x

2

and

∀y

1 , y

2 ⊆ |V | y

1 ∼

V

y

2 ⇒ y

1 ∩ |U | ∼

U

y

2 ∩ |U | .

Observethat

⊑

apartialorder relationandlet

PerC

bethepartially ordered lassofPER-obje tsorderedby

⊑

.

Oneofthemainfeaturesofthisdenitionisthatlinearnegationis ovariant withrespe tto thesubobje tpartial order.