The Legendre-Fenchel transform from a category theoretic perspective

0.3 MB·English

by  Simon Willerton

The Legendre-Fenchel transform from a
category theoretic perspective
SimonWillerton
5
1 Abstract
0
2 TheLegendre-Fencheltransformisaclassicalpieceofmathematicswith
manyapplications. Inthispaperweshowhowitarisesinthecontextof
n
categorytheoryusingcategoriesenrichedovertheextendedrealnumbers
a
R:=[ ∞,+∞]. AkeyingredientisPavlovic’s“nucleusofaprofunctor”
J −
construction.Thepairingbetweenavectorspaceanditsdualcanbeviewed
5
asanR-profunctor; theconstructionofthenucleusofthisprofunctoris
1
theconstructionofalotofthetheoryoftheLegendre-Fencheltransform.

] Forarelationbetweensetsviewedasa true,false -valuedprofunctor,the
T construction of the nucleus is the cons{truction o}f the Galois connection
C associatedtotherelation.
. Oneinsightgivenbythisapproachisthattherelevantstructureonthe
h
functionspacesinvolvedintheLegendre-Fencheltransformissomething
t
a like a metric but is asymmetric and can take negative values. This ‘R-
m structure’isaconsiderablerefinementoftheusualpartialorderonreal-
[ valuedfunctionspaceanditallowsanaturalinterpretationofToland-Singer
duality and of the two tropical module structures on the set of convex

1 functions.
v
1
9
Contents
7
3
0
1 Introduction 1
.
1
0 2 Categoriesenrichedoverposets 7
5
1
3 Thenucleusofaprofunctor 22
:
v
i 4 TheGaloiscorrespondencefromarelation 25
X
r 5 TheLegendre-Fencheltransform 26
a
1 Introduction
One role of category theory is as a kind of metamathematics: it gives a lan-
guagefordescribinghowseeminglydifferentconstructionsindisparateparts
ofmathematicsarereallythesame. Untilthe1970s,areassuchaslogic,algebra,
algebraicgeometryandalgebraictopologyweretheareasinwhichcategory
1
theoryhadmostreach,butthenLawvere[10]showedusingenrichedcategory
theorythatcategorytheorywasentwinedwithmetricspacetheory.
FollowingLawvere[11]andtakingenrichedcategoriesseriously,wewill
seeherehow,fromthepairingbetweenavectorspaceanditsdual,thenucleus
ofaprofunctorconstruction[17]givesrisetoalargeamountofthetheoryaround
theLegendre-Fencheltransformandconvexfunctions. Wewillalsoseehow
theGaloiscorrespondencearisingfromarelationbetweensetsisanexampleof
thisnucleusfromaprofunctorconstruction;otherexamplesofthisconstruction
includetheIsbellcompletionofacategory,theDedekind-MacNeillecompletion
for posets, the fuzzy concept lattice arising from a fuzzy context [2, 17], the
tropicalspanofatropicalmatrix[5]andthedirectedtightspanforgeneralized
metricspaces[23].
1.1 Thebasicidea
Here we will recall the construction of Galois connection from a relation —
which is a fundamental construction across mathematics — and also recall
theconstructionoftheLegendre-Fencheltransformandthenbrieflyexplain
howtheyhaveacommonrefinedgeneralizationinthenucleusofaprofunctor
construction.
1.1.1 RelationsandGaloisconnections.
Suppose you have two sets G and M and a relation I between them. Two
examplestobearinmindherearethefollowing:
1. fromalgebraicgeometry,G =Cn, M =C[x ,...x ]andxIpif p(x) =0;
1 n
2. from number theory, G is a field L equipped with a subfield K L,
⊂
M =Aut(L,K)thefieldautomorphismsfixingK,and(cid:96)Iϕif ϕ((cid:96)) = (cid:96).
Wewrite (G)fortheposetofsubsetsofGorderedbysubsetinclusion. Simi-
P
larly,wewrite (M)op fortheposetofsubsetsof Mwiththeoppositeorder.
P
FromIwegetapairoforderpreservingfunctions
I∗: (G)(cid:29) (M)op: I ,
P P ∗
where
I∗(S)(cid:66) m M sImforalls S ,
{ ∈ | ∈ }
I (T)(cid:66) g G gItforallt T .
∗ { ∈ | ∈ }
TheseformaGaloisconnectioninthat
S I (T) I∗(S) T,
⊆ ∗ ⇐⇒ ⊇
asbothsidesaresayingthateveryelementofTisrelatedtoeveryelementofS.
Wethengetaclosureoperationoneachpowerset:
I I∗: (G) (G), I∗I : (M)op (M)op.
∗ P → P ∗ P → P
Theseareidempotent—(I I )2 =I I and(I I )2 =I I —andtheysatisfy
∗ ∗ ∗ ∗
S I I∗(S)andT I∗I (∗T)forall∗S G,T ∗M,thust∗heymeritthename
⊂ ∗ ⊃ ∗ ⊂ ⊂
2
closureoperations. Theclosedsubsetsarethosewhicharefixedbytheclosure
operations,i.e.thosethatareequaltotheirclosure.Thetotalityofclosedsubsets
givestwoposets (G)and (G). TheGaloisconnectionrestrictstoaGalois
Pcl Pcl
correspondence,i.e.aduality,betweentheclosedsubsetsofGandtheclosed
subsetsof M:
I∗: Pcl(G)(cid:27)Pcl(M)op: I∗.
Wecanlookatwhatthisgivesintheexamplesmentionedabove.
1. Inthealgebraicgeometryexample,theclosedsubsetsofCn areprecisely
thealgebraicsets(thatisthosedefinedbyasetofpolynomials)andthe
closedsubsetsofC[x ,...,x ]arepreciselytheradicalideals. TheGalois
1 n
correspondenceisthefundamentalcorrespondenceofalgebraicgeometry.
2. Inthenumbertheoryexample,providedtheextensionisGalois,theclosed
subsetsofLaretheintermediatefieldsbetweenLandK,whiletheclosed
subsetsofAut(L,K)arepreciselytheKrull-closedsubgroups. TheGalois
correspondenceistheclassicalGaloiscorrespondence.
1.1.2 TheLegendre-Fencheltransform.
SupposethatVisarealvectorspace,V#isitslineardualandFun(V,R)denotes
thespaceoffunctionsfromV totheextendedrealnumbersR := [ ∞,+∞].
−
Thereisastandardpairoftransformsbetweenfunctionspaces
L∗: Fun(V,R)(cid:29)Fun(V#,R): L ,
∗
givenby
L∗(f)(k)(cid:66)sup(cid:8) k,x f(x)(cid:9), L (g)(x)(cid:66) sup(cid:8) k,x g(k)(cid:9).
(cid:104) (cid:105)− ∗ (cid:104) (cid:105)−
x V k V#
∈ ∈
ThetransformL istheLegendre-FencheltransformandwewillrefertoL
∗
asthereverseLegendre-Fencheltransform.1 Thefunctionspaceseachhavea∗
partialorderonthemusingthetotalorderonR,for f , f : V R,
1 2
→
f (cid:60) f f (x)(cid:62) f (x) forallx V.
1 2 1 2
⇐⇒ ∈
WeusetheoppositeorderforthefunctionsonV#. Withrespecttotheseorders
thetwotransformsformaGaloisconnection,sowegetaclosureoperationon
eachfunctionspace.
L L∗: Fun(V,R) Fun(V,R); L∗L : Fun(V#,R) Fun(V#,R).
∗ → ∗ →
Theseareidempotentand,forinstance,L L∗(f) (cid:60) f. Infact,L L∗(f)isthe
∗ ∗
pointwise-smallest,lowersemi-continuous,convexfunctionwhichdominates
f;thelower-semicontinouspartisonlyrelevantwhenthefunctiontakesinfinite
values. Now we can look at the functions fixed by these closure operators
andthesearepreciselythelower-semicontinuousconvexfunctions. Fromthis
it follows that the Lengendre-Fenchel transform and its reverse restrict to a
1InSection5below,IwillreverttothestandardnotationofwritingL∗(f)as f(cid:63)andsimilarly
L (g)asg(cid:63).
∗
3
bijection, actually a Galois correspondence: using Cvx to denote the lower
semi-continuous,convexfunctionswehave
L∗: Cvx(V,R)(cid:27)Cvx(V#,R) : L .
∗
ThisisknownasLegendre-Fenchelduality. Itcropsupinmanyareasofmathe-
maticsforinstanceinmathematicalphysicstotranslatebetweenHamiltonian
andLagrangianformalisms(see,e.g.,[1]),inoptimizationtheory(see,e.g.,[18])
andinlargedeviationtheory(see,e.g.,[22]).
1.1.3 Thegeneralconstruction
Thetwoconstructionspresentedareclearlysimilar,theybothinvolveaGalois
connection to construct a duality (or a Galois correspondence if you prefer).
However,thesimilaritygoesmuchdeeperanditcanbeilluminatedinthelight
ofenrichedcategorytheory,asbothoftheseconstructionsareexamplesofwhat,
followingPavlovic[17],wewillcallthenucleusofaprofunctorconstruction;
ShenandZhang[19]alsoconsideredtheconstruction. Thetermsusedinthe
followingparagraphwillbeexplainedinSection2.
Fortheconstructionyoustartwithaprofunctorbetweentwosetsorenriched
categories,inthefirstcasethisistherelationbetweenGand M,thoughtofa
functionG M true,false ,andinthesecondcasethisisthetautological
× → { }
pairing V V# R R between the vector space and its dual. You then
× → ⊂
considerthecategoriesofpresheavesonyouroriginalcategories,inthefirst
casethisleadstotheposetsofsubsetsof G and M andinthesecondcaseto
the spaces of functions on V and V#. From the profunctor you construct an
enrichedadjunctionbetweenthepresheafcategories. Intheonecaseyouget
the Galois connection between the sets of subsets, in the other case you get
theLegendre-Fencheltransformanditsreverse. Thenucleusisthenextracted
asthepartsofthepresheafcategoriesonwhichtheadjunctionrestrictstoan
equivalence. In the one case you get the Galois correspondence, and often
aninterestingduality,intheothercaseyougettheLegendre-Fenchelduality
betweenconvexfunctions.
Herethesetofclassicaltruthvalues true,false andthesetofextendedreal
{ }
numbersRbothformexamplesofwhatacategorytheoristmightcallacomplete
andcocomplete, closedmonoidalposet; otherfolkmightcallsuchathinga
commutativequantaleoracomplete,commutative,idempotentsemiring. In
thispaperwepresentthetheoryofcategoriesandprofunctorsenrichedover
suchthingspayingparticularattentiontothesetwomainexamples.
1.2 Whatwegain
Afairquestiontoaskatthispointis“WhatdowegainfromviewingLegendre-
Fenchel duality as a profunctor nucleus?” It could be a case of translating
somethingfamiliartoanunfamiliarlanguagefornorealgain. Infact,wegain
a few things. On the one hand we see that from a category theory point of
view,theLegendre-Fencheltransformisinevitableonceyouhavethepairing
betweenavectorspaceanditsdual. Youcanseehowmuchofthetheoryisjust
formalnonsenseandhowmuchisspecifictothecaseinhand.Astheprofunctor
nucleusarisesinvariousplaces,suchasintightspansformetricspaces,thereis
thepotentialforbuildingbridgesbetweendifferentdisciplines.
4
Ontheotherhand,onamoreconcretelevel,afundamentalinsightthatwe
gainisthatthesetoffunctionsFun(V,R)shouldbeconsiderednotasaposet
butasanR-space,somethinglikeanasymmetricmetricspacewithpossibly
negative distances. There is an R-metric, like a very generalized notion of
distance,d: Fun(V,R) Fun(V,R) Rgivenby
× →
d(f , f )(cid:66)sup f (x) f (x) ,
1 2 2 1
{ − }
x V
∈
wheresomecareisneededtounderstandthedifferencebetweeninfinitequan-
tities, for example ( ∞) ( ∞) = ∞ and (+∞) (+∞) = ∞, see Sec-
− − − − − −
tion2.2.2fordetails. Thissatisfiesthetriangleinequalityandd(f, f) = 0if f
takesafinitevaluesomewhere, otherwised(f, f) = ∞. However, itisnot
−
symmetric.
UnderlyinganysuchR-metricisapreorder(cid:60),definedasfollows:
f (cid:60) f 0(cid:62)d(f , f ).
1 2 1 2
⇐⇒
Unpacking the definition this gives the usual partial order on the space of
functionsdescribedabove. SothisR-metriconFun(V,R)isarefinementofthe
usualpartialorder.
With respect to this R-metric, we find (Theorem 3) that the Legendre-
Fenchel transform is a distance non-increasing function so that d(f , f ) (cid:62)
1 2
d(L∗(f1),L∗(f2)),inotherwords,
sup f2(x) f1(x) (cid:62) sup L∗(f1)(k) L∗(f2)(k) .
{ − } { − }
x V k V#
∈ ∈
We also learn (Theorem 2) that the Legendre-Fenchel transform and its re-
verseformanadjunctionwhichmeansthatd(L∗(f),g) =d(f,L (g)),inother
∗
words,
sup L∗(f)(k) g(k) =sup L (g)(x) f(x) .
{ − } { ∗ − }
k V# x V
∈ ∈
ThisisarefinementofthestatementthattheLegendre-Fencheltransformand
itsreverseformaGaloisconnection,i.e.that f (cid:60)L (g)ifandonlyifg(cid:60)L∗(f).
∗
Inthecasethatitisrestrictedtoconvexfunctionsthenwefind(Theorem5)
that the Legendre-Fenchel transform is actually a distance preserving map,
meaningd(L∗(f1),L∗(f2)) =d(f1, f2),inotherwords,
sup L∗(f1)(k) L∗(f2)(k) =sup f2(x) f1(x) .
{ − } { − }
k V# x V
∈ ∈
ThisisnotpartofthestandardtreatmentoftheLegendre-Fencheltransform,
butitisknownasToland-Singerduality. Itisarefinementofthestandardorder
theoreticstatementthatforconvexfunctions f and f wehave f (cid:60) f
1 2 1 2
L∗(f2)(cid:60)L∗(f1). ThefactthatitisstrongerthanthatisillustratedinFigur⇐e1⇒.
InthefirstpictureofFigure1weseetwoconvexfunctions f , f : R R.
1 2
→
Neither function dominates the other so the fact that the Legendre-Fenchel
transform is order reversing tells us nothing about the transforms of these
functions. We see, however, that d(f , f ) = 1 and d(f , f ) = 3, so Toland-
1 2 2 1
Singer duality tells us that d(L∗(f1),L∗(f2)) = 1 and d(L∗(f2),L∗(f1)) = 3
whichweclearlyseeinthesecondpicture. Thedistanceonthefirstpictureis
5
3 f1
3
L∗(f2)
2
2
1 1
f2 x L∗(f1) k
-1 1 2 3 1 1
−
1
-1 −
Figure1: IllustratingToland-Singerduality. NoteL∗(f1)(k) =L∗(f2)(k) = +∞
when k >1.
| |
themaximumclimbbetweengraphsandthedistanceonthesecondpictureis
themaximumfall.
The nucleus of a profunctor is complete and cocomplete in a categorical
sense. Whatthatmeanshereinparticularisthatthespaceofconvexfunctions
hasproducts,coproducts,tensorsandcotensors(Theorem7).
Theproductoftwoconvexfunctionsisjusttheirpointwisemaximum:
f f (x)(cid:66)max(f (x), f (x)).
1 2 1 2
(cid:117)
The coproduct is the lower semi-continuous, convex hull of their pointwise
minimum:
f1 f2 (cid:66)L L∗(min(f1, f2)).
(cid:116) ∗
Intheordertheoreticworldthesearetheleastupperboundandgreatestlower
boundsofthefunctions,butinthismetricworldtheysatisfystrongerproperties,
namely
d(f, f f ) =max(d(f, f ),d(f, f ))
1 2 1 2
(cid:117)
d(f f , f) =max(d(f , f),d(f , f))
1 2 1 2
(cid:116)
The tensor product and cotensor product (cid:116) are actions of the monoid
(cid:12)
(R,+)onthesetofconvexfunctions: fora R, f Cvx(V,R)andx V,
∈ ∈ ∈
(a(cid:116) f)(x) = f(x) a; (a f)(x) = f(x)+a.
− (cid:12)
Theproduct andcotensor(cid:116)makethesetofconvexfunctions f Cvx(V,R)
(cid:117) ∈
intoamoduleoverthetropicalsemiring(R,min,+),thatisRwithminimum
asadditionandusualadditionasmultiplication. Similarly,thecoproduct and
(cid:116)
tensor makethesetofconvexfunctionsCvx(V,R)intoamoduleoverthe
(cid:12)
tropicalsemiring(R,min,+)inadifferentway.
1.3 Synopsis
ThebulkofthepaperisSection2inwhichwegothroughalotofessentiallystan-
dardcategorytheoreticnotionssuchaspresheaves,adjunctions,profunctors,
6
completenessandcocompleteness,butwedothisinthenon-standardcontext
ofcategoriesenrichedovertheextendedrealnumbersR; thisisintendedto
besuitableforthoseinterestedintheLegendre-Fencheltransformwithlittle
background in category theory. The nucleus of a profunctor is described in
Section 3 by analogy with constructing a pair of adjoint linear maps from a
matrix. InSection4weseehowtheusualtheoryofGaloisconnectionsfrom
relations arises by considering the case of categories enriched over classical
truthvalues. Thepay-offcomesinSection5inwhichtheprecedingtheoryis
puttogetherforR-categoriesandalotofthetheoryoftheLegendre-Fenchel
transformdropout,suchasToland-Singerdualityandthetwotropicalmodule
structuresonthesetofconvexfunctions.
1.4 Furtherquestions
Thereareseveralquestionsthatthisworknaturallyleadsto: herearetwoof
them.
What happens if we work over Z = Z ∞,+∞ ? Discrete convex
• ∪{− }
analysisseemstobemoresubtle(see[16]and[6])anditwouldbeworth
seeingifthisapproachleadstoanythinginteresting.
Fenchel Duality (see Theorem 31.1 of [18]) involves both convex and
•
concavefunctions;isitpossibletoformulatethisincategoricaltermsand
doesitgeneralizetoothersituations?
1.5 Acknowledgements
IwouldliketothankTomLeinster,MarkKambites,MarianneJohnson,Hiroshi
Hirai,SoichiroFujiiandBruceBartlettforusefulcommentsandconversations.
I would also like to thank contributors to the n-Category Cafe´ where earlier
piecesofthisworkwasposted.
2 Categories enriched over posets
Inthissectionthetheoryofcategoriesenrichedovermonoidalposetsispre-
sented with particular emphasis placed on enriching over the posets R and
Truth. Theintentionisthatthisshouldbeaccessibletoareaderwithaninterest
intheLegendre-Fencheltransformbutwithnotmuchbackgroundincategory
theory.
2.1 Quantales,thincategoriesandidempotentsemirings
Inthispaperweconsidercategoriesenrichedovermonoidalposets. Thestruc-
tureofamonoidalposetcropsupindifferentareaswithdifferentnamesand
differentfeaturesemphasised,soitisworthexplainingherehowtheseconnect.
Ourmainexamples—theextendedrealnumbersRwiththeorder(cid:62)andthe
setofclassicaltruthvaluesTruthwithentailmentastheorder—aregivenafter
thedefinitions.
7
2.1.1 Commutativequantales
In concise terms, a commutative quantale is a complete lattice which has a
commutative, unital, associative, binary operation which distributes over
arbitraryjoins. Unpackingthisdefinition,wehave⊗aposet2 (Q,(cid:88))suchthat
(cid:86) (cid:87)
everysubsetofW Qhasbothameet xandajoin x;inparticular,
thesesatisfy,foral⊂lw W, x∈W x∈W
∈
(cid:86) x(cid:88)w and w(cid:88)(cid:87) x.
x W x W
∈ ∈
When(cid:88)is(cid:54)thenthemeet(cid:86)istheinfimum,andwhenitis(cid:62)thenthemeet(cid:86)
isthesupremum.
Thisposetisequippedwithamonoidstructure : Q Q Qwithunit
⊗ × →
1 Qandwhichsatisfiesdistributivityoverjoins:
∈
q (cid:87) x = (cid:87) (q x).
⊗ x∈W x∈W ⊗
Fromthisdatawecandefinearesiduationmap[ , ]: Q Q Qby
− − × →
[b,c](cid:66)(cid:87)a:a b(cid:88)ca.
⊗
Thiswillsatisfytheadjunctionproperty
a b(cid:88)c a(cid:88)[b,c].
⊗ ⇐⇒
2.1.2 Idempotentsemirings
Acommutativeidempotentsemiringisacommutativesemiringinwhichthe
additionisidempotent.Firstly,acommutativesemiring,basicallyaringwithout
negatives,consistsoftwocommutativemonoidstructures(S, ,0)and(S, ,1)
⊕ ⊗
onthesamesetSsuchthat distributesover . Theadditionisidempotentif
⊗ ⊕
s s = s foralls S.
⊕ ∈
ThisallowsustodefineapartialorderonSby a (cid:88) bifandonlyif a b = b.
(cid:87) ⊕
Then becomesthejoin ,and distributesoverit. Byinductionthismeans
⊕ ⊗
thatwehavejoinsforarbitraryfinitesubsets,andthatmultiplicationdistributes
overthesefinitejoins. WesaythatSiscompleteifarbitrarysubsetshavejoins
andmultiplicationdistributesoverarbitraryjoins. Arbitrarymeetscanthenbe
definedintermsofthejoins: forasubsetW S defineW(cid:48) := y S y (cid:88)
xforallx W then(cid:86) x = (cid:87) y. Again⊂theresiduationca{nb∈ede|fined
by ∈ } x∈W y∈W(cid:48)
[b,c](cid:66) (cid:77) a.
a:a b(cid:88)c
⊗
Onecanseeinthismannerthatacomplete,commutative,idempotentsemiring
isthesamethingasacommutativequantale.
2Iamassiduouslynotpreferringeither(cid:54)or(cid:62)asmyrelation.Here(cid:88)canbepronouncedas‘is
relatedto’.
8
2.1.3 Monoidalthincategories
Athincategoryisacategoryinwhichthereisatmostonemorphismbetween
eachpairofobjects. Workinginathincategoryismuchsimplerthanworking
inanarbitrarycategoryas,forinstance,alldiagramscommute. Havingathin
categoryisthesameashavingatransitive,reflexiverelationontheobjectsof
thecategory,thecorrespondenceisdefinedvia
a(cid:88)b thereisamorphisma b.
⇐⇒ →
This is not quite the same thing as having a partial order, however, as anti-
symmetrymightfail: wecouldhavedistinctobjectsaandbwhichhavearrows
a b and b a. Such a relation is called a preorder. The condition that a
→ →
thincategoryactuallycorrespondstoapartialorderispreciselytheskeletal
conditionwhichisthatallisomorphismsareequalities. Wecanturnanythin
categoryintoaskeletalcategorybytakingisomorphismclassesofobjects.
Amonoidalcategoryisacategory withafunctor : together
C ⊗ C×C → C
with some associativity and unitality constraints. In the case of a skeletal,
thin category the constraints are just that the product should be associative
and unital on the set of objects. The monoidal structure is symmetric if this
productiscommutative. Themonoidalstructureisclosedifthereisafunctor
[ , ]: op whichisadjointto inthesensethat
− − V ×V → V ⊗
thereisamorphisma b c thereisamorphisma [b,c].
⊗ → ⇐⇒ →
Asyoushouldbeabletotell,thisisthesameastheconditionontheresiduation
foraquantale. Therearenotionsofcategoricalproductsandcoproducts,for
thin categories these amount to precisely meets and joins for the associated
preorders. Asthemonoidalproducthasarightadjointitwillautomatically
distributeoverjoins. Beingcompleteandcocompleteinthecategoricalsense
amounts,forathincategory,tohavingarbitraryproductsandcoproducts.
Fromthisconcisediscussionitishopefullyclearthatacompleteandcocom-
plete,skeletal,closed,symmetricmonoidalthincategoryisthesamethingasa
commutativequantale.
2.2 Ourmainexamples
Themainexamplesoftheabovestructuresthatweareinterestedinhereare
truth values and extended real numbers. We will get to Galois connections
usingthefirstandtheLegendre-Fencheltransformusingthesecond.
2.2.1 Truth
WetakeTruthtobethecategorywithtwoobjectstrueandfalseandasingle
non-identityarrowfalse true. Themonoidalproductistakentobe‘logical
→
and’&,andtheresiduationisimplication,so[a,b] = (a b). Theorderonthe
⇒
set true,false canbeinterpretedas‘entailment’ ,sotheonlynon-reflexive
{ } (cid:96)
relationshipisfalse true. Themeetisthesameastheproduct,namelylogical
(cid:96)
and,butisoftenthoughtofas‘forall’. Thejoinis‘logicalor’,oftenthoughtof
as‘thereexists’.
9
2.2.2 R
Todefinethestructureontheextendedrealnumbers,westartwithRequipped
with the order (cid:62), so that meet (cid:86) is supremum and join (cid:87) is infimum, this
hasusualadditionasthemonoidalproductandsubtractionastheresiduation
because
a+b(cid:62)c a(cid:62)c b.
⇐⇒ −
We make this structure complete by adding a maximum element +∞ and a
minimumelement ∞todefineR. Thentherequirementthat+distributes
−
overinfimaensuresthatthereisauniqueextensionof+toR. Finallywecan
usetheformulaforresiduationtoextendsubtractiontoR:
c b = inf a.
− a+b(cid:62)c
ThearithmeticissummarizedinthefollowingtablesliftedfromtheBachelor’s
ThesisofFujii[6].
y y
x+y ∞ t +∞ y x ∞ t +∞
− − −
∞ ∞ ∞ +∞ ∞ ∞ +∞ +∞
x −s −∞ s−+t +∞ x −s −∞ t s +∞
+∞ −+∞ +∞ +∞ +∞ −∞ −∞ ∞
− − −
Itisimportanttoobservethatthereareafewsubtletiesinthesedefinitions,so
theydon’tbehavequitehowyoumightnaivelyexpect.Forinstance,subtracting
+∞isnotthesameasadding ∞:
−
(+∞)+( ∞) = (+∞); (+∞) (+∞) = ( ∞).
− − −
Asimilarstructure,butwiththeoppositeorderwasconsideredbyLawvere
in[12];thisstructurewasconsideredfromtheidempotentsemiringperspective
in[4]. Whilstthisarithmeticoftheinfinitemightseemalittlestrange,itisthe
appropriatestructurefortheLegendre-Fencheltransform.
As an idempotent semiring, R has + as its multiplication and min as its
addition.
2.3 EnrichedstructuresforTruthandR
Wewillgothroughalotofthetheoryofcategoriesenrichedoverourtwomain
examples. Thestandardreferencesformuchofthegeneraltheoryofenriched
categoriesare[8]and[3].
2.3.1 Categories
Recallthatanordinarysmallcategory canbedefinedtoconsistofasetof
C
objectsob( )suchthat
C
foreveryc,c(cid:48) thereisaset (c,c(cid:48)) ob(Set)calledthehom-set;
• ∈ C C ∈
foreveryc,c(cid:48),c(cid:48)(cid:48) thereisafunction (c,c(cid:48)) (c(cid:48),c(cid:48)(cid:48)) (c,c(cid:48)(cid:48)),this
• ∈ C C ×C → C
iscalledcomposition;
10