Table Of ContentQuotients in monadic programming:
Projective algebras are equivalent to coalgebras
7
1
0 DuskoPavlovic∗ Peter-Michael Seidel
2
b UniversityofHawaii,HonoluluHI, USA
e
F
7
] Abstract
O
L Inmonadicprogramming,datatypesarepresentedasfreealgebras,generated
bydatavalues,andbythealgebraicoperationsandequationscapturingsomecom-
.
s putational effects. These algebras are free in the sense that they satisfy just the
c
equations imposed by their algebraic theory, and remain free of any additional
[
equations. Theconsequence isthattheydonotadmitquotient types. Thisis,of
2 course, ofteninconvenient. Wheneveracomputationinvolvesdatawithmultiple
v representatives, and they need to be identified according to some equations that
1
arenotsatisfiedbyalldata,themonadicprogrammerhastoleavetheuniverseof
0
freealgebras,andresorttoexplicitdestructors.Wecharacterizethesituationwhen
6
these destructors are preserved under all operations, and the resulting quotients
7
of freealgebras are alsotheir subalgebras. Such quotients are calledprojective.
0
. Althoughpopularinuniversalalgebra,projectivealgebrasdidnotattractmuchat-
1
tentioninthemonadicsetting,wheretheyturnouttohaveasurprisingavatar:for
0
anygivenmonad,asuitablecategoryofprojectivealgebrasisequivalentwiththe
7
categoryofcoalgebrasforthecomonadinducedbyanymonadresolution. Fora
1
monadic programmer, this equivalence provides a convenient way to implement
:
v polymorphicquotientsascoalgebras. Thedualcorrespondenceofinjectivecoal-
i gebrasandallalgebrasleadstoadifferentfamilyofquotienttypes,whichseems
X
to have a different family of applications. Both equivalences also entail several
r
a generalcorollariesconcerningmonadicityandcomonadicity.
1 Introduction
1.1 The story
Background:Monadicprogramming. Monadsareoneoffunctionalprogrammers’
favoritetools,andpossiblyoneofthemostpopularcategoricalconcepts[14,13].Asa
typeconstructor,amonadgivesrisetodatatypesthatcapturenotonlythedatavalues,
butalsosomecomputationaleffectsofinterest[35,42]. While thisisachievedusing
∗SupportedbyAFOSRandNSF.
1
averysimpleandconvenientsetoftools,thehistoryoftheunderlyingideasisconvo-
luted, and the conceptualand technicalbackgroundof monadic programmingcovers
enoughterritoryofalgebraandofcategorytheorytoconcealmanymathematicalmys-
teries.
The conceptual origin of monadic programming was probably the idea that data
structures can be captured as algebraic theories, which goes back to the early days
ofsemanticsofcomputation[20,21]. Thetechnicaloriginofmonadicprogramming
wasthentheideathatalgebraictheoriescanbecapturedasmonads,whichgoesback
even further, to the early days of category theory [33, 34, Ch. I]. The upshot of the
view of data-structures-as-algebraic-theories is that computational datatypes, as do-
mainsofinductiveandrecursivedefinitions,canbeviewedasinitial,orfreealgebras,
implementinginductionasauniversalproperty. Theupshotoftheviewofalgebraic-
theories-as-monadsinthiscontextisthefactthatmonadsencapsulateandhidebehind
theirstandardstructure1thediverseandoftenbewilderingsetsofalgebraicoperations,
andmakethemavailableonlythroughuniformlystructuredmonadiccombinators.
The mainfeatureof computationalmonadsis thustheir succinctandelegantren-
dering of inductive datatypes as free algebras. But this main feature is perhaps also
theirmainlimitation:thequotientsoffreealgebrasarenotfreealgebras.
Problem: Quotient types. Whenevera data value can be given by differentrepre-
sentatives,itsdatatypeisaquotient. E.g.,eachrationalnumbercanberepresentedby
infinitelymanyfractions(3 = 39 = 273 = ···), so thedatatypeof rationalsisa quo-
7 91 637
tientofthedatatypeoforderedpairsofintegers. Setsareaquotientofbags,bagsare
aquotientoflists, andsoon. Identifyingtheequivalentrepresentativescanbeahard
andimportantcomputationaltask, tackledin typetheoryfromthe outset, goingback
to Martin-Löf, and still further back to Leibniz. Different applications often justify
differentimplementations[1,6],whichvaryfromsimplycarryingexplicitequivalence
relationswithsetoids[11,31],throughcarryingcoherentequivalenceswithgroupoids
[25,24,3],allthewaytotherichstructureofhomotopytypes[8,40],wheretheprob-
lemofquotientsintypetheoryandtheproblemofinvariantsingeometryseemtobe
solvingeachother.
Thebasicideaofmonadicprogramming,topresentdatatypesasfreealgebras,pre-
cludesdirectimplementationsofquotienttypes,sinceaquotientofafreealgebraisin
generalnotfree.Thisisoftenviewedasafeature,sincepolymorphismrequiresthatall
datasatisfyexactlythesameequations,whichforalgebrasmeansthattheyshouldsat-
isfyjusttheequationsimposedbytheiralgebraictheory,andnoadditionalequations.
Whennecessary, the additionalequationscanbe imposedbyexplicitdestructors, but
thepolymorphicconstructionsgenerallydonotcarryovertosuchquotients,unlessthe
destructorspreservethem. Underwhichconditionsdoesthathappen?
Solution: Projective algebras. In the present paper, we study a special family of
quotientsoffreealgebras: thosethatalsohappentobetheirsubalgebras. Thismeans
thattheycan be implementednotonlyby imposingadditionalequations, butalso by
1Godementintroducedmonadsunderthenamestandardconstruction,standardizingthesheafcohomol-
ogyconstruction[19].
2
adjoiningsuitableoperations,calledprojectors,asdescribedbelow. Analgebrawhich
isbothaquotientandasubalgebraofafreealgebra,i.e. itsretract,issaidtobeprojec-
tive[22,§82]2. Sinceprojectorsinducepreciselythequotientsthatarepreservedbyall
functors[36], thisaproachseemsnecessaryandsufficientforextendingpolymorphic
constructionsfromfreealgebrastotheirquotients.Theequivalenceofprojectivealge-
brasfora monadandallcoalgebrasforanyofthe correspondingcomonads,claimed
inthetitleofthepaper,suggestsalinkbetweentheproblemofpolymorphicquotients
in monadic programmingand the ideas of comonadic programming, put forward by
severalauthors[17, 41, 2]. Theduallinkofinjectivealgebrasfora comonadandall
algebras for any of the corresponding monads suggests a link between polymorphic
quotientsof cofree coalgebrasand unrestrictedquotientsof algebras. We proceedto
workouttheselinks.
Prerequisites
This is a paper about categorical semantics of computation, so the prerequisites are
mostlycategorical. ThemainbackgrounddefinitionsarereproducedintheAppendix.
Averysuccinctoverviewoftheunderlyingconceptscanbefoundin[28,Sec.I.3].
1.2 Motivatingexample
LetCbeacartesianclosedcategory,i.e. givenwithanadjunction
A×
C ⊥ C (1)
A⇒
foreveryobjectA. FixanobjectS asastatespace,andconsiderthemonad
←−
S : C →− C (2)
X 7→ S ⇒(S ×X)
inducedbytheadjunction(1)instantiatedtoS. Asexplainedin[35], thecategoryof
←−
free S-algebrasC←− capturescomputationswith explicitstate, orwith side effects. A
S
computationovertheinputsoftypeA,theoutputsoftypeB,andthestatesoftypeS is
presentedasafreealgebrahomomorphism f ∈ C←−(A,B),whichcanbeconveniently
S
viewed as a C-morphism in the form A −−−f→ S ⇒ (S × B) [32]. This computation
f(a)
thusmaps everyinputa ∈ A to a functionS −−−−→ S × B, determiningat each state
s ∈ S anextstate,andanoutput. Equivalently,suchmorphismscanbeviewedinthe
f′
transposedformS ×A−−−−→S ×B,assigningtoeachstateandeveryinputanextstate
and an output. This transposed form of homomorphismsbetween free algebras will
turnouttobemoreconvenientforthepurposesofthispaper. Inthecase ofthestate
←−
monad S,suchhomomorphismscaptureMealymachines[15,23,26,Sec.2.7(a)].
2Thenameisborrowedfromtheoryofmodules,wheretheretractsoffreemodulesarealsocalledpro-
jective.
3
Towards a more concrete example, consider the following model of data release
f
policiesfrom[39]. SupposethataMealymachineS ×A→− S ×Bmodelsadatabase.
S areitsstates,Aaretheinputs(insertions,queries,...) thattheusersmayenter,and
g
Baretheoutputssuppliedbythedatabase.AstatelessmapA→− Bcanbethoughtofa
rudimentarydeterministicchannel,justmappingdataoftypeAtodataoftypeB.Since
therearemultipleusers,theremaybeprivacypolicies,andauthorizationpoliciesthat
ψ
needto be implemented. A privacypolicycanbe viewed asa mapS ×B −→ S ×B,
whichprojectsanyoutputb ∈ Bofthedatabaseatastate s ∈ S toasanitized,public
componentψ(s,b)ofthestateandtheoutputofthedatabase,andfiltersoutallprivate
ϕ
data.AnauthorizationpolicycanbesimilarlyviewedasamapS ×A→− S ×A,which
projects any database state s ∈ S, and any user input a ∈ A (including the relevant
credentials)to the authorizedcomponentϕ(s,a) of the state and the input. Since the
publiccomponentsshouldnotcontainanyprivateresiduetobefilteredoutinasecond
runofψ,andtheauthorizedcomponentsshouldnotcontainanyunauthorizedresidue,
thepoliciesshouldbeidempotent,i.e. satisfytheequations
ϕ◦ϕ=ϕ ψ◦ψ=ψ
Such equations define projectors. There are at least two different ways to interpret
projectors as policies. One is to view them as policy specifications. The database
f ψ ϕ
S ×A→− S ×BthenimplementsthepoliciesS ×B−→S ×BandS ×A→− S ×Aifit
satisfiestheequation
f = ψ◦ f ◦ϕ (3)
whichiseasilyseentobeequivalenttothepairofequations
ψ◦ f = f = f ◦ϕ (4)
In other words, a compliant database only ever supplies public data, and only ever
permitsauthorizedrequests.
A different view to interpret projectors as policies is to view them as policy im-
f
plementations. The database S × A →− S × B then does not implement the policies
ϕ
itself,butneedstobeprecomposedwiththeauthorizationpolicyS ×A →− S ×Aand
ψ
postcomposedwiththeprivacypolicyS ×B −→ S ×B. However,sinceeachofthese
f
policiesregulatesthesamedatarelease,thedatabaseS ×A→− S ×Bisconsistentwith
ψ ϕ
thepoliciesS ×B−→S ×BandS ×A→− S ×Aifitsatisfiestheequation
ψ◦ f = f ◦ϕ (5)
It is obvious that compliance implies consistency. A consistent database, however,
doesnothavetobecompliant. Thereasonisthataconsistentdatabasedoesnothave
toimplementthepoliciesitself,butitrequiresseparatepolicyimplementationsatthe
inputandattheoutput. Onitsown,suchadatabasemayacceptunauthorizedrequests
and it may supply private data. Its consistency means that if we make sure that no
4
unauthorizedrequestsaresubmitted,thenwecanbesurethatnoprivateresponseswill
besupplied,andviceversa. Moreprecisely,adatabase f isconsistentwiththepolicies
ϕ and ψ whenever a request consistent with ϕ results in a response consistent with
ψ, andallresponsesconsistentwith ψ canbeobtainedonrequestsconsistentwith ϕ.
Sincethetwopoliciesthuspreciselyenforceeachotheronadatabaseconsistentwith
them,theyimplementthesame datareleaseprocessonthisdatabase, whichcanthus
beimplementedeitherasanauthorizationpolicy,orasaprivacypolicy.
Lifting this distinction between compliance and consistency from databases and
statemonads,toadistinctionbetweentwotypesofhomomorphismsbetweenprojec-
tivealgebras,wearriveatthemainresultsofthepaper.
1.3 The setting and theresult
EveryadjunctionF∗ ⊣ F :B→− Adetermines
∗
←−
• themonad F =F F∗ :A→− A,withtheinducedcategoriesof
∗
– freealgebrasA←−,
F
– projectivealgebrasA(cid:9),and
←−
F
←−
– allalgebrasAF,
andontheotherhand
→−
• thecomonad F = F∗F :B→− B,withtheinducedcategoriesof
∗
– cofreecoalgebrasB→−,
F
– injectivecoalgebrasB(cid:9),and
→−
F
→−
– allcoalgebrasBF.
←−
Theotherwayaround,givenamonad F : A→− A, anyadjunctionF∗ ⊣ F : B→− A
∗
←− ←− →−
such that F = F F∗ is a resolution of F; and given a comonad F : B →− B, any
∗
→− →−
adjunction such that F = F∗F is a resolution of F. Each of the above categories
∗
defined for a monad (resp. for a comonad) gives its resolution. The definitions are
standard,andcanbefoundintheAppendix. Inthispaper,weintroducethecategories
of
• projectivealgebraswithconsistentmorphismsA",and
F
• injectivecoalgebraswithconsistentmorphismsB#.
F
Weprovethefollowingequivalencesofcategories
←− →−
AF ≃B# BF ≃A" (6)
F F
5
undertheassumptionthatthecategoriesAandBareCauchy3complete,whichmeans
thattheysplitidempotents. Thisisaverymildassumption,sincetheCauchycomple-
tion,theweakestofallcategoricalcompletions,iseasytoconstructforanycategory,
asspelledoutinProp.2.2.
Overviewofthe paper
Sec. 2 begins with a discussion about projectors, and analyzes projectors over free
algebras,whichdetermineprojectivealgebras. InSec.3westate andprovethemain
theorem,showingthatprojectorsoverfreealgebrascorrespondtocoalgebras.Wealso
statethedualversion,whichsaysthatprojectorsovercofreecoalgebrascorrespondto
algebras. In Sec. 4, we return to the motivating example from Sec. 1.2, and analyze
thedifferentcategoricalformalisationsofdatareleasepolicies. Sec.5closesthepaper
withcommentsabouttherelatedpastwork,andaboutthefuturework.
2 Projectors over algebras and coalgebras
2.1 Projectors ingeneral
Consideranequalizerandcoequalizerdiagram
ϕ
E i A A q Q
id
foranarbitraryendomorphismϕ.Intuitively,theequalizerEconsistsofthefixedpoints
ofϕ, whereasthecoequalizerQ isthequotientwhereeachelementof Aisidentified
with all of its direct and inverse images along ϕ, which together form its orbit. The
obviousmapE→− Qmapseachfixedpointintoauniqueorbit;butsomeorbitsmaynot
containanyfixedpoints.Weareinterestedinthesituationwheneachorbitdoescontain
a fixedpoint, so thateachequivalenceclassfrom Q hasa canonicalrepresentativein
E. This means that the iterated applications of ϕ push each element of A along its
orbittowardsafixedpoint. Itcanbeshownthatthissituationischaracterizedbythe
requirementthatthefollowingcountablyextendeddiagramcommutes
ϕ
ϕ ϕ
A A A ···
id
In terms of elements, this means that for every x ∈ A there is some n ∈ N such that
ϕn+1(x) = ϕn(x). Inotherwords,ϕthusequipsAwiththestructureofaforest,where
theequivalenceclassesthatformQarethecomponenttrees,andtheelementsofEare
theirroots.Projectorsarethespecialcaseofthissituation,wherealreadythediagram
ϕ
ϕ
A A A
id
3ThehabitofattributingcategoricalconceptstoXIXcenturymathematiciansandphilosophershasstyled
theterminologyinsomepartsofcategorytheory.
6
commutes. The forest thus reduces to a shrub, where each component may branch
at the rootas wide as it likes, but can onlygrow one layer tall, and cannotgrowany
furtherbranches.Thefollowingsummarizes[7,Sec.IV.7.5].
Proposition2.1. If the endomorphism A →−ϕ A satisfies ϕ◦ϕ = ϕ, then for any pair
q
A B,thefollowingstatementsareequivalent:
i
(a) iisanequalizerandqisacoequalizerofϕandid
B
i q
ϕ
A A
id
(b) i◦q=ϕandq◦i=id.
ϕ
Definition 2.1. An endomorphism A →− A is called a projector (or idempotent) if
q
ϕ◦ϕ = ϕ. Its splitting is an object B with a pair of arrows A B such that
i
i◦q=ϕandq◦i=id. Moresuccinctly,wewriteA#ϕ B.
Since the projectors and their splittings are defined by equations, every functor
mustpreservethem.Sinceasplittingconsistsofanequalizerandacoequalizer,itisan
equalizerandacoequalizerthatmustbepreservedbyallfunctors.
Definition 2.2. A categorical property is called absolute when it is preserved by all
functors. A category that has all absolute limits and absolute colimits is said to be
Cauchycomplete.
It follows from the results of [36], as well as from the differentapproach in [16,
Sec.I.6.5],thatallabsolutelimitsandcolimitsboildowntosplittings.
Proposition2.2. ForanycategoryCthefollowingstatementsareequivalent
(a) CisCauchycomplete,
(b) allprojectorssplitinC,
(c) theobviousembeddingC֒→C(cid:9) isanequivalence,where
|C(cid:9)| = {A→−ϕ A|ϕ◦ϕ=ϕ}
a
A∈|C|
f
A B
C(cid:9)(A→−ϕ A,B−→ψ B) = f ∈C(A,B)(cid:12)(cid:12) ϕ ψ
(cid:12) f
(cid:12) A B
7
TheabsolutecompletionC(cid:9)issometimescalledKaroubienvelopeofC[7,Sec.IV.7.5].
Two categoriesare Morita equivalent when their Cauchy completionsare equivalent
[16,Thm.7.9.4]. Notethattheconditionψ◦ f ◦ϕ = f isequivalentwiththerequire-
mentthatboth f ◦ϕ= f andψ◦ f = f arevalid.
Assumption. In the rest of this paper, we assume that each of the categories under
consideration is Cauchy complete, i.e. that projectors split in it. Any category C
that does not fulfill this assumption should be replaced by its Karoubi envelopeC(cid:9),
describedinProp.2.2(c).
2.2 Projectivealgebras overfree algebras
Inhomologicalalgebra,projectivemodulesareusuallydefinedasdirectsummandsof
freemodules. In the terminologyofthe precedingsection, thismeansthattheyarise
bysplittingtheprojectorsoverfreemodules. Itisnaturaltodefineprojectivealgebras
inasimilarway:asprojectorsoverfreealgebras[22,§82].
Intheusual(Kleisli)viewofthecategoryoffreealgebras[32],reproducedinthe
Appendix,amorphism f ∈A←−(A,B)isamorphismA→−f ←F−BinA,anditscomposition
withg∈A←−(B,C),whichisBF→−g ←F−CinAisdefinedby
F
g◦←− f = A−−→f ←F−B−←−F−→g ←F−←F−C −−µ→←F−C ∈ A←−(A,C)
F F
Aprojectorϕ∈A←−(A,A)overthefreealgebrageneratedbyAisthusanA-morphism
F
A →−ϕ ←F−Asuchthatϕ◦←F− ϕ = µA◦←F−ϕ◦ϕ = ϕ. Astheyexpand,thecalculationswith
projectorsintheKleisliformofcategoryA←−dogetincreasinglyclumsy.
←− F
Whenamonad F : A→− AisinducedbyanadjunctionF∗ ⊣ F : B→− Asothat
∗
←−
F = F F∗,thenthecategoryoffreealgebrascanbeequivalentlydefinedby
∗
|A←−| = |A|
F
A←−(X,Y) = B(F∗X,F∗Y)
F
Itiseasytocheckthatthenaturalbijections
A(X,F F∗Y) (cid:27) B(F∗X,F∗Y)
∗
←−
maptheKleislicompositionofthemorphismsinA(X,FY)totheordinarycomposition
←−
oftheiradjunctiontransposesinB(F∗X,F∗Y). Ifthehomomorphismsbetweenfree F-
algebrasarepresentedastheelementsofB(F∗X,F∗Y),thenaprojectorϕ ∈ A←−(X,X)
F
is just a B-morphism F∗X →−ϕ F∗X such that ϕ◦ϕ = ϕ. The category of projective
←−
F-algebrasand compliant homomorphisms(explainedin Sec. 1.2) is thusdefined as
8
follows:
|A(cid:9)| = ϕ∈B(F∗X,F∗X) ϕ◦ϕ=ϕ
←F− Xa∈|A|n (cid:12)(cid:12)(cid:12) o
(cid:12)
F∗X h F∗Y
A(cid:9)←F−(ϕ,ψ) = h∈B(F∗X,F∗Y)(cid:12)(cid:12) ϕ ψ
(cid:12)(cid:12) F∗X h F∗Y
whereψ ∈ B(F∗C,F∗C) is anotherprojectivealgebra, viewedas a projectoroverthe
freealgebrageneratedbyC ∈|A|.
2.3 Projectivealgebras among allalgebras
←− ←−
Thecategoryoffree F-algebrasA←− embedsfullyandfaithfullyintothecategoryAF
←− F
ofall F-algebrasbythefunctor
←−
A M AF (7)
←−
F
definedby
X ∈ A←− F∗X→−h F∗Y ∈A←−(X,Y)
(cid:12) F(cid:12) (cid:18) (cid:19) F
(cid:18)←F−←F−X→−µ ←F−(cid:12)(cid:12) X(cid:19)(cid:12)(cid:12)∈(cid:12)A←F−(cid:12) and (cid:18)←F−X −F−∗→h ←F−Y(cid:19) ∈ A←F− µX,µY
(cid:12) (cid:12) (cid:0) (cid:1)
(cid:12) (cid:12)
whereF hisanalgebrahomomorphismbecauseµ= F ε,andthenaturalityofεthus
∗ ∗
←− ←−
impliesF h◦µ =µ ◦ FF h. SincetheprojectorsinAF splitwhenevertheysplitin
∗ X Y ∗
A,asassumedhere,theembeddingMhasauniqueextensionM(cid:9),
A(cid:9) M(cid:9) A←F−
←−
F
(8)
M
A
←−
F
←−
whichmapseachϕ∈|A(cid:9)|toasplittingαoftheprojectorF ϕ∈AF(µ ,µ )
←− ∗ X X
F
←F−←F−X ←F−q ←F−A ←F−i ←F−←F−X
(9)
µ α µ
←F−X q A i ←F−X
F∗ϕ
←− ←− α ←−←− µ
An F-algebraFA−→Aisthusprojectiveifitisaretractofsomefreealgebra FFX→−
←− ←−
FX, i.e. if there are an X ∈ |A| and some homomorphismsq ∈ AF(µ ,α) and i ∈
X
←−
AF(α,µ )suchthatq◦i=id.
X
9
←− α
Itturnsout,however,thateachprojectivealgebra FA−→Aisnotjustaretractofa
freealgebraoversomeX ∈|A|,butaretractofthefreealgebraoveritsowncarrierA.
←− α
Proposition 2.3. An algebra FA −→ A is projective if and only if there is a unique
←−
algebrahomomorphismα∈AF(α,µ )suchthatα◦α=id.
A
←− ←− ←−←− ←− ←−
FA Fα FFA Fα FA
α µA α
←−
A FA A
α α
←− ←−
Proof. Toconstructα ∈ AF(α,µ ),extendthealgebrahomomorphismq ∈ AF(µ ,α)
A X
←− ←− ←− ←− ←−
to F(q◦η)∈AF(µ ,µ )andprecomposewithi∈AF(α,µ ).Henceα= Fq◦Fη◦i∈
X A X
←−
AF(α,µ ),asdisplayedinthemiddlerowofthefollowingdiagram.
A
←F−A ←F−i ←F−←F−X ←F−←F−η ←F−←F−←F−X ←F−←F−q ←F−←F−A
α µ µ µ
←− ←F−η ←−←− ←−
A FX FFX FA
i ←−
Fq
µ α
id
←−
FX A
q
←−
Thecommutativityoftheupperthreesquaresimpliesthatαisan F-algebrahomomor-
phism. The commutativityof the lower square and the triangle implies that α◦α =
q◦i=id.
←− α
Toseethatαisunique,i.e. thatitistheonlywaytodisplay FA−→Aasasubalge-
←−←− µ ←−
braof FFA→− FA,notethatthecompositehomomorphismα◦αisaprojectoronthe
freealgebra←F−←F−A→−µ ←F−A,andthat←F−A։α Aisthesplittingofthisprojector.
←−←− ←− ←− ←− ←−←−
Fα Fα
FFA FA FFA
µ α µ
←− ←−
FA A FA
α α
Whilethesplittingofaprojectorisuniqueuptoanisomorphism,fixingonecomponent
ofthesplittingdeterminestheotheroneon-the-nose:sinceαisanepi,α ◦α=α ◦α
0 1
impliesthatα =α . (cid:3)
0 1
The precedingpropositionthussays thateveryprojectivealgebra←F−A ։α A hasa
uniqueembeddingA α ←F−A intothefreealgebraµ overitscarrier A. With noloss
A
10
