THE CATEGORICAL THEORY OF SELF-SIMILARITY
This paper is dediated to Joahim Lambek,
whose work was the inspiration for the following results.
PETER HINES
ABSTRACT. WedemonstratehowtheidentityNN
=
N in amonoidalategory
allows us to onstrut afuntor from the full subategory generated by N and to
theendomorphismmonoid oftheobjetN. This providesaategorialfoundationfor
one-objetanaloguesofthesymmetrimonoidalategories usedbyJ.-Y.Girardin his
Geometry of Interation series of papers, and expliitly desribed in terms of inverse
semigrouptheoryin[6,11℄.
This funtor also allows the onstrution of one-objetanalogues of other ategorial
strutures. Wegive the exampleof one-objetanaloguesof the ategorialtrae, and
ompatlosedness. Finally,wedemonstratehowtheategorialtheoryofself-similarity
an be related to the algebrai theory (as presented in [11℄), and Girard'sdynamial
algebra,byonsideringone-objetanaloguesofprojetionsandinlusions.
1. Introdution
It is well-known, [12℄, that any one-objet monoidal ategory is an abelian monoid with
respet totwo operationsthat satisfy the interhangelaw (ab)Æ(d)=(aÆ)(bÆd);
that is, allthe anonial isomorphismsfor the tensor are identities. However, non-trivial
one-objet analogues of the anonial assoiativity and ommutativity morphisms for
symmetri monoidal ategories have been impliitly onstruted by Girard for his work
onlinearlogi,[4,5℄,andhavebeenstudiedintermsof inversesemigroupsin[6℄and[11℄.
The absene of aunit allows fora more interesting theory.
Wedemonstrate insetion2 howthese examplesarise naturallyfromthe assumption
of self-similarity. This is motivated by the examples of either the intuitive onstrution
of the Cantorset, or bijetionsbetween N and N N. Wedene aself-similar objet of a
symmetrimonoidalategorytobeanobjetN satisfyingN
=
NN. Inthe ategorial
ase, this gives rise to examples of struture-preserving maps between ategories and
monoids. In setion 3, we then demonstrate how these maps allow us to onstrut
one-objet analogues of the ategorial trae and ompat losedness. Finally, in setion
4, these ategorial onstrutions are related to the algebrai approah to self-similarity
desribed in[11℄, and Girard'sdynamialalgebra,byonsidering one-objet analoguesof
projetions and inlusions.
Reeivedbytheeditors1998Otober6and,in revisedform,1999September13.
Publishedon1999November30.
1991MathematisSubjetClassiation: 18D10,20M18.
Key wordsand phrases: Monoidal Categories, Categorial Trae, Compat Closure, Linear Logi,
InverseSemigroups.
2. Self-similarity and tensors
2.1. Definition. Let (M;)bea symmetri monoidalategoryand denotethe
assoia-tivity, ommutativity, and left and right unit isomorphisms by t
A;B;C ;s
A;B ;
A ;
A
respe-tively. We say that an objet N of M is self-similar if N
=
N N; that is, there exist
morphisms : N N ! N and d : N ! N N that satisfy d = 1
NN
and d = 1
N .
We all these morphismsthe ompression and division morphisms of N.
Forthepurposesofthis paper, wewillrequirethe additionalonditionthatthetensor
isnot strit at N, so t
N;N;N
is not the identity map. We alsoassumethat the
ommu-tativityisnotstrit,toavoidstudyingdegenerateases. Westudythe fullsubategoryof
M given by all objets onstruted fromN and , whih we referto asN
. We assume
throughout that N 6=I.
The subategory N
has allthe strutureof amonoidalategory apartfromthe unit
objet;weuse the awkwardtermunitless monoidal ategories 1
forstruturesofthis form.
The followingis then immediate:
2.2. Lemma. All objets of N
are isomorphi to N, by maps d
X
: N ! X and
X :
X !N.
Proof. We (indutively) dene the isomorphisms by:
d
N =1
N =
N
:N !N,
d
UV =(d
U d
V
)d:N !UV
UV =(
U
V
):UV !N.
Firstnotethat =
NN
and d=d
NN
. The uniqueness ofthesemapsthen follows from
therequirementthatisnotstritatN,andhenethatt
X ;Y;Z
isnevertheidentitymap,
foranyX;Y;Z 2Ob(N
). Thisensures thatthe setofobjetsisindiretorrespondene
tothesetof(non-empty)binarybraketingsofasinglesymbol,andthetensororresponds
to braketing two objets together. This an be ompared to the objets and tensor of
MaLane's freemonoidalategoryonone generator,asusedintheproofofhiselebrated
oherene theorem. Werefer to[12℄for this onstrution and proof.
Let U and V be objets of N
satisfying d
U
U = 1
U
and d
V
V = 1
V
. (This holds
triviallyfor U =V =N). Then
d
UV
UV =(d
U d
V )d(
U
V )=(d
U d
V )1
NN (
U
V )
=(d
U
U d
V
V )=(1
U 1
V )=1
UV :
Therefore, by indution, d
X
X =1
X
for allobjetsX. Similarly,
X d
X =1
N .
1
Thisis admittedlyanabuseofnotation, inasmuh asamonoid withoutanidentity isasemigroup.
Thisallowsustoonstrutafuntorfromthe ategoryN totheendomorphismmonoid
of N (onsidered asa one-objetategory), asfollows:
2.3. Proposition. Let N be a self-similar objet of a symmetri monoidal ategory,
(M;). The map :N
!M(N;N), dened by (f)=
Y fd
X
for all f :X !Y and
(X)=N for all X 2Ob(N
) isa funtor.
Proof. Forall f :X !Y and g :Y !Z in N
, by denition,
(g)(f)=
Z gd
Y
Y fd
X =
Z g1
Y f
X
=(gf):
Therefore, preserves omposition,and itisimmediate fromthe denitionthat (1
X )=
1
N
for allobjets X. Hene is a funtor.
These denitions allow us to onstrut a monoid homomorphism that has a very lose
onnetion with the tensor, asfollows:
2.4. Lemma. Let N be a self-similar objet of a symmetri monoidal ategory (M;),
and let : M(N;N)M(N;N) ! M(N;N) be dened by f g = (f g)d, for all
f;g :N !N. Then
(i) is a monoid homomorphism,
(ii) (f g)=(f)(g) for allf :U !X and g :V !Y, for U;V;X;Y 2Ob(N
).
Proof.
(i)By denition, (11)=d=1
N ,and
(f g)(hk)=(f g)d(hk)d=(fhgk)d=fhgk:
Hene is amonoid homomorphism.
(ii) Bydenition (f)(g)=((f)(g))d. Hene, by denition of ,
(f)(g)=(
X fd
U
Y gd
V
)d=(
X
Y
)(f g)(d
U d
V )d
and so, by denition of
XY
and d
UV ,
(f)(g)=
XY
(f g)d
UV
=(f g):
Ourlaimisthatthis monoidhomomorphism(whih,in[6℄, isreferredtoasthe
internal-isationof the tensor) gives the endomorphismmonoidof N the struture ofa one-objet
(unitless) symmetri monoidalategory, with asthe tensor.
2.5. Definition. The one-objet analoguesof the axioms for a tensor are as follows:
There existspeialelements s and t (the analogues of theommutativityand assoiativity
2. t(u(vw))=((uv)w)t,
3. t 2
=(t1)t(1t),
4. tst=(s1)t(1s),
5. s 2
=1,
6. t has an inverse, t 1
, satisfying tt 1
=1=t 1
t.
These are just the axioms for a (unitless) symmetri monoidal ategory with the objet
subsripts erased.
The homomorphism satises the above onditions,as follows:
2.6. Theorem. LetN beaself-similarobjetof asymmetrimonoidalategory(M;),
andletbeasdenedin Lemma2.4ofSetion2. Thenthereexistdistinguishedelements
s and t of the endomorphism monoidof N satisfying 1. to 6. above.
Proof. Dene s = (s
NN
), where s
X ;Y
is the family of ommutativity morphisms for
(M;). Similarly, dene t = (t
N;N;N
) and t 1
= (t 1
N;N;N
). Note that, for the
isomor-phisms of Lemma 1to bewell-dened,we require that t
N;N;N 6=1
N(NN)
Then (t
X ;Y;Z
)=(t
N;N;N
)=t, by the naturality oft
X ;Y;Z
,and the fatthat (1
X )=
1,for all X;Y;Z 2Ob(N
). Similarly,(s
X ;Y
)=s, for allX;Y 2Ob(N
).
The onditions 1. to 4. follow immediately, by applying to the axioms 1. to 4.
respetively.
1. s(ba)=(ab)s,
2. t(a(b)) =((ab))t,
3. The MaLane Pentagon:
t
(AB);C ;D t
A;B;(CD) =(t
A;B;C 1
D )t
A;(BC);D (1
A t
B;C ;D ):
4. The Commutativity Hexagon:
t
C ;A;B s
(AB);C t
A;B;C =(s
A;C 1
B )t
A;C ;B (1
A s
B;C );
and by the naturality of the anonial isomorphisms for asymmetri monoidalategory.
For example,applying to the MaLane pentagon gives
(t
(AB);C ;D t
A;B;(CD)
)=((t
A;B;C 1
D )t
A;(BC);D (1
A t
B;C ;D ));
and so
(t
(AB);C ;D )(t
A;B;(CD)
)=((t
A;B;C
)(1
D ))(t
A;(BC);D )((1
A
)(t
B;C ;D )):
Therefore, tt=(t1)t(1t), whih is ondition 3. above. Conditions 5. and 6. follow
by applying to the equations s
A;B s
B;A
=1 and t
X ;Y;Z t
1
X ;Y;Z
Hene, the endomorphism monoid of a self-similar objet of a symmetri monoidal
ate-goryis a unitlessone-objet symmetrimonoidalategory.
2.7. Definition. We dene a monoidal funtor between unitless symmetri monoidal
ategories (C;) and (D;) to be a pair (F;m), where F :C!D is a funtor, and m
is a naturaltransformation with omponents m
A;B
:F(A)F(B)!F(AB).
Withthis terminology,the followingis immediate
2.8. Corollary. Let N be a self-similar objet of a symmetri monoidal ategory M.
Then is a monoidal funtor from N
to M(N;N), onsidered a one-objet unitless
symmetri monoidal ategory.
MonoidsM thathavemonoidhomomorphismsfromMM toM,togetherwithanalogues
of the assoiativity and ommutativity elements satisfying 1. to 6. above have already
been onstruted, underthe name(strong) M{monoidsin[6℄andGirard Monoidsin[11℄.
We demonstrate how the two examples given are examples of this onstrution in the
ategory of partialinjetive maps.
2.9. Examples of self-similarity. Any innite objet in the ategory Set is
self-similar with respet to two distint tensors; Set is a symmetri monoidal ategory with
respet toboth the Cartesian produt of sets, XY =f(x;y): x2 X;y 2Yg and the
oprodut,AtB =f(a;0):a2Ag[f(b;1):b 2Bg. This then gives usthe following:
2.10. Proposition. The set of natural numbers, N is a self-similar objet of Set, with
respet to both and t.
Proof. The elementarytheory ofinnitesets givestheexistene ofbijetionsN !NN,
and N !N tN. Therefore, our resultfollows.
M-monoidstruturesanthenbederivedfromspeiexamplesofbijetions':NtN !N
and :N N ! N. The examples given in[6,11℄ are onstruted fromthe funtion
'(n;i)=2n+i n2N ; i2f0;1g
whihisa bijetionfromNtN toN This isthen used toonstrut abijetionfromNN
toN by denoting '(n;i) by '
i
(n), and dening
(x;y)=' y
1 ('
0 (x)):
Similarresultsholdfortheategoryofrelations,andtheategoryofpartialinjetivemaps
{interestingly,algebrairepresentations of theseM-monoidstrutures are given interms
of inverse semigroup theory and polyyli monoids (see Setion 4 for the onstrutions
of these). We refer to [11℄ for the algebraitheory of inverse semigroups, and to [6, 11℄
3. Monoid analogues of other ategorial strutures
Afuntorfromasymmetrimonoidalategorytotheendomorphismmonoidofanobjet
(viewed as a unitless one-objet symmetri monoidal ategory), that preserves both the
ategorystruture and the symmetri monoidalstruture, mightbeexpeted topreserve
otherategorialproperties. Wedemonstratehowtheassumptionofself-similarityallows
ustoonstrutone-objetanaloguesoftheategorialtrae(asdesribedbyJoyal,Street,
and Verity in[8℄) and, insome ases, one-objetanalogues ofompat losedness.
These spei examples were motivated by Abramsky's analysis of the Geometry of
Interation,[1℄,andthe paperofAbramskyandJagadeesan, [2℄,whereitisdemonstrated
thatthe system of[1℄isequivalenttotheonstrutionof [8℄|hene demonstratingthat
the dynamis of the Geometry of Interation system is given by a ategorial trae, and
ompat losedness. The motivation for onstruting one-objet analogues of ompat
losedness and the ategorial trae then omes from Girard's omment in [4℄ where he
states that his system `forgets types' | the usual representation of types being objets
in aategory.
3.1. The ategorial trae.
3.2. Definition. Let (M;;s;t;;;I) be a symmetri monoidal ategory. A trae on
it isdenedin [8℄ tobea family of funtions, Tr U
A;B
:M(AU;BU)!M(A;B), that
are natural in A;B and U, and satisfy the following:
1. Given f :XI !Y I, then Tr I
X ;Y
(f)=f 1
:X!Y.
2. Given f :A(UV)!B(U V), then
Tr UV
A;B
(f)=Tr U
A;B (Tr
V
AU;BU (t
B;U;V ft
1
A;U;V )):
3. Given f :AU !BU, and g :C !D, then
Tr U
A;B
(f)g =Tr U
AC ;BD (t
BDU (1
B s
D;U )t
1
BUD
(f g)t
AUC (1
A s
C ;U )t
1
ACU )
4. Tr U
U;U (s
U;U )=1
U .
LetN beaself-similarobjetofasymmetrimonoidalategory. ThenN
isatraed
symmetri monoidal ategory (although axiom 1 is not appliable, due to the absene
of the unit objet). We an use the funtor to onstrut one-objet analogues of the
ategorial trae, asfollows:
3.3. Lemma. Let N be a self-similar objet of a traed symmetri monoidal ategory
(M;), and denote the ompression and division morphisms by and d respetively.
Then the map trae : M(N;N) ! M(N;N) dened by trae(f) = Tr N
N;N
(df) satises
trae((F))=(Tr U
(F))for allF 2N
Proof. trae((F))=Tr N
N;N
(d(F))=Tr N
N;N (d
YU Fd
XU
) bydenitionof traeand
. However,
YU =(
Y
U
)andd
XU =(d
X d
U
)d, bydenition of the divisionand
ompression morphisms. Therefore
trae((F))=Tr N
N;N (d(
Y
U )F(d
X d
U )d)
=Tr N
N;N ((
Y
U )F(d
X d
U ))=
Y (Tr
N
X ;Y ((1
Y
U )F(1
X d
U )))d
X ;
by the naturality of the trae inX and Y, and so
trae((F))=
Y Tr
U
X ;Y (F(1
X d
U
U ))d
X
=
Y Tr
U
X ;Y (F(1
X 1
U ))d
X =
Y Tr
U
X ;Y (F)d
X ;
by the naturality of the trae inU. Therefore trae((F))=(Tr U
X ;Y
(F)),bythe
deni-tion of .
This then satises one-objet analogues of the ategorial trae, as follows:
3.4. Theorem. Let N be a self-similar objet of a traed symmetri monoidal ategory
(M;), let ;s;t;t 1
be as dened in Theorem 2.6 of Setion 2, and let the map trae
be as dened above. Then
(i) trae(f)=trae(trae(tft 1
)),
(ii) trae(f)g =trae(t(1s)t 1
(f g)t(1s)t 1
),
(iii) trae(s)=1,
(iv) trae((h1)f(g1))=h(trae(f))g,
(v) trae(f(1g))=trae((1g)f).
Proof. These followimmediatelyfrom the above Lemma, the axioms for the ategorial
trae, and the fat that isa monoidhomomorphism.
Speiexamplesanbeonstrutedatanyountableset,inboththeategoryofrelations
(whih is demonstrated in [8℄ to be a traed symmetri monoidal ategory), and in the
sub-ategory of partial injetive maps (whih is demonstrated in [6℄ to be losed under
the same ategorial trae).
3.5. ompat losedness.
3.6. Definition. AompatlosedategoryMisasymmetrimonoidalategorywhere
for every objetA, there exists a leftdual, A _
. Theleft dual has the following properties:
For every A2 Ob(M),there exist two morphisms, the ounit map
A :A
_
A!I, and
the unit map
A
:I !AA _
, that satisfy the followingoherene onditions:
1.
A (1
A
A )t
1
AA _
A (
A 1
A )
1
A =1
A
2.
A _(
A 1
A _)t
A _
AA _(1
A _
A )
1
_ =1
Werefer to[9℄ forthe detailsof the oherene theorem for ompat losed ategories.
The point of the main onstrution of [8℄ was to show that every traed symmetri
monoidal ategory is a monoidal full subategory of a ompat losed ategory, whih
gives a struture theorem for traed symmetri monoidalategories. In light of this, we
would again expet that the assumption of self-similarity at an objet (together with
possible extra onditions) should allow us to onstrut one-objet analogues of ompat
losedness. However, thisraisesthe followingproblem: thedistinguishedmapsthatmake
asymmetrimonoidalategoryintoaompatlosedategory(theunitsandounits)are
dened intermsof the unitobjet. Howeverthe funtorofProposition2.3of Setion2
givesone-objet analogues of unitless symmetri monoidalategories. This motivates an
alternative haraterisation of ompat losed ategories.
In many appliations of ompat losed ategories (for example, the onstrution of
a anonial trae ona ompat losed ategory, [8℄), the and maps always appear in
onjuntion with the unit isomorphisms ; and 1
; 1
respetively. For example, [8℄
uses morphisms between X and X(AA _
) for allX and A dened by (1
X
A )
1
X ,
and similarlyfor and. Wetakemapsof this formasprimitive;this givesthefollowing
alternative set of axioms for a ompat losed ategory:
3.7. Definition. For every objet A 2 Ob(M), there exists a dual objet A _
, together
with morphisms
XA
:X!(AA _
)X,
Æ
XA
:X(A _
A)!X,
that are natural in X and satisfy the following axioms
1. Æ
AA t
1
AA _
A
AA =1
A ,
2. Æ
A _
A s
A _
A;A _t
A _
AA _s
AA _
;A _
A _
A =1
A _,
3. (
AA _
IA 1
X )
1
X =
XA ,
4.
X (1
X Æ
IA
1
A _
A )=Æ
XA .
Thenaturalityof
XA andÆ
XA
inX anbewrittenexpliitlyas,((1
A 1
A
_)f)
XA =
YA
f and fÆ
XA =Æ
YA
(f (1
A _
1
A
)) for allf :X !Y.
The proof of the equivalene of this set of axioms with the usual set is postponed
until the Appendix. However, note that the unit element is not entral to the above
haraterisation of ompat losedness. This allows us to dene one-objet (unitless)
analogues of these axioms for M-monoids,as follows:
3.8. Definition. An M-monoid is said to satisfy the one-objet analogues of ompat
losedness if there exist distinguished elements ; 1
and Æ;Æ 1
thatsatisfy
2. Æsts=1,
3. (1a) =a,
4. aÆ =Æ(a1).
Axioms 1. and 2. are one-objet analogues of axioms 1. and 2. of Denition 3.7, and
axioms 3. and 4. are one-objet analogues of the naturality onditions.
3.9. Theorem. Let (M;) be a ompat losed ategory, and let N be a self-dual
self-similarobjetofM. ThenthereexistelementsandÆthatsatisfytheone-objetanalogues
of the alternative axioms for a ompat losed ategory.
Proof. Dene = (
NN
) and Æ = (Æ
NN
). The naturality of
XN
and Æ
XN in X
allows us to dedue that (
XN
)=(
NN
),and similarly,(
XN
)=(
NN
). Then, as
N _
=N, applying tothe alternative axioms for aompat losed ategorywill give 1.
and2. ofthe aboveDenition,andapplyingtotheexpliitdesriptionofthenaturality
onditions (and using the fat that (11) = 1), will give us 3. and 4. of the above
Denition.
4. Relating the ategorial and algebrai theories of self-similarity
Werelatethe ategorialapproahtoself-similaritygiven abovetothe inversesemigroup
theoreti approah given in [11℄, where self-similarity is studiedalgebraiallyin terms of
polyyli monoids. These were introduedin [13℄, asfollows:
4.1. Definition. P
X
, the polyyli monoid on the set X, is dened to be the inverse
monoid (with a zero, for n 2) generated by a set of ountable ardinality, X, say
fp
0 ;:::p
n 1
g, in ase jXj = n, or fp
0 ;p
1
;:::g for ountably innite X, subjet to the
relationsp
i p
1
j =Æ
ij
, where p 1
is the generalised inverse 2
of p
i .
It willbe onvenient to denote the polyylimonoid on the set fp;qg by P
2
{ this is
followingtheonventionofGirard,whouseselementsofaC
algebrasatisfyingthe above
axioms (whih he refers toasthe dynamial algebra)inhisworkonlinearlogi, [4,5℄. In
[3℄, thisis demonstratedtobearepresentation ofthe freeontratedsemigroupring over
a monoid,whih is triviallythe polyylimonoidon two generators
The polyyli monoids appear to be the algebrai representation of self-similarity.
It is demonstrated in [11℄ that the polyyli monoid on two generators an also be
onstrutedintermsofthe inverse monoidofpartialisomorphismsof theCantorset, and
in[7℄that anembeddingof apolyylimonoidintoa ring Rgivesthe ring isomorphism
M
n (R )
=
R , for all n 2 N. They also have a lose onnetion with the algebrai theory
of tilings; we againrefer to[11℄.
2
Notethattheexisteneofgeneralisedinversesisasigniantlyweakeronditionthangroup-theoreti
inverses. Intheinversesemigrouptheoreti ase, foreveryelement a,there exists auniquegeneralised
inverse,whihsatisesaa 1
a=aand a 1
aa 1
=a 1
. However,weannot,in general,dedueaa 1
=
4.2. Definition. LetN beaself-similarobjetofasymmetrimonoidalategory(M;),
andletM(N;N)haveazero,whihwedenote0. Wesaythatmaps
1 ;
2
2M(NN;N)
are left and right projetion maps and i
1 ;i
2
2M(N;N N) are their inlusionmaps, if
they satisfy 1. 1 i 1 =1 N and 2 i 2 =1 N , 2. i 1 1 =1 N
0 and i
2
2
=01
N .
4.3. Theorem. LetN beaself-similarobjetof asymmetrimonoidalategory(M;).
IfN hasprojetionandinlusionmaps,thenthereexistsanembeddingofP
2
intoM(N;N).
Proof. Aspreservesomposition,(
1 )(i
1 )=1
N
=(
2 )(i
2
). Similarly,(i
1 )(
1 )=
10,and (i
2 )(
2
)=01,sine(0)=0. Hene, asisasemigrouphomomorphism,
(i 1 )( 1 )(i 2 )( 2
)=0=(i
2 )( 2 )(i 1 )( 1 ): Therefore, ( 1 )(i 2
) = 0 = (
2 )(i
1
), and so (
1 );(
2
) satisfy the axioms for the
generatorsofP
2
,and(i
1 );(i
2
)satisfytheonditionsfortheirgeneralisedinverses. Also,
by [13℄, the polyyli monoidsare ongruene-free, hene the onlyquotient of P
2 is the
trivialmonoid. Therefore, these elements generate anembedding of P
2
intoM(N;N).
From the above, we are in a position to prove the onverse to the main result of [7℄,
where it is proved that an embedding of P
2
into the multipliative monoid of a ring R
denes a familyof isomorphisms R
= M
n
(R )for allpositiveintegers n. Fora unitalring
R , wedene the ategoryMat
R
thathas naturalnumbers asobjets, and ba matries
as morphisms from a to b. It is immediate that this ategory is a symmetri monoidal
ategory, with the tensor t given by AtB =
A 0
0 B !
.
4.4. Theorem. Let R be a unital ring, and assume there exists 2 Mat
R
(2;1), d 2
Mat
R
(1;2)suhthatthemap:Mat
R
(2;2)!R ,denedby(X)=Xd,isaninjetive
ring homomorphism. Then P
2
is embedded in R .
Proof. First note that the ondition onR is equivalent to statingthat 1 is a self-similar
objetoftheategoryMat
R
. WedemonstratethatMat
R
hasinlusionsandprojetions.
Dene
1
=(1 0);
2
=(01) ; i
1 = 1 0 ! ; i 2 = 0 1 ! :
Then it isimmediate from the denition of ompositionin Mat
R that 1 i 1 =1 R = 2 i 2 , and i 1 1 = 1 0 0 0 ! ;i 2 2 = 0 0 0 1 ! : Therefore, i 1 1
= 1t0 and i
2
2
= 0t1, and so Mat
R
has projetions and inlusions.
Therefore, by Theorem4.3above,there exists anembeddingofP
2
5. Researh questions
Thispaperisbynomeansaompleteaountofthetheoryofself-similarity. Thefollowing
points are among the many itraises:
Although we have onentrated on pratial appliations, presumably there is a
reasonable denition of the free self-similar monoidal ategory, analogously to the
MaLanedenitionofthefreemonoidalategoryononegenerator[12℄. Thisshould
allowfortheformulationandproofofaoherenetheoremforself-similarategories.
This is relatedto the seond point:
The requirementthat the tensor isnot stritly assoiativeatthe objetN isnot
needed -in [6℄,self-similarityisused toonstrut one-objetanalogues ofmonoidal
ategoriesinthestritlyassoiativease. However, this isdoneinanadhoway. It
isalsouriousthatthe monoidatN satisesone-objetanaloguesof theaxiomsfor
anon-stritmonoidalategory. Clearly thereis someunderlyingtheorythat overs
allases.
In at least some of [4℄, Girard uses aweaker ondition to our self-similaritythat is
equivalentto maps :N N !N and d :N !N N satisfyingd=1
NN and
d< 1
N
. Although this does not allow the denition of a funtor from a ategory
toa monoid,itallows afuntion that preserves omposition and mapsidentities at
objetstoidempotentsofamonoid. Thislearlydenesafuntorfromthe ategory
N
to the Karoubi envelope of the monoid at N. However, the omputational or
logialinterpretationof this remains unlear.
The onstrution of a ompat losed ategory from a traed symmetri monoidal
ategory, as a `dualising' proess is given in [8℄, and the logial and omputational
signianeofthis hasbeenwellstudied. Itwouldbeinteresting tobeabletouse a
similarproesstogofromtraedM-monoidstoompatlosedM-monoids,without
passing thougha many-objet ategoryin anintermediate stage.
For monoidal ategories with additional ategorial struture X, we would like to
saythatthe funtorallowsustodeneone-objetanaloguesofX-ategories. The
problem is to make this preise, and relate it to the diret limitonstrution used
toonstrut one-objet analogues of Cartesian losed ategories, asfound in[10℄.
6. Aknowledgements
Speial thanks are due to Ross Street, for reading a preliminary version of this paper
and helping melarify some denitions, and toSamson Abramsky, for anexplanation of
Appendix: Equivalene of axioms for ompat losedness
We demonstrate that the alternative axioms for ompat losed ategories given in
Def-inition 3.7 are equivalent to the usual set. We refer to the usual set of axioms, as found
in Denition3.6, as A1 and the alternativeset of axioms asA2.
6.1. Theorem. A symmetri monoidal ategory thatsatises theaxioms A1 also
satis-es the axioms A2, and vie versa.
Proof. ()) Let (M;;;) be a symmetri monoidal ategory satisfying A1. For all
X;A 2 Ob(M), we dene
XA = ( A 1 X ) 1 X and Æ XA = X (1 X A ). Consider
arbitrary f : X ! Y. By the denition of , and the naturality of
X
in X, we an
dedue that(1
AA
_f)
XA =
YA
f,andso
XA
isnaturalinX. Similarly,bydenition
of Æ, and the naturality of
X
inX, Æ
XA
is naturalin X.
Also,by denition of and Æ,
1. Æ AA t 1 AA _ A AA = A (1 A A )t 1 AA _ A ( A 1 A ) 1 A =1 A
, by axiom 1of A1.
2. Bydenition, Æ
A _ A s A _ A;A _t A _ AA _s AA _ ;A _Æ A _ A = A _(1 A _ A )s A _ A;A _t A _ AA _s AA _ ;A _( A 1 _ A ) 1 A _:
Bythe naturality of s
XY
in X and Y this isequal to
A _s I;A _( A 1 A _)t A _ AA _(1 A _ A )s A _ ;I 1 A _: Also, as X s IX = X
, this is equal to
A _( A 1 A _)t A _ AA _(1 A _ A ) 1 A
_, whih is
1
A _,
by axiom2 of A1.
3. Bythe naturality of
Z
inZ,and the denition of
XA , ( AA _ 1 X )( IA 1 X ) 1 X =( AA _ 1 X )( A 1 I )( 1 I 1 X )= ( A 1 X )( I 1 I 1 X ) 1 X =( A 1 X )(1 I 1 X ) 1 X =( A 1 X ) 1 X = XA
4. In asimilar way to3,
X (1 X Æ IA )(1 X 1 A _ A )= X (1 X A )=Æ XA
by the denition of and the naturality of
Z in Z. Therefore, AX and Æ AX
aremorphismsthatare naturalinX,andsatisfytheaxiomsA2.
Therefore, every symmetrimonoidalategory satisfyingA1 also satisesA2.
(() Let (M;;;Æ) be a symmetri monoidal ategory satisfying A2. We dene
mor-phisms
A :(A
_
A)!I ;
A
:I !(AA _ ) by A = AA _ IA and A = Æ IA 1 A _ A
1. A (1 A A )t AA _ A ( A 1 A ) A = A;A t AA _ A Æ AA =1 A
, by axiom1 of A2.
2. Bydenition of and ,
A _(
A _1
A _)t A _ AA _(1 A _ A ) 1 A _ = A _(Æ IA _ 1 AA _ 1 A _ j )t A _ AA _(1 A _ AA _ IA ) 1 A _: However, A _ = A _s IA _ and 1 A _ =s IA _ 1 A _
. Therefore, this is equalto
A _s IA (Æ IA _ 1 AA _ 1 A _ j )t A _ AA _(1 A _ AA _ IA )s IA _ 1 A _;
and by the naturality of s
XY
in X and Y,the aboveis equalto
A _(1 A _ Æ IA _ 1 AA _)s A _ A;A _t A _ AA _s AA _ ;A _( AA _ IA 1 A _) 1 A _:
Then by axioms 3 and 4 of A2, this is Æ
A _ A s A _ A;A _t A _ AA _s AA _ ;A _ A _ A
, whih is
1
A _,
by axiom2 of A2.
Therefore, the axioms A1 are satised, and this ompletes our proof.
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Shool of Informatis,
University of Wales,Bangor,
Gwynedd, U.K. LL57 1UT
Email: max003bangor.a.uk
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