Directory UMM :Journals:Journal_of_mathematics:TAC:
Theory and Appli
ations of Categories,
Vol. 6, No. 3, 1999, pp. 33{46.
THE CATEGORICAL THEORY OF SELF-SIMILARITY
This paper is dedi
ated to Joa
him Lambek,
whose work was the inspiration for the following results.
PETER HINES
ABSTRACT. We demonstrate how the identity N
N
= N in a monoidal
ategory
allows us to
onstru
t a fun
tor from the full sub
ategory generated by N and
to
the endomorphism monoid of the obje
t N . This provides a
ategori
al foundation for
one-obje
t analogues of the symmetri
monoidal
ategories used by J.-Y. Girard in his
Geometry of Intera
tion series of papers, and expli
itly des
ribed in terms of inverse
semigroup theory in [6, 11℄.
This fun
tor also allows the
onstru
tion of one-obje
t analogues of other
ategori
al
stru
tures. We give the example of one-obje
t analogues of the
ategori
al tra
e, and
ompa
t
losedness. Finally, we demonstrate how the
ategori
al theory of self-similarity
an be related to the algebrai
theory (as presented in [11℄), and Girard's dynami
al
algebra, by
onsidering one-obje
t analogues of proje
tions and in
lusions.
1. Introdu
tion
It is well-known, [12℄, that any one-obje
t monoidal
ategory is an abelian monoid with
respe
t to two operations that satisfy the inter
hange law (a b) Æ (
d) = (a Æ
) (b Æ d);
that is, all the
anoni
al isomorphisms for the tensor are identities. However, non-trivial
one-obje
t analogues of the
anoni
al asso
iativity and
ommutativity morphisms for
symmetri
monoidal
ategories have been impli
itly
onstru
ted by Girard for his work
on linear logi
, [4, 5℄, and have been studied in terms of inverse semigroups in [6℄ and [11℄.
The absen
e of a unit allows for a more interesting theory.
We demonstrate in se
tion 2 how these examples arise naturally from the assumption
of self-similarity. This is motivated by the examples of either the intuitive
onstru
tion
of the Cantor set, or bije
tions between N and N N . We de ne a self-similar obje
t of a
symmetri
monoidal
ategory to be an obje
t N satisfying N
= N
N . In the
ategori
al
ase, this gives rise to examples of stru
ture-preserving maps between
ategories and
monoids. In se
tion 3, we then demonstrate how these maps allow us to
onstru
t oneobje
t analogues of the
ategori
al tra
e and
ompa
t
losedness. Finally, in se
tion
4, these
ategori
al
onstru
tions are related to the algebrai
approa
h to self-similarity
des
ribed in [11℄, and Girard's dynami
al algebra, by
onsidering one-obje
t analogues of
proje
tions and in
lusions.
Re
eived by the editors 1998 O
tober 6 and, in revised form, 1999 September 13.
Published on 1999 November 30.
1991 Mathemati
s Subje
t Classi
ation: 18D10, 20M18.
Key words and phrases: Monoidal Categories, Categori
al Tra
e, Compa
t Closure, Linear Logi
,
Inverse Semigroups.
Peter Hines 1999. Permission to
opy for private use granted.
33
Theory and Appli
ations of Categories, Vol. 6, No. 3
34
2. Self-similarity and tensors
M
( ;
) be a symmetri
monoidal
ategory and denote the asso
ia-
2.1. Definition. Let
M
tivity,
ommutativity, and left and right unit isomorphisms by
tively. We say that an obje
t
morphisms
N ! N
:N
N
of
self-similar
if
A;B;C
;s
A;B
; ;
A
A
respe
-
; that is, there exist
d : N ! N
N that satisfy d
= 1
ompression and division morphisms of N .
and
We
all these morphisms the
is
t
N
= N
N
N
N
and
d = 1
N.
For the purposes of this paper, we will require the additional
ondition that the tensor
is not stri
t at N , so t
is not the identity map. We also assume that the
ommutativity is not stri
t, to avoid studying degenerate
ases. We study the full sub
ategory of
given by all obje
ts
onstru
ted from N and
, whi
h we refer to as
. We assume
throughout that N 6= I .
The sub
ategory
has all the stru
ture of a monoidal
ategory apart from the unit
obje
t; we use the awkward term unitless monoidal
ategories1 for stru
tures of this form.
The following is then immediate:
are isomorphi
to N , by maps d : N ! X and
:
2.2. Lemma. All obje
ts of
N;N;N
M
X
N
N
!N
N
.
X
X
roof. We (indu
tively) de ne the isomorphisms by:
P
! N,
d )d : N ! U
V
) : U
V ! N.
d =1 =
:N
N
N
N
d
= (d
U
V
U
=
(
U
V
U
V
V
First note that
=
and d = d
. The uniqueness of these maps then follows from
the requirement that
is not stri
t at N , and hen
e that t
is never the identity map,
for any X; Y; Z 2 Ob(
). This ensures that the set of obje
ts is in dire
t
orresponden
e
to the set of (non-empty) binary bra
ketings of a single symbol, and the tensor
orresponds
to bra
keting two obje
ts together. This
an be
ompared to the obje
ts and tensor of
Ma
Lane's free monoidal
ategory on one generator, as used in the proof of his
elebrated
oheren
e theorem. We refer to [12℄ for this
onstru
tion and proof.
Let U and V be obje
ts of
satisfying d
= 1 and d
= 1 . (This holds
trivially for U = V = N ). Then
N
N
N
N
N
X;Y ;Z
N
V
U
V
U
U
V
d )d
(
) = (d
d )1
= (d
d
) = (1
1 ) = 1
d
= (d
U
U
U
U
V
U
U
V
V
V
U
U
V
V
U
N
V
V
(
N
V
U
V
)
:
Therefore, by indu
tion, d
= 1 for all obje
ts X . Similarly,
d = 1 .
X
1 This
X
X
X
X
N
is admittedly an abuse of notation, inasmu
h as a monoid without an identity is a semigroup.
However, the natural alternatives, `multipli
ative
ategory', or `semigroup
ategory' have already been
used in the work of Hyland and DePaiva, and Lawson respe
tively.
Theory and Appli
ations of Categories, Vol. 6, No. 3
35
This allows us to
onstru
t a fun
tor from the
ategory N
to the endomorphism monoid
of N (
onsidered as a one-obje
t
ategory), as follows:
Let N be a self-similar obje
t of a symmetri
monoidal
ategory,
(M;
). The map : N
! M(N; N ), de ned by (f ) =
Y fdX for all f : X ! Y and
(X ) = N for all X 2 Ob(N
) is a fun
tor.
Proof. For all f : X ! Y and g : Y ! Z in N , by de nition,
2.3. Proposition.
(g ) (f ) =
Z gdY
Y fdX =
Z g 1Y f
X = (gf ):
Therefore, preserves
omposition, and it is immediate from the de nition that (1X ) =
1N for all obje
ts X . Hen
e is a fun
tor.
These de nitions allow us to
onstru
t a monoid homomorphism that has a very
lose
onne
tion with the tensor, as follows:
Let N be a self-similar obje
t of a symmetri
monoidal
ategory (M;
),
and let : M(N; N ) M(N; N ) ! M(N; N ) be de ned by f g =
(f
g )d, for all
f; g : N ! N . Then
(i) is a monoid homomorphism,
(ii) (f
g ) = (f ) (g ) for all f : U ! X and g : V ! Y , for U; V; X; Y 2 Ob(N
).
2.4. Lemma.
P
roof.
By de nition, (1 1) =
d = 1N , and
(i)
(f
g)(h k) =
(f
g)d
(h
k)d =
(fh
gk)d = fh gk:
Hen
e is a monoid homomorphism.
By de nition (f ) (g ) =
( (f )
(g ))d. Hen
e, by de nition of ,
(ii)
(f ) (g ) =
(
X fdU
Y gdV )d =
(
X
Y )(f
g )(dU
dV )d
and so, by de nition of
X
Y and dU
V ,
(f ) (g ) =
X
Y (f
g )dU
V = (f
g ):
Our
laim is that this monoid homomorphism (whi
h, in [6℄, is referred to as the internalisation of the tensor) gives the endomorphism monoid of N the stru
ture of a one-obje
t
(unitless) symmetri
monoidal
ategory, with as the tensor.
The one-obje
t analogues of the axioms for a tensor are as follows:
There exist spe
ial elements s and t (the analogues of the
ommutativity and asso
iativity
elements respe
tively) that satisfy:
2.5. Definition.
1. s(u v ) = (v u)s,
Theory and Appli
ations of Categories, Vol. 6, No. 3
36
2. t(u (v w)) = ((u v ) w)t,
3. t2 = (t 1)t(1 t),
4. tst = (s 1)t(1 s),
5. s2 = 1,
6. t has an inverse, t 1 , satisfying tt
= 1 = t 1 t.
1
These are just the axioms for a (unitless) symmetri
monoidal
ategory with the obje
t
subs
ripts erased.
The homomorphism satis es the above
onditions, as follows:
2.6. Theorem. Let N be a self-similar obje
t of a symmetri
monoidal
ategory (
;
),
and let be as de ned in Lemma 2.4 of Se
tion 2. Then there exist distinguished elements
s and t of the endomorphism monoid of N satisfying 1. to 6. above.
Proof. De ne s = (s
), where s
is the family of
ommutativity morphisms for
(M;
). Similarly, de ne t = (t
) and t 1 = (t 1 ). Note that, for the isomorphisms of Lemma 1 to be well-de ned, we require that t
6= 1
(
)
Then (t
) = (t
) = t, by the naturality of t
, and the fa
t that (1 ) =
1, for all X; Y; Z 2 Ob(
). Similarly, (s ) = s, for all X; Y 2 Ob(
).
The
onditions 1. to 4. follow immediately, by applying to the axioms 1. to 4.
respe
tively.
M
NN
X;Y
N;N;N
N;N;N
N;N;N
N
X;Y;Z
N;N;N
N
N
N
N
X;Y ;Z
X;Y
X
1. s(b
a) = (a
b)s,
2. t(a
(b
)) = ((a
b)
)t,
3. The Ma
Lane Pentagon:
t(
A
B );C;D
t
A;B;(C
D)
= (t
A;B;C
1
D
)t
A;(B
C );D
(1
A
t
B;C;D
):
4. The Commutativity Hexagon:
t
C;A;B
s(
A
B );C
t
= (s
A;B;C
A;C
1
B
)t
A;C;B
s
(1
A
B;C
);
and by the naturality of the
anoni
al isomorphisms for a symmetri
monoidal
ategory.
For example, applying to the Ma
Lane pentagon gives
(t(
A
B );C;D
t
A;B;(C
) ) = ((t
D
A;B;C
1
D
)t
A;(B
C );D
(1
A
t
B;C;D
));
and so
(t(
A
B );C;D
) (t
A;B;(C
) ) = ( (t
A;B;C
D
) (1 )) (t
D
A;(B
C );D
)( (1 ) (t
A
B;C;D
)):
Therefore, tt = (t 1)t(1 t), whi
h is
ondition 3. above. Conditions 5. and 6. follow
by applying to the equations s s
= 1 and t
t 1 = 1, whi
h follow from the
de nition of a symmetri
monoidal
ategory.
A;B
B;A
X;Y ;Z
X;Y ;Z
Theory and Appli
ations of Categories, Vol. 6, No. 3
37
Hen
e, the endomorphism monoid of a self-similar obje
t of a symmetri
monoidal
ategory is a unitless one-obje
t symmetri
monoidal
ategory.
2.7. Definition. We de ne a monoidal fun
tor between unitless symmetri
monoidal
ategories (C;
) and (D; ) to be a pair (F; m), where F : C ! D is a fun
tor, and m
is a natural transformation with
omponents m : F (A) F (B ) ! F (A
B ).
A;B
With this terminology, the following is immediate
Let N be a self-similar obje
t of a symmetri
monoidal
ategory M.
Then is a monoidal fun
tor from N
to M(N; N ),
onsidered a one-obje
t unitless
symmetri
monoidal
ategory.
2.8. Corollary.
Monoids M that have monoid homomorphisms from M M to M , together with analogues
of the asso
iativity and
ommutativity elements satisfying 1. to 6. above have already
been
onstru
ted, under the name (strong) M{monoids in [6℄ and Girard Monoids in [11℄.
We demonstrate how the two examples given are examples of this
onstru
tion in the
ategory of partial inje
tive maps.
2.9. Examples of self-similarity. Any in nite obje
t in the
ategory Set is selfsimilar with respe
t to two distin
t tensors; Set is a symmetri
monoidal
ategory with
respe
t to both the Cartesian produ
t of sets, X Y = f(x; y ) : x 2 X; y 2 Y g and the
oprodu
t, A t B = f(a; 0) : a 2 Ag [ f(b; 1) : b 2 B g. This then gives us the following:
2.10. Proposition.
Set, with
Proof.
! N N,
respe
t to both
and N
The set of natural numbers, N is a self-similar obje
t of
and t.
The elementary theory of in nite sets gives the existen
e of bije
tions N
Therefore, our result follows.
! N t N.
M-monoid stru
tures
an then be derived from spe
i
examples of bije
tions ' : N tN ! N
and : N N ! N . The examples given in [6, 11℄ are
onstru
ted from the fun
tion
'(n; i) = 2n + i
n 2 N ; i 2 f0; 1g
whi
h is a bije
tion from N t N to N This is then used to
onstru
t a bije
tion from N N
to N by denoting '(n; i) by ' (n), and de ning
i
(x; y ) = '1 ('0 (x)):
y
Similar results hold for the
ategory of relations, and the
ategory of partial inje
tive maps
{ interestingly, algebrai
representations of these M-monoid stru
tures are given in terms
of inverse semigroup theory and poly
y
li
monoids (see Se
tion 4 for the
onstru
tions
of these). We refer to [11℄ for the algebrai
theory of inverse semigroups, and to [6, 11℄
for expli
it des
riptions of M-monoids in terms of the inverse monoid of partial inje
tions
on the natural numbers.
Theory and Appli
ations of Categories, Vol. 6, No. 3
38
3. Monoid analogues of other
ategori
al stru
tures
A fun
tor from a symmetri
monoidal
ategory to the endomorphism monoid of an obje
t
(viewed as a unitless one-obje
t symmetri
monoidal
ategory), that preserves both the
ategory stru
ture and the symmetri
monoidal stru
ture, might be expe
ted to preserve
other
ategori
al properties. We demonstrate how the assumption of self-similarity allows
us to
onstru
t one-obje
t analogues of the
ategori
al tra
e (as des
ribed by Joyal, Street,
and Verity in [8℄) and, in some
ases, one-obje
t analogues of
ompa
t
losedness.
These spe
i
examples were motivated by Abramsky's analysis of the Geometry of
Intera
tion, [1℄, and the paper of Abramsky and Jagadeesan, [2℄, where it is demonstrated
that the system of [1℄ is equivalent to the
onstru
tion of [8℄ | hen
e demonstrating that
the dynami
s of the Geometry of Intera
tion system is given by a
ategori
al tra
e, and
ompa
t
losedness. The motivation for
onstru
ting one-obje
t analogues of
ompa
t
losedness and the
ategori
al tra
e then
omes from Girard's
omment in [4℄ where he
states that his system `forgets types' | the usual representation of types being obje
ts
in a
ategory.
3.1. The
ategori
al tra
e.
Let (M;
; s; t; ; ; I ) be a symmetri
monoidal
ategory. A tra
e on
it is de ned in [8℄ to be a family of fun
tions, T r : M (A
U; B
U ) ! M (A; B ), that
are natural in A; B and U , and satisfy the following:
3.2. Definition.
U
A;B
1. Given f : X
I
2.
! Y
I , then T r (f ) = f
Given f : A
(U
V ) ! B
(U
V ), then
I
X;Y
T r
(f ) = T r
U
V
A;B
U
A;B
(T r
V
A
U;B
(t
U
B;U;V
ft
1
A;U;V
)):
3. Given f : A
U
Tr
4. T r
U
U
A;B
U
U;U
! B
U , and g : C ! D, then
(f )
g = T r
(t
(1
s )t 1
!Y.
:X
1
(s
U;U
A
C;B
D
BDU
B
D;U
BU D
(f
g )t
AU C
(1
A
)=1 .
U
s
C;U
)t
1
AC U
)
N
Let N be a self-similar obje
t of a symmetri
monoidal
ategory. Then
is a tra
ed
symmetri
monoidal
ategory (although axiom 1 is not appli
able, due to the absen
e
of the unit obje
t). We
an use the fun
tor to
onstru
t one-obje
t analogues of the
ategori
al tra
e, as follows:
Let N be a self-similar obje
t of a tra
ed symmetri
monoidal
ategory
( ;
), and denote the
ompression and division morphisms by
and d respe
tively.
Then the map tra
e : (N; N ) ! (N; N ) de ned by tra
e(f ) = T r (df
) satis es
tra
e( (F )) = (T r (F )) for all F 2
(X
U; Y
U ).
M
3.3. Lemma.
U
M
X;Y
M
N
N
N;N
Theory and Appli
ations of Categories, Vol. 6, No. 3
39
roof. tra
e( (F )) = T r (d (F )
) = T r (d
F d
) by de nition of tra
e and
. However,
=
(
) and d
= (d
d )d, by de nition of the division and
ompression morphisms. Therefore
N
N;N
P
Y
U
Y
N
N;N
U
X
tra
e( (F ))
= Tr
N
N;N
((
Y
)F (d
U
= Tr
X
d
U
N
N;N
U
Y
X
(d
(
Y
U
X
U
U
)) =
(T r
Y
)F (d
U
N
X;Y
X
((1
d
U
Y
U
)d
)
)F (1
X
d
U
)))d
X
;
by the naturality of the tra
e in X and Y , and so
tra
e( (F ))
=
=
T rX;Y (F (1X
U
Y
Y
T rX;Y (F (1X
U
1
U
))d =
X
by the naturality of the tra
e in U . Therefore
tion of .
d
U
U ))dX
T rX;Y (F )dX ;
U
Y
tra
e( (F ))
= (T r
U
X;Y
(F )), by the de ni-
This then satis es one-obje
t analogues of the
ategori
al tra
e, as follows:
be a self-similar obje
t of a tra
ed symmetri
monoidal
ategory
(M;
), let ; s; t; t be as de ned in Theorem 2.6 of Se
tion 2, and let the map tra
e
be as de ned above. Then
(i) tra
e(f ) = tra
e(tra
e(tf t 1 )),
(ii) tra
e(f ) g = tra
e(t(1 s)t 1 (f g )t(1 s)t 1 ),
(iii) tra
e(s) = 1,
(iv) tra
e((h 1)f (g 1)) = h(tra
e(f ))g ,
(v) tra
e(f (1 g )) = tra
e((1 g )f ).
3.4. Theorem.
Let
1
N
roof. These follow immediately from the above Lemma, the axioms for the
ategori
al
tra
e, and the fa
t that is a monoid homomorphism.
P
Spe
i
examples
an be
onstru
ted at any
ountable set, in both the
ategory of relations
(whi
h is demonstrated in [8℄ to be a tra
ed symmetri
monoidal
ategory), and in the
sub-
ategory of partial inje
tive maps (whi
h is demonstrated in [6℄ to be
losed under
the same
ategori
al tra
e).
3.5.
ompa
t
losedness.
A
ompa
t
losed
ategory M is a symmetri
monoidal
ategory where
for every obje
t A, there exists a left dual, A_ . The left dual has the following properties:
For every A 2 Ob(M), there exist two morphisms, the
ounit map : A_
A ! I , and
the unit map : I ! A
A_ , that satisfy the following
oheren
e
onditions:
3.6. Definition.
A
A
)t 1 _ (
1 ) 1 = 1
_ (
1 _ )t _ _ (1 _
) _1 = 1_
1.
A (1A
2.
A
A
A
AA
A
A
A
A
AA
A
A
A
A
A
A
A
Theory and Appli
ations of Categories, Vol. 6, No. 3
40
We refer to [9℄ for the details of the
oheren
e theorem for
ompa
t
losed
ategories.
The point of the main
onstru
tion of [8℄ was to show that every tra
ed symmetri
monoidal
ategory is a monoidal full sub
ategory of a
ompa
t
losed
ategory, whi
h
gives a stru
ture theorem for tra
ed symmetri
monoidal
ategories. In light of this, we
would again expe
t that the assumption of self-similarity at an obje
t (together with
possible extra
onditions) should allow us to
onstru
t one-obje
t analogues of
ompa
t
losedness. However, this raises the following problem: the distinguished maps that make
a symmetri
monoidal
ategory into a
ompa
t
losed
ategory (the units and
ounits) are
de ned in terms of the unit obje
t. However the fun
tor of Proposition 2.3 of Se
tion 2
gives one-obje
t analogues of unitless symmetri
monoidal
ategories. This motivates an
alternative
hara
terisation of
ompa
t
losed
ategories.
In many appli
ations of
ompa
t
losed
ategories (for example, the
onstru
tion of
a
anoni
al tra
e on a
ompa
t
losed
ategory, [8℄), the and maps always appear in
onjun
tion with the unit isomorphisms ; and 1 ; 1 respe
tively. For example, [8℄
uses morphisms between X and X
(A
A_ ) for all X and A de ned by (1
) 1 ,
and similarly for and . We take maps of this form as primitive; this gives the following
alternative set of axioms for a
ompa
t
losed
ategory:
3.7. Definition. For every obje
t A 2 Ob(
), there exists a dual obje
t A_ , together
X
X A
ÆX A
! (A
A_ )
X
: X
(A_
A) ! X
:X
,
,
that are natural in
X
and satisfy the following axioms
1.
1
ÆAA tAA
_ A AA
2.
ÆA_ A sA_
A;A_ tA_ AA_ sA
A_ ;A_ A_ A
3.
(
4.
X (1X
A
X
M
with morphisms
A
A
_ I A
Æ
=1
1
IA
X
A,
)
1
X
A_1
A )
=
=Æ
=1
A
_,
XA,
XA.
The naturality of and Æ in X
an be written expli
itly as, ((1
1 _ )
f ) =
f and f Æ
= Æ (f
(1 _
1 )) for all f : X ! Y .
The proof of the equivalen
e 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
hara
terisation of
ompa
t
losedness. This allows us to de ne one-obje
t (unitless)
analogues of these axioms for M-monoids, as follows:
XA
YA
XA
YA
XA
A
A
A
XA
A
3.8. Definition. An M-monoid is said to satisfy the one-obje
t analogues of
ompa
t
losedness if there exist distinguished elements
1.
Æt 1
= 1,
; 1
and
Æ; Æ 1
that satisfy
Theory and Appli
ations of Categories,
2.
3.
4.
Vol. 6, No. 3
41
= 1,
(1 a) = a,
aÆ = Æ (a 1).
Æsts
Axioms 1. and 2. are one-obje
t analogues of axioms 1. and 2. of De nition 3.7, and
axioms 3. and 4. are one-obje
t analogues of the naturality
onditions.
M
Let ( ;
) be a
ompa
t
losed
ategory, and let N be a self-dual selfsimilar obje
t of . Then there exist elements and Æ that satisfy the one-obje
t analogues
of the alternative axioms for a
ompa
t
losed
ategory.
M
3.9. Theorem.
roof. De ne = (NN ) and Æ = (ÆNN ). The naturality of XN and ÆXN in X
allows us to dedu
e that (XN ) = (NN ), and similarly, (XN ) = (NN ). Then, as
N _ = N , applying to the alternative axioms for a
ompa
t
losed
ategory will give 1.
and 2. of the above De nition, and applying to the expli
it des
ription of the naturality
onditions (and using the fa
t that (1 1) = 1), will give us 3. and 4. of the above
De nition.
P
4. Relating the
ategori
al and algebrai
theories of self-similarity
We relate the
ategori
al approa
h to self-similarity given above to the inverse semigroup
theoreti
approa
h given in [11℄, where self-similarity is studied algebrai
ally in terms of
poly
y
li
monoids. These were introdu
ed in [13℄, as follows:
the poly
y
li
monoid on the set X , is de ned to be the inverse
monoid (with a zero, for n 2) generated by a set of
ountable
ardinality, X , say
fp0; : : : pn 1g, in
ase jX j = n, or fp0; p1; : : :g for
ountably in nite X , subje
t to the
relations pi pj 1 = Æij , where p 1 is the generalised inverse2 of pi .
4.1. Definition.
PX ,
It will be
onvenient to denote the poly
yli
monoid on the set fp; qg by P2 { this is
following the
onvention of Girard, who uses elements of a C algebra satisfying the above
axioms (whi
h he refers to as the dynami
al algebra) in his work on linear logi
, [4, 5℄. In
[3℄, this is demonstrated to be a representation of the free
ontra
ted semigroup ring over
a monoid, whi
h is trivially the poly
y
li
monoid on two generators
The poly
y
li
monoids appear to be the algebrai
representation of self-similarity.
It is demonstrated in [11℄ that the poly
y
li
monoid on two generators
an also be
onstru
ted in terms of the inverse monoid of partial isomorphisms of the Cantor set, and
in [7℄ that an embedding of a poly
y
li
monoid into a ring R gives the ring isomorphism
Mn ( R )
= R, for all n 2 N. They also have a
lose
onne
tion with the algebrai
theory
of tilings; we again refer to [11℄.
2 Note
that the existen
e of generalised inverses is a signi
antly weaker
ondition than group-theoreti
inverses. In the inverse semigroup theoreti
ase, for every element a, there exists a unique generalised
inverse, whi
h satis es aa 1 a = a and a 1 aa 1 = a 1 . However, we
annot, in general, dedu
e aa 1 =
1 = a 1 a. We refer to [11℄ for a
omprehensive a
ount of the theory of inverse semigroups.
Theory and Appli
ations of Categories, Vol. 6, No. 3
M
42
Let N be a self-similar obje
t of a symmetri
monoidal
ategory ( ;
),
and let (N; N ) have a zero, whi
h we denote 0. We say that maps 1 ; 2 2 (N
N; N )
are left and right proje
tion maps and i1 ; i2 2 (N; N
N ) are their in
lusion maps, if
they satisfy
M
M
4.2. Definition.
M
1. 1 i1 = 1N and 2 i2 = 1N ,
2. i1 1 = 1N
0 and i2 2 = 0
1N .
M
M
Let N be a self-similar obje
t of a symmetri
monoidal
ategory ( ;
).
If N has proje
tion and in
lusion maps, then there exists an embedding of P2 into (N; N ).
4.3. Theorem.
roof. As preserves
omposition, (1 ) (i1 ) = 1N = (2 ) (i2 ). Similarly, (i1 ) (1 ) =
1 0, and (i2 ) (2 ) = 0 1, sin
e (0) = 0. Hen
e, as is a semigroup homomorphism,
P
(i1 ) (1 ) (i2 ) (2 ) = 0 = (i2 ) (2 ) (i1 ) (1 ):
Therefore, (1 ) (i2 ) = 0 = (2 ) (i1 ), and so (1 ); (2 ) satisfy the axioms for the
generators of P2 , and (i1 ); (i2 ) satisfy the
onditions for their generalised inverses. Also,
by [13℄, the poly
y
li
monoids are
ongruen
e-free, hen
e the only quotient of P2 is the
trivial monoid. Therefore, these elements generate an embedding of P2 into (N; N ).
M
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 P2 into the multipli
ative monoid of a ring R
de nes a family of isomorphisms R
= Mn (R) for all positive integers n. For a unital ring
R, we de ne the
ategory
R that has natural numbers as obje
ts, and b a matri
es
as morphisms from a to b. It is immediate that this
ategory
is a symmetri
monoidal
!
A 0
ategory, with the tensor t given by A t B =
.
0 B
Mat
Let R be a unital ring, and assume there exists
2 Mat (2; 1), d 2
MatR(1; 2) su
h that the map : MatR(2; 2) ! R, de ned by (X ) =
Xd, isRan inje
tive
4.4. Theorem.
ring homomorphism. Then P2 is embedded in R.
roof. First note that the
ondition on R is equivalent to stating that 1 is a self-similar
obje
t of the
ategory
R . We demonstrate that
R has in
lusions and proje
tions.
De ne
!
!
1
0
1 = (1 0) ; 2 = (0 1) ; i1 =
; i2 =
:
0
1
P
Mat
Mat
Mat
Then it is immediate from the de nition of
omposition in
R that 1 i1 = 1R = 2 i2 ,
and
!
!
0 0
1 0
:
; i2 2 =
i1 1 =
0 0
0 1
Mat
Therefore, i1 1 = 1 t 0 and i2 2 = 0 t 1, and so
R has proje
tions and in
lusions.
Therefore, by Theorem 4.3 above, there exists an embedding of P2 into the multipli
ative
monoid of R.
Theory and Appli
ations of Categories, Vol. 6, No. 3
43
5. Resear
h questions
This paper is by no means a
omplete a
ount of the theory of self-similarity. The following
points are among the many it raises:
Although we have
on
entrated on pra
ti
al appli
ations, presumably there is a
reasonable de nition of the free self-similar monoidal
ategory, analogously to the
Ma
Lane de nition of the free monoidal
ategory on one generator [12℄. This should
allow for the formulation and proof of a
oheren
e theorem for self-similar
ategories.
This is related to the se
ond point:
The requirement that the tensor
is not stri
tly asso
iative at the obje
t N is not
needed - in [6℄, self-similarity is used to
onstru
t one-obje
t analogues of monoidal
ategories in the stri
tly asso
iative
ase. However, this is done in an ad ho
way. It
is also
urious that the monoid at N satis es one-obje
t analogues of the axioms for
a non-stri
t monoidal
ategory. Clearly there is some underlying theory that
overs
all
ases.
In at least some of [4℄, Girard uses a weaker
ondition to our self-similarity that is
equivalent to maps
: N
N ! N and d : N ! N
N satisfying d
= 1N
N and
d < 1N . Although this does not allow the de nition of a fun
tor from a
ategory
to a monoid, it allows a fun
tion that preserves
omposition and maps identities at
obje
ts to idempotents of a monoid. This
learly de nes a fun
tor from the
ategory
N
to the Karoubi envelope of the monoid at N . However, the
omputational or
logi
al interpretation of this remains un
lear.
The
onstru
tion of a
ompa
t
losed
ategory from a tra
ed symmetri
monoidal
ategory, as a `dualising' pro
ess is given in [8℄, and the logi
al and
omputational
signi
an
e of this has been well studied. It would be interesting to be able to use a
similar pro
ess to go from tra
ed M-monoids to
ompa
t
losed M-monoids, without
passing though a many-obje
t
ategory in an intermediate stage.
For monoidal
ategories with additional
ategori
al stru
ture X , we would like to
say that the fun
tor allows us to de ne one-obje
t analogues of X -
ategories. The
problem is to make this pre
ise, and relate it to the dire
t limit
onstru
tion used
to
onstru
t one-obje
t analogues of Cartesian
losed
ategories, as found in [10℄.
6. A
knowledgements
Spe
ial thanks are due to Ross Street, for reading a preliminary version of this paper
and helping me
larify some de nitions, and to Samson Abramsky, for an explanation of
the
omputational signi
an
e. Similarly, thanks are due to John Fountain for a
riti
al
reading of [6℄, and Mark Lawson, who helped develop the algebrai
theory of self-similarity.
Theory and Appli
ations of Categories, Vol. 6, No. 3
44
Appendix: Equivalen
e of axioms for
ompa
t
losedness
We demonstrate that the alternative axioms for
ompa
t
losed
ategories given in Definition 3.7 are equivalent to the usual set. We refer to the usual set of axioms, as found
in De nition 3.6, as A1 and the alternative set of axioms as A2.
A symmetri
monoidal
ategory that satis es the axioms A1 also satis es the axioms A2, and vi
e versa.
6.1. Theorem.
roof. ()) Let (M;
; ; ) be a symmetri
monoidal
ategory satisfying A1. For all
X; A 2 Ob(M), we de ne
= (
1 ) 1 and Æ
= (1
). Consider
arbitrary f : X ! Y . By the de nition of , and the naturality of in X , we
an
dedu
e that (1
_
f ) = f , and so is natural in X . Similarly, by de nition
of Æ , and the naturality of in X , Æ is natural in X .
Also, by de nition of and Æ ,
P
XA
A
X
XA
X
X
X
A
X
A
XA
A
Y A
XA
X
1.
1
ÆAA tAA
_ A AA
= (1
A
2. By de nition, Æ
A
A
XA
A
1
)t
_ A (A
AA
1
1
)
A
= 1 , by axiom 1 of A1.
A
=
_ A sA_
A;A_ tA_ AA_ sA
A_ ;A_ ÆA_ A
A_ (1A_
By the naturality of s
A
A
A_ sI ;A_ (A
A;A
1
A
A
1
_:
A
this is equal to
Y
_ )tA_ AA_ (1A_
A_ (A
Also, as s = , this is equal to
1 _ , by axiom 2 of A1.
IX
1_ )
_ tA_ AA_ sA
A_ ;A_ (A
)s _
in X and
XY
X
A
X
1
A
A
)s
A
_ ;I A_1 :
_ )tA_ AA_ (1A_
A
) _1 , whi
h is
A
A
3. By the naturality of in Z , and the de nition of
Z
(
A
(
A
1
X
A
1
_
)(
I
X
XA
)(
IA
1
I 1
X
1
)
X
1
X
)
1
X
= (
A
= (
A
1
X
_
A
)(1
I
1
1
X
)(
X
)
,
1 )( 1
1
1
= (
1 )
A
I
A
X
X
I
X
)=
1
X
=
XA
4. In a similar way to 3,
X (1X
Æ
IA
)(1
X
A
1
_
A )
= (1
X
X
A
)=Æ
XA
by the de nition of and the naturality of in Z .
Z
Therefore, and Æ are morphisms that are natural in X , and satisfy the axioms A2.
Therefore, every symmetri
monoidal
ategory satisfying A1 also satis es A2.
(() Let (M;
; ; Æ ) be a symmetri
monoidal
ategory satisfying A2. We de ne morphisms
: (A_
A) ! I ; : I ! (A
A_ )
AX
AX
A
by =
_
the axioms A1:
A
A
A
IA
and
A
A
=Æ
IA
1
A_
A .
We now
he
k the that these morphisms satisfy
Theory and Appli
ations of Categories, Vol. 6, No. 3
1.
A (1A
A
)t
1
1
_ A (A
AA
A
1
)
=
A
45
t 1 _ A ÆAA
A;A AA
= 1 , by axiom 1 of A2.
A
2. By de nition of and ,
1 _ )t _
_
1 _ j )t _
A_ (A_
1
A_ (ÆI A_ A
A
However,
A_
=
A
_ sI A_
1
A_ sI A (ÆI A_ A
A_
XY
A_ (1A_
Æ
1
A
_
=s
1
AA
A
A
and
and by the naturality of s
A
A
IA
_ (1A_
AA
) _1 =
_
_
_ (1A
_ A_1 .
in X and
Y,
A
A
A
IA
A
A
A
_:
_ I A )sI A_ A_1 ;
the above is equal to
1
_ A
)sA_
A;A_ tA_ AA_ sA
A_ ;A_ (A
A_ I A
A_
IA
Then by axioms 3 and 4 of A2, this is
1 _ , by axiom 2 of A2.
1
)
Therefore, this is equal to
_ j )tA_ AA_ (1A_
A
A
1
_ )A_1 :
A
ÆA_ A sA_
A;A_ tA_ AA_ sA
A_ ;A_ A_ A ,
whi
h is
A
Therefore, the axioms A1 are satis ed, and this
ompletes our proof.
Referen
es
[1℄ S. Abramsky, Retra
ing some paths in Pro
ess algebra, CONCUR
96.
[2℄ S. Abramsky, R. Jagadeesan, New Foundations for the Geometry of Intera
tion, Pro
.
Seventh IEEE Symposium on Logi
in Computer S
ien
e, 211-222 (1992)
[3℄ V. Danos, L. Regnier, Lo
al and Asyn
hronous Beta Redu
tion, (an analysis of Girard's exe
ution formula), Pro
. Eighth Ann. IEEE Symposium on Logi
in Computer
[4℄
[5℄
[6℄
S
ien
e, IEEE Comp. So
. Press 296-306 (1993)
J-Y. Girard, Geometry of intera
tion 1, Pro
eedings Logi
Colloquium '88, (1989)
221-260, North Holland.
J-Y. Girard, Geometry of intera
tion 2, Pro
eedings of COLOG 88, Martin-Lof &
Mints, 76-93 SLNCS 417
P.M. Hines, The Algebra of Self-similarity and its Appli
ations, PhD thesis, University of Wales (1997)
[7℄ P. Hines, M.V. Lawson, An appli
ation of poly
y
li
monoids to rings,
Forum 56 (1998), 146{149.
[8℄ A. Joyal, R. Street, D. Verity, Tra
ed Monoidal
ategories, Math.
So
., (1996) 425-446
Semigroup
Pro
. Camb. Phil.
Theory and Appli
ations of Categories, Vol. 6, No. 3
46
[9℄ G. Kelly, M. Laplaza, Coheren
e for
ompa
t
losed
ategories, Journal
of Pure and
Applied Algebra 19 (1980) 193-213
[10℄ J. Lambek, P.J. S
ott, Introdu
tion to Higher Order Categori
al Logi
, Cambridge
Studies in Advan
ed Mathemati
s, Cambridge University Press (1986)
[11℄ M.V. Lawson, Inverse Semigroups:
Singapore (1998)
the theory of partial symmetries,
World S
ienti
,
[12℄ S. Ma
Lane, Categories for the Working Mathemati
ian, Springer-Verlag New York,
(1971)
[13℄ M. Nivat, J.-F. Perrot, Une Generalisation du Monoide bi
y
lique,
C. R. A
ad. S
.
Paris, t.271 (1970)
[14℄ M. Petri
h, Inverse Semigroups, Pure and Applied Mathemati
s, Wiley-Inters
ien
e
Series, John Wiley and Sons (1984)
S
hool of Informati
s,
University of Wales, Bangor,
Gwynedd, U.K. LL57 1UT
Email: max003bangor.a
.uk
This arti
le may be a
essed via WWW at http://www.ta
.mta.
a/ta
/ or by anonymous ftp at ftp://ftp.ta
.mta.
a/pub/ta
/html/volumes/6/n3/n3.fdvi,psg
THEORY AND APPLICATIONS OF CATEGORIES (ISSN 1201-561X) will disseminate arti
les that
signi
antly advan
e the study of
ategori
al algebra or methods, or that make signi
ant new
ontributions to mathemati
al s
ien
e using
ategori
al methods. The s
ope of the journal in
ludes: all areas of
pure
ategory theory, in
luding higher dimensional
ategories; appli
ations of
ategory theory to algebra,
geometry and topology and other areas of mathemati
s; appli
ations of
ategory theory to
omputer
s
ien
e, physi
s and other mathemati
al s
ien
es;
ontributions to s
ienti
knowledge that make use of
ategori
al methods.
Arti
les appearing in the journal have been
arefully and
riti
ally refereed under the responsibility
of members of the Editorial Board. Only papers judged to be both signi
ant and ex
ellent are a
epted
for publi
ation.
The method of distribution of the journal is via the Internet tools WWW/ftp. The journal is ar
hived
ele
troni
ally and in printed paper format.
Individual subs
ribers re
eive (by e-mail) abstra
ts of arti
les as
they are published. Full text of published arti
les is available in .dvi and Posts
ript format. Details will
be e-mailed to new subs
ribers and are available by WWW/ftp. To subs
ribe, send e-mail to ta
mta.
a
in
luding a full name and postal address. For institutional subs
ription, send enquiries to the Managing
Editor, Robert Rosebrugh, rrosebrughmta.
a.
Information for authors. The typesetting language of the journal is TEX, and LATEX is the
preferred
avour. TEX sour
e of arti
les for publi
ation should be submitted by e-mail dire
tly to an
appropriate Editor. They are listed below. Please obtain detailed information on submission format and
style les from the journal's WWW server at URL http://www.ta
.mta.
a/ta
/ or by anonymous ftp
from ftp.ta
.mta.
a in the dire
tory pub/ta
/info. You may also write to ta
mta.
a to re
eive
details by e-mail.
Subs
ription information.
Editorial board.
John Baez, University of California, Riverside: baezmath.u
r.edu
Mi
hael Barr, M
Gill University: barrbarrs.org
Lawren
e Breen, Universite de Paris 13: breenmath.univ-paris13.fr
Ronald Brown, University of North Wales: r.brownbangor.a
.uk
Jean-Lu
Brylinski, Pennsylvania State University: jlbmath.psu.edu
Aurelio Carboni, Universita dell Insubria:
arbonifis.uni
o.it
P. T. Johnstone, University of Cambridge: ptjpmms.
am.a
.uk
G. Max Kelly, University of Sydney: maxkmaths.usyd.edu.au
Anders Ko
k, University of Aarhus: ko
kimf.au.dk
F. William Lawvere, State University of New York at Bu alo: wlawverea
su.buffalo.edu
Jean-Louis Loday, Universite de Strasbourg: lodaymath.u-strasbg.fr
Ieke Moerdijk, University of Utre
ht: moerdijkmath.ruu.nl
Susan Nie eld, Union College: niefielsunion.edu
Robert Pare, Dalhousie University: parems
s.dal.
a
Andrew Pitts, University of Cambridge: ap
l.
am.a
.uk
Robert Rosebrugh, Mount Allison University: rrosebrughmta.
a
Jiri Rosi
ky, Masaryk University: rosi
kymath.muni.
z
James Stashe , University of North Carolina: jds
harlie.math.un
.edu
Ross Street, Ma
quarie University: streetmath.mq.edu.au
Walter Tholen, York University: tholenmathstat.yorku.
a
Myles Tierney, Rutgers University: tierneymath.rutgers.edu
Robert F. C. Walters, University of Sydney: walters bmaths.usyd.edu.au
R. J. Wood, Dalhousie University: rjwoodms
s.dal.
a
ations of Categories,
Vol. 6, No. 3, 1999, pp. 33{46.
THE CATEGORICAL THEORY OF SELF-SIMILARITY
This paper is dedi
ated to Joa
him Lambek,
whose work was the inspiration for the following results.
PETER HINES
ABSTRACT. We demonstrate how the identity N
N
= N in a monoidal
ategory
allows us to
onstru
t a fun
tor from the full sub
ategory generated by N and
to
the endomorphism monoid of the obje
t N . This provides a
ategori
al foundation for
one-obje
t analogues of the symmetri
monoidal
ategories used by J.-Y. Girard in his
Geometry of Intera
tion series of papers, and expli
itly des
ribed in terms of inverse
semigroup theory in [6, 11℄.
This fun
tor also allows the
onstru
tion of one-obje
t analogues of other
ategori
al
stru
tures. We give the example of one-obje
t analogues of the
ategori
al tra
e, and
ompa
t
losedness. Finally, we demonstrate how the
ategori
al theory of self-similarity
an be related to the algebrai
theory (as presented in [11℄), and Girard's dynami
al
algebra, by
onsidering one-obje
t analogues of proje
tions and in
lusions.
1. Introdu
tion
It is well-known, [12℄, that any one-obje
t monoidal
ategory is an abelian monoid with
respe
t to two operations that satisfy the inter
hange law (a b) Æ (
d) = (a Æ
) (b Æ d);
that is, all the
anoni
al isomorphisms for the tensor are identities. However, non-trivial
one-obje
t analogues of the
anoni
al asso
iativity and
ommutativity morphisms for
symmetri
monoidal
ategories have been impli
itly
onstru
ted by Girard for his work
on linear logi
, [4, 5℄, and have been studied in terms of inverse semigroups in [6℄ and [11℄.
The absen
e of a unit allows for a more interesting theory.
We demonstrate in se
tion 2 how these examples arise naturally from the assumption
of self-similarity. This is motivated by the examples of either the intuitive
onstru
tion
of the Cantor set, or bije
tions between N and N N . We de ne a self-similar obje
t of a
symmetri
monoidal
ategory to be an obje
t N satisfying N
= N
N . In the
ategori
al
ase, this gives rise to examples of stru
ture-preserving maps between
ategories and
monoids. In se
tion 3, we then demonstrate how these maps allow us to
onstru
t oneobje
t analogues of the
ategori
al tra
e and
ompa
t
losedness. Finally, in se
tion
4, these
ategori
al
onstru
tions are related to the algebrai
approa
h to self-similarity
des
ribed in [11℄, and Girard's dynami
al algebra, by
onsidering one-obje
t analogues of
proje
tions and in
lusions.
Re
eived by the editors 1998 O
tober 6 and, in revised form, 1999 September 13.
Published on 1999 November 30.
1991 Mathemati
s Subje
t Classi
ation: 18D10, 20M18.
Key words and phrases: Monoidal Categories, Categori
al Tra
e, Compa
t Closure, Linear Logi
,
Inverse Semigroups.
Peter Hines 1999. Permission to
opy for private use granted.
33
Theory and Appli
ations of Categories, Vol. 6, No. 3
34
2. Self-similarity and tensors
M
( ;
) be a symmetri
monoidal
ategory and denote the asso
ia-
2.1. Definition. Let
M
tivity,
ommutativity, and left and right unit isomorphisms by
tively. We say that an obje
t
morphisms
N ! N
:N
N
of
self-similar
if
A;B;C
;s
A;B
; ;
A
A
respe
-
; that is, there exist
d : N ! N
N that satisfy d
= 1
ompression and division morphisms of N .
and
We
all these morphisms the
is
t
N
= N
N
N
N
and
d = 1
N.
For the purposes of this paper, we will require the additional
ondition that the tensor
is not stri
t at N , so t
is not the identity map. We also assume that the
ommutativity is not stri
t, to avoid studying degenerate
ases. We study the full sub
ategory of
given by all obje
ts
onstru
ted from N and
, whi
h we refer to as
. We assume
throughout that N 6= I .
The sub
ategory
has all the stru
ture of a monoidal
ategory apart from the unit
obje
t; we use the awkward term unitless monoidal
ategories1 for stru
tures of this form.
The following is then immediate:
are isomorphi
to N , by maps d : N ! X and
:
2.2. Lemma. All obje
ts of
N;N;N
M
X
N
N
!N
N
.
X
X
roof. We (indu
tively) de ne the isomorphisms by:
P
! N,
d )d : N ! U
V
) : U
V ! N.
d =1 =
:N
N
N
N
d
= (d
U
V
U
=
(
U
V
U
V
V
First note that
=
and d = d
. The uniqueness of these maps then follows from
the requirement that
is not stri
t at N , and hen
e that t
is never the identity map,
for any X; Y; Z 2 Ob(
). This ensures that the set of obje
ts is in dire
t
orresponden
e
to the set of (non-empty) binary bra
ketings of a single symbol, and the tensor
orresponds
to bra
keting two obje
ts together. This
an be
ompared to the obje
ts and tensor of
Ma
Lane's free monoidal
ategory on one generator, as used in the proof of his
elebrated
oheren
e theorem. We refer to [12℄ for this
onstru
tion and proof.
Let U and V be obje
ts of
satisfying d
= 1 and d
= 1 . (This holds
trivially for U = V = N ). Then
N
N
N
N
N
X;Y ;Z
N
V
U
V
U
U
V
d )d
(
) = (d
d )1
= (d
d
) = (1
1 ) = 1
d
= (d
U
U
U
U
V
U
U
V
V
V
U
U
V
V
U
N
V
V
(
N
V
U
V
)
:
Therefore, by indu
tion, d
= 1 for all obje
ts X . Similarly,
d = 1 .
X
1 This
X
X
X
X
N
is admittedly an abuse of notation, inasmu
h as a monoid without an identity is a semigroup.
However, the natural alternatives, `multipli
ative
ategory', or `semigroup
ategory' have already been
used in the work of Hyland and DePaiva, and Lawson respe
tively.
Theory and Appli
ations of Categories, Vol. 6, No. 3
35
This allows us to
onstru
t a fun
tor from the
ategory N
to the endomorphism monoid
of N (
onsidered as a one-obje
t
ategory), as follows:
Let N be a self-similar obje
t of a symmetri
monoidal
ategory,
(M;
). The map : N
! M(N; N ), de ned by (f ) =
Y fdX for all f : X ! Y and
(X ) = N for all X 2 Ob(N
) is a fun
tor.
Proof. For all f : X ! Y and g : Y ! Z in N , by de nition,
2.3. Proposition.
(g ) (f ) =
Z gdY
Y fdX =
Z g 1Y f
X = (gf ):
Therefore, preserves
omposition, and it is immediate from the de nition that (1X ) =
1N for all obje
ts X . Hen
e is a fun
tor.
These de nitions allow us to
onstru
t a monoid homomorphism that has a very
lose
onne
tion with the tensor, as follows:
Let N be a self-similar obje
t of a symmetri
monoidal
ategory (M;
),
and let : M(N; N ) M(N; N ) ! M(N; N ) be de ned by f g =
(f
g )d, for all
f; g : N ! N . Then
(i) is a monoid homomorphism,
(ii) (f
g ) = (f ) (g ) for all f : U ! X and g : V ! Y , for U; V; X; Y 2 Ob(N
).
2.4. Lemma.
P
roof.
By de nition, (1 1) =
d = 1N , and
(i)
(f
g)(h k) =
(f
g)d
(h
k)d =
(fh
gk)d = fh gk:
Hen
e is a monoid homomorphism.
By de nition (f ) (g ) =
( (f )
(g ))d. Hen
e, by de nition of ,
(ii)
(f ) (g ) =
(
X fdU
Y gdV )d =
(
X
Y )(f
g )(dU
dV )d
and so, by de nition of
X
Y and dU
V ,
(f ) (g ) =
X
Y (f
g )dU
V = (f
g ):
Our
laim is that this monoid homomorphism (whi
h, in [6℄, is referred to as the internalisation of the tensor) gives the endomorphism monoid of N the stru
ture of a one-obje
t
(unitless) symmetri
monoidal
ategory, with as the tensor.
The one-obje
t analogues of the axioms for a tensor are as follows:
There exist spe
ial elements s and t (the analogues of the
ommutativity and asso
iativity
elements respe
tively) that satisfy:
2.5. Definition.
1. s(u v ) = (v u)s,
Theory and Appli
ations of Categories, Vol. 6, No. 3
36
2. t(u (v w)) = ((u v ) w)t,
3. t2 = (t 1)t(1 t),
4. tst = (s 1)t(1 s),
5. s2 = 1,
6. t has an inverse, t 1 , satisfying tt
= 1 = t 1 t.
1
These are just the axioms for a (unitless) symmetri
monoidal
ategory with the obje
t
subs
ripts erased.
The homomorphism satis es the above
onditions, as follows:
2.6. Theorem. Let N be a self-similar obje
t of a symmetri
monoidal
ategory (
;
),
and let be as de ned in Lemma 2.4 of Se
tion 2. Then there exist distinguished elements
s and t of the endomorphism monoid of N satisfying 1. to 6. above.
Proof. De ne s = (s
), where s
is the family of
ommutativity morphisms for
(M;
). Similarly, de ne t = (t
) and t 1 = (t 1 ). Note that, for the isomorphisms of Lemma 1 to be well-de ned, we require that t
6= 1
(
)
Then (t
) = (t
) = t, by the naturality of t
, and the fa
t that (1 ) =
1, for all X; Y; Z 2 Ob(
). Similarly, (s ) = s, for all X; Y 2 Ob(
).
The
onditions 1. to 4. follow immediately, by applying to the axioms 1. to 4.
respe
tively.
M
NN
X;Y
N;N;N
N;N;N
N;N;N
N
X;Y;Z
N;N;N
N
N
N
N
X;Y ;Z
X;Y
X
1. s(b
a) = (a
b)s,
2. t(a
(b
)) = ((a
b)
)t,
3. The Ma
Lane Pentagon:
t(
A
B );C;D
t
A;B;(C
D)
= (t
A;B;C
1
D
)t
A;(B
C );D
(1
A
t
B;C;D
):
4. The Commutativity Hexagon:
t
C;A;B
s(
A
B );C
t
= (s
A;B;C
A;C
1
B
)t
A;C;B
s
(1
A
B;C
);
and by the naturality of the
anoni
al isomorphisms for a symmetri
monoidal
ategory.
For example, applying to the Ma
Lane pentagon gives
(t(
A
B );C;D
t
A;B;(C
) ) = ((t
D
A;B;C
1
D
)t
A;(B
C );D
(1
A
t
B;C;D
));
and so
(t(
A
B );C;D
) (t
A;B;(C
) ) = ( (t
A;B;C
D
) (1 )) (t
D
A;(B
C );D
)( (1 ) (t
A
B;C;D
)):
Therefore, tt = (t 1)t(1 t), whi
h is
ondition 3. above. Conditions 5. and 6. follow
by applying to the equations s s
= 1 and t
t 1 = 1, whi
h follow from the
de nition of a symmetri
monoidal
ategory.
A;B
B;A
X;Y ;Z
X;Y ;Z
Theory and Appli
ations of Categories, Vol. 6, No. 3
37
Hen
e, the endomorphism monoid of a self-similar obje
t of a symmetri
monoidal
ategory is a unitless one-obje
t symmetri
monoidal
ategory.
2.7. Definition. We de ne a monoidal fun
tor between unitless symmetri
monoidal
ategories (C;
) and (D; ) to be a pair (F; m), where F : C ! D is a fun
tor, and m
is a natural transformation with
omponents m : F (A) F (B ) ! F (A
B ).
A;B
With this terminology, the following is immediate
Let N be a self-similar obje
t of a symmetri
monoidal
ategory M.
Then is a monoidal fun
tor from N
to M(N; N ),
onsidered a one-obje
t unitless
symmetri
monoidal
ategory.
2.8. Corollary.
Monoids M that have monoid homomorphisms from M M to M , together with analogues
of the asso
iativity and
ommutativity elements satisfying 1. to 6. above have already
been
onstru
ted, under the name (strong) M{monoids in [6℄ and Girard Monoids in [11℄.
We demonstrate how the two examples given are examples of this
onstru
tion in the
ategory of partial inje
tive maps.
2.9. Examples of self-similarity. Any in nite obje
t in the
ategory Set is selfsimilar with respe
t to two distin
t tensors; Set is a symmetri
monoidal
ategory with
respe
t to both the Cartesian produ
t of sets, X Y = f(x; y ) : x 2 X; y 2 Y g and the
oprodu
t, A t B = f(a; 0) : a 2 Ag [ f(b; 1) : b 2 B g. This then gives us the following:
2.10. Proposition.
Set, with
Proof.
! N N,
respe
t to both
and N
The set of natural numbers, N is a self-similar obje
t of
and t.
The elementary theory of in nite sets gives the existen
e of bije
tions N
Therefore, our result follows.
! N t N.
M-monoid stru
tures
an then be derived from spe
i
examples of bije
tions ' : N tN ! N
and : N N ! N . The examples given in [6, 11℄ are
onstru
ted from the fun
tion
'(n; i) = 2n + i
n 2 N ; i 2 f0; 1g
whi
h is a bije
tion from N t N to N This is then used to
onstru
t a bije
tion from N N
to N by denoting '(n; i) by ' (n), and de ning
i
(x; y ) = '1 ('0 (x)):
y
Similar results hold for the
ategory of relations, and the
ategory of partial inje
tive maps
{ interestingly, algebrai
representations of these M-monoid stru
tures are given in terms
of inverse semigroup theory and poly
y
li
monoids (see Se
tion 4 for the
onstru
tions
of these). We refer to [11℄ for the algebrai
theory of inverse semigroups, and to [6, 11℄
for expli
it des
riptions of M-monoids in terms of the inverse monoid of partial inje
tions
on the natural numbers.
Theory and Appli
ations of Categories, Vol. 6, No. 3
38
3. Monoid analogues of other
ategori
al stru
tures
A fun
tor from a symmetri
monoidal
ategory to the endomorphism monoid of an obje
t
(viewed as a unitless one-obje
t symmetri
monoidal
ategory), that preserves both the
ategory stru
ture and the symmetri
monoidal stru
ture, might be expe
ted to preserve
other
ategori
al properties. We demonstrate how the assumption of self-similarity allows
us to
onstru
t one-obje
t analogues of the
ategori
al tra
e (as des
ribed by Joyal, Street,
and Verity in [8℄) and, in some
ases, one-obje
t analogues of
ompa
t
losedness.
These spe
i
examples were motivated by Abramsky's analysis of the Geometry of
Intera
tion, [1℄, and the paper of Abramsky and Jagadeesan, [2℄, where it is demonstrated
that the system of [1℄ is equivalent to the
onstru
tion of [8℄ | hen
e demonstrating that
the dynami
s of the Geometry of Intera
tion system is given by a
ategori
al tra
e, and
ompa
t
losedness. The motivation for
onstru
ting one-obje
t analogues of
ompa
t
losedness and the
ategori
al tra
e then
omes from Girard's
omment in [4℄ where he
states that his system `forgets types' | the usual representation of types being obje
ts
in a
ategory.
3.1. The
ategori
al tra
e.
Let (M;
; s; t; ; ; I ) be a symmetri
monoidal
ategory. A tra
e on
it is de ned in [8℄ to be a family of fun
tions, T r : M (A
U; B
U ) ! M (A; B ), that
are natural in A; B and U , and satisfy the following:
3.2. Definition.
U
A;B
1. Given f : X
I
2.
! Y
I , then T r (f ) = f
Given f : A
(U
V ) ! B
(U
V ), then
I
X;Y
T r
(f ) = T r
U
V
A;B
U
A;B
(T r
V
A
U;B
(t
U
B;U;V
ft
1
A;U;V
)):
3. Given f : A
U
Tr
4. T r
U
U
A;B
U
U;U
! B
U , and g : C ! D, then
(f )
g = T r
(t
(1
s )t 1
!Y.
:X
1
(s
U;U
A
C;B
D
BDU
B
D;U
BU D
(f
g )t
AU C
(1
A
)=1 .
U
s
C;U
)t
1
AC U
)
N
Let N be a self-similar obje
t of a symmetri
monoidal
ategory. Then
is a tra
ed
symmetri
monoidal
ategory (although axiom 1 is not appli
able, due to the absen
e
of the unit obje
t). We
an use the fun
tor to
onstru
t one-obje
t analogues of the
ategori
al tra
e, as follows:
Let N be a self-similar obje
t of a tra
ed symmetri
monoidal
ategory
( ;
), and denote the
ompression and division morphisms by
and d respe
tively.
Then the map tra
e : (N; N ) ! (N; N ) de ned by tra
e(f ) = T r (df
) satis es
tra
e( (F )) = (T r (F )) for all F 2
(X
U; Y
U ).
M
3.3. Lemma.
U
M
X;Y
M
N
N
N;N
Theory and Appli
ations of Categories, Vol. 6, No. 3
39
roof. tra
e( (F )) = T r (d (F )
) = T r (d
F d
) by de nition of tra
e and
. However,
=
(
) and d
= (d
d )d, by de nition of the division and
ompression morphisms. Therefore
N
N;N
P
Y
U
Y
N
N;N
U
X
tra
e( (F ))
= Tr
N
N;N
((
Y
)F (d
U
= Tr
X
d
U
N
N;N
U
Y
X
(d
(
Y
U
X
U
U
)) =
(T r
Y
)F (d
U
N
X;Y
X
((1
d
U
Y
U
)d
)
)F (1
X
d
U
)))d
X
;
by the naturality of the tra
e in X and Y , and so
tra
e( (F ))
=
=
T rX;Y (F (1X
U
Y
Y
T rX;Y (F (1X
U
1
U
))d =
X
by the naturality of the tra
e in U . Therefore
tion of .
d
U
U ))dX
T rX;Y (F )dX ;
U
Y
tra
e( (F ))
= (T r
U
X;Y
(F )), by the de ni-
This then satis es one-obje
t analogues of the
ategori
al tra
e, as follows:
be a self-similar obje
t of a tra
ed symmetri
monoidal
ategory
(M;
), let ; s; t; t be as de ned in Theorem 2.6 of Se
tion 2, and let the map tra
e
be as de ned above. Then
(i) tra
e(f ) = tra
e(tra
e(tf t 1 )),
(ii) tra
e(f ) g = tra
e(t(1 s)t 1 (f g )t(1 s)t 1 ),
(iii) tra
e(s) = 1,
(iv) tra
e((h 1)f (g 1)) = h(tra
e(f ))g ,
(v) tra
e(f (1 g )) = tra
e((1 g )f ).
3.4. Theorem.
Let
1
N
roof. These follow immediately from the above Lemma, the axioms for the
ategori
al
tra
e, and the fa
t that is a monoid homomorphism.
P
Spe
i
examples
an be
onstru
ted at any
ountable set, in both the
ategory of relations
(whi
h is demonstrated in [8℄ to be a tra
ed symmetri
monoidal
ategory), and in the
sub-
ategory of partial inje
tive maps (whi
h is demonstrated in [6℄ to be
losed under
the same
ategori
al tra
e).
3.5.
ompa
t
losedness.
A
ompa
t
losed
ategory M is a symmetri
monoidal
ategory where
for every obje
t A, there exists a left dual, A_ . The left dual has the following properties:
For every A 2 Ob(M), there exist two morphisms, the
ounit map : A_
A ! I , and
the unit map : I ! A
A_ , that satisfy the following
oheren
e
onditions:
3.6. Definition.
A
A
)t 1 _ (
1 ) 1 = 1
_ (
1 _ )t _ _ (1 _
) _1 = 1_
1.
A (1A
2.
A
A
A
AA
A
A
A
A
AA
A
A
A
A
A
A
A
Theory and Appli
ations of Categories, Vol. 6, No. 3
40
We refer to [9℄ for the details of the
oheren
e theorem for
ompa
t
losed
ategories.
The point of the main
onstru
tion of [8℄ was to show that every tra
ed symmetri
monoidal
ategory is a monoidal full sub
ategory of a
ompa
t
losed
ategory, whi
h
gives a stru
ture theorem for tra
ed symmetri
monoidal
ategories. In light of this, we
would again expe
t that the assumption of self-similarity at an obje
t (together with
possible extra
onditions) should allow us to
onstru
t one-obje
t analogues of
ompa
t
losedness. However, this raises the following problem: the distinguished maps that make
a symmetri
monoidal
ategory into a
ompa
t
losed
ategory (the units and
ounits) are
de ned in terms of the unit obje
t. However the fun
tor of Proposition 2.3 of Se
tion 2
gives one-obje
t analogues of unitless symmetri
monoidal
ategories. This motivates an
alternative
hara
terisation of
ompa
t
losed
ategories.
In many appli
ations of
ompa
t
losed
ategories (for example, the
onstru
tion of
a
anoni
al tra
e on a
ompa
t
losed
ategory, [8℄), the and maps always appear in
onjun
tion with the unit isomorphisms ; and 1 ; 1 respe
tively. For example, [8℄
uses morphisms between X and X
(A
A_ ) for all X and A de ned by (1
) 1 ,
and similarly for and . We take maps of this form as primitive; this gives the following
alternative set of axioms for a
ompa
t
losed
ategory:
3.7. Definition. For every obje
t A 2 Ob(
), there exists a dual obje
t A_ , together
X
X A
ÆX A
! (A
A_ )
X
: X
(A_
A) ! X
:X
,
,
that are natural in
X
and satisfy the following axioms
1.
1
ÆAA tAA
_ A AA
2.
ÆA_ A sA_
A;A_ tA_ AA_ sA
A_ ;A_ A_ A
3.
(
4.
X (1X
A
X
M
with morphisms
A
A
_ I A
Æ
=1
1
IA
X
A,
)
1
X
A_1
A )
=
=Æ
=1
A
_,
XA,
XA.
The naturality of and Æ in X
an be written expli
itly as, ((1
1 _ )
f ) =
f and f Æ
= Æ (f
(1 _
1 )) for all f : X ! Y .
The proof of the equivalen
e 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
hara
terisation of
ompa
t
losedness. This allows us to de ne one-obje
t (unitless)
analogues of these axioms for M-monoids, as follows:
XA
YA
XA
YA
XA
A
A
A
XA
A
3.8. Definition. An M-monoid is said to satisfy the one-obje
t analogues of
ompa
t
losedness if there exist distinguished elements
1.
Æt 1
= 1,
; 1
and
Æ; Æ 1
that satisfy
Theory and Appli
ations of Categories,
2.
3.
4.
Vol. 6, No. 3
41
= 1,
(1 a) = a,
aÆ = Æ (a 1).
Æsts
Axioms 1. and 2. are one-obje
t analogues of axioms 1. and 2. of De nition 3.7, and
axioms 3. and 4. are one-obje
t analogues of the naturality
onditions.
M
Let ( ;
) be a
ompa
t
losed
ategory, and let N be a self-dual selfsimilar obje
t of . Then there exist elements and Æ that satisfy the one-obje
t analogues
of the alternative axioms for a
ompa
t
losed
ategory.
M
3.9. Theorem.
roof. De ne = (NN ) and Æ = (ÆNN ). The naturality of XN and ÆXN in X
allows us to dedu
e that (XN ) = (NN ), and similarly, (XN ) = (NN ). Then, as
N _ = N , applying to the alternative axioms for a
ompa
t
losed
ategory will give 1.
and 2. of the above De nition, and applying to the expli
it des
ription of the naturality
onditions (and using the fa
t that (1 1) = 1), will give us 3. and 4. of the above
De nition.
P
4. Relating the
ategori
al and algebrai
theories of self-similarity
We relate the
ategori
al approa
h to self-similarity given above to the inverse semigroup
theoreti
approa
h given in [11℄, where self-similarity is studied algebrai
ally in terms of
poly
y
li
monoids. These were introdu
ed in [13℄, as follows:
the poly
y
li
monoid on the set X , is de ned to be the inverse
monoid (with a zero, for n 2) generated by a set of
ountable
ardinality, X , say
fp0; : : : pn 1g, in
ase jX j = n, or fp0; p1; : : :g for
ountably in nite X , subje
t to the
relations pi pj 1 = Æij , where p 1 is the generalised inverse2 of pi .
4.1. Definition.
PX ,
It will be
onvenient to denote the poly
yli
monoid on the set fp; qg by P2 { this is
following the
onvention of Girard, who uses elements of a C algebra satisfying the above
axioms (whi
h he refers to as the dynami
al algebra) in his work on linear logi
, [4, 5℄. In
[3℄, this is demonstrated to be a representation of the free
ontra
ted semigroup ring over
a monoid, whi
h is trivially the poly
y
li
monoid on two generators
The poly
y
li
monoids appear to be the algebrai
representation of self-similarity.
It is demonstrated in [11℄ that the poly
y
li
monoid on two generators
an also be
onstru
ted in terms of the inverse monoid of partial isomorphisms of the Cantor set, and
in [7℄ that an embedding of a poly
y
li
monoid into a ring R gives the ring isomorphism
Mn ( R )
= R, for all n 2 N. They also have a
lose
onne
tion with the algebrai
theory
of tilings; we again refer to [11℄.
2 Note
that the existen
e of generalised inverses is a signi
antly weaker
ondition than group-theoreti
inverses. In the inverse semigroup theoreti
ase, for every element a, there exists a unique generalised
inverse, whi
h satis es aa 1 a = a and a 1 aa 1 = a 1 . However, we
annot, in general, dedu
e aa 1 =
1 = a 1 a. We refer to [11℄ for a
omprehensive a
ount of the theory of inverse semigroups.
Theory and Appli
ations of Categories, Vol. 6, No. 3
M
42
Let N be a self-similar obje
t of a symmetri
monoidal
ategory ( ;
),
and let (N; N ) have a zero, whi
h we denote 0. We say that maps 1 ; 2 2 (N
N; N )
are left and right proje
tion maps and i1 ; i2 2 (N; N
N ) are their in
lusion maps, if
they satisfy
M
M
4.2. Definition.
M
1. 1 i1 = 1N and 2 i2 = 1N ,
2. i1 1 = 1N
0 and i2 2 = 0
1N .
M
M
Let N be a self-similar obje
t of a symmetri
monoidal
ategory ( ;
).
If N has proje
tion and in
lusion maps, then there exists an embedding of P2 into (N; N ).
4.3. Theorem.
roof. As preserves
omposition, (1 ) (i1 ) = 1N = (2 ) (i2 ). Similarly, (i1 ) (1 ) =
1 0, and (i2 ) (2 ) = 0 1, sin
e (0) = 0. Hen
e, as is a semigroup homomorphism,
P
(i1 ) (1 ) (i2 ) (2 ) = 0 = (i2 ) (2 ) (i1 ) (1 ):
Therefore, (1 ) (i2 ) = 0 = (2 ) (i1 ), and so (1 ); (2 ) satisfy the axioms for the
generators of P2 , and (i1 ); (i2 ) satisfy the
onditions for their generalised inverses. Also,
by [13℄, the poly
y
li
monoids are
ongruen
e-free, hen
e the only quotient of P2 is the
trivial monoid. Therefore, these elements generate an embedding of P2 into (N; N ).
M
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 P2 into the multipli
ative monoid of a ring R
de nes a family of isomorphisms R
= Mn (R) for all positive integers n. For a unital ring
R, we de ne the
ategory
R that has natural numbers as obje
ts, and b a matri
es
as morphisms from a to b. It is immediate that this
ategory
is a symmetri
monoidal
!
A 0
ategory, with the tensor t given by A t B =
.
0 B
Mat
Let R be a unital ring, and assume there exists
2 Mat (2; 1), d 2
MatR(1; 2) su
h that the map : MatR(2; 2) ! R, de ned by (X ) =
Xd, isRan inje
tive
4.4. Theorem.
ring homomorphism. Then P2 is embedded in R.
roof. First note that the
ondition on R is equivalent to stating that 1 is a self-similar
obje
t of the
ategory
R . We demonstrate that
R has in
lusions and proje
tions.
De ne
!
!
1
0
1 = (1 0) ; 2 = (0 1) ; i1 =
; i2 =
:
0
1
P
Mat
Mat
Mat
Then it is immediate from the de nition of
omposition in
R that 1 i1 = 1R = 2 i2 ,
and
!
!
0 0
1 0
:
; i2 2 =
i1 1 =
0 0
0 1
Mat
Therefore, i1 1 = 1 t 0 and i2 2 = 0 t 1, and so
R has proje
tions and in
lusions.
Therefore, by Theorem 4.3 above, there exists an embedding of P2 into the multipli
ative
monoid of R.
Theory and Appli
ations of Categories, Vol. 6, No. 3
43
5. Resear
h questions
This paper is by no means a
omplete a
ount of the theory of self-similarity. The following
points are among the many it raises:
Although we have
on
entrated on pra
ti
al appli
ations, presumably there is a
reasonable de nition of the free self-similar monoidal
ategory, analogously to the
Ma
Lane de nition of the free monoidal
ategory on one generator [12℄. This should
allow for the formulation and proof of a
oheren
e theorem for self-similar
ategories.
This is related to the se
ond point:
The requirement that the tensor
is not stri
tly asso
iative at the obje
t N is not
needed - in [6℄, self-similarity is used to
onstru
t one-obje
t analogues of monoidal
ategories in the stri
tly asso
iative
ase. However, this is done in an ad ho
way. It
is also
urious that the monoid at N satis es one-obje
t analogues of the axioms for
a non-stri
t monoidal
ategory. Clearly there is some underlying theory that
overs
all
ases.
In at least some of [4℄, Girard uses a weaker
ondition to our self-similarity that is
equivalent to maps
: N
N ! N and d : N ! N
N satisfying d
= 1N
N and
d < 1N . Although this does not allow the de nition of a fun
tor from a
ategory
to a monoid, it allows a fun
tion that preserves
omposition and maps identities at
obje
ts to idempotents of a monoid. This
learly de nes a fun
tor from the
ategory
N
to the Karoubi envelope of the monoid at N . However, the
omputational or
logi
al interpretation of this remains un
lear.
The
onstru
tion of a
ompa
t
losed
ategory from a tra
ed symmetri
monoidal
ategory, as a `dualising' pro
ess is given in [8℄, and the logi
al and
omputational
signi
an
e of this has been well studied. It would be interesting to be able to use a
similar pro
ess to go from tra
ed M-monoids to
ompa
t
losed M-monoids, without
passing though a many-obje
t
ategory in an intermediate stage.
For monoidal
ategories with additional
ategori
al stru
ture X , we would like to
say that the fun
tor allows us to de ne one-obje
t analogues of X -
ategories. The
problem is to make this pre
ise, and relate it to the dire
t limit
onstru
tion used
to
onstru
t one-obje
t analogues of Cartesian
losed
ategories, as found in [10℄.
6. A
knowledgements
Spe
ial thanks are due to Ross Street, for reading a preliminary version of this paper
and helping me
larify some de nitions, and to Samson Abramsky, for an explanation of
the
omputational signi
an
e. Similarly, thanks are due to John Fountain for a
riti
al
reading of [6℄, and Mark Lawson, who helped develop the algebrai
theory of self-similarity.
Theory and Appli
ations of Categories, Vol. 6, No. 3
44
Appendix: Equivalen
e of axioms for
ompa
t
losedness
We demonstrate that the alternative axioms for
ompa
t
losed
ategories given in Definition 3.7 are equivalent to the usual set. We refer to the usual set of axioms, as found
in De nition 3.6, as A1 and the alternative set of axioms as A2.
A symmetri
monoidal
ategory that satis es the axioms A1 also satis es the axioms A2, and vi
e versa.
6.1. Theorem.
roof. ()) Let (M;
; ; ) be a symmetri
monoidal
ategory satisfying A1. For all
X; A 2 Ob(M), we de ne
= (
1 ) 1 and Æ
= (1
). Consider
arbitrary f : X ! Y . By the de nition of , and the naturality of in X , we
an
dedu
e that (1
_
f ) = f , and so is natural in X . Similarly, by de nition
of Æ , and the naturality of in X , Æ is natural in X .
Also, by de nition of and Æ ,
P
XA
A
X
XA
X
X
X
A
X
A
XA
A
Y A
XA
X
1.
1
ÆAA tAA
_ A AA
= (1
A
2. By de nition, Æ
A
A
XA
A
1
)t
_ A (A
AA
1
1
)
A
= 1 , by axiom 1 of A1.
A
=
_ A sA_
A;A_ tA_ AA_ sA
A_ ;A_ ÆA_ A
A_ (1A_
By the naturality of s
A
A
A_ sI ;A_ (A
A;A
1
A
A
1
_:
A
this is equal to
Y
_ )tA_ AA_ (1A_
A_ (A
Also, as s = , this is equal to
1 _ , by axiom 2 of A1.
IX
1_ )
_ tA_ AA_ sA
A_ ;A_ (A
)s _
in X and
XY
X
A
X
1
A
A
)s
A
_ ;I A_1 :
_ )tA_ AA_ (1A_
A
) _1 , whi
h is
A
A
3. By the naturality of in Z , and the de nition of
Z
(
A
(
A
1
X
A
1
_
)(
I
X
XA
)(
IA
1
I 1
X
1
)
X
1
X
)
1
X
= (
A
= (
A
1
X
_
A
)(1
I
1
1
X
)(
X
)
,
1 )( 1
1
1
= (
1 )
A
I
A
X
X
I
X
)=
1
X
=
XA
4. In a similar way to 3,
X (1X
Æ
IA
)(1
X
A
1
_
A )
= (1
X
X
A
)=Æ
XA
by the de nition of and the naturality of in Z .
Z
Therefore, and Æ are morphisms that are natural in X , and satisfy the axioms A2.
Therefore, every symmetri
monoidal
ategory satisfying A1 also satis es A2.
(() Let (M;
; ; Æ ) be a symmetri
monoidal
ategory satisfying A2. We de ne morphisms
: (A_
A) ! I ; : I ! (A
A_ )
AX
AX
A
by =
_
the axioms A1:
A
A
A
IA
and
A
A
=Æ
IA
1
A_
A .
We now
he
k the that these morphisms satisfy
Theory and Appli
ations of Categories, Vol. 6, No. 3
1.
A (1A
A
)t
1
1
_ A (A
AA
A
1
)
=
A
45
t 1 _ A ÆAA
A;A AA
= 1 , by axiom 1 of A2.
A
2. By de nition of and ,
1 _ )t _
_
1 _ j )t _
A_ (A_
1
A_ (ÆI A_ A
A
However,
A_
=
A
_ sI A_
1
A_ sI A (ÆI A_ A
A_
XY
A_ (1A_
Æ
1
A
_
=s
1
AA
A
A
and
and by the naturality of s
A
A
IA
_ (1A_
AA
) _1 =
_
_
_ (1A
_ A_1 .
in X and
Y,
A
A
A
IA
A
A
A
_:
_ I A )sI A_ A_1 ;
the above is equal to
1
_ A
)sA_
A;A_ tA_ AA_ sA
A_ ;A_ (A
A_ I A
A_
IA
Then by axioms 3 and 4 of A2, this is
1 _ , by axiom 2 of A2.
1
)
Therefore, this is equal to
_ j )tA_ AA_ (1A_
A
A
1
_ )A_1 :
A
ÆA_ A sA_
A;A_ tA_ AA_ sA
A_ ;A_ A_ A ,
whi
h is
A
Therefore, the axioms A1 are satis ed, and this
ompletes our proof.
Referen
es
[1℄ S. Abramsky, Retra
ing some paths in Pro
ess algebra, CONCUR
96.
[2℄ S. Abramsky, R. Jagadeesan, New Foundations for the Geometry of Intera
tion, Pro
.
Seventh IEEE Symposium on Logi
in Computer S
ien
e, 211-222 (1992)
[3℄ V. Danos, L. Regnier, Lo
al and Asyn
hronous Beta Redu
tion, (an analysis of Girard's exe
ution formula), Pro
. Eighth Ann. IEEE Symposium on Logi
in Computer
[4℄
[5℄
[6℄
S
ien
e, IEEE Comp. So
. Press 296-306 (1993)
J-Y. Girard, Geometry of intera
tion 1, Pro
eedings Logi
Colloquium '88, (1989)
221-260, North Holland.
J-Y. Girard, Geometry of intera
tion 2, Pro
eedings of COLOG 88, Martin-Lof &
Mints, 76-93 SLNCS 417
P.M. Hines, The Algebra of Self-similarity and its Appli
ations, PhD thesis, University of Wales (1997)
[7℄ P. Hines, M.V. Lawson, An appli
ation of poly
y
li
monoids to rings,
Forum 56 (1998), 146{149.
[8℄ A. Joyal, R. Street, D. Verity, Tra
ed Monoidal
ategories, Math.
So
., (1996) 425-446
Semigroup
Pro
. Camb. Phil.
Theory and Appli
ations of Categories, Vol. 6, No. 3
46
[9℄ G. Kelly, M. Laplaza, Coheren
e for
ompa
t
losed
ategories, Journal
of Pure and
Applied Algebra 19 (1980) 193-213
[10℄ J. Lambek, P.J. S
ott, Introdu
tion to Higher Order Categori
al Logi
, Cambridge
Studies in Advan
ed Mathemati
s, Cambridge University Press (1986)
[11℄ M.V. Lawson, Inverse Semigroups:
Singapore (1998)
the theory of partial symmetries,
World S
ienti
,
[12℄ S. Ma
Lane, Categories for the Working Mathemati
ian, Springer-Verlag New York,
(1971)
[13℄ M. Nivat, J.-F. Perrot, Une Generalisation du Monoide bi
y
lique,
C. R. A
ad. S
.
Paris, t.271 (1970)
[14℄ M. Petri
h, Inverse Semigroups, Pure and Applied Mathemati
s, Wiley-Inters
ien
e
Series, John Wiley and Sons (1984)
S
hool of Informati
s,
University of Wales, Bangor,
Gwynedd, U.K. LL57 1UT
Email: max003bangor.a
.uk
This arti
le may be a
essed via WWW at http://www.ta
.mta.
a/ta
/ or by anonymous ftp at ftp://ftp.ta
.mta.
a/pub/ta
/html/volumes/6/n3/n3.fdvi,psg
THEORY AND APPLICATIONS OF CATEGORIES (ISSN 1201-561X) will disseminate arti
les that
signi
antly advan
e the study of
ategori
al algebra or methods, or that make signi
ant new
ontributions to mathemati
al s
ien
e using
ategori
al methods. The s
ope of the journal in
ludes: all areas of
pure
ategory theory, in
luding higher dimensional
ategories; appli
ations of
ategory theory to algebra,
geometry and topology and other areas of mathemati
s; appli
ations of
ategory theory to
omputer
s
ien
e, physi
s and other mathemati
al s
ien
es;
ontributions to s
ienti
knowledge that make use of
ategori
al methods.
Arti
les appearing in the journal have been
arefully and
riti
ally refereed under the responsibility
of members of the Editorial Board. Only papers judged to be both signi
ant and ex
ellent are a
epted
for publi
ation.
The method of distribution of the journal is via the Internet tools WWW/ftp. The journal is ar
hived
ele
troni
ally and in printed paper format.
Individual subs
ribers re
eive (by e-mail) abstra
ts of arti
les as
they are published. Full text of published arti
les is available in .dvi and Posts
ript format. Details will
be e-mailed to new subs
ribers and are available by WWW/ftp. To subs
ribe, send e-mail to ta
mta.
a
in
luding a full name and postal address. For institutional subs
ription, send enquiries to the Managing
Editor, Robert Rosebrugh, rrosebrughmta.
a.
Information for authors. The typesetting language of the journal is TEX, and LATEX is the
preferred
avour. TEX sour
e of arti
les for publi
ation should be submitted by e-mail dire
tly to an
appropriate Editor. They are listed below. Please obtain detailed information on submission format and
style les from the journal's WWW server at URL http://www.ta
.mta.
a/ta
/ or by anonymous ftp
from ftp.ta
.mta.
a in the dire
tory pub/ta
/info. You may also write to ta
mta.
a to re
eive
details by e-mail.
Subs
ription information.
Editorial board.
John Baez, University of California, Riverside: baezmath.u
r.edu
Mi
hael Barr, M
Gill University: barrbarrs.org
Lawren
e Breen, Universite de Paris 13: breenmath.univ-paris13.fr
Ronald Brown, University of North Wales: r.brownbangor.a
.uk
Jean-Lu
Brylinski, Pennsylvania State University: jlbmath.psu.edu
Aurelio Carboni, Universita dell Insubria:
arbonifis.uni
o.it
P. T. Johnstone, University of Cambridge: ptjpmms.
am.a
.uk
G. Max Kelly, University of Sydney: maxkmaths.usyd.edu.au
Anders Ko
k, University of Aarhus: ko
kimf.au.dk
F. William Lawvere, State University of New York at Bu alo: wlawverea
su.buffalo.edu
Jean-Louis Loday, Universite de Strasbourg: lodaymath.u-strasbg.fr
Ieke Moerdijk, University of Utre
ht: moerdijkmath.ruu.nl
Susan Nie eld, Union College: niefielsunion.edu
Robert Pare, Dalhousie University: parems
s.dal.
a
Andrew Pitts, University of Cambridge: ap
l.
am.a
.uk
Robert Rosebrugh, Mount Allison University: rrosebrughmta.
a
Jiri Rosi
ky, Masaryk University: rosi
kymath.muni.
z
James Stashe , University of North Carolina: jds
harlie.math.un
.edu
Ross Street, Ma
quarie University: streetmath.mq.edu.au
Walter Tholen, York University: tholenmathstat.yorku.
a
Myles Tierney, Rutgers University: tierneymath.rutgers.edu
Robert F. C. Walters, University of Sydney: walters bmaths.usyd.edu.au
R. J. Wood, Dalhousie University: rjwoodms
s.dal.
a