On closed categories of functors - University of Rochester Mathematics
ON CLOSED CATEGORIES
OF FUNCTORS
Brian Day Received November 7, 19~9 The purpose of the present paper is to develop in further detail the remarks, extensions
concerning
the relationship
to closed structures
"Enriched
functor categories"
on functor categories,
| 1] §9.
is familiar with the basic results including the representation
of Kan functor
It is assumed that the reader
of closed category theory,
theorem.
mentioned below, the terminology
made in
Apart from some minor changes
and notation employed are those of
|i], |3], and |5]. Terminology A closed category will be called a normalised normalisation. normalised
Throughout
V in the sense of Eilenberg and Kelly |B| closed category, V: V o ÷ En6 being the this paper V is taken to be a fixed
symmetric monoidal
closed category
(Vo, @, I, r, £, a, c,
V, |-,-|, p) with V ° admitting all small limits colimits
(direct limits).
V
closed category
(with possibly
choice has been made of it.
those hypotheses
large domain)
"natural"
In short, we place on
sets Ens as a ground category and are closed categories.
As in [i], an end in B of a V-functor T: A°P@A ÷ B is a
Y-natural
property
and
exists
which both allow it to replace the cartesian
of (small)
satisfied by most
limits)
It is further supposed that if the limit
or colimit in ~o of a functor then a definite
(inverse
family mA: K ÷ T(AA) of morphisms
that the family B(1,mA):
in B o with the
B(BK) ÷ B(B,T(AA))
in V o is
-2-
universally
V-natural in A for each B 6 B; then an end in V turns out
to be simply a family sA: K ~ T(AA) universally
V-natural in A.
of morphisms
in V o which is
The dual concept is called a coend.
From [i] we see that the choice of limits and collmlts made in V o determines
a definite
end and coend of each V-functor
T: A°P®A ÷ V for which such exist. SA: fAT(AA)
~ T(AA)
now construct,
These are denoted by
and sA: T(AA) ÷ fAT(AA)
respectively.
for each pair A, B of V-categories
definite
V-category
[A,B] having
objects,
and having [A,B](S,T)
V-functors
VSA: V;AB(SA,TA)
than the family
An element
under the projections
÷ Bo(SA,TA) , to a V-natural
mA: SA ÷ TA in the sense of [3].
family of morphlsms
It is convenient
{a A} of its components,
of V-functors
to call a, rather
a V-natural transformation
from S to T; for then the underlying ordinary the category
with A small, a
S, T, ...: A + B as its
= ~AB(SA,TA).
E V;AB(SA,TA ) clearly corresponds,
We can
category
[A,B] o i_~s
and V-natural transformations.
Limits and colimits
in the functor category
always be computed evaluationwlse,
[A,B] wlll
so that the choice of limits and
colimlts made in V fixes a choice in [A,V]
for each small
V-category
A.
of cotensoring
and
Included in this rule are the concepts
tensoring,which
were seen in |5] to behave
like limits and collmits
respectively. In order to replace the category normalised
closed category
terminology.
A V-monoidal
of sets by the given
V, we shall "lift" most of the usual category ~ is a V-category
A together with
a M-functor @: A@A + A, an object Y • A, and V-natural isomorphisms a: (A@B)@C a A@(B@C),
~: A @ I a A,snd r: Y@A a A, satisfying
usual coherence
for a monoidal
axioms
and MC3 of [3]. (chosen) right
If, furthermore, M-adJoints
the
category - namely axioms MC2
-@A and A@-: A ÷ A both have
for each A • A, then ~ is called a
M-biclosed category
(see Lambek |8]).
V-monoidal
(A, @, I~ r, A, a) is a V-natural isomorphism
category
c: A@B ~ B@A satisfying the coherence
A M-symmetry
for a
axioms MC6 and MC7 of [3].
Finally we come to the concept of a M-symmetric-monoidal-closed category which can be described
simply as a M-biclosed category
with a M-symmetry;
we do not insist on a "M-normalisation"
of this structure.
An obvious example of such a category
itself, where ~ is taken to be the M-functor Ten:
as part is M
M@M ÷ M defined in
[ 3] Theorem 111.6.9. We note here that, the symbol
S: A ÷ C and T: B ÷ P,
S@T may have two distinct meanings.
the canonical pair)
for M-functors
M-functor
A®B
~
C@D which sends the object
(A,B) • A@B to the object
Proposition
III.3.2.
In general
it is
(ordered
(SA,TB) • C@D, as defined in [3]
When C and P are both M, however,
we shall
also use S@T to denote the composite S@T A@B
Ten ~
V@M
~
The context always clearly indicates Henceforth we work enti~el~ unqualified words
"cate~ory"~
etc. mean "M-category",
V.
the meaning. over M and suppose that the
"functor"~
"M-functor",
"natural transformation"~
"V-natural
transformatlon"~
etc.
-4 -
i.
Introduction
Let A be a small category subcategory
of |A,V],
identifying
and regard
A 6 A °p with the left r e p r e s e n t e d
functor LA: A ÷ V in the usual way. canonical
expansion
(adequacy) biclosed
A °p as a full
For each S 6 |A,V]
we have the
(see |l]) S ~ ~AsA@LA w h i c h asserts
the density
of A °p in |A,V].
category V then,
S@T of @: [A,V]®|A,V]
If [A,V|
in view of this expansion,
÷ |A,V]
the values LA@T, b e c a u s e
has the structure
at (S,T) is e s s e n t i a l l y
-@T has a right adJoint.
of a
the value d e t e r m i n e d by
These in turn are
d e t e r m i n e d by the values LA@L B, b e c a u s e each L A @ - has a right adJoint.
Writing
is e s s e n t i a l l y
P(ABC)
for (LA@LB)(c),
d e t e r m i n e d by the functor P: A ° P @ A ° P @ A ~ V, in the
same way that the m u l t i p l i c a t i o n by structure
we see that the functor
in a linear algebra
is d e t e r m i n e d
constants.
These c o n s i d e r a t i o n s a premonoldal
structure
on A.
suggest what is called in section This
consists
of functors
P: A ° P @ A ° P @ A
÷ V and J: A ÷ V, t o g e t h e r with certain n a t u r a l
isomorphisms
corresponding
rlght-ldentlty structure
morphlsms,
is a special
down, we collect
to a s s o c l a t i v i t y , w h i c h satisfy
case.
Before
left-ldentity,
suitable
attempting
in section 2 the properties
3
axioms;
and
a monoldal
to write the axioms
of ends and coends
that
we shall need. The main aim of this paper is to show that, premonoidal
structure
canonical b i c l o s e d
on a small category
structure
from a
A, there results
on the functor category
tA,V];
a this is
-5-
done in section 3. [A,V]
As one would expect, biclosed structures on
correspond biJectlvely to premonoldal structures on A to
within "isomorphism".
However we do not formally prove this
assertion, which would require the somewhat lengthy introduction of premonoldal functors to make it clear what "isomorphism" was intended. The concluding sections contain descriptions of some commonly occurring types of premonoidal structure on a (possibly large) category A.
The case in which the premonoidal structure is
actually monoidal is discussed in section 4.
In section 5 we
provide the data for a premonoidal structure which arises when the hom-obJects of A are comonoids
(@-coalgebras)
in V in a natural way.
In both cases the tensor-product and internal-hom formulas given in section 3 for the biclosed structure on [A,V| may be simplified to allow comparison with the corresponding formulas for some well-known examples of closed functor categories.
2.
Induced Natural Transformations
Natural transformations,
in both the ordinary and
extraordinary senses, are treated in [2]
and [3].
Our applications
of the rules governing their composition with each other
(and with
functors) are quite straightforward and will not be analysed in detail. The following dualisable lemmas on induced naturality are expressed in terms of coends. Lemma 2.1.
Let T: A°P~A@B + C be a functor and let
aAB: T(AAB) ÷ SB be a coend over A for each B E B.
Then there
-6-
exists a unique functor S: B ~ C makin~ the family ~AB natural in B. Proof.
For each pair B, B' E B consider the diagram SBB,
B(BB')
T(AA-)BB'
>
I
C(SB SB')
C(~,I)
C(T(AAB),T(AAB'))
~
C(T(AAB),SB')
,
C(l,~) Because C(s,l) is an end and C(I,a).T(AA-)BB , is natural in A we can define SBB , to be the unique morphism making this diagram commute.
The functor axioms VFI' and VF2' of [ 3| are easily
verified for this definition of S using the fact that C(~,I) is an end.
S is then the unique functor making ~AB natural in B. Lemma 2.2.
Let T: A°P@A@B * C and S,R: B ÷ C be functors,
let aAB: T(AAB) + SB be a coend over A, natural in B, and let
8AB:
T(AAB) + RB be natural in A and B.
Then the induced family
YB: SB ÷ RB is natural in B. Proof.
For each pair B, B' E B consider the
diagram
-7-
C(l,~) C(T(AAB),T(AAB'))
~
C(T(AAB),SB')
1
T(
~,i)
"~
C(l,y) \
/ B(BB')
\
,.
SBB, -.
~,
.~- C(SB,SB')
2
C(RB,RB')
~
C(T(AAB),RB')
C(SB,RB')
c(y,x) The commutatlvity the naturallty commutes
i and that of the exterior
in B of ~ and 8 respectively.
hence,
each pair B,B'
of region
because
C(e,1)
Region
is an end, region
express
2 clearly
3 commutes
for
6 B, as required.
By similar arguments Lemma 2.3.
we obtain
Let T: A°P@A@B°P@B
÷ C and S: B°P®B ÷ C ,,=
be functors t let eABB':
T(AABB')
÷ S(BB')
be a coend over A,
natural
in B and B', and let 8AB: T(AABB)
÷ C be natural
in A
and B.
T_hen the induced
÷ C is natural
in B.
Lemma 2.4, functors~ 8AB: T(AA)
YB: C ~ R(BB)
YB: S(BB)
Let T: A°P@A ÷ C and R: B°P@B ÷ C be
let CA: T(AA) * R(BB)
family
÷ C be a coend over A, and let
be natural
is natural
in B,
in A and B.
Then the induced
family
-8-
Let A b e whose
a category
coend sA: T(AA-)
and let T(AA-) be a functor into
÷ fAT(AA-)
over A E A exists
of the e x t r a variables
"-".
canonically
in these e x t r a variables.
functorlal
case where T(AA-) (with different this n o t a t i o n
Then, by Lemma 2.1,
~ S(A-)@T(A-)
variances
in A) we will
to sA: S(A-)@R(A-)
repeated
dummy variable
indicate
the domain of ~'. expressions
the dummy variables example,
V
abbreviate
S(A-)2R(A-)
to
formed entirely by the
to introduce
which we do not formalise
an e x p r e s s i o n
In the special
S and R into
the following
completely.
e x p r e s s i o n N w h i c h is formed by one or more uses corresponds
is
÷ S(A-)2R(A-) , leaving the
use of 2, it is convenient
considerations
for all values
fAT(AA-)
frequently
A in the e x p r e s s i o n
In order to handle repeated
for functors
V
To each
of 2, there
N in which each 2 is replaced by @,
in N b e c o m i n g r e p e a t e d
if N is (RA2S(AB))2T(BC)
variables
for functors
in N; for
R: A ÷ V,
S: A°P@B ~ V, and T: B°P@c ÷ V, then N is (RA@S(AB))@T(BC). Moreover,
there
is a canonical
q = qN: N ÷ N defined, occurrences
as follows,
of @ in N. - -
natural t r a n s f o r m a t i o n by i n d u c t i o n
If N contains
- -
on the n u m b e r of
no occurrence
of @ then
m
N ~ N and qN = i; otherwise N = N'@N" and qN is the composite N'@N"
~ N'@N" q'@q"
In the above example, (RA@S(AB))@T(BC)
~ N'@N". s
q is the composite ~ (RA2S(AB))@T(BC)
s@l and this is n a t u r a l
~ (RA2S(AB))@T(BC) s
in A, B,snd C; we say that the variables
and B are "summed out" by q.
A
The path qN: N ÷ N is in fact all those
variables
in N w h i c h
Le_~..
are
summed
~
qN:
in all the r e p e a t e d N ÷ N.
transformation
Then
result three
This
of @ in N.
is trivial;
V and let
is~
in p a r t i c u l a r ,
in N which
are
for a unique
is by i n d u c t i o n If N contains
othe~ise
summed
out
natural
N = N'@N"
on the n ~ b e r
no o c c u r r e n c e
of of @ the
and we can factor
f in
steps: q'@l
N
as g ' q N
over
g: N ÷ M.
Proo=~f. occurrences
f factors
into
which
variables
coend
out by qN:
Let M be a functor
f: N ÷ M be a n a t u r a l _ t r a n s f o r m a t i o n natural
a multiple
= N'@N"
l@q" )
N'@N"
s >
N'@N"
~ N'@N"
= N
M
First
consider
the t r a n s f o ~
tensor-hom
adJunction
hypothesis
and routine
w(f):
isomorphism naturality
N' ÷ [N",M] ~ = Vp of
of f under
V.
considerations,
By the i n d u c t i o n the d i a g r ~
q' N'
~ N'
[ N",M]
commutes
for a unique
morphlsm
~(f'): N' ÷ |N",M]
the
where
-10-
f': N'®N" by q'.
÷ M is n a t u r a l
Similarly
f' factors
f": N'@N"
~ M which
by e i t h e r
of q' or q".
as g.s
for a unique
remaining
in all the v a r i a b l e s
variables
as f".(l@q")
is n a t u r a l
not
summed
for a unique
in all the variables
Finally,
because
g: N ÷ M w h i c h
in N and M by Lemmas
morphism
not
s is a coend,
is n a t u r a l
out
summed
out
f" factors
in all the
2.2,
2.3,
and 2.4.
m
Combining
these
steps•
in the r e q u i r e d
we have
that
for a path q': N' ~ N'
is d e n o t e d part
by n.
of the
three
the
induced
f in Lemma induced
qN = s(q'@q")
cases
transformation
n: N ÷ N' is a n a t u r a l
form
g: N ÷ N'
are a n e c e s s a r y
category
and we c o n s i d e r
below.
if n: N ÷ N' is a n a t u r a l
from the coherent
2.5 is of the
transformations
of a p r e m o n o i d a l
special
First, entirely
Such
concept
relevant
through
manner.
W h e n the t r a n s f o r m a t i o n q'.n
f factors
isomorphism
data isomorphisms
isomorphism
a, r,
and is called
constructed
£, c of V then
an induced
m
coherence Lemma that the
isomorphism.
2.5,
and the o r i g i n a l
induced induced
determined shall
not
coherence coherence
coherence
isomorphisms isomorphism
by the a r r a n g e m e n t label
such
Secondly, natural
In view of the u n i q u e n e s s of a, r,
assertion
of
£, c, it is clear
are coherent.
In other words,
n: N + N' is completely
of @ in N and N';
consequently
we
isomorphisms. w h e n n z h@k:
transformations
S(A-)@R(A-)
÷ S'(A-)@R'(A-)
h: S ~ S' and k: R ÷ R',
let us w r i t e
for
-11
h@._k for h@k.
-
This not only makes the symbol @ En~-functorial
whenever it is defined on objects, but also makes the coend sA: S(A-)@R(A-) ÷ S(A-)@_R(A-) End-natural in S and R.
Under
reasonable conditions the same observations can be made at the V-level. If we restrict our attention to functors into V with small domains then the functors themselves may be regarded as extra variables. A and B small.
For example,
let T: A°P@A@B + V be a functor with
Then fAT(AAB) is canonically functorial in T and B
for we can write T(AAB) = F(AATB) where F is the composite
A°P@A@([A°P@A@B,V]@B) ~
[A°P@A@B,V]@(A°P@A@B) > V, E
and where E is the evaluation functor defined in [i]
§4.
if S(A-) and R(A-) are functors into V with small domains
Similarly, (and
different variances in A) then S(A-)@__R(A-) is functorial in S and R in a unique way that makes sA: S(A-)@R(A-) ÷ S(A-)@R(A-) natural in S and R. Lastly, let S(A-) be a functor into V which is covariant in A E A.
As part of the data for S, we have a family of morphisms
SAB: A(AB) ÷ [S(A-),S(B-)]
which is natural in A and B and also in
the extra variables in S.
Transforming this family by the
tensor-hom adJunction of V, we get a transformation w-I(SAB):
A(AB)@S(A-) + S(B-) which is natural in A and B and the
extra variables in S.
As a result of the generalised "higher"
representation theorem (see [I], §3 and §5), this induces the Yoneda isomorphism
-12-
YS,B: A(AB)@S(A-)
÷ S(B-).
By Lemma 2.2, we then have Lemma 2.6.
The Yoneda isomorphism YS,A is natural in
A and in the extra variables
in S; if the domain of S is small
then it is natural in S. The following diagram lemmas for the Yoneda isomorphism y are all proved using [3] Proposition as the representatlon
theorem.
in their most convenient Lemma 2.~.
II.7.4 which we shall refer to
These lemmas are presented here
forms for application
in sections
3 and 4.
Given functors S: A ~ V and T: A °p * V
for which SA@_TA exlsts m the following diagram commutes:
(A(AB)O_SA)O3B
> SBe_TB
ZEk
SA@(A(AB)e_TB)
> SAe_TA • icy
Proof.
Replacing @ by @,
we obtain a new diagram:
y by its definition,
etc.,
-13 -
(A(AB) @SA)~TB
\
s@l
\
\ SB@TB
(A(AB) Q_SA)@TB Y@I
\
y~_l ( A(AB)@SA) @_TB
"~ SB@TB
SA@( A( AB )@_TB )
SA@TA
~II
l@y SA@ ( A( AB )@__TB)
SA@(A(AB)@TB)
.
By Lemma
s(s@l)
2.5,
is a coend over A and B hence
to prove that the exterior A, B E A.
SA@TA
This is easily
of this new diagram
it suffices
commutes
seen to be so on applying
the
for all
-14-
representation
theorem;
put B m A and compose both exterior
legs with
(I@SA)@TA
~ (A(AA)@SA)@TA; (JA®I)@I
the resulting diagram commutes, Lemma 2.8.
hence the original one does.
Given functors
S: A°P@B ÷ V and T: B °p ÷ Y
for which S(AC)0_TC exists for each A E A, the followin~ diagram commutes
for each A E A:
A(AB)@(S(BC)®_TC)
S(AC)e_TC
(A(AB)®_S(BC))@_TC , Proof.
Again replacing @ by @,
etc., we obtain a new diagram:
y by its definition,
-15 -
A(AB)~(BC)emc)
A(AB)e(s (BC)_eTc)
~,
~"
S(AC)_OTC ~
A AB )_( aS BC ))_ ~ ((
//
I
S(AC)eTC
~ j~
(A(AB)_eS(BC)~ / / / /
(*(AB)eS(BC))~C In this
diagram the region
transform
labelled
1 commutes b e c a u s e
of t h e d i a g r a m
S(-C)eTC A(AB)
>
[ S(BC)®TC,S(AC)~'C]
!
S(-C)eTC
I [1,s]
[S(BC)e_TC,S(AC)eTC! ~ [ s,1]
[S(BC)®TO,S(AO)®TC]
it
is
the
-16 -
which expresses in A.
the naturality
Hence, because s(l@s)
of s = sC: S(AC)@TC ÷ S(AC)@TC
is a coend over B and C by Lemma
2.5, it suffices to prove that the exterior of the new diagram commutes
for all A, B E A and C E 8.
consequence
of the representation
The remaining
Again this is a simple
theorem.
lemmas are obtained by the same type of
argument. ~
.
for which TCSS(AC) commutes
Given functors S: A°P@B ~ V and T: B °p ~ V exists for each A E A, the following diagram
for each A E A: A(AB)@(TCe_S(BC))
III
~~'TCQS(AC) /
j J
ley
TCO_(A (AB)®_S (BC))
Lemma 2.10.
For any functors S; A ~ 8 and T: B °p ÷ V
the followln~ diagram commutes
for each A E A:
l~y A(AB)@(8(SB,C)®_TC)
> A(AB)%TSB
8(SA,C)~3C
~ TSA Y
-17 -
Lemma 2.11 diasram
commutes
For an~ functor T: A@B + V the following
for all B, D E A:
ley A(AB)@_(B(CD)@_T(AC) )
~
A(AB)@_T(AD)
S(CD )@_T(BC)-
~
T(BD)
.
Y 3.
Premonoidal
We emphaslse all concepts monoidal
over
again that,
are relative
closed category Definition
V consists
a category a functor
Categories unless
otherwise
to the given normallsed
3.1
A premonoidal
category
of
A, P: A°P@A°P@A
÷ V,
isomorphisms
I = AAB: JX@P(XAB)
÷ A(AB),
P = PAB: JX@P(AXB)
~ A(AB),
a = aABCD: satisfying
P(ABX)@P(XCD)
the following
symmetric
V.
a functor J: A * V, and natural
indicated,
÷ P(BCX)@_P(AXD),
two axioms:
P = (A,P,J,A,p,~)
-18 PC1.
For all A,B,C E A, the following diagram commutes: l@a > JXQ_(P(XBY)Q_P(AYC))
JX@_(P(AXY)@_P(YBC) ) -
(JX~P(AXY))~P(YBC)
(JX~_P (XBY)) ~P (AYC)
p@l
A(AY)~_P(YBC)
A(BY)~P(AYC)
P(ABC) PC2.
.
For all A,B,C,D,E E A, the following diagram commutes: P(ABX )0_(P (XCY) ~_P(YDE) )
P(ABX)O(P(CDY)~P(XYE))
(P(ABX)@P(XCY))@P(YDE)
P(CDY)@(P(ABX)@P(XYE))
( P(BCX)~P(AX¥))~_P(YDE)
P(CDY)~(P(BYX)@P(~E))
~II
P (BCX)@( P(AXY) ~_P(YDE) )
P (BCX)@(P(XDY) ~_P(AYE))
(P(CDX)~P(BXY))~P(AYE)
(P(BCX)@P(XDY))@P(AYE)
-19 -
Remark requisite
3.2
It is assumed
~'s exist for the g i v e n
hypothesis
on V, when
P(AB-): A ÷
V and J:
A is small.
in the d e f i n i t i o n
A, P, and J.
|A,V].
to a b l c l o s e d
for all A,B 6 A.
of this s e c t i o n we will suppose
A is small and show that each p r e m o n o i d a l "extends"
They do so, by
They also exist w h e n e v e r
A ÷ V are r e p r e s e n t a b l e
In the r e m a i n d e r
that the
structure
[P,V]
structure
that
P on A
on the functor category
For the m o n o i d a l part define a t e n s o r - p r o d u c t
*: [ A , V ] ~ [ A , V ] (3.1)
÷ [A,V]
by
S*T = fAsA@;BTB@P(AB-)
for all S,T E [A,V];
= SA@(TB@P(AB-))
this e x p r e s s i o n
is c a n o n i c a l l y
in S and T by the c o n s i d e r a t i o n s
of s e c t i o n 2.
J 6 [A,V]
of.,
be the i d e n t l t y - o b J e c t
Isomorphisms respective
£* = ~ :
functorlal
Next,
let
and define n a t u r a l
J*T ÷ T and r* = r~: T.J ÷ J as the
composites
J,T = J X @ ( T A @ P ( X A - ) )
a (JX@P(XA-))@_TA
A(A-)@TA k@l
~ T y
and T.J = T A @ ( J X @ P ( A X - ) )
m (JX@P(AX-))@_TA > T.
A(A-)@_TA p®l Lastly,
define a natural
as the composite
y isomorphism
a* = a~ST:
(R,S),T ÷ R*(S,T)
-
20
-
(R*S)*T = (RA@(SB@P(ABX)))@(TC~P(XC-)) RA@(SB@(TC@(P(ABX)@P(XC-)))) RA@(SB~(TC@(P(BCX)@P(AX-)))) i®(1®(1®~)) RA~((SB@(TC@P(BCX)))@P(AX-)) = R,(S,T). Then £*,
r*, and a* are natural by Lemmas Theorem
monoidal
category admittln5 Proof
category,
[P,V]
3.3
First,
= ([A,VJ,*,J,£*,r*,a*)
to show that [P,V]
is a monoidal
we need to prove PC1 ~ MC2 and PC2 ~ MC3.
commutes by PCI;
definitions
of *,r*,a*,
by the naturality isomorphlsms
diagram
(3.2) in which the
and £*;
4, 5, 6, 7, 8, and 9 commute of the induced coherence
(Lemma 2.5 and the succeeding remarks);
commute by Lemma 2.7; and 12 commutes by Lemma 2.9.
isomorphlsms
The proof
and coherence
of * and a*, uses
of the induced coherence
involved.
To complete the structure on [A,V] biclosed
I0 and Ii
a diagram that is too large for the space
available but, apart from the definitions only the naturality
The first
l, 2, and 3 commute by the
and coherence
of PC2 ~ MC3 requires
is a
a biclosed structure.
of these is obtained by considering exterior
2.5 and 2.6.
category,
consider the composite
to that of a
isomorphism:
®l
r~@l=r~l ~ -
il~
i®(i®y)
10
SX@(TC@P(XCD )
~/(A(AX)~SA)@(TCgP(XCD))4
1
(SA@(JB@P(ABX)))@(TC@P(XCD})
( (JB@_P(ABX))@__SA@ ) _(TC@__P(XCD) )
SAg(TCg(A(AX)~P(XCD)))
®l
v
®i
v
6
SA@(JB@(TC~(P(ABX)@P(XCD))))
SAe(TC@((JB@P(ABX))@P(XCD)))
SA@(TC@(JB@(P(ABX)®P(XCD)) )
/
t -
/
3
l
9
v v
®
F-J
I® I®
SAe(TCe(A(CX)~P(AXD)))
SA@_((A (CX)@_TC)@_P(AXD))
SA@(TX@P(AXD))
i~( ~ *~l )
i
SA@_(((JB@_P(BCX))®_TC)@_P(AXD))
~r SA@((JB~(TC~P(BCX)))@P(AXD))
~Jl
SA@_(JB@_(TC@_(P(BCX)@_P(AXD)))
SA@_(TC@_((JB@P (BCX))@_P(AXD)))
SA@(TC@(JB@(P(BCX)~P(AXD))))
~> SAe(TCeP(ACD))-
P(BCX)@P(AXD)
P(BAX)@P(XCD)
P(ACX)@P(BXD)
P(BCX)@P(XAD)
~
P(CAX)@P(BXD)
.
o@i This definition does not, of course,
require
A to be small. It remains to be shown that [ P,V] admits a symmetric monoidal
closed structure whenever
P has a symmetry.
For this,
define a natural isomorphism c * = C~T: SwT * T~S as the composite
-
S*T
=
24
-
SA@(TB@P(AB-)) ~ TB@(SA®P(AB-)) TB@(SA®P(BA-)) = T*S.
i®(l~c) Again, the naturallty of e* is a consequence of Lemma 2.5. If e is a symmetry for P then c* is a s~etr~
for [P,V]. P~oof
To prove PC3 ~ MC6 consider diagram (3.5): TB~(SA@P(BAC))
TB@(SA@P(ABC))
SA@(TB@_P(BAC))
~
2
(3.5) II~
TB@(SA@P(ABC))
SA~(TB~P(ABC))
u
I@(I~o)
SA~(TB~P(ABC))
Region 1 commutes by PC3, and region 2 commutes by the naturallty of the Induoed coherence isomorphism involved; hence the exterior commutes and so, by definition of c*, Me6 is satisfied.
To prove
PC4 ~ MC7 consider diagram (3.6), in which the exterior commutes by PC4; i, 2, and 3 commute by the definitions of , and c*; 4, 5, and 6 commute by the definition of a*; and 7, 8, 9, and i0 commute by the naturality and coherence of induced coherence Isomorphlsms.
.i-
/
a •
SBO_((TC~_(~A03 (ACX)) ) ~_P(BXD))
Imc m
aM
le(1®(1~))
c**l
SB@_((RAQ_(TC~P(ACX)) )~_P(BXD))
RA@(SB@(TC~(P(ACX)~P(BXD))))
®I
v
I
(SB@(RA@P(BAX)))@(TC@P(XCD))
(l@(l@~))@_l I
(RA~(SB@P(ABX)))~(TC@P(XCD))
RA@(SB@(TC@(P(BAX)@__P(XCD))))
®I
,-4
®I
®I
v
®I ~D
(SB~(RA~P(ABX)))@(TC~P(XCD))
RA@(SB~(TC~(P(ABX)~P(XCD))))
I
!~
OF FUNCTORS
Brian Day Received November 7, 19~9 The purpose of the present paper is to develop in further detail the remarks, extensions
concerning
the relationship
to closed structures
"Enriched
functor categories"
on functor categories,
| 1] §9.
is familiar with the basic results including the representation
of Kan functor
It is assumed that the reader
of closed category theory,
theorem.
mentioned below, the terminology
made in
Apart from some minor changes
and notation employed are those of
|i], |3], and |5]. Terminology A closed category will be called a normalised normalisation. normalised
Throughout
V in the sense of Eilenberg and Kelly |B| closed category, V: V o ÷ En6 being the this paper V is taken to be a fixed
symmetric monoidal
closed category
(Vo, @, I, r, £, a, c,
V, |-,-|, p) with V ° admitting all small limits colimits
(direct limits).
V
closed category
(with possibly
choice has been made of it.
those hypotheses
large domain)
"natural"
In short, we place on
sets Ens as a ground category and are closed categories.
As in [i], an end in B of a V-functor T: A°P@A ÷ B is a
Y-natural
property
and
exists
which both allow it to replace the cartesian
of (small)
satisfied by most
limits)
It is further supposed that if the limit
or colimit in ~o of a functor then a definite
(inverse
family mA: K ÷ T(AA) of morphisms
that the family B(1,mA):
in B o with the
B(BK) ÷ B(B,T(AA))
in V o is
-2-
universally
V-natural in A for each B 6 B; then an end in V turns out
to be simply a family sA: K ~ T(AA) universally
V-natural in A.
of morphisms
in V o which is
The dual concept is called a coend.
From [i] we see that the choice of limits and collmlts made in V o determines
a definite
end and coend of each V-functor
T: A°P®A ÷ V for which such exist. SA: fAT(AA)
~ T(AA)
now construct,
These are denoted by
and sA: T(AA) ÷ fAT(AA)
respectively.
for each pair A, B of V-categories
definite
V-category
[A,B] having
objects,
and having [A,B](S,T)
V-functors
VSA: V;AB(SA,TA)
than the family
An element
under the projections
÷ Bo(SA,TA) , to a V-natural
mA: SA ÷ TA in the sense of [3].
family of morphlsms
It is convenient
{a A} of its components,
of V-functors
to call a, rather
a V-natural transformation
from S to T; for then the underlying ordinary the category
with A small, a
S, T, ...: A + B as its
= ~AB(SA,TA).
E V;AB(SA,TA ) clearly corresponds,
We can
category
[A,B] o i_~s
and V-natural transformations.
Limits and colimits
in the functor category
always be computed evaluationwlse,
[A,B] wlll
so that the choice of limits and
colimlts made in V fixes a choice in [A,V]
for each small
V-category
A.
of cotensoring
and
Included in this rule are the concepts
tensoring,which
were seen in |5] to behave
like limits and collmits
respectively. In order to replace the category normalised
closed category
terminology.
A V-monoidal
of sets by the given
V, we shall "lift" most of the usual category ~ is a V-category
A together with
a M-functor @: A@A + A, an object Y • A, and V-natural isomorphisms a: (A@B)@C a A@(B@C),
~: A @ I a A,snd r: Y@A a A, satisfying
usual coherence
for a monoidal
axioms
and MC3 of [3]. (chosen) right
If, furthermore, M-adJoints
the
category - namely axioms MC2
-@A and A@-: A ÷ A both have
for each A • A, then ~ is called a
M-biclosed category
(see Lambek |8]).
V-monoidal
(A, @, I~ r, A, a) is a V-natural isomorphism
category
c: A@B ~ B@A satisfying the coherence
A M-symmetry
for a
axioms MC6 and MC7 of [3].
Finally we come to the concept of a M-symmetric-monoidal-closed category which can be described
simply as a M-biclosed category
with a M-symmetry;
we do not insist on a "M-normalisation"
of this structure.
An obvious example of such a category
itself, where ~ is taken to be the M-functor Ten:
as part is M
M@M ÷ M defined in
[ 3] Theorem 111.6.9. We note here that, the symbol
S: A ÷ C and T: B ÷ P,
S@T may have two distinct meanings.
the canonical pair)
for M-functors
M-functor
A®B
~
C@D which sends the object
(A,B) • A@B to the object
Proposition
III.3.2.
In general
it is
(ordered
(SA,TB) • C@D, as defined in [3]
When C and P are both M, however,
we shall
also use S@T to denote the composite S@T A@B
Ten ~
V@M
~
The context always clearly indicates Henceforth we work enti~el~ unqualified words
"cate~ory"~
etc. mean "M-category",
V.
the meaning. over M and suppose that the
"functor"~
"M-functor",
"natural transformation"~
"V-natural
transformatlon"~
etc.
-4 -
i.
Introduction
Let A be a small category subcategory
of |A,V],
identifying
and regard
A 6 A °p with the left r e p r e s e n t e d
functor LA: A ÷ V in the usual way. canonical
expansion
(adequacy) biclosed
A °p as a full
For each S 6 |A,V]
we have the
(see |l]) S ~ ~AsA@LA w h i c h asserts
the density
of A °p in |A,V].
category V then,
S@T of @: [A,V]®|A,V]
If [A,V|
in view of this expansion,
÷ |A,V]
the values LA@T, b e c a u s e
has the structure
at (S,T) is e s s e n t i a l l y
-@T has a right adJoint.
of a
the value d e t e r m i n e d by
These in turn are
d e t e r m i n e d by the values LA@L B, b e c a u s e each L A @ - has a right adJoint.
Writing
is e s s e n t i a l l y
P(ABC)
for (LA@LB)(c),
d e t e r m i n e d by the functor P: A ° P @ A ° P @ A ~ V, in the
same way that the m u l t i p l i c a t i o n by structure
we see that the functor
in a linear algebra
is d e t e r m i n e d
constants.
These c o n s i d e r a t i o n s a premonoldal
structure
on A.
suggest what is called in section This
consists
of functors
P: A ° P @ A ° P @ A
÷ V and J: A ÷ V, t o g e t h e r with certain n a t u r a l
isomorphisms
corresponding
rlght-ldentlty structure
morphlsms,
is a special
down, we collect
to a s s o c l a t i v i t y , w h i c h satisfy
case.
Before
left-ldentity,
suitable
attempting
in section 2 the properties
3
axioms;
and
a monoldal
to write the axioms
of ends and coends
that
we shall need. The main aim of this paper is to show that, premonoidal
structure
canonical b i c l o s e d
on a small category
structure
from a
A, there results
on the functor category
tA,V];
a this is
-5-
done in section 3. [A,V]
As one would expect, biclosed structures on
correspond biJectlvely to premonoldal structures on A to
within "isomorphism".
However we do not formally prove this
assertion, which would require the somewhat lengthy introduction of premonoldal functors to make it clear what "isomorphism" was intended. The concluding sections contain descriptions of some commonly occurring types of premonoidal structure on a (possibly large) category A.
The case in which the premonoidal structure is
actually monoidal is discussed in section 4.
In section 5 we
provide the data for a premonoidal structure which arises when the hom-obJects of A are comonoids
(@-coalgebras)
in V in a natural way.
In both cases the tensor-product and internal-hom formulas given in section 3 for the biclosed structure on [A,V| may be simplified to allow comparison with the corresponding formulas for some well-known examples of closed functor categories.
2.
Induced Natural Transformations
Natural transformations,
in both the ordinary and
extraordinary senses, are treated in [2]
and [3].
Our applications
of the rules governing their composition with each other
(and with
functors) are quite straightforward and will not be analysed in detail. The following dualisable lemmas on induced naturality are expressed in terms of coends. Lemma 2.1.
Let T: A°P~A@B + C be a functor and let
aAB: T(AAB) ÷ SB be a coend over A for each B E B.
Then there
-6-
exists a unique functor S: B ~ C makin~ the family ~AB natural in B. Proof.
For each pair B, B' E B consider the diagram SBB,
B(BB')
T(AA-)BB'
>
I
C(SB SB')
C(~,I)
C(T(AAB),T(AAB'))
~
C(T(AAB),SB')
,
C(l,~) Because C(s,l) is an end and C(I,a).T(AA-)BB , is natural in A we can define SBB , to be the unique morphism making this diagram commute.
The functor axioms VFI' and VF2' of [ 3| are easily
verified for this definition of S using the fact that C(~,I) is an end.
S is then the unique functor making ~AB natural in B. Lemma 2.2.
Let T: A°P@A@B * C and S,R: B ÷ C be functors,
let aAB: T(AAB) + SB be a coend over A, natural in B, and let
8AB:
T(AAB) + RB be natural in A and B.
Then the induced family
YB: SB ÷ RB is natural in B. Proof.
For each pair B, B' E B consider the
diagram
-7-
C(l,~) C(T(AAB),T(AAB'))
~
C(T(AAB),SB')
1
T(
~,i)
"~
C(l,y) \
/ B(BB')
\
,.
SBB, -.
~,
.~- C(SB,SB')
2
C(RB,RB')
~
C(T(AAB),RB')
C(SB,RB')
c(y,x) The commutatlvity the naturallty commutes
i and that of the exterior
in B of ~ and 8 respectively.
hence,
each pair B,B'
of region
because
C(e,1)
Region
is an end, region
express
2 clearly
3 commutes
for
6 B, as required.
By similar arguments Lemma 2.3.
we obtain
Let T: A°P@A@B°P@B
÷ C and S: B°P®B ÷ C ,,=
be functors t let eABB':
T(AABB')
÷ S(BB')
be a coend over A,
natural
in B and B', and let 8AB: T(AABB)
÷ C be natural
in A
and B.
T_hen the induced
÷ C is natural
in B.
Lemma 2.4, functors~ 8AB: T(AA)
YB: C ~ R(BB)
YB: S(BB)
Let T: A°P@A ÷ C and R: B°P@B ÷ C be
let CA: T(AA) * R(BB)
family
÷ C be a coend over A, and let
be natural
is natural
in B,
in A and B.
Then the induced
family
-8-
Let A b e whose
a category
coend sA: T(AA-)
and let T(AA-) be a functor into
÷ fAT(AA-)
over A E A exists
of the e x t r a variables
"-".
canonically
in these e x t r a variables.
functorlal
case where T(AA-) (with different this n o t a t i o n
Then, by Lemma 2.1,
~ S(A-)@T(A-)
variances
in A) we will
to sA: S(A-)@R(A-)
repeated
dummy variable
indicate
the domain of ~'. expressions
the dummy variables example,
V
abbreviate
S(A-)2R(A-)
to
formed entirely by the
to introduce
which we do not formalise
an e x p r e s s i o n
In the special
S and R into
the following
completely.
e x p r e s s i o n N w h i c h is formed by one or more uses corresponds
is
÷ S(A-)2R(A-) , leaving the
use of 2, it is convenient
considerations
for all values
fAT(AA-)
frequently
A in the e x p r e s s i o n
In order to handle repeated
for functors
V
To each
of 2, there
N in which each 2 is replaced by @,
in N b e c o m i n g r e p e a t e d
if N is (RA2S(AB))2T(BC)
variables
for functors
in N; for
R: A ÷ V,
S: A°P@B ~ V, and T: B°P@c ÷ V, then N is (RA@S(AB))@T(BC). Moreover,
there
is a canonical
q = qN: N ÷ N defined, occurrences
as follows,
of @ in N. - -
natural t r a n s f o r m a t i o n by i n d u c t i o n
If N contains
- -
on the n u m b e r of
no occurrence
of @ then
m
N ~ N and qN = i; otherwise N = N'@N" and qN is the composite N'@N"
~ N'@N" q'@q"
In the above example, (RA@S(AB))@T(BC)
~ N'@N". s
q is the composite ~ (RA2S(AB))@T(BC)
s@l and this is n a t u r a l
~ (RA2S(AB))@T(BC) s
in A, B,snd C; we say that the variables
and B are "summed out" by q.
A
The path qN: N ÷ N is in fact all those
variables
in N w h i c h
Le_~..
are
summed
~
qN:
in all the r e p e a t e d N ÷ N.
transformation
Then
result three
This
of @ in N.
is trivial;
V and let
is~
in p a r t i c u l a r ,
in N which
are
for a unique
is by i n d u c t i o n If N contains
othe~ise
summed
out
natural
N = N'@N"
on the n ~ b e r
no o c c u r r e n c e
of of @ the
and we can factor
f in
steps: q'@l
N
as g ' q N
over
g: N ÷ M.
Proo=~f. occurrences
f factors
into
which
variables
coend
out by qN:
Let M be a functor
f: N ÷ M be a n a t u r a l _ t r a n s f o r m a t i o n natural
a multiple
= N'@N"
l@q" )
N'@N"
s >
N'@N"
~ N'@N"
= N
M
First
consider
the t r a n s f o ~
tensor-hom
adJunction
hypothesis
and routine
w(f):
isomorphism naturality
N' ÷ [N",M] ~ = Vp of
of f under
V.
considerations,
By the i n d u c t i o n the d i a g r ~
q' N'
~ N'
[ N",M]
commutes
for a unique
morphlsm
~(f'): N' ÷ |N",M]
the
where
-10-
f': N'®N" by q'.
÷ M is n a t u r a l
Similarly
f' factors
f": N'@N"
~ M which
by e i t h e r
of q' or q".
as g.s
for a unique
remaining
in all the v a r i a b l e s
variables
as f".(l@q")
is n a t u r a l
not
summed
for a unique
in all the variables
Finally,
because
g: N ÷ M w h i c h
in N and M by Lemmas
morphism
not
s is a coend,
is n a t u r a l
out
summed
out
f" factors
in all the
2.2,
2.3,
and 2.4.
m
Combining
these
steps•
in the r e q u i r e d
we have
that
for a path q': N' ~ N'
is d e n o t e d part
by n.
of the
three
the
induced
f in Lemma induced
qN = s(q'@q")
cases
transformation
n: N ÷ N' is a n a t u r a l
form
g: N ÷ N'
are a n e c e s s a r y
category
and we c o n s i d e r
below.
if n: N ÷ N' is a n a t u r a l
from the coherent
2.5 is of the
transformations
of a p r e m o n o i d a l
special
First, entirely
Such
concept
relevant
through
manner.
W h e n the t r a n s f o r m a t i o n q'.n
f factors
isomorphism
data isomorphisms
isomorphism
a, r,
and is called
constructed
£, c of V then
an induced
m
coherence Lemma that the
isomorphism.
2.5,
and the o r i g i n a l
induced induced
determined shall
not
coherence coherence
coherence
isomorphisms isomorphism
by the a r r a n g e m e n t label
such
Secondly, natural
In view of the u n i q u e n e s s of a, r,
assertion
of
£, c, it is clear
are coherent.
In other words,
n: N + N' is completely
of @ in N and N';
consequently
we
isomorphisms. w h e n n z h@k:
transformations
S(A-)@R(A-)
÷ S'(A-)@R'(A-)
h: S ~ S' and k: R ÷ R',
let us w r i t e
for
-11
h@._k for h@k.
-
This not only makes the symbol @ En~-functorial
whenever it is defined on objects, but also makes the coend sA: S(A-)@R(A-) ÷ S(A-)@_R(A-) End-natural in S and R.
Under
reasonable conditions the same observations can be made at the V-level. If we restrict our attention to functors into V with small domains then the functors themselves may be regarded as extra variables. A and B small.
For example,
let T: A°P@A@B + V be a functor with
Then fAT(AAB) is canonically functorial in T and B
for we can write T(AAB) = F(AATB) where F is the composite
A°P@A@([A°P@A@B,V]@B) ~
[A°P@A@B,V]@(A°P@A@B) > V, E
and where E is the evaluation functor defined in [i]
§4.
if S(A-) and R(A-) are functors into V with small domains
Similarly, (and
different variances in A) then S(A-)@__R(A-) is functorial in S and R in a unique way that makes sA: S(A-)@R(A-) ÷ S(A-)@R(A-) natural in S and R. Lastly, let S(A-) be a functor into V which is covariant in A E A.
As part of the data for S, we have a family of morphisms
SAB: A(AB) ÷ [S(A-),S(B-)]
which is natural in A and B and also in
the extra variables in S.
Transforming this family by the
tensor-hom adJunction of V, we get a transformation w-I(SAB):
A(AB)@S(A-) + S(B-) which is natural in A and B and the
extra variables in S.
As a result of the generalised "higher"
representation theorem (see [I], §3 and §5), this induces the Yoneda isomorphism
-12-
YS,B: A(AB)@S(A-)
÷ S(B-).
By Lemma 2.2, we then have Lemma 2.6.
The Yoneda isomorphism YS,A is natural in
A and in the extra variables
in S; if the domain of S is small
then it is natural in S. The following diagram lemmas for the Yoneda isomorphism y are all proved using [3] Proposition as the representatlon
theorem.
in their most convenient Lemma 2.~.
II.7.4 which we shall refer to
These lemmas are presented here
forms for application
in sections
3 and 4.
Given functors S: A ~ V and T: A °p * V
for which SA@_TA exlsts m the following diagram commutes:
(A(AB)O_SA)O3B
> SBe_TB
ZEk
SA@(A(AB)e_TB)
> SAe_TA • icy
Proof.
Replacing @ by @,
we obtain a new diagram:
y by its definition,
etc.,
-13 -
(A(AB) @SA)~TB
\
s@l
\
\ SB@TB
(A(AB) Q_SA)@TB Y@I
\
y~_l ( A(AB)@SA) @_TB
"~ SB@TB
SA@( A( AB )@_TB )
SA@TA
~II
l@y SA@ ( A( AB )@__TB)
SA@(A(AB)@TB)
.
By Lemma
s(s@l)
2.5,
is a coend over A and B hence
to prove that the exterior A, B E A.
SA@TA
This is easily
of this new diagram
it suffices
commutes
seen to be so on applying
the
for all
-14-
representation
theorem;
put B m A and compose both exterior
legs with
(I@SA)@TA
~ (A(AA)@SA)@TA; (JA®I)@I
the resulting diagram commutes, Lemma 2.8.
hence the original one does.
Given functors
S: A°P@B ÷ V and T: B °p ÷ Y
for which S(AC)0_TC exists for each A E A, the followin~ diagram commutes
for each A E A:
A(AB)@(S(BC)®_TC)
S(AC)e_TC
(A(AB)®_S(BC))@_TC , Proof.
Again replacing @ by @,
etc., we obtain a new diagram:
y by its definition,
-15 -
A(AB)~(BC)emc)
A(AB)e(s (BC)_eTc)
~,
~"
S(AC)_OTC ~
A AB )_( aS BC ))_ ~ ((
//
I
S(AC)eTC
~ j~
(A(AB)_eS(BC)~ / / / /
(*(AB)eS(BC))~C In this
diagram the region
transform
labelled
1 commutes b e c a u s e
of t h e d i a g r a m
S(-C)eTC A(AB)
>
[ S(BC)®TC,S(AC)~'C]
!
S(-C)eTC
I [1,s]
[S(BC)e_TC,S(AC)eTC! ~ [ s,1]
[S(BC)®TO,S(AO)®TC]
it
is
the
-16 -
which expresses in A.
the naturality
Hence, because s(l@s)
of s = sC: S(AC)@TC ÷ S(AC)@TC
is a coend over B and C by Lemma
2.5, it suffices to prove that the exterior of the new diagram commutes
for all A, B E A and C E 8.
consequence
of the representation
The remaining
Again this is a simple
theorem.
lemmas are obtained by the same type of
argument. ~
.
for which TCSS(AC) commutes
Given functors S: A°P@B ~ V and T: B °p ~ V exists for each A E A, the following diagram
for each A E A: A(AB)@(TCe_S(BC))
III
~~'TCQS(AC) /
j J
ley
TCO_(A (AB)®_S (BC))
Lemma 2.10.
For any functors S; A ~ 8 and T: B °p ÷ V
the followln~ diagram commutes
for each A E A:
l~y A(AB)@(8(SB,C)®_TC)
> A(AB)%TSB
8(SA,C)~3C
~ TSA Y
-17 -
Lemma 2.11 diasram
commutes
For an~ functor T: A@B + V the following
for all B, D E A:
ley A(AB)@_(B(CD)@_T(AC) )
~
A(AB)@_T(AD)
S(CD )@_T(BC)-
~
T(BD)
.
Y 3.
Premonoidal
We emphaslse all concepts monoidal
over
again that,
are relative
closed category Definition
V consists
a category a functor
Categories unless
otherwise
to the given normallsed
3.1
A premonoidal
category
of
A, P: A°P@A°P@A
÷ V,
isomorphisms
I = AAB: JX@P(XAB)
÷ A(AB),
P = PAB: JX@P(AXB)
~ A(AB),
a = aABCD: satisfying
P(ABX)@P(XCD)
the following
symmetric
V.
a functor J: A * V, and natural
indicated,
÷ P(BCX)@_P(AXD),
two axioms:
P = (A,P,J,A,p,~)
-18 PC1.
For all A,B,C E A, the following diagram commutes: l@a > JXQ_(P(XBY)Q_P(AYC))
JX@_(P(AXY)@_P(YBC) ) -
(JX~P(AXY))~P(YBC)
(JX~_P (XBY)) ~P (AYC)
p@l
A(AY)~_P(YBC)
A(BY)~P(AYC)
P(ABC) PC2.
.
For all A,B,C,D,E E A, the following diagram commutes: P(ABX )0_(P (XCY) ~_P(YDE) )
P(ABX)O(P(CDY)~P(XYE))
(P(ABX)@P(XCY))@P(YDE)
P(CDY)@(P(ABX)@P(XYE))
( P(BCX)~P(AX¥))~_P(YDE)
P(CDY)~(P(BYX)@P(~E))
~II
P (BCX)@( P(AXY) ~_P(YDE) )
P (BCX)@(P(XDY) ~_P(AYE))
(P(CDX)~P(BXY))~P(AYE)
(P(BCX)@P(XDY))@P(AYE)
-19 -
Remark requisite
3.2
It is assumed
~'s exist for the g i v e n
hypothesis
on V, when
P(AB-): A ÷
V and J:
A is small.
in the d e f i n i t i o n
A, P, and J.
|A,V].
to a b l c l o s e d
for all A,B 6 A.
of this s e c t i o n we will suppose
A is small and show that each p r e m o n o i d a l "extends"
They do so, by
They also exist w h e n e v e r
A ÷ V are r e p r e s e n t a b l e
In the r e m a i n d e r
that the
structure
[P,V]
structure
that
P on A
on the functor category
For the m o n o i d a l part define a t e n s o r - p r o d u c t
*: [ A , V ] ~ [ A , V ] (3.1)
÷ [A,V]
by
S*T = fAsA@;BTB@P(AB-)
for all S,T E [A,V];
= SA@(TB@P(AB-))
this e x p r e s s i o n
is c a n o n i c a l l y
in S and T by the c o n s i d e r a t i o n s
of s e c t i o n 2.
J 6 [A,V]
of.,
be the i d e n t l t y - o b J e c t
Isomorphisms respective
£* = ~ :
functorlal
Next,
let
and define n a t u r a l
J*T ÷ T and r* = r~: T.J ÷ J as the
composites
J,T = J X @ ( T A @ P ( X A - ) )
a (JX@P(XA-))@_TA
A(A-)@TA k@l
~ T y
and T.J = T A @ ( J X @ P ( A X - ) )
m (JX@P(AX-))@_TA > T.
A(A-)@_TA p®l Lastly,
define a natural
as the composite
y isomorphism
a* = a~ST:
(R,S),T ÷ R*(S,T)
-
20
-
(R*S)*T = (RA@(SB@P(ABX)))@(TC~P(XC-)) RA@(SB@(TC@(P(ABX)@P(XC-)))) RA@(SB~(TC@(P(BCX)@P(AX-)))) i®(1®(1®~)) RA~((SB@(TC@P(BCX)))@P(AX-)) = R,(S,T). Then £*,
r*, and a* are natural by Lemmas Theorem
monoidal
category admittln5 Proof
category,
[P,V]
3.3
First,
= ([A,VJ,*,J,£*,r*,a*)
to show that [P,V]
is a monoidal
we need to prove PC1 ~ MC2 and PC2 ~ MC3.
commutes by PCI;
definitions
of *,r*,a*,
by the naturality isomorphlsms
diagram
(3.2) in which the
and £*;
4, 5, 6, 7, 8, and 9 commute of the induced coherence
(Lemma 2.5 and the succeeding remarks);
commute by Lemma 2.7; and 12 commutes by Lemma 2.9.
isomorphlsms
The proof
and coherence
of * and a*, uses
of the induced coherence
involved.
To complete the structure on [A,V] biclosed
I0 and Ii
a diagram that is too large for the space
available but, apart from the definitions only the naturality
The first
l, 2, and 3 commute by the
and coherence
of PC2 ~ MC3 requires
is a
a biclosed structure.
of these is obtained by considering exterior
2.5 and 2.6.
category,
consider the composite
to that of a
isomorphism:
®l
r~@l=r~l ~ -
il~
i®(i®y)
10
SX@(TC@P(XCD )
~/(A(AX)~SA)@(TCgP(XCD))4
1
(SA@(JB@P(ABX)))@(TC@P(XCD})
( (JB@_P(ABX))@__SA@ ) _(TC@__P(XCD) )
SAg(TCg(A(AX)~P(XCD)))
®l
v
®i
v
6
SA@(JB@(TC~(P(ABX)@P(XCD))))
SAe(TC@((JB@P(ABX))@P(XCD)))
SA@(TC@(JB@(P(ABX)®P(XCD)) )
/
t -
/
3
l
9
v v
®
F-J
I® I®
SAe(TCe(A(CX)~P(AXD)))
SA@_((A (CX)@_TC)@_P(AXD))
SA@(TX@P(AXD))
i~( ~ *~l )
i
SA@_(((JB@_P(BCX))®_TC)@_P(AXD))
~r SA@((JB~(TC~P(BCX)))@P(AXD))
~Jl
SA@_(JB@_(TC@_(P(BCX)@_P(AXD)))
SA@_(TC@_((JB@P (BCX))@_P(AXD)))
SA@(TC@(JB@(P(BCX)~P(AXD))))
~> SAe(TCeP(ACD))-
P(BCX)@P(AXD)
P(BAX)@P(XCD)
P(ACX)@P(BXD)
P(BCX)@P(XAD)
~
P(CAX)@P(BXD)
.
o@i This definition does not, of course,
require
A to be small. It remains to be shown that [ P,V] admits a symmetric monoidal
closed structure whenever
P has a symmetry.
For this,
define a natural isomorphism c * = C~T: SwT * T~S as the composite
-
S*T
=
24
-
SA@(TB@P(AB-)) ~ TB@(SA®P(AB-)) TB@(SA®P(BA-)) = T*S.
i®(l~c) Again, the naturallty of e* is a consequence of Lemma 2.5. If e is a symmetry for P then c* is a s~etr~
for [P,V]. P~oof
To prove PC3 ~ MC6 consider diagram (3.5): TB~(SA@P(BAC))
TB@(SA@P(ABC))
SA@(TB@_P(BAC))
~
2
(3.5) II~
TB@(SA@P(ABC))
SA~(TB~P(ABC))
u
I@(I~o)
SA~(TB~P(ABC))
Region 1 commutes by PC3, and region 2 commutes by the naturallty of the Induoed coherence isomorphism involved; hence the exterior commutes and so, by definition of c*, Me6 is satisfied.
To prove
PC4 ~ MC7 consider diagram (3.6), in which the exterior commutes by PC4; i, 2, and 3 commute by the definitions of , and c*; 4, 5, and 6 commute by the definition of a*; and 7, 8, 9, and i0 commute by the naturality and coherence of induced coherence Isomorphlsms.
.i-
/
a •
SBO_((TC~_(~A03 (ACX)) ) ~_P(BXD))
Imc m
aM
le(1®(1~))
c**l
SB@_((RAQ_(TC~P(ACX)) )~_P(BXD))
RA@(SB@(TC~(P(ACX)~P(BXD))))
®I
v
I
(SB@(RA@P(BAX)))@(TC@P(XCD))
(l@(l@~))@_l I
(RA~(SB@P(ABX)))~(TC@P(XCD))
RA@(SB@(TC@(P(BAX)@__P(XCD))))
®I
,-4
®I
®I
v
®I ~D
(SB~(RA~P(ABX)))@(TC~P(XCD))
RA@(SB~(TC~(P(ABX)~P(XCD))))
I
!~
