Theory and Applications of Categories, Vol. 20, No. 11, 2008, pp. 307–333.
STAR-AUTONOMOUS FUNCTOR CATEGORIES
J.M. EGGER
Abstract. We construct a star-autonomous structure on the functor category KJ ,
where J is small, K is small-complete, and both are star-autonomous. A weaker result,
that KJ admits a linear distributive structure, is also shown under weaker hypotheses.
The latter leads to a deeper understanding of the notion of linear functor.
1. Introduction
This paper is, to a large extent, a continuation of [6]. In that paper it is shown inter alia
that the eighteen coherence axioms required of a linear functor J −→ K can, in the case
J = 1, be neatly summarised using such familiar notions as (co)actions of (co)monoids,
and (co)equivariance. Briefly, linear functors 1 −→ K are in bijective correspondence with
cyclic nuclear monoids in K. That observation brought fresh intuition to the concept of
linear functor: just as a monoidal functor can be thought of as a generalised monoid, so
a linear functor should be thought of as a generalised cyclic nuclear monoid.
But there exists a partial converse to the idea that a monoidal functor is a generalised
monoid: under certain circumstances it is possible to endow a functor category KJ with
a convolution tensor product [5]; in these cases, monoidal functors J −→ K may be
regarded as monoids in KJ . Here we show that, under comparable circumstances, it is
possible to endow KJ with a linear distributive structure so that linear functors J −→ K
may be regarded as cyclic nuclear monoids in KJ .
Moreover, we show that if J and K also admit duals—i.e., if they are star-autonomous
categories—then so is KJ . In fact, the formulae describing the (right- and left-) duals of an
object X of KJ are very intuitive: they are its two “de Morgan duals”, X ∗ (p) = (X(∗p))∗
and ∗X(p) = ∗(X(p∗ )).
While this paper follows organically from its predecessor, the real impetus for continuing this vein of research came from collaboration with David Kruml [7]. In order to
show that Kruml’s notion of Girard couple of quantales (which was defined in response to
concrete considerations arising in quantale theory) is an instance of a very general concept
in [6], the author was led to construct a star-autonomous structure on Sup→ . This having
been achieved, with the aid of the usual star-autonomous structures on Sup and →, it
was natural to try to understand the full extent of the latter construction.
Received by the editors 2007-12-18 and, in revised form, 2008-06-26.
Transmitted by Robert Paré. Published on 2008-07-03.
2000 Mathematics Subject Classification: 18D10,18D15.
Key words and phrases: Linear distributive categories, star-autonomous categories, functor categories.
c J.M. Egger, 2008. Permission to copy for private use granted.
°
307
308
1.1.
J.M. EGGER
Remarks.
X
1.1.1. Throughout this paper our focus will be on functors J −→ K where J is small
and K large. In this context, it is useful to think of J as a category of indices; we shall
therefore write X• in place of the more usual X(•).
1.1.2. We shall assume the reader familiar with the general theory of linear distributive
categories and linear functors [3, 4]. We recall that, just as a boolean algebra can be
(equivalently) defined either as a Heyting algebra whose negation operation is involutive
or as a distributive lattice in which each element has a complement, so a star-autonomous
category can be (essentially equivalently) defined either as a monoidal closed category
equipped with a dualising object or as a linear distributive category in which every object
has a left dual and a right dual. In the present paper, we shall (for reasons explained in
Section 3) take the latter approach; but we shall insist that duals be functorially specified,
as occurs tautologically in the former approach.
1.1.3. We do not assume symmetry; in this context, it is convenient to take (co)closed
to mean both left- and right-(co)closed.
2. Preliminaries
2.1. Definition. A category K is said to have (co)limits of size J if J is small and K
small-(co)complete with respect to some reasonable notion of small (including finite and
countable, as well as set-sized).
A monoidal category (K; ⊗, i) is said to have distributive (co)limits of size J if it has
(co)limits of size J and if these are distributed by ⊗.
× , eJ ) and (K; ∩
× , eK ) be monoidal categories, and suppose that
2.2. Theorem. Let (J ; ∩
the latter has distributive colimits of size J .
× C , E) such
Then the functor category KJ can be equipped with a monoidal structure (∩
ζ
ϑ
× C Y −→ Z are in bijective correspondence
that natural transformations E −→ Z and X ∩
ζ̂
with arrows eK −→ ZeJ and natural transformations of the form
× Yn
Xm ∩
ϑ̂m,n
/ Zm×
∩n
respectively.
× C , E) if and only if (M, µ̂, η̂) is a monoidal
In particular, (M, µ, η) is a monoid in (KJ ; ∩
× , eJ ) −→ (K; ∩
× , eK ).
functor (J ; ∩
2.3. Remark. The hypothesis that K has colimits of size J implies that J can be made
into a K-category; one sets the hom-K-object Jm, nK to be the appropriate co-power of
eK —i.e., that corresponding to the hom-set [m, n]. The tensor product constructed in
Theorem 2.2 can then be understood as a convolution tensor product in the sense of
Day [5]. But the observation that convolution tensor product represents a very natural
309
STAR-AUTONOMOUS FUNCTOR CATEGORIES
multicategory structure on KJ , although widely known, appears to have remained folklore.
We shall therefore only sketch a proof of Theorem 2.2, primarily to settle notation.
2.4.
2.4.1.
Lemmata.
× C Y )r is defined as the colimit of
(X ∩
/J ×J
× ↓ r
∩
X ×Y
×
∩
/K×K
/K
—more colloquially, we write
× C Y )r = colim Xp ∩
× Yq .
(X ∩
p×
∩q→r
λ
× q −→ r, we write $λ for the corresponding coprojection Xp ∩
× Yq −→
Given an arrow p ∩
× C Y )r .
(X ∩
2.4.2.
× C Y )r
(X ∩
× C Y )ψ
(X ∩
/ (X ∩
× C Y )r0 is defined by
$λ
/ (X ∩
× Yq
× C Y )r
Xp ∩
SSSS
SSSS
S
× C Y )ψ
(X ∩
$ψ◦λSSSSSS)
²
× C Y )r0
(X ∩
×C B
and A ∩
×C ϑ
ζ∩
/X ∩
× Y by
× Bq
Ap ∩
× ϑq
ζp ∩
$λ
²
× Yq
Xp ∩
/ (A ∩
× C B)r
²
$λ
× C ϑ)r
(ζ ∩
/ (X ∩
× C Y )r
λ
× q −→ r.
for all p ∩
2.4.3.
The (inverse of the) associativity isomorphism is defined by
× (Yk ∩
× Zp )
Xj ∩
× $ λ0
id ∩
α
²
²
× (Y ∩
× C Z)q
Xj ∩
$ λ1
× id
$id ∩
× C Y )j ×
× Zp
(X ∩
∩k ∩
²
× C (Y ∩
× C Z))r
(X ∩
/ (Xj ∩
× Yk ) ∩
× Zp
²
(α)r
$ λ2
/ ((X ∩
×C Y ) ∩
× C Z)r
—where λ0 , λ1 are arbitrary and λ2 is the composite
× k) ∩
× p
(j ∩
α
/j ∩
× (k ∩
× p)
× λ0
id ∩
/j ∩
× q
λ1
/ r.
310
J.M. EGGER
eK /
/1
2.4.4. Er is defined to be the colimit of eJ ↓ r
K. Note that eJ ↓ r
is just the hom-set [eJ , r] regarded as a discrete category; hence, Er is just a copower of
ζ
eK . Given an arrow eJ −→ r we write $ζ for the corresponding coprojection eK −→ Er .
Eψ
/ Er0 is defined by
Er
$ζ
/ Er
eK NN
NNN
NNN
$ψ◦ζ NNNNNN
'
²
Eψ
Er0
ζ
for all eJ −→ r.
2.4.5.
The (inverse of the) right unit law is defined by
υr
Xr
υ
/ (X ∩
× C E)r
O
$υ
²
× eK
Xr ∩
× $id
id ∩
/ Xr ∩
× Ee
J
and the (inverse of the) left unit law is defined symmetrically.
2.5. Example. As noted in [7, Remark 4], a unital couple of quantales is essentially
× , ). It can therefore also be
the same thing as a monoidal functor (→; ∧, >) −→ (Sup; ∩
y!
x!
⊥
→
× C , E), where E = ( −→ ) and (x1 −→ x0 ) ∩
× C (y1 −→
regarded as a monoid in (Sup ; ∩
y0 ) is the dotted arrow in the diagram below.
× id
x! ∩
GF
× y1
x0 ∩
O
× y!
id ∩
× y0
x0 ∩
p.o.
× id
x! ∩
/p
O
/ x1 ∩
× y0
ED²
/ x1 ∩
× y1
l6
l
l
ll
lll
llid
l
×
l
∩ y!
l
→
× C , E) bears little relation to (Sup(Set ); ∩
× , Set→ ).
The reader is cautioned that (Sup→ ; ∩
→
It follows from [8, Proposition VI.2.1] that Sup(Set ) is (equivalent to) the subcategory
of Sup→ which has: for objects, surjective sup- and inf-homomorphisms; for arrows,
commutative squares satisfying the Beck-Chevalley condition. But this is far from being
a monoidal subcategory of Sup→ , since it does not even contain the object E.
STAR-AUTONOMOUS FUNCTOR CATEGORIES
311
3. Plan
The main objective of this paper, as indicated in the title and abstract, is to construct a
meaningful star-autonomous structure on the functor category KJ , where J and K are
themselves endowed with star-autonomous structures and K has colimits of size J .
× K distributes all colimits; hence we can apply Theorem 2.2
Under these hypotheses, ∩
× C , E) on KJ . Since such a convolution tensor product
to obtain a monoidal structure (∩
is necessarily closed, it would suffice to find a suitable dualising object. The author,
however, frankly balked at the prospect of computing the double dual of an object in
KJ and observed that, under the same hypotheses, K also has limits of size J which are
× K . Thus, we can also invoke the dual of Theorem 2.2 to obtain a second
distributed by ∪
× C , D) on KJ . Moreover, we already have natural candidates for the
monoidal structure (∪
(right- and left-) duals of an object of KJ , as described in Section 1.
× C , E, ∪
× C , D), and then
Thus we opt instead to construct linear distributions for (KJ , ∩
∗
∗
to find linear adjoints overlying X, X, and X . Unsurprisingly, the first half of this
project can be carried out under somewhat weaker hypotheses; so, as a bonus, we obtain
a more general theorem which turns out to be of some interest in its own right.
3.1.
Remark. We shall need to consider the common composite of a naturality square
× (Yq ∪
× Zt )
Xp ∩
× (Yβ ∪
× id)
id ∩
/ Xp ∩
× (Ys ∪
× Zt )
κ
²
× Yq ) ∪
× Zt
(Xp ∩
²
× Yβ ) ∪
× id
(id ∩
κ
/ (Xp ∩
× Yq ) ∪
× Zt
[β]
× (Yq ∪
× Zt ) _ _ _ _/ (Xp ∩
× Ys ) ∪
× Zt .
sufficiently often that it will merit abbreviation to Xp ∩
× C and ∪
×C .
Henceforth, we also omit the subscripts on eJ , eK , ∩
4. Linear distributive functor categories
× , e, ∪
× , d) is called bilinear if ∩
× is closed
We recall that a linear distributive category (J ; ∩
× is coclosed. Any star-autonomous category is bilinear with, for instance, the right
and ∪
× z∗ , but not vice versa.
coclosed structure defined by x ; z = x ∩
Although we are dealing with arbitrary (i.e., not necessarily symmetric) bilinear categories, our arguments shall concentrate on their left closed and right coclosed structures
(−◦ and ;, respectively). In order to distinguish between the units and counits of the
× ( ) a x −◦ ( ) and ( ) ; z a ( ) ∪
× z—we write η, ε for those
two adjunctions at hand—x ∩
of the former and η, ε for those of the latter.
Finally, we recall that in a bilinear category, there is a canonical isomorphism (x −◦
∼
× z −→ x −◦ (y ∪
× z). [In the star-autonomous case, this is just the associativity
y) ∪
∼
∗ ×
× z −→ x∗ ∪
× (y ∪
× z).]
isomorphism (x ∪ y) ∪
312
4.1.
J.M. EGGER
× , e, ∪
× , d) be a bilinear category. Then the Beck transform of
Definition. Let (J ; ∩
β
α
× q −→ s ∪
× t in J is the map q ; t −→ p −◦ s obtained by transposition as
a map p ∩
follows:
× q −→ s ∪
× t
p∩
× t)
q −→ p −◦ (s ∪
× t
q −→ (p −◦ s) ∪
q ; t −→ p −◦ s
—equivalently, it is the composite depicted below.
q;t
η ; id
²
p −◦ s
O
ε
× q)) ; t
(p −◦ (p ∩
(id −◦ α) ; id
/ (p −◦ (s ∪
× t)) ; t
∼
/ ((p −◦ s) ∪
× t) ; t
× , e, ∪
× , d) is bilinear, that (K; ∩
× , e, ∪
× , d) is linear dis4.2. Theorem. Suppose that (J ; ∩
×
tributive, that K has limits and colimits of size J , and that these are distributed by ∪
× , respectively.
and ∩
Then there exists a natural transformation, κ, such that for all arrows
× q
p∩
λ
/r
ρ
/s∪
× t
in J , the diagram
κr
$λkkkkkkk
k
kkk
kkk
/ ((X ∩
× Y) ∪
× Z)r
SSS
SSS πρ
SSS
SSS
SS)
SSS
SSS
SSS
SS
×
id ∩ πη SSS)
5
kkk
kkk
k
k
kkk × id
kkk $ε ∪
× (Y ∪
× Z))r
(X 5 ∩
× (Y ∪
× Z)
Xp ∩
q
× Y) ∪
× Zt
(X ∩
s
[β]
× (Yq;t ∪
× Zt ) _ _ _ _ _ _/ (Xp ∩
× Yp−◦s ) ∪
× Zt
Xp ∩
(where β denotes the Beck transform of ρ ◦ λ) commutes in K.
Proof. The diagram contained in the statement of the theorem amounts to a definition
of the components of κ if we can demonstrate the appropriate conicity and coconicity
criteria. It suffices to show the latter, as the former follows by a dual argument.
Therefore, let
× χ
φ∩
/ p0 ∩
× q S
× q0
p∩
S
j
SSS
SSS
SS
λ SSS)
r
j
jjjj
j
j
j
ju jjj λ0
313
STAR-AUTONOMOUS FUNCTOR CATEGORIES
ρ /
× ↓ r and r
× t an object of r ↓ ∪
× . Let β be the Beck transform
be an arrow in ∩
s∪
0
0
of ρ ◦ λ and β be that of ρ ◦ λ . Then the commutativity of the diagrams
χ ; id
q;t
β
/ q0 ; t
²
²
p −◦ s o
β0
× (φ −◦ id)
id ∩
p0 −◦ s
φ −◦ id
× id
φ∩
× (p0 −◦ s)
p∩
/ p0 ∩
× (p0 −◦ s)
²
²
× (p −◦ s)
p∩
ε
/s
ε
together with the naturality of κ and η combine to prove that
× (Y ∪
× Z)
Xφ ∩
χ
× (Y ∪
× Z)
Xp ∩
q
× πη
id ∩
²
/ Xp0 ∩
× (Y ∪
× Z) 0
q
× (Yχ;id ∪
× id)
Xφ ∩
× (Yq;t ∪
× Zt )
Xp ∩
× πη
id ∩
²
/ Xp0 ∩
× (Yq 0 ;t ∪
× Zt )
κ
²
× Yq;t ) ∪
× Zt
(Xp ∩
²
× Yχ;id ) ∪
× id
(id ∩
/ (Xp ∩
× Yq 0 ;t ) ∪
× Zt
× Yβ ) ∪
× id
(id ∩
× id) ∪
× id
(Xφ ∩
× Yβ 0 ) ∪
× id
(id ∩
²
/ (Xp0 ∩
× Yq 0 ;t ) ∪
× Zt
× Yβ 0 ) ∪
× id
(id ∩
²
κ
²
/ (Xp0 ∩
× Yp−◦s ) ∪
× Zt o
× Yp0 −◦s ) ∪
× Zt
× Yp0 −◦s ) ∪
× Zt
(Xp ∩
(Xp ∩
× id) ∪
× id
× Yφ−◦id ) ∪
× id
(Xφ ∩
(id ∩
. (X ∩
× Y) ∪
× Zt p
s
× id
$ε ∪
× id
$ε ∪
commutes as well.
Now it remains to show that κ is natural in r, and that κ is natural in X, Y, Z. For
the former, it suffices to show that for all arrows
λ
× q
p∩
ψ
/r
/ r0
ρ
/s∪
× t
in J , the composites
$ψ◦λ
× (Y ∪
× Z)q
Xp ∩
$λ
²
× (Y ∪
× Z))r
(X ∩
× (Y ∪
× Z))ψ
(X ∩
%
/ (X ∩
× (Y ∪
× Z))r0
κr
²
× Y) ∪
× Z)r
((X ∩
× Y) ∪
× Z)ψ
((X ∩
κr 0
²
/ ((X ∩
× Y) ∪
× Z)r0
²
πρ◦ψ
πρ
× Y )s ∪
× Zt
0 (X ∩
314
J.M. EGGER
agree. But since (ρ ◦ ψ) ◦ λ = ρ ◦ (ψ ◦ λ), the top-right and bottom-left composites of this
diagram are, by definition, equal.
The remaining naturality is left as an exercise to the reader.
Symmetrically, we can also define a natural transformation κ of the correct type.
4.3. Theorem. Under the same hypotheses as Theorem 4.2, κ and κ satisfy the neces× , E, ∪
× , D) a linear distributive category.
sary coherence conditions to make (KJ ; ∩
Proof. We shall only prove one of ten axioms, namely
× (X ∩
× (Y ∪
× Z))
W ∩
α
/ (W ∩
× X) ∩
× (Y ∪
× Z)
× κ
id ∩
²
× ((X ∩
× Y) ∪
× Z)
W ∩
(ld1 )
κ
κ
²
× (X ∩
× Y )) ∪
× Z
(W ∩
× id
α∪
²
/ ((W ∩
× X) ∩
× Y) ∪
× Z.
Three more coherence pentagons follow immediately (by symmetry); the remaining two—
× and ∪
× , and therefore involve components of both κ and κ—can
those which alternate ∩
be proven in a similar fashion, as well as the four coherence triangles.
λ1
× q −→ r,
To verify the rth component of (ld1 ), it suffices to show that, for all arrows j ∩
ρ
λ0
× p −→ q, and r −→ s ∪
× t in J , the composites
k∩
× (Xk ∩
× (Y ∪
× Z)p )
Wj ∩
× (Xk ∩
× (Y ∪
× Z)p )
Wj ∩
× $ λ0
id ∩
²
× $ λ0
id ∩
²
× (X ∩
× (Y ∪
× Z))q
Wj ∩
× (X ∩
× (Y ∪
× Z))q
Wj ∩
$ λ1
²
× (X ∩
× (Y ∪
× Z)))r
(W ∩
$ λ1
²
× (X ∩
× (Y ∪
× Z)))r
(W ∩
²
× id) ◦ κ ◦ (id ∩
× κ)
(α ∩
× X) ∩
× Y) ∪
× Z)r
(((W ∩
²
πρ
× (X ∩
× Y ))s ∪
× Zt
(W ∩
²
κ◦α
× X) ∩
× Y) ∪
× Z)r
(((W ∩
²
πρ
× (X ∩
× Y ))s ∪
× Zt
(W ∩
× ( ) preserves colimits.]
agree. [Here we use that the functor Wj ∩
Let βn denote the Beck transform of ρn ◦ λn for n ∈ {0, 1, 2}, where λ0 , λ1 , ρ1 are
η
× t, λ2 is as in Lemma 2.4.3, and
as above, ρ0 is the coevaluation map q −→ (q ; t) ∪
× k) ∩
× (k −◦ (j −◦ s)) −→ s and ξ˜ its
ρ1 = ρ = ρ2 ; also let ξ denote the obvious arrow (j ∩
× k) −◦ s.
transpose—i.e., the canonical isomorphism (k −◦ (j −◦ s)) −→ (j ∩
315
STAR-AUTONOMOUS FUNCTOR CATEGORIES
Then the two composites depicted above can be rewritten as
× (Xk ∩
× (Y ∪
× Z)p )
Wj ∩
³
´
× id ∩
× πη
id ∩
× (Xk ∩
× (Y ∪
× Z)p )
Wj ∩
α
²
²
× (Xk ∩
× (Yp;t ∪
× Zt ))
Wj ∩
Â
× Xk ) ∩
× (Y ∪
× Z)p
(Wj ∩
Â
² [β0 ]
¡
× (Xk ∩
× Yk−◦(q;t) ) ∪
× Zt
Wj ∩
²
¢
× X)j ×
× (Y ∪
× Z)p
(W ∩
∩k ∩
Â
Â
¡
¡
¡
πη
² [id −◦ β1 ]
²
¢
× (Xk ∩
× Yk−◦(j−◦s) ) ∪
× Zt
Wj ∩
× id
α∪
²
× X)j ×
× (Yp;t ∪
× Zt )
(W ∩
∩k ∩
Â
Â
Â
¢
Â
[β2 ]
× Xk ) ∩
× Yk−◦(j−◦s) ∪
× Zt
(Wj ∩
× X)j ×
(W ∩
∩k
Â
Â
× id) ∪
× id
($id ∩
²
¢
× Yk−◦(j−◦s) ∪
× Zt
∩
¡
²
¢
× X)j ×
× Y(j ×
× Zt
(W ∩
∩k ∩
∩k)−◦s ∪
× id
$ξ ∪
²
× id
$id ∩
²
× X) ∩
× Y )s ∪
× Zt
((W ∩
$ε
× X) ∩
× Y )s ∪
× Zt
((W ∩
respectively, as demonstrated in Figures A.1 and A.2.
Now it is easy to see that the diagram
β2
p;t
β0
/ (j ∩
× k) −◦ s
O
ξ˜
²
k −◦ (q ; t)
id −◦ β1
/ k −◦ (j −◦ s)
commutes. This, together with the fact that K itself satisfies (ld1 ), entails that the
diagram
α
× (Xk ∩
× (Yp;t ∪
× Zt ))
Wj ∩
Â
Â
¡
× [β0 ]
² id ∩
× (Xk ∩
× Yk−◦(q;t) ) ∪
× Zt
Wj ∩
¢
Â
(∗)
Â
¡
² [id −◦ β1 ]
¢
× (Xk ∩
× Yk−◦(j−◦s) ) ∪
× Zt
Wj ∩
× id
α∪
/ (Wj ∩
× Xk ) ∩
× (Yp;t ∪
× Zt )
Â
Â
² [β2 ]
¡
¢
× Xk ) ∩
× Y(j ×
× Zt
(Wj ∩
∩k)−◦s ∪
O¡
¢
× Y
×
id ∩
ξ̃ ∪ id
¡
¢
/ (Wj ∩
× Xk ) ∩
× Yk−◦(j−◦s) ∪
× Zt
commutes. Hence the two composites we are interested in do coincide, as demonstrated
in Figure A.3.
316
J.M. EGGER
4.4. Remark. If K is also bilinear, then the distributivity hypotheses of Theorems 4.2
and 4.3 are superfluous. Moreover, in this case KJ will be bilinear too, since the results
op
of [5] can be applied to both KJ and (KJ )op = (Kop )J . It therefore seems that bilinear
categories provide a more natural setting for these results than the more general one in
which we stated them.
On the other hand, a more general theorem—one which imposed no completeness
conditions on K—could undoubtably be proven if we were willing to adopt the more
general framework of polycategories. [A linear distributive category is essentially the same
thing as a representable polycategory.] Even in this context, however, the bilinearity of
J appears to remain crucial.
× C and
4.5. Example. There is a linear distributive category structure on Sup→ with: ∩
y!
x!
!
× C (y0 −→ y1 ) the dotted arrow in
E as in Example 2.5, D = ( −→ ), and (x0 −→ x1 ) ∪
the diagram below.
id yl!lllll
l
lll
lll
×
∪
× y0
x0 ∪
@A
x ×y
6 0 ∪O 1
× id
x! ∪
/ x1 ∪
× y1
O
× y!
id ∪
p.b.
/ x1 ∪
× y
BCO 0
/p
× id
x! ∪
5. Star-autonomous functor categories
We now return to the case of greatest interest; throughout this section, J and K will
denote star-autonomous categories such that K has colimits of size J . We shall not
always pedantically distinguish between, for instance, e ; s and s∗ .
5.1.
Definitions.
X
X∗
∗X
5.1.1. Given a functor J −→ K, the functors J −→ K and J −→ K are defined by
the formulae
(X ∗ )p = (X∗ p )∗ ,
(∗X)p = ∗(Xp∗ ).
ρ
× t in J , we define e
5.1.2. Given an arrow e −→ s ∪
in K constructed as follows:
× t
e −→ s ∪
∗
s −→ t
X∗s −→ Xt
× Xt .
e −→ (X∗ s )∗ ∪
τρ
/ (X ∗ )s ∪
× Xt to be the arrow
317
STAR-AUTONOMOUS FUNCTOR CATEGORIES
λ
× (X∗ q )∗
× q −→ d in J we define Xp ∩
5.1.3. Dually, given an arrow p ∩
the arrow in K constructed as follows:
γλ
/ d to be
× q −→ d
p∩
p −→ ∗q
Xp −→ X∗q
× (X∗ q )∗ −→ d.
Xp ∩
τ
× X whose compo5.2. Theorem. There exists a natural transformation E −→ X ∗ ∪
nents are defined by
e
@A
$ζ
τr
/ Er
πρ
/ (X ∗ ∪
× X)r
/ (X ∗ )s ∪
× Xt .
BCO
τρ◦ζ
γ
× X ∗ −→ D satisfying the dual diagram.
Dually, there exists a natural transformation X ∩
The proof of this theorem is merely a case of verifying the (co)conicity criteria, which
we leave as an exercise to the reader.
γ
τ
× X and X ∩
× X ∗ −→ D
5.3. Theorem. The natural transformations E −→ X ∗ ∪
defined above form a linear adjunction in KJ . [Thus X ∗ is the right dual of X.]
Proof. The statement of the theorem means that the composites
X∗
X
υ
²
× E
X∩
× X∗
E∩
× τ
id ∩
²
× (X ∗ ∪
× X)
X∩
υ
²
× id
τ∩
²
× X) ∩
× X∗
(X ∗ ∪
κ
²
∗
× X ) ∪
× X
(X ∩
×
X∗ ∪
κ
²
× X∗ )
(X ∩
× id
γ∪
²
× X
D∪
²
× D
X∗ ∪
²
υ
X
× γ
id ∪
²
X
υ
∗
equal the identity natural transformations on X and X ∗ respectively.
318
J.M. EGGER
As demonstrated in Figure A.4, the rth component of the left-hand composite equals
υ
Xr
/ Xr ∩
× e
× τη
id ∩
/ Xr ∩
× ((X ∗ )r∗ ∪
× Xr )
Xr o
× Xr o
d∪
υ
²
κ
× (X ∗ )r∗ ) ∪
× Xr .
(Xr ∩
× id
γε ∪
[The Beck transform of the composite
υ
× e
r∩
υ
/r
/d∪
× r
equals the canonical isomorphism e ; r −→ r −◦ d(= r∗ ) —but in keeping with our
η
× r to subsume
declared intention to minimise pedantry, we have allowed e −→ (e ; r) ∪
the latter.]
An analogous reduction applies to the right-hand composite; thus the statement of
the theorem is equivalent to the assertion that
τη
× Xr
e −→ (X ∗ )r∗ ∪
γε
× (X ∗ )r∗ −→ d
Xr ∩
consitute a linear adjunction in K.
This is essentially tautologous since (X ∗ )r∗ is, by definition, (X∗(r∗ ) )∗ which is isomorη
ε
× r and r ∩
× r∗ −→ d are the
phic to (Xr )∗ . Moreover, since the transposes of e −→ r∗ ∪
identities on r and r∗ respectively, it follows (again by definition) that τ η and γε equal the
composites
e
× (X ∗ )r∗
Xr ∩
η
/ (Xr )∗ ∪
× Xr
∼
/ Xr ∩
× (Xr )∗
∼
/ (X ∗ )r∗ ∪
× Xr
ε
/d
respectively.
A symmetric argument shows that we have a linear adjunction overlying ∗X and X
which demonstrates that ∗X is the left dual of X.
5.4. Corollary. If J and K are star-autonomous categories such that K has (co)limits
of size J , then KJ is star-autonomous as well.
× , E, ∪
× and D
5.5. Example. There is a star-autonomous structure on Sup→ , with ∩
∗
x!
x
!
x0∗ ).
as in Example 4.5 and the dual of an object (x0 −→ x1 ) given by (x1∗ −→
STAR-AUTONOMOUS FUNCTOR CATEGORIES
319
5.6. Remark. One case of interest is when J and K happen to be compact—i.e., when
× = ∪
× . In this case, one might expect KJ to be compact also; but
they have d = e and ∩
this is not necessarily the case.
Unfortunately, the most natural example of this phenomenon occurs in the more general polycategorical setting referred to in the second half of Remark 4.4. We shall briefly
sketch this idea before discussing how it can be modified into an example which does
satisfy the hypotheses of this section.
5.7. Example. Let G denote the ordered group of integers regarded as a compact closed
× = + = ∪
× , e = 0 = d, and
category in the usual way—i.e., as a posetal category with ∩
∗
∗
x = −x = x; let V denote the usual compact closed category of finite-dimensional kvector spaces and k-linear transformations. Then, although V does not have limits of size
G, we see that the units of polycategorical structure of V G are representable, and that
they are different.
½
½
k if n ≥ 0
k if n ≤ 0
En =
Dn =
0 otherwise
0 otherwise
[In fact, it is possible to carve out an interesting full subpolycategory of V G which is
entirely representable: let us say that an object X of V G is stable if all but finitely many
of the transition maps Xn −→ Xn+1 are isomorphisms; then the full subcategory of stable
objects of V G does indeed form a star-autonomous category which is not compact.]
Now if we could replace V by a (non-trivial) compact closed category with all countable
limits, then the same argument would apply; embarrassingly, the author does not know
whether such a category exists. Alternatively, we could replace G by any finite compact
closed category containing an object r with a different number of arrows i −→ r and
r −→ i. Such categories are known to exist—the author is indebted to Robin Houston
for providing us with an example.
6. Linear functors
We now reap the reward for having pursued the greater generality advocated in section
3.
We recall that a linear functor [4] between linear distributive categories consists of a
monoidal functor and a comonoidal functor together with a set of four natural transformations called linear strengths and linear costrengths satisfying a number of coherence
conditions. Our notation will be as follows: if T is a linear functor, then its monoidal
part will be denoted (∀T ; µ, η), its comonoidal part (∃T ; δ, ε), and its linear strengths and
costrengths as below.
× ∃T (y)
∀T (x) ∩
× ∀T (y) o
∃T (x) ∪
σ (`)
ϑ(`)
/ ∃T (x ∩
× y) o
× y)
∀T (x ∪
σ (r)
ϑ(r)
× ∀T (y)
∃T (x) ∩
/ ∀T (x) ∪
× ∃T (y)
320
J.M. EGGER
This use of quantifier symbols is intended as a mnemonic aid; for instance, σ (`) should be
seen as corresponding to the tautology (∀t: T.x(t)) ∧ (∃t: T.y(t)) ⇒ (∃t: T.x(t) ∧ y(t)).
6.1. Theorem. Suppose that I is an arbitrary linear distributive category, and that J
and K satisfy the hypotheses of Theorem 4.2. Then there is a natural bijection between
linear functors I −→ KJ and linear functors I × J −→ K.
6.2. Lemma. Suppose that J and K satisfy the hypotheses of Theorem 4.2. Then the
linear distributions of J , K and KJ are related by the fact that the diagram
× Y) ∪
× Z)p×
((X ∩
∩(q×
∪r)
hh4
κ(p×∩q)×∪hrhhhhhhhh
hhh
hhhh
VVVV
VVVV((X ∩
× Z)
VVVV × Y ) ∪
κ
VVVV
VV*
× (Y ∪
× Z))p×
(X ∩
∩(q×
V ∪r)
O
VVVV
VVV(X
× (Y ∪
× Z))
VVVV∩
κ
VVVV
VV*
$id
× (Y ∪
× Z) ×
Xp ∩
q ∪r
× πid
id ∩
× Y) ∪
× Z)(p×
((X ∩
∩q)×
∪r
hh4
κ(p×∩q)×∪hrhhhhhhhh
hhh
hhhh
× Y) × ∪
× Zr
(X ∩
p∩q
× (Y ∪
× Z))(p×
(X ∩
∩q)×
∪r
²
O
× id
$id ∪
κ
× (Yq ∪
× Zr )
Xp ∩
/ (Xp ∩
× Yq ) ∪
× Zr
commutes.
Proof. The top diamond of the diagram above is a naturality square.
Note that by the definition contained in Lemma 2.4.2, the diagram
× (Y ∪
× Z))p×
(X ∩
∩(q×
W∪r)
O
WWWWW
WWW(X
× (Y ∪
× Z))
WWWW∩
κ
$id
WWWWW
WWW+
/ (X ∩
× (Y ∪
× Z) ×
× (Y ∪
× Z))(p×
Xp ∩
∩q)×
∪r
q ∪r
$
κ
commutes.
Now, by the definition contained in Theorem 4.2, we also have that
× (Y ∪
× Z))(p×
(X ∩
∩q)×
∪r
O
κ(p×∩q)×∪r
/ ((X ∩
× Y) ∪
× Z)(p×
∩q)×
∪r
$κ
× (Y ∪
× Z) ×
Xp ∩
q ∪r
id
×
∩
πη
²
× (Y(q×
Xp ∩
∪r);r
πid
²
²
πid
× Y) × ∪
× Zr
(X ∩
p∩q
O
× id
$ε ∪
[β]
× Zr ) _ _ _ _ _ _ _/ (Xp ∩
× Yp−◦(p×
× Zr
∪
∩q) ) ∪
κ
× q) ∪
× r.
× (q ∪
× r) −→ (p ∩
commutes, where β is the Beck transform of p ∩
321
STAR-AUTONOMOUS FUNCTOR CATEGORIES
But in any bilinear category, the latter equals
ε
× r) ; r
(q ∪
η
/q
/ p −◦ (p ∩
× q)
and so [β] may be factorised as follows:
/ (Xp ∩
× Y(q×
× Zr
∪r);r ) ∪
× (Y(q×
× Zr )
Xp ∩
∪r);r ∪
× (Yε ∪
× id)
id ∩
×
Xp ∩
²
²
/ (Xp ∩
× Yq ) ∪
× Zr
×
(Yq ∪
Zr )
κ
²
²
× Yη ) ∪
× id
(id ∩
/ (Xp ∩
× Yp−◦(p×
× Xr .
∩q) ) ∪
× (Yp−◦(p×
× Zr )
Xp ∩
∩q) ∪
Now regarding the triangle identities
× q T
p∩
TT
× η
id ∩
/p∩
× (p −◦ (p ∩
× q))
TTTT
TTTT
TTT
id TTTTTT*
²
η
/ ((q ∪
× r) ; r) ∪
× r
TTTT
TTTT
× id
TT
ε∪
id TTTTTT) ²
× r T
q∪
TT
ε
× q
p∩
× r
q∪
× ↓ (p ∩
× q) and (q ∪
× r) ↓ ∪
× , we conclude that
as arrows in, respectively, ∩
× Yη
id ∩
/ Xp ∩
× Yp−◦(p×
× Yq
Xp ∩
∩q)
TTTT
TTTT
TT
$ε
$id TTTTTT)
²
× Y )p×
(X ∩
∩q
πη
/ Y(q×r);r ∪
× Zr
∪
TTTT
TTTT
× id
T
Yε ∪
πid TTTTTTT*
²
× Z)q×
(Y ∪
∪rT
× Zr
Yq ∪
commute.
Proof of Theorem 6.1.
T
(⇒). Given a linear functor I −→ KJ , we define functors I × J
obvious way:
∀T̂ (x, p) = [∀T (x)]p
∃T̂ (x, p) = [∃T (x)]p .
∀T̂ , ∃T̂
/ K in the
[Note that we apply the notational convention of Remark 1.1.1 only to J , since I is not
assumed to be small.]
η
Now we know, from Theorem 2.2, that E −→ ∀T (e) corresponds to a unique arrow
η̂
µ
× ∀T (y) −→ ∀T (x ∩
× y)
e −→ [∀T (e)]e = ∀T̂ (e, e) and that each component of µ, ∀T (x) ∩
corresponds to a transformation (natural in p and q) of the form
× [∀T (y)]
× ∀ (y, q) = [∀T (x)] ∩
∀T̂ (x, p) ∩
T̂
p
q
µ̂
/ [∀T (x ∩
× y, p ∩
× q)
× y)] × = ∀ (x ∩
T̂
p∩q
322
J.M. EGGER
—the naturality of µ (in x and y) naturally entails the same for µ̂. We take it as read
that (∀T̂ ; µ̂, η̂) form a monoidal functor I × J −→ K.
Similarly, σ (`) and σ (r) give rise to natural transformations
σ̂ (`)
× ∃ (y, q)
∀T̂ (x, p) ∩
T̂
/ ∃ (x ∩
× y, p ∩
× q) o
T̂
σ̂ (r)
× ∀ (y, q)
∃T̂ (x, p) ∩
T̂
and, dually δ, ε, ϑ(`) , ϑ(r) give rise to natural transformations δ̂, ε̂, ϑ̂(`) , ϑ̂(r) of the correct
T̂
form. Thus, we have all the data necessary for a linear functor I × J −→ K; all that
remains to show are the coherence conditions.
Again, we shall prove only one of these, as an illustration:
× ∀ (y ∪
× z, q ∪
× r)
∀T̂ (x, p) ∩
T̂
µ̂
²
× (y ∪
× z), p ∩
× (q ∪
× r))
∀T̂ (x ∩
× ϑ̂(`) /
id ∩
× (∃ (y, q) ∪
× ∀ (z, r))
∀T̂ (x, p) ∩
T̂
T̂
(lf11
ˆ)
²
× ∃ (y, q)) ∪
× ∀ (z, r)
(∀T̂ (x, p) ∩
T̂
T̂
∀T̂ (κ, κ)
²
²
× y) ∪
× z, (p ∩
× q) ∪
× r)
∀T̂ ((x ∩
ϑ(`)
κ
× id
σ (`) ∪
/ ∃ (x ∩
× y, p ∩
× q) ∪
× ∀ (z, r)
T̂
T̂
× (q ∪
× r) and (p ∩
× q) ∪
× r components of the
—to show that this holds, we select the p ∩
corresponding axiom for T :
× ∀T (y ∪
× z)
∀T (x) ∩
µ
× ϑ(`) /
id ∩
× (∃T (y) ∪
× ∀T (z))
∀T (x) ∩
²
× (y ∪
× z))
∀T (x ∩
κ
²
× ∃T (y)) ∪
× ∀T (z)
(lf11 ) (∀T (x) ∩
∀T (κ)
²
× y) ∪
× z)
∀T ((x ∩
²
ϑ(`)
× id
σ (`) ∪
/ ∃T (x ∩
× y) ∪
× ∀T (z)
κ
× q) ∪
× r −→
which, together with the naturality squares corresponding to the arrow (p ∩
× (q ∪
× r), prove that the bottom rectangle, and therefore the whole of, Figure A.5
p ∩
commutes.
(⇐). We hold it self-evident that the construction of the data of T̂ from those of T is
reversible—i.e., that the only non-trivial part of the converse has to do with the coherence
conditions.
Suppose that (lf11
ˆ ) holds; to show that the rth component of (lf11 ) holds, we must
ρ
λ
× q −→ r and r −→ s ∪
× t, and show that the
(yet again) consider arbitrary arrows p ∩
323
STAR-AUTONOMOUS FUNCTOR CATEGORIES
composites
× [∀T (y ∪
× z)]
[∀T (x)]p ∩
q
× [∀T (y ∪
× z)]
[∀T (x)]p ∩
q
$λ
²
$λ
²
× ∀T (y ∪
× z)]
[∀T (x) ∩
r
£ (`)
¤
ϑ ◦ ∀T (κ) ◦ µ r
× ∀T (y ∪
× z)]
[∀T (x) ∩
r
£ (`)
¤
× id) ◦ κ ◦ (id ∩
× ϑ(`) )
(σ ∪
r
× y) ∪
× ∀T (z)]
[∃T (x ∩
r
× y) ∪
× ∀T (z)]
[∃T (x ∩
r
²
²
²
πρ
× y)] ∪
× [∀T (z)]
[∃T (x ∩
s
t
²
πρ
× y)] ∪
× [∀T (z)]
[∃T (x ∩
s
t
coincide. The procedure seen in the proofs of Theorems 4.3 and 5.3 must be followed once
more: each of these composites can be rewritten (expanded) according to the definitions
of the terms involved; to show that their expanded versions of these composites coincide
then quickly follows from the hypothesis, and the commutativity of a certain diagram in
J . We refer to Figures A.6 and A.7 for some of details—the reader who has been alert
up until this point should be able to fill in what remains.
Considering the case I = 1, and applying [6, Theorem 2.4], we obtain the following.
6.3. Corollary. Under the hypotheses of Theorem 4.2, there exists a natural bijection
between linear functors J −→ K and cyclic nuclear monoids in KJ .
We recall that a Frobenius monoid is defined in [6] to be a certain kind of cyclic
nuclear monoid; it therefore seems to make sense to distinguish those linear functors
which correspond to Frobenius monoids.
6.4. Definition. Let J , K be arbitrary linear distributive categories; then a Frobenius
monoidal functor J −→ K is a linear functor such that ∀T = ∃T , σ (`) = µ = σ (r) and
ϑ(`) = δ = ϑ(r) .
F
Equivalently, it is a functor J −→ K equipped with both a monoidal structure (µ, η
× , e), and a comonoidal structure (δ, ε with respect to ∪
× , d) satisfying the
with respect to ∩
diagrams below.
× (y ∪
× z)) o
F (x ∩
µx,y×∪z
× F (y ∪
× z)
F (x) ∩
× δy,z
id ∩
/ F (x) ∩
× (F (y) ∪
× F (z))
F (κ)
²
× y) ∪
× z)
F ((x ∩
× y) ∩
× z) o
F ((x ∪
F (κ)
δx×∩y,z
µx×∪y,z
²
× (y ∩
× z))
F (x ∪
δx,y×∩z
/ F (x ∩
× y) ∪
× F (z) o
× y) ∩
× F (z)
F (x ∪
/ F (x) ∪
× F (y ∩
× z) o
²
κ
× id
µx,y ∪
× id
δx,y ∩
× F (y)) ∪
× F (z)
(F (x) ∩
/ (F (x) ∪
× F (y)) ∩
× F (z)
²
× µy,z
id ∪
κ
× (F (y) ∩
× F (z))
F (x) ∪
324
J.M. EGGER
× , , ∪
× , op )
6.5. Example. We have previously shown that a Frobenius monoid in (Sup; ∩
amounts to a quantale equipped with a dualising element [6, Theorem 4.1]. A Frobenius
× , E, ∪
× , D) —equivalently, a Frobenius functor (→; ∧, >, ∨, ⊥) −→
monoid in (Sup→ ; ∩
op
× , , ∪
× ,  )—similarly amounts to a couple of quantales equipped with a dualising
(Sup; ∩
element in the sense of [7, Definition 1]. [Note that that definition requires a dualising
φ
element of a couple of quantales C −→ Q to be an element of C; therefore, it determines
a unique square
/ op
C
φ
²
²
Q
!
!
/
φ
in Sup, which is to say, a unique arrow (C −→ Q) −→ D in Sup→ .]
Finally we note that the construction, given in [7, Theorem 15], of a single Frobenius
φ
quantale G from a Frobenius couple of quantales C −→ Q can be derived from the fact
that the functor
Sup→
φ
(C −→ Q)
/ Sup
7→
{(α, β) ∈ C × Q | φ(α) ≤ β}
admits a Frobenius monoidal structure.
7. Future Work
A definition of Girard monoid (or, symmetric Frobenius monoid) which could be interpreted in any symmetric linear distributive category was given in [6]; it was further shown
× , , ∪
× , op ) amounts to a quantale equipped with a cyclic
that a Girard monoid in (Sup; ∩
dualising element. It is not hard to show that the star-autonomous structure we have
constructed on Sup→ is also symmetric, nor that a dualising element of a couple of quanφ
tales C −→ Q is cyclic if and only if the corresponding Frobenius monoid in Sup→ is a
Girard monoid; thus Kruml’s notion of a Girard couple of quantales does fit into the more
general framework.
But it has become clear to the author, as a result of joint research with M.B. McCurdy
into cyclic star-autonomous categories [9, 2], that the notion of Girard monoid can, and
should, be studied in even greater generality than that of symmetric linear distributive
categories, and so we have decided to postpone a fuller discussion of Girard monoids (and
the associated Girard functors) until all the pre-requisites for the more general notion are
in place.
It is anticipated that the sequel to this paper will also address the observation, made
in [6, Remark 3.3], that there exists a more general notion than that of cyclic nuclear
STAR-AUTONOMOUS FUNCTOR CATEGORIES
325
monoid, in which we do not have M ∗ ∼
= ∗M . A study of the latter is made more urgent
by the linear distributive structure of KJ , which implies that there is also a more general
concept of “morphism of linear distributive category” than that of linear functor.
Acknowledgements. The author would like to thank: the reviewer for several helpful
comments; Robin Cockett for the suggestion to revise Theorem 6.1 to cover the case I 6= 1;
Robin Houston, again, for helping resolve the dilemma of Example 5.7; David Kruml for
his continuing collaboration; Bob Paré, and many others, for moral support; Phil Scott
for providing office space at the University of Ottawa.
A. Large diagrams
See following pages.
²
Â
Â
Â
²
Â
Â
²
`
´
(Wj ×
∩Xk )×
∩Yk−◦(j−◦s) ×
∪Zt
α×
∪id
`
´
Wj ×
∩(Xk ×
∩Yk−◦(j−◦s) ) ×
∪Zt
[id −◦ β1 ]
`
´
Wj ×
∩ (Xk ×
∩Yk−◦(q;t) )×
∪Zt
id×
∩[β0 ]
id×
∩ ($ε ×
∪id)
(nat)
id×
∩ ($ε ×
∪id)
($id ×
∩id) ×
∪id
(2.4.3)op
$ε ×
∪id
(4.2)
$λ1
(2.4.2)
$λ1
Figure A.1:
/ (Wj ×
∩(X ×
∩Y )j−◦s ) ×
∪Zt
²
/ Wj ×
∩ ((X ×
∩Y )q;t ×
∪Zt )
Â
Â
 [β1 ]
²
id×
∩πη
²
Wj ×
∩((X ×
∩Y )×
∪Z)q
(4.2)
id×
∩κq
/ Wj ×
∩(X ×
∩(Y ×
∪Z))q
²
Â
id×
∩$λ0
Wj ×
∩ (Xk ×
∩(Yp;t ×
∪Zt ))
“
”
id×
∩ id×
∩πη
Wj ×
∩ (Xk ×
∩(Y ×
∪Z)p )
²
²
/ (W ×
∩(X ×
∩Y ))s ×
∪Zt
πρ1
((W ×
∩(X ×
∩Y ))×
∪Z)r
κr
²
/ (W ×
∩((X ×
∩Y )×
∪Z))r
(id×
∩ κ)r
/ (W ×
∩(X ×
∩(Y ×
∪Z)))r
αs ×
∪id
(2.4.2)op
(α×
∪id)r
´
/ `(W ×
∩X)j×
∩Yk−◦(j−◦s) ×
∪Zt
∩k ×
$ξ ×
∪id
²
/ ((W ×
∩X)×
∩Y )s ×
∪Zt
O
πρ1
/ (((W ×
∩X)×
∩Y )×
∪Z)r
326
J.M. EGGER
²
²
× (X ∩
× (Y ∪
× Z)))r
(W ∩
$λ1
× (X ∩
× (Y ∪
× Z))q
Wj ∩
× $λ0
id ∩
× (Xk ∩
× (Y ∪
× Z)p )
Wj ∩
αr
(2.4.3)
α
²
²
Figure A.2:
× X) ∩
× Y) ∪
× Z)r
(((W ∩
κr
²
/ ((W ∩
× X) ∩
× (Y ∪
× Z))r
$λ2
× X)j ×
× (Y ∪
× Z)p
(W ∩
∩k ∩
× id
$id ∩
/ (Wj ∩
× Xk ) ∩
× (Y ∪
× Z)p
πρ2
(4.2)
πη
²
/ ((W ∩
× X) ∩
× Y )s ∪
× Zt
$ε
× X)j ×
× Y(j ×
× Zt
(W ∩
∩k ∩
∩k)−◦s ∪
/ (W ∩
× X)j ×
× (Yp;t ∪
× Zt )
∩k ∩
Â
Â
 [β2 ]
Â
²
¡
¢
STAR-AUTONOMOUS FUNCTOR CATEGORIES
327
²
Â
Â
 [k −◦ β1 ]
Â
Â
²
`
´
(Xk ×
∩ Yk−◦(q;t) ) ×
∪ Zt
²
`
´
Wj ×
∩ (Xk ×
∩ Yk−◦(j−◦s) ) ×
∪ Zt
Wj ×
∩
∩ [β0 ]
 id ×
Â
Â
Wj ×
∩ (Xk ×
∩ (Yp;t ×
∪ Zt ))
“
”
id ×
∩ id ×
∩ πη
Wj ×
∩ (Xk ×
∩ (Y ×
∪ Z)p )
α×
∪ id
(∗)
α
(nat)
α
Figure A.3:
($id ×
∩ id) ×
∪ id
×
²
((W ×
∩ X) ×
∩ Y )s ×
∪ Zt
$ξ ×
∪ id
o
BC
$id ∩ id
/ (W ×
/ (Wj ×
∩ X)j×
∩ (Y ×
∩ Xk ) ×
∩ (Y ×
∪ Z)p
∪ Z)p
∩k ×
WWWWW
WWWWW
WWWW$Wid ×
∩π
WWWWWη
id ×
∩ πη
id ×
∩ πη
WWWWW
WWWWW
WW+
²
²
$id ×
∩ id
/ (Wj ×
/ (W ×
∩ X)j×
∩ (Yp;t ×
∩ Xk ) ×
∩ (Yp;t ×
∪ Zt )
∪ Zt )
∩k ×
Â
Â
Â
Â
(nat)
 [β2 ]
 [β2 ]
²
²
($id ×
∩ id) ×
`
´
`
´
∪ id
/
×
×
×
×
×
(W ∩ X)j×
(Wj ∩ Xk ) ∩ Y(j×
∪ Zt
∪ Zt
∩k ∩ Y(j×
∩k)−◦s ×
O ∩k)−◦s
O
3
ED
ggg
g
g
g
g
“
”
“
”
ggg
g
g
g
g
id ×
∩ Yξ̃ ×
id ×
∩ Yξ̃ ×
∪ id
∪ id
“ ggggg ”
ggggg$id ×
g
×
∩
Y
id
g
∪
g
g
ξ̃
g´gg
´
/ `(Wj ×
/ `(W ×
$ε ×
∩ Xk ) ×
∩ Yk−◦(j−◦s) ×
∩ X)j×
∩ Yk−◦(j−◦s) ×
∪ id
∪ Zt
∪ Zt
∩k ×
328
J.M. EGGER
GF
Xr
υ
× e
Xr O ∩
υr
(2.4.5)
× $id
id ∩
²
/ (X ∩
× E)r
$υ
/ Xr ∩
× Ee
× τ)
(id ∩
r
(2.4.2)
× τe
id ∩
²
²
²
Figure A.4:
Xr o
υr
× X)r
(D ∪
× id)
(γ ∪
r
× X∗ ) ∪
× X)r
((X ∩
κr
²
/ (X ∩
× (X ∗ ∪
× X))r
$υ
/ Xr ∩
× (X ∗ ∪
× X)e
× τη
id ∩
υ
(2.4.5)op
(2.4.2)op
πυ
πυ
(4.2)
× πη
id ∩
²
²
× Xr o
d∪
× id
πid ∪
²
/ Dd ∪
× Xr
× id
γd ∪
²
/ (X ∩
× X ∗ )d ∪
× Xr
× id
$ε ∪
× X ∗ r∗ ) ∪
× Xr
(Xr ∩
κ
ED²
/ Xr ∩
× (X ∗ r∗ ∪
× Xr )
BC
ED
× id
γε ∪
STAR-AUTONOMOUS FUNCTOR CATEGORIES
329
µ̂
²
²
[· · ·] κ
²
²
/ [∀T (x∩
× (y ∪
× z))] × ×
p∩(q ∪r)
[µ]p×∩(q×∪r)
× ∀T (y ∪
× z)] × ×
[∀T (x)∩
p∩(q ∪r)
$id
@A
× y)∪
× z)] × ×
[∀T ((x∩
(p∩q)∪r
× y)∪
× z)] × ×
[∀T ((x∩
p∩(q ∪r)
£
¤
∀T (κ) p×∩(q×∪r)
@A
GF
GF
× [∀T (y ∪
× z)] ×
[∀T (x)]p ∩
q ∪r
²
²
²
[· · ·] κ
× ∃T (y))∪
× ∀T (z)] × ×
[(∀T (x)∩
p∩(q ∪r)
× (∃T (y)∪
× ∀T (z))] × ×
¤ / [∀T (x)∩
p∩(q ∪r)
× ϑ(`)
id∩
£
¤
×
×
p∩(q ∪r)
κ p×∩(q×∪r)
$id
(6.2)
× πid
id∩
Figure A.5:
ϑ̂(`)
πid
× ∃T (y))∪
× ∀T (z)] × ×
[(∀T (x)∩
(p∩q)∪r
£ (`)
¤
× id
σ ∪
£ (`) ¤
(2.4.2)op
(p×
∩q)×
∪r
²
ϑ (p×∩q)×∪r
πid
/ [∃T (x∩
× y)∪
× ∀T (z)] × ×
(p∩q)∪r
£
(2.4.2)
£ ¤
× ϑ(`)
id∩
q×
∪r
/ [∀T (x)] ∩
× [∃T (y)∪
× ∀T (z)] ×
p
q ∪r
× ϑ̂(`)
id∩
´
²
σ (`)
¤
p×
∩q
× id
∪
/ [∃T (x∩
× y)] × ∪
× [∀T (z)] o
p∩q
r
O
BC
²
£
× id
$id ∪
× [∃T (y)]
×
[∀T (x)]p ∩
q ∪ [∀T (z)]r
²
/ [∀T (x)∩
× ∃T (y)] × ∪
× [∀T (z)]
p∩q
r
³
κ
³ ED²
´
/ [∀T (x)] ∩
× [∃T (y)] ∪
× [∀T (z)]
p
q
r
BC
ED
σ̂ (`)
330
J.M. EGGER
[∀T (x)]p ×
∩ [∀T (y×
∪z)](q;t)×
∪t
Figure A.6:
[∀T ((x×
∩y)×
∪z)]s×
∪t
VVVV
i4
VVVVid×
iiii
i
VV∩Vϑ̂V(`)
i
i
VVVV
ii
i
i
i
VVV+
iiii
“
”
/
/
×
×
×
×
[∀T (x)]p ∩ [∀T (y ∪z)]q
[∀T (x)]p ×
∩ [∃T (y)]q;t ×
∪ [∀T (z)]t
h
i [∀T (x)]p ∩ [∃T (y)∪∀T (z)]q
×
id
∩
π
(`)
η
Â
id×
∩ ϑ
lll
q
µ̂ lll
 [β]
l
$λ
$λ
l
l
l
l
Â
l
l
l
²
ul
²
²
“
”
/ [∀T (x)×
[∀T (x×
∩(y×
[∀T (x)×
∩∀T (y×
∩ (∃T (y)×
∪z))]p×
[∀T (x)]p ×
∩ [∃T (y)]p−◦s ×
∪z)]r h
∪∀T (z))]r
∪ [∀T (z)]t
∩q
i
SSS
(`)
VVVV
id×
∩ϑ
SSS
VVVVσ̂(`)×
h
i
r
SSS
VVV∪Vid
κ
[µ]r
×
$
id
S
∪
ε
VVVV
SSS
r
[· · ·]λ
VVV+
SS)
²
²
²
πρ
/ [∀T (x)×
[∃T (x×
∩y)]p×
∪(∀T (z))t
[(∀T (x)×
∩∃T (y)) ×
∩∃T (y)]s ×
[∀T (x×
∩(y×
∪∀T (z)]r
∪(∀T (z))t
∪z))]r
∩(p−◦s) ×
g
g
g
g
h
i
h
i
h
i
ggggg
σ (`) ×
σ (`) ×
∀T ( κ )
∪id
∪id
h
i
gg[·gg· ·]gε ×
s
r
r
g
g
∪id
(`)
g
g
ϑ
²
sgg
²
²
πρ
r
/ [∃T (x×
/ [∃T (x×
∩y)]s ×
[∀T ((x×
∩y)×
∩y)×
∪ [∀T (z)]t
∪z)]r
∪∀T (z)]r
UUUU
gg3
UUUU
ggggg
g
g
UUUU
g
g
UUUU
ggggϑ̂(`)
[· · ·]ρ
U*
ggggg
id×
∩ [· · ·]η
STAR-AUTONOMOUS FUNCTOR CATEGORIES
331
²
²
²
× y) ∪
× z)]
[∀T ((x ∩
r
× (y ∪
× z))]
[∀T (x ∩
r
£
¤
∀T (κ) r
[· · ·]λ
× (y ∪
× z))] ×
[∀T (x ∩
p∩q
µ̂
× [∀T (y ∪
× z)]
[∀T (x)]p ∩
q
[· · ·]ρ
[· · ·]id×∩η
× [· · ·]
id ∩
η
µ̂
[· · ·]ε×∪id
Figure A.7:
/ [∀T ((x ∩
× y) ∪
× z)] ×
s∪t
²
× y) ∪
× z)] ×
[∀T ((x ∩
(p∩(p−◦s))×
∪t
/ [∀T (x ∩
× (y ∪
× z))] ×
p∩((q;t)×
∪t)
Â
£
¤
 ∀ (κ)
T
[β]
²
²
/ [∀T (x)] ∩
× [∀T (y ∪
× z)]
p
(q;t)×
∪t
ϑ̂
(`)
ϑ̂(`)
× ϑ̂(`)
id ∩
× id
σ̂ (`) ∪
× id
[· · ·]ε ∪
/ [∃T (x ∩
× y)] ∪
× [∀T (z)]
s
t
²
/ [∃T (x ∩
× y)] ×
×
p∩(p−◦s) ∪ [∀T (z)]t
²
× [∃T (y)]
×
[∀T (x)]p ∩
p−◦s ∪ [∀T (z)]t
/ [∀T (x)] ∩
× [∃T (y)]
× [∀T (z)]
∪
p
t
 q;t
Â
² [β]
³
´
332
J.M. EGGER
STAR-AUTONOMOUS FUNCTOR CATEGORIES
333
References
[1] Michael Barr. ∗-autonomous categories, volume 752 of Lecture Notes in Mathematics.
Springer, Berlin, 1979. With an appendix by Po Hsiang Chu.
[2] Richard F. Blute, François Lamarche, and Paul Ruet. Entropic Hopf algebras and
models of non-commutative logic. Theory Appl. Categ., 10:No. 17, 424–460 (electronic), 2002.
[3] J. R. B. Cockett and R. A. G. Seely. Weakly distributive categories. J. Pure Appl.
Algebra, 114(2):133–173, 1997.
[4] J. R. B. Cockett and R. A. G. Seely. Linearly distributive functors. J. Pure Appl.
Algebra, 143(1-3):155–203, 1999. Special volume on the occasion of the 60th birthday
of Professor Michael Barr (Montreal, QC, 1997).
[5] Brian Day. On closed categories of functors. In Reports of the Midwest Category
Seminar, IV, Lecture Notes in Mathematics, Vol. 137, pages 1–38. Springer, Berlin,
1970.
[6] J.M. Egger. The Frobenius relations meet linear distributivity. Submitted to Proceedings of CT’06 (White Point), volume 19 of Theory and Applications of Categories,
2007.
[7] J.M. Egger and David Kruml. Girard couples of quantales. Appl. Categ. Structures,
16, 2008.
[8] André Joyal and Myles Tierney. An extension of the Galois theory of Grothendieck.
Mem. Amer. Math. Soc., 51(309):vii+71, 1984.
[9] Kimmo I. Rosenthal. Quantales and their applications, volume 234 of Pitman Research
Notes in Mathematics Series. Longman, Essex, 1990.
Laboratory for Foundations of Computer Science
School of Informatics, University of Edinburgh
Edinburgh EH8 9AB, United Kingdom
Email: jeffegger@yahoo.ca
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