A short introduction

Quantaloids and their calculus

Monoidal structure for quantaloids
Extending the monoidal structure to Q-categories
Calculus in the monoidal quantaloid
where is the physics?
So why monoidal quantaloids?
References

Monoidal Quantaloids as a framework

for (bi)categorical QM
Part I: Introduction

1
Emmanuel Galatoulas

1
Department of Mathematics
University of Athens

Category Theory 2010

Genova, Italy

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Emmanuel Galatoulas Monoidal Quantaloids
 A short introduction
Quantaloids and their calculus
Monoidal structure for quantaloids
Extending the monoidal structure to Q-categories
Calculus in the monoidal quantaloid
where is the physics?
So why monoidal quantaloids?
References

1 A short introduction

2 Quantaloids and their calculus

3 Monoidal structure for quantaloids

4 Extending the monoidal structure to Q-categories

5 Calculus in the monoidal quantaloid

6 where is the physics?

7 So why monoidal quantaloids?

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Emmanuel Galatoulas Monoidal Quantaloids
 A short introduction
Quantaloids and their calculus
Monoidal structure for quantaloids
Extending the monoidal structure to Q-categories
Calculus in the monoidal quantaloid
where is the physics?
So why monoidal quantaloids?
References

A brief prehistory on the subject

• the notion of a monoidal quantaloid is implicit in the pioneering

work of Carboni, Walters et al on cartesian bicategories (part I)
• Andrew Pitt in his ”Applications of Sup-lattice enriched category
theory to Sheaf theory” introduced cartesian Suplattice-enriched
categories and its application in the distributive categories of
relations
• from Quantales to Quantaloids (Mulvey and Rosenthal)
• Garraway’s work on Involutive quantaloids and Semi-quantaloids
• lot of activity on enrichement over commutative quantales (metric
spaces etc.)
• Stubbe systematised in a series of papers quantaloidal calculus and
enrichement
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Emmanuel Galatoulas Monoidal Quantaloids
 A short introduction
Quantaloids and their calculus
Monoidal structure for quantaloids
Extending the monoidal structure to Q-categories
Calculus in the monoidal quantaloid
where is the physics?
So why monoidal quantaloids?
References

on the other hand the notion of monoidal bicategory is not new either
(and quantaloids are particularly simple bicategories):
• monoidal bicategories eg. in Street’s and Day’s work, Gray monoids
and so on, however the presentation is rather ”awkward”
• perhaps the most ”elementary” presentation of monoidal
bicategories is quite recent: in Robin Houston’s PhD thesis (”Linear
Logic w/o units”)

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Emmanuel Galatoulas Monoidal Quantaloids
 A short introduction
Quantaloids and their calculus
Monoidal structure for quantaloids
Extending the monoidal structure to Q-categories
Calculus in the monoidal quantaloid
where is the physics?
So why monoidal quantaloids?
References

what is a quantaloid?

• A Sup enriched category

• a locally small complete and cocomplete partially ordered bicategory
with colimits stable under composition in both sides
• a quantaloid is biclosed: both pre-composing and post-composing
with an arrow f : u → v have right adjoints:

− ◦ f ⊣ {f , −} and f ◦ − ⊣ [f , −]

All these make a quantaloid an appropriate basis for enrichement

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Emmanuel Galatoulas Monoidal Quantaloids
 A short introduction
Quantaloids and their calculus
Monoidal structure for quantaloids
Extending the monoidal structure to Q-categories
Calculus in the monoidal quantaloid
where is the physics?
So why monoidal quantaloids?
References

The quantaloid Dist(Q)

has objects the categories enriched over Q, namely:
• Q-typed sets of objects with type: ta = u ∈ Q
• homarrows A(a, a′ ) : ta′ → ta satisfying:
unit-inequalities: 1ta ≤ A(a, a)
transitivity: A(a, a′ ) ◦ A(a′ , a′′ ) ≤ A(a, a′′ )
and arrows the distributors between Q-categories, namely families of
Q-arrows satisfying:

B(b ′ , b) ◦ Φ(b, a) ≤ Φ(b ′ , a)

Φ(b, a) ◦ A(a, a′ ) ≤ Φ(b, a′ )

Dist(Q) is the universal direct-sum and split-monad completion of Q in

QUANT.
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Emmanuel Galatoulas Monoidal Quantaloids
 A short introduction
Quantaloids and their calculus
Monoidal structure for quantaloids
Extending the monoidal structure to Q-categories
Calculus in the monoidal quantaloid
where is the physics?
So why monoidal quantaloids?
References

The locally preordered 2-category Cat(Q)

has objects the categories enriched over Q and arrows functors between
them, ie. an object mapping F : A0 → B0 satisfying:
type equalities t(Fa) = ta, ∀a ∈ A
action inequalities A(a′ , a) ≤ B(Fa′ , Fa)
Dist(Q) and Cat(Q) relate by means of this 2-functor:

Cat(Q) −→ Dist(Q) : F 7−→ B(−, F−)

each functor F : A → B induces a pair of adjoint distributors
B(−, F−) ⊣ B(F−, −) which we denote as F, F respectively

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Emmanuel Galatoulas Monoidal Quantaloids
 A short introduction
Quantaloids and their calculus
Monoidal structure for quantaloids
Extending the monoidal structure to Q-categories
Calculus in the monoidal quantaloid
where is the physics?
So why monoidal quantaloids?
References

the advantage of being bicategorical!

is that the bicategorical notions and machinery are directly available in

the quantaloidal calculus, namely things like:
• adjoints
• Kan extensions
• equivalent categories
and so on

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Emmanuel Galatoulas Monoidal Quantaloids
 A short introduction
Quantaloids and their calculus
Monoidal structure for quantaloids
Extending the monoidal structure to Q-categories
Calculus in the monoidal quantaloid
where is the physics?
So why monoidal quantaloids?
References

The main definition

A monoidal quantaloid is a quantaloid Q endowed with a tensor product,
namely a quantaloid homomorphism (ie. a Sup-enriched functor)
⊗ : Q × Q → Q satisfying:
interchange law: for appropriate arrows
f : u → v , g : v → w , f ′ : u ′ → v ′ , g ′ : v ′ → w ′ we get:
(g ⊗ g ′ ) ◦ (f ⊗ f ′ ) = (g ◦ f ) ⊗ (g ′ ◦ f ′ ) (1)
tensor preserves suprema:
∨ ∨ ∨
(fi ⊗ gj ) = ( fi ) ⊗ ( gj ) (2)
i,j i j

associativity: isos αABC : ((A⊗B) ⊗ C)(((∼ =, A ⊗ (B⊗C)), ),((

,[),u))nit for ⊗:] a (unique up to iso) object i such that:
ρu : u ⊗ i ∼
= u and λu : i ⊗ u ∼=u . . . .... .... .... . . . . .
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Emmanuel Galatoulas Monoidal Quantaloids
 A short introduction
Quantaloids and their calculus
Monoidal structure for quantaloids
Extending the monoidal structure to Q-categories
Calculus in the monoidal quantaloid
where is the physics?
So why monoidal quantaloids?
References

diagrammatically:

f ⊗f ′ 1 v ⊗ v′ g ⊗g ′
u
f /v g
/w

′ f′ / v′ g′
/ w′ u ⊗ u′ w9 ⊗ w ′
u

(g ◦f )⊗(g ′ ◦f ′ )

and:
∨
∨ i,j (fi ⊗gj )
i fi
u /v
'
∨
gj
u ⊗ u′ = v8 ⊗ v ′
u′
j
/ v′
∨ ∨
( i fi )⊗( j gj )

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Emmanuel Galatoulas Monoidal Quantaloids
 A short introduction
Quantaloids and their calculus
Monoidal structure for quantaloids
Extending the monoidal structure to Q-categories
Calculus in the monoidal quantaloid
where is the physics?
So why monoidal quantaloids?
References

a couple of remarks

the definition is a direct adaptation of the generic definition of a

monoidal bicategory as a homomorphism B × B → B from which for
instance the ”interchange law” obtains immediately.
Also the normality of the tensor means that:

1u⊗v = 1u ⊗ 1v

for all objects u, v in Q which is really the most crucial ingredient in

extending the monoidal structure to Q-categories.
What enables the simplification of the generic definition is really the
locally partially ordered structure of Q: 2-cells are just inequalities, there
are no non-identical isomorphic 1-cells and all diagrams involving 2-cells
automatically commute.
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Emmanuel Galatoulas Monoidal Quantaloids
 A short introduction
Quantaloids and their calculus
Monoidal structure for quantaloids
Extending the monoidal structure to Q-categories
Calculus in the monoidal quantaloid
where is the physics?
So why monoidal quantaloids?
References

Two very useful lemmas

Lemma (tensor ”respects” order)

Given arrows f ≤ g : u → v and h ≤ k : u ′ → v ′ then:

f ⊗h ≤g ⊗h

A similar result holds for composition:

Lemma (composition respects order)
Given arrows f ≤ g : u → v and k : w → u, h : v → z then

f ◦ k ≤ g ◦ k and h ◦ f ≤ h ◦ g

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Emmanuel Galatoulas Monoidal Quantaloids
 A short introduction
Quantaloids and their calculus
Monoidal structure for quantaloids
Extending the monoidal structure to Q-categories
Calculus in the monoidal quantaloid
where is the physics?
So why monoidal quantaloids?
References

extend ⊗ to Dist(Q)

our main goal here is to extend the monoidal structure of the ”basis”
quantaloid Q to the quantaloid of Q-categories Dist(Q).
What should be an appropriate notion of tensor for Q-categories?

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Emmanuel Galatoulas Monoidal Quantaloids
 A short introduction
Quantaloids and their calculus
Monoidal structure for quantaloids
Extending the monoidal structure to Q-categories
Calculus in the monoidal quantaloid
where is the physics?
So why monoidal quantaloids?
References

Defining the tensor for Q-categories

Given Q-categories A and B enriched over a monoidal quantaloid
(Q, ⊗, i) we define:
the tensor A⊗B: to be the Q-typed set having:
objects: pairs (a, b) of objects a ∈ A0 and b ∈ B0
with type t(a, b) = ta ⊗ tb
homarrows: for objects (a, b), (a′ , b ′ ) in A⊗B the
homarrow (A⊗B)((a, b), (a′ , b ′ )) is given by
the formula:
(A⊗B)((a, b), (a′ , b ′ )) = A(a, a′ ) ⊗ B(b, b ′ )
the tensor Φ ⊗ Ψ: of Φ : A −→ C and Ψ : B −→ D between Q-categories
we define their tensor Φ ⊗ Ψ : A⊗B → C⊗D to be
determined by the following family of Q-arrows:
(Φ⊗Ψ)((c, d), (a, b)) = Φ(c, a)⊗Ψ(d, b) : ta⊗tb → tc⊗td
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Emmanuel Galatoulas Monoidal Quantaloids
 A short introduction
Quantaloids and their calculus
Monoidal structure for quantaloids
Extending the monoidal structure to Q-categories
Calculus in the monoidal quantaloid
where is the physics?
So why monoidal quantaloids?
References

see the diagrams:

A(a,a′ )
ta′ / ta
(A⊗B)((a,b),(a′ ,b ′ ))
ta′ ⊗ tb ′ / ta⊗tb
:=A(a,a′ )⊗B(b,b ′ )
tb′ / tb
B(b,b ′ )

Φ(c,a)
ta / tc
(Φ⊗Ψ)((c,d),(a,b))
ta ⊗ tb / tc ⊗ td
:=Φ(c,a)⊗Ψ(d,b)
tb / td
Ψ(d,b)

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Emmanuel Galatoulas Monoidal Quantaloids
 A short introduction
Quantaloids and their calculus
Monoidal structure for quantaloids
Extending the monoidal structure to Q-categories
Calculus in the monoidal quantaloid
where is the physics?
So why monoidal quantaloids?
References

Outline of the proof of the main theorem

In order to show that the structure defined in the aforegiven definition

does endow Dist(Q) with a tensor someone should prove that:
• the Q-typed set A⊗B is a Q-category
• the tensor of distributors Φ ⊗ Ψ is a distributor between the
corresponding Q-categories
• there exists a Q-category I which is the unit for the tensor
• the tensor is associative
• the tensor thus defined is indeed a quantaloidal homomorphism
⊗ : Dist(Q) × Dist(Q) → Dist(Q) satisfying the ”interchange” and
the ”preservation” properties

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Emmanuel Galatoulas Monoidal Quantaloids
 A short introduction
Quantaloids and their calculus
Monoidal structure for quantaloids
Extending the monoidal structure to Q-categories
Calculus in the monoidal quantaloid
where is the physics?
So why monoidal quantaloids?
References

The lemmas

This is indeed what we prove in a series of lemmas. The less trivial is

perhaps to construct the isomorphisms satisfied by the unit:
Lemma (the unit of (Dist(Q), ⊗):)
The unit for the tensor in Dist(Q) is the Q-category I having the single
object {∗i } of type i (i being the unit for the tensor in Q itself) and the
single morphism I(∗i , ∗i ) = 1i = 1, is the unique (up to isomorphism)
unit for the tensor in Dist(Q) with the (natural) isomorphisms
νA : A⊗I ∼= A and µA : I⊗A ∼ = A provided by appropriate distributors

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Emmanuel Galatoulas Monoidal Quantaloids
 A short introduction
Quantaloids and their calculus
Monoidal structure for quantaloids
Extending the monoidal structure to Q-categories
Calculus in the monoidal quantaloid
where is the physics?
So why monoidal quantaloids?
References

the main theorem

After having proved all the corresponding lemmas it follows that:

Theorem (the monoidal quantaloid (Dist(Q), ⊗, I))
If Q is a monoidal quantaloid, then (Dist(Q), ⊗, I), with the tensor
structure defined as in the Definition is also monoidal

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Emmanuel Galatoulas Monoidal Quantaloids
 A short introduction
Quantaloids and their calculus
Monoidal structure for quantaloids
Extending the monoidal structure to Q-categories
Calculus in the monoidal quantaloid
where is the physics?
So why monoidal quantaloids?
References

Tensor of presheaves
Lemma (tensor of presheaves)
Given presheaves ϕ : ∗u −→ A and ψ : ∗v −→ B their tensor is also a
\
presheaf ϕ ⊗ ψ : u ⊗ v −→ A⊗B of type t(ϕ ⊗ ψ) = tϕ ⊗ tψ = u ⊗ v
Lemma (right lifting and extension of tensors)
Given presheaves ϕ : ∗u −→ A, ϕ′ : ∗v −→ A and ψ : ∗u −→ B and
ψ ′ : ∗v −→ B the right lifting [ϕ ⊗ ψ, ϕ′ ⊗ ψ ′ ] of their tensor satisfies the
inequality:
[ϕ, ϕ′ ] ⊗ [ψ, ψ ′ ] ≤ [ϕ ⊗ ψ, ϕ′ ⊗ ψ ′ ]
and given presheaves ϕ : ∗u −→ A, ϕ′ : ∗u −→ C and ψ : ∗v −→ B,
ψ ′ : ∗v −→ C the right extension of the tensors satisfies:

{ϕ, ϕ′ } ⊗ {ψ, ψ ′ } ≤ {ϕ ⊗ ψ, ϕ′ ⊗ ψ ′ }
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Emmanuel Galatoulas Monoidal Quantaloids
 A short introduction
Quantaloids and their calculus
Monoidal structure for quantaloids
Extending the monoidal structure to Q-categories
Calculus in the monoidal quantaloid
where is the physics?
So why monoidal quantaloids?
References

the speciality of Cauchy presheaves

Cauchy presheaves (and distributors in general) are just the left adjoint
ones (usually also called maps). They determine a full subcategory of the
free cocompletion PA which is the Cauchy completion Acc of A. They
behave quite nicely:
Lemma (tensor of Cauchy presheaves)
Given Cauchy presheaves ϕ : ∗u −→ A and ψ : ∗v −→ B on A and B
respectively, their tensor ϕ ⊗ ψ is also Cauchy, its adjoint given by
ϕ⊗ψ =ϕ⊗ψ
which hold in general:
Theorem (tensor of adjoint distributors)
Given Cauchy distributors Φ : A −→ C and Φ : B −→ D their tensor is
also Cauchy, its adjoint be given by:

Φ⊗Ψ=Φ⊗Ψ .
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Emmanuel Galatoulas Monoidal Quantaloids

 A short introduction
Quantaloids and their calculus
Monoidal structure for quantaloids
Extending the monoidal structure to Q-categories
Calculus in the monoidal quantaloid
where is the physics?
So why monoidal quantaloids?
References

Theorem (lifting/extension of the tensor of Cauchy presheaves)

If the presheaves ϕ, ψ of the previous lemma are Cauchy, then the
corresponding inequalities saturate to equalities, namely:

[ϕ, ϕ′ ] ⊗ [ψ, ψ ′ ] = [ϕ ⊗ ψ, ϕ′ ⊗ ψ ′ ]
{ϕ, ϕ′ } ⊗ {ψ, ψ ′ } = {ϕ ⊗ ψ, ϕ′ ⊗ ψ ′ }

which gives also as a direct consequence that for ϕ, ψ Cauchy presheaves

∥ϕ ⊗ ψ∥ = ∥ϕ∥ ⊗ ∥ψ∥

where ∥ϕ∥ denotes the lifting [ϕ, ϕ]

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Emmanuel Galatoulas Monoidal Quantaloids
 A short introduction
Quantaloids and their calculus
Monoidal structure for quantaloids
Extending the monoidal structure to Q-categories
Calculus in the monoidal quantaloid
where is the physics?
So why monoidal quantaloids?
References

why bicategories?

In the so called dagger compact closed categories model of Quantum

Information/Computation someone works in a (symmetric) monoidal
category and this explains why the ”dagger” structure has to be imposed.
There is not a notion of an adjoint arrow in a monoidal category!
However a monoidal categrory is actually equivalent to (the suspension
of) a single-object bicategory. Arrows in V are the 1-cells and the
monoidal structure is the composition.
This motivates us to work directly with bicategories and with quantaloids
in particular!

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Emmanuel Galatoulas Monoidal Quantaloids
 A short introduction
Quantaloids and their calculus
Monoidal structure for quantaloids
Extending the monoidal structure to Q-categories
Calculus in the monoidal quantaloid
where is the physics?
So why monoidal quantaloids?
References

So recall that in the DCCCs models a state for instance is defined as an

arrow f : I → A where I is the unit of the tensor. And that the scalars is
precisely the (commutative in this case) category of the endoarrows
s : I → I (since V is self-enriched!)
Is there an analogue in the monoidal quantaloid Dist(Q)? Of course! It
is the presheaves ϕ : I → A.
Mind though that seen as a bicategory a monoidal category is
single-object (or untyped) whereas a quanaloid is a many-object
bicategory (so from this point of view the closest analogue to V would be
a quantale!)
This motivates the following terminology:
• we call (all) presheaves over a Q-category A states (those of type i
are the unital ones)
• we call the category PA the free state space over A

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Emmanuel Galatoulas Monoidal Quantaloids
 A short introduction
Quantaloids and their calculus
Monoidal structure for quantaloids
Extending the monoidal structure to Q-categories
Calculus in the monoidal quantaloid
where is the physics?
So why monoidal quantaloids?
References

this ”interpetation” means amongst other that:

• A quantum mechanical system A over the quantaloid Q is a
Q-enriched category, ie. an object of Dist(Q) (or Cat(Q) for that
matter!)
• The Yoneda Lemma expresses the natural embedding of any system
A embedds naturally and fully faithfully into its free state space PA
yA : A ,→ PA
′
• homarrows A(a, a ) represent transition amplitudes for transitions
a → a′ in the system A.
• homarrows in the free state space PA represent transition amplitude
between states of the system A:
PA(ϕ, ψ) = [ϕ, ψ] = ∥ϕ → ψ∥
Notice that for states of the same type these ”amplitudes” take
values in the quantale Q(u, u) . . . . . . . . . . . . . . . . . . . .
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Emmanuel Galatoulas Monoidal Quantaloids
 A short introduction
Quantaloids and their calculus
Monoidal structure for quantaloids
Extending the monoidal structure to Q-categories
Calculus in the monoidal quantaloid
where is the physics?
So why monoidal quantaloids?
References

• By the Yoneda Lemma, every presheaf is a colimit of representables:

colim(ϕ, yA ) = ∆ϕ

which motivates to interpet representables as ”eigenstates” thus

making the Yoneda Lemma sort of the categorical analogue of the
Spectral Theorem. Colimits of representables are the analogue of
the quantum superpositions.
• in fact we can calculate the components (projections) of a state ϕ
along any eigenstate A(−, a) = ea : they are simply:
PA(A(−, a), ϕ) = ϕ(a)
• the tensor category A⊗B is then the composite system and the
corresponding state space is just P(A⊗B).

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Emmanuel Galatoulas Monoidal Quantaloids
 A short introduction
Quantaloids and their calculus
Monoidal structure for quantaloids
Extending the monoidal structure to Q-categories
Calculus in the monoidal quantaloid
where is the physics?
So why monoidal quantaloids?
References

a setting to understand disentanglement

As an illustration of the plausibility of our approach let us present here

some thoughts on (dis)entanglement!
Let us first ask: what should we expect an entangled state over A⊗B to
be in Dist(Q)?
Answer: It would be a state (ie. a presheaf) ϕ : ∗u → A⊗B which cannot
be ”analysed” into a tensor ϕa ⊗ ϕb of states over A and B!
Now let us try to ”formalise” it a bit! To this end the following
Q-categories are naturally involved:

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Emmanuel Galatoulas Monoidal Quantaloids
 A short introduction
Quantaloids and their calculus
Monoidal structure for quantaloids
Extending the monoidal structure to Q-categories
Calculus in the monoidal quantaloid
where is the physics?
So why monoidal quantaloids?
References

The categories P(A⊗B) and PA⊗PB

The tensor of state spaces PA⊗PB has by definition:
objects : pairs ϕA , ϕB of presheaves over the systems A and B
respectively with type: t(ϕA , ϕB ) = tϕA ⊗ tϕB
homarrows: for pairs of presheaves (ϕA , ϕB ) and (ψA , ψB ) we have:

(PA⊗PB)((ϕA , ϕB ), (ψA , ψB )) =
PA(ϕA , ψA ) ⊗ PB(ϕB , ψB ) =
[ϕA , ψA ] ⊗ [ϕB , ψB ]

On the other hand consider P(A⊗B), ie. the category of (contravariant)

presheaves over the tensor category A⊗B having:
objects: presheaves ϕ : ∗u −→ A⊗B
homarrows: (P(A⊗B))(ϕ, ψ) = [ϕ, ψ] . . . . . . . . . . . . . . . . . . . .
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Emmanuel Galatoulas Monoidal Quantaloids
 A short introduction
Quantaloids and their calculus
Monoidal structure for quantaloids
Extending the monoidal structure to Q-categories
Calculus in the monoidal quantaloid
where is the physics?
So why monoidal quantaloids?
References

the (Dis)entanglement functor

Let us then introduce the assignment D : PA⊗PB −→ P(A⊗B) acting as

follows:
D : (ϕA , ϕB ) 7→ ϕA ⊗ ϕB
ie. mapping the object (ϕA , ϕB ), a pair of states over A and B, to their
tensor ϕA ⊗ ϕB , which is indeed a state over A⊗B)
In the first place we prove:
Theorem (D is a functor)
The assignment defined D : PA⊗PB −→ P(A⊗B) above is functorial.
We call it the disentanglement functor.

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Emmanuel Galatoulas Monoidal Quantaloids
 A short introduction
Quantaloids and their calculus
Monoidal structure for quantaloids
Extending the monoidal structure to Q-categories
Calculus in the monoidal quantaloid
where is the physics?
So why monoidal quantaloids?
References

but there is more! D is a Kan extension!

Lemma (The left Kan extension ⟨yA⊗B , yA ⊗yB ⟩:)

The left Kan extension ⟨yA⊗B , yA ⊗yB ⟩ given by the colimit
colim(yA ⊗yB , yA⊗B ) of the ”tensor” Yoneda embedding yA⊗B weighted
by the right adjoint distributor induced by yA ⊗yB , always exists and it is
just the (dis)entanglement functor itself:

colim(yA ⊗yB , yA⊗B ) ∼

=D

where the functors involved are:

yA⊗B : A⊗B → P(A⊗B)

yA ⊗yB : A⊗B → PA⊗PB
. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
Emmanuel Galatoulas Monoidal Quantaloids
 A short introduction
Quantaloids and their calculus
Monoidal structure for quantaloids
Extending the monoidal structure to Q-categories
Calculus in the monoidal quantaloid
where is the physics?
So why monoidal quantaloids?
References

the left Kan extension ⟨yA ⊗yB , yA⊗B ⟩

However the left Kan extenstion ⟨yA ⊗yB , yA⊗B ⟩ does not always exist! In
fact its existence imposes a ”disentanglement” condition:
Lemma
⟨yA ⊗yB , yA⊗B ⟩ exists iff colim(ϕ, yA ⊗yB ) exists, ie. iff there exist a
(constant) functor ∆(ϕA ,ϕB ) : ∗tϕ −→ PA⊗PB picking out the object
(ϕA , ϕB ) of A⊗B, necessarily of type: t(ϕA , ϕB ) = tϕA ⊗ tϕB = tϕ such
that:
∀ψA , ψB ∈ P(A⊗B) : ∆(ϕA ,ϕB ) = [ϕ, yA ⊗yB ]
or equivalently:

[ϕA , ψA ] ⊗ [ϕB , ψB ] = [ϕ, ψA ⊗ ψB ]

When the left Kan extension is well defined we denote it as:

⟨yA ⊗yB , yA⊗B ⟩ = y⊗ . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
Emmanuel Galatoulas Monoidal Quantaloids
 A short introduction
Quantaloids and their calculus
Monoidal structure for quantaloids
Extending the monoidal structure to Q-categories
Calculus in the monoidal quantaloid
where is the physics?
So why monoidal quantaloids?
References

the disentanglement adjunction: y⊗ ⊣ D

Therefore if the ”disentanglement” condition is satisfied for all ϕ, ie . if
the left Kan extension y⊗ exists then we have the adjunction y⊗ ⊣ D:

P(A⊗B)
(P(A⊗B))(yA⊗B (−,−),−)=yA⊗B
7 C O

yA⊗B

 y⊗
A⊗B D=⟨yA⊗B ,yA⊗yB ⟩ ⟨yA⊗yB ,yA⊗B ⟩=y⊗
Y D

yA⊗yB
'  
(PA⊗PB)((yA⊗yB )(−,−),(−,−))=yA⊗yB
PA⊗PB
and we get the factorization: yA⊗B ∼
= D ◦ (yA ⊗yB. ).. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
Emmanuel Galatoulas Monoidal Quantaloids
 A short introduction
Quantaloids and their calculus
Monoidal structure for quantaloids
Extending the monoidal structure to Q-categories
Calculus in the monoidal quantaloid
where is the physics?
So why monoidal quantaloids?
References

Disentanglement as the equivalence P(A⊗B) ≃ PA⊗PB

However we want disentanglement to mean: for every state ϕ on A⊗B
there is a (unique) pair of presheaves (ϕA , ϕB ) on A and B respectively
which is precisely the colim(ϕ, yA ⊗yB ) such that ϕ = ϕA ⊗ ϕB . This
means that the adjunction y⊗ ⊣ D must really be an equivalence!
Definition
The system A⊗B is disentangled (or decomposable or separable) iff D has
the left adjoint y⊗ = ⟨yA ⊗yB , yA⊗B ⟩ and the unit and counit of the
adjunction y⊗ ⊣ D determine this equivalence in Cat(Q):

y⊗
/ #
P(A⊗B) o PA⊗PB
d D

D
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. . . . . . . . . . . . . . . . . . . .
Emmanuel Galatoulas Monoidal Quantaloids
 A short introduction
Quantaloids and their calculus
Monoidal structure for quantaloids
Extending the monoidal structure to Q-categories
Calculus in the monoidal quantaloid
where is the physics?
So why monoidal quantaloids?
References

Prospectus

• how far the analogy with DCCCs can go? This seems to involve an
involution on the quanaloids as well
• applications to quantales (quantales are single-object quantaloids)
• applications to Q-orders (or Q-sheaves) (Ord(Q) is basically
Map(Dist(Qsi )) and Qsi hence Dist(Qsi ) are also monoidal)
• and of course how do Cartesian quantaloids fit in this framework!
For instance: how does our ”disentanglement” condition in terms of
the equivalence P(A⊗B) ≃ PA⊗PB relate to the
Frobenius-separability condition in the context of cartesian
bicategories?

. . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . .
Emmanuel Galatoulas Monoidal Quantaloids
 A short introduction
Quantaloids and their calculus
Monoidal structure for quantaloids
Extending the monoidal structure to Q-categories
Calculus in the monoidal quantaloid
where is the physics?
So why monoidal quantaloids?
References

Some references:

Only indicative!
[1] P.Katis, Sabadini, RFC Walters, Bicategories of Processes, Journal of Pure and
Applied Algebra, vol. 115. pp. 141-178, 1997
[2] I.Stubbe, Categories enriched over quantaloids: categories, distributors and
functors, Theory and Applications of Categories 14, 1-45.
[3] H.Heymans and I.Stubbe, On principally generated Q-modules in general, and
skew local homeomorphism in particular
[4] B.Coecke, D.Palvovic: Quantum measurements without sums
[5] Carboni A., Walters RFC: Cartesian Bicategories I
[6] D. Garraway: Sheaves for an Involutive Quantaloid, Cahiers de Top. et Geom.
Diff. Categoriques, vol. XLVI-4, 2005
[7] Street Day Monoidal bicategories and Hopf algebroids

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Emmanuel Galatoulas Monoidal Quantaloids