Categorical models of computation: partially traced categories and presheaf models of quantum computation
aa r X i v : . [ m a t h . C T ] J a n CATEGORICAL MODELS OF COMPUTATION:PARTIALLY TRACED CATEGORIES AND PRESHEAFMODELS OF QUANTUM COMPUTATION
ByOctavio Malherbe, B.Sc., M.Sc.Thesis submitted to theFaculty of Graduate and Postdoctoral StudiesUniversity of Ottawain partial fulfillment of the requirements for thePhD degree in theOttawa-Carleton Institute for Graduate Studies and Research in Mathematics and Statisticsc (cid:13) bstract
This dissertation has two main parts. The first part deals with questions relatingto Haghverdi and Scott’s notion of partially traced categories. The main result isa representation theorem for such categories: we prove that every partially tracedcategory can be faithfully embedded in a totally traced category. Also conversely,every monoidal subcategory of a totally traced category is partially traced, so thischaracterizes the partially traced categories completely. The main technique we useis based on Freyd’s paracategories, along with a partial version of Joyal, Street, andVerity’s Int construction. Along the way, we discuss some new examples of partiallytraced categories, mostly arising in the context of quantum computation.The second part deals with the construction of categorical models of higher-orderquantum computation. We construct a concrete semantic model of Selinger and Val-iron’s quantum lambda calculus, which has been an open problem until now. We dothis by considering presheaf categories over appropriate base categories arising fromfirst-order quantum computation. The main technical ingredients are Day’s convolu-tion theory and Kelly and Freyd’s notion of continuity of functors. We first give anabstract description of the properties required of the base categories for the modelconstruction to work; then exhibit a specific example of base categories satisfyingthese properties. ii cknowledgements
I want first to express my very deep gratitude to Phil Scott and Peter Selinger. Thisthesis would never have come into existence without their advice and encouragement.I have also benefited from stimulating discussions with Sergey Slavnov, Benoˆıt Val-iron and Mark Weber. I am grateful to my examiners, Richard Blute, Robin Cockett,Pieter Hofstra and Benjamin Steinberg, for their helpful comments and useful sug-gestions on this work. I am in addition particularly indebted to the University ofOttawa and Dalhousie University. Finally, I would like to thank In´es and Reina forsupporting me throughout my time as a student.iii o Magdalena iv ontents Abstract iiAcknowledgements iiiiv1 Introduction 12 Some mathematical background 4
FinSet → C + . . . . . . . . . . . . . . . . . . . . . . . 192.6 Affine monoidal categories . . . . . . . . . . . . . . . . . . . . . . . . 212.7 Traced monoidal categories . . . . . . . . . . . . . . . . . . . . . . . . 242.8 Graphical language . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.9 Compact closed categories . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊗ . . . . . . . . 434.2.4 Partial trace in the category Vect . . . . . . . . . . . . . . . . 464.2.5 Completely positive maps with ⊕ . . . . . . . . . . . . . . . . 574.3 Partial trace in a monoidal subcategory of a partially traced category 614.4 Another partial trace on completely positive maps with ⊕ . . . . . . 704.5 Partial trace on superoperators with ⊕ and ⊗ . . . . . . . . . . . . . 71 C , A ] Γ . . . . . . . . . . . . . . . . . . . . 1416.10 Day’s reflection theorem . . . . . . . . . . . . . . . . . . . . . . . . . 1496.11 Application of Day’s reflection theorem to presheaves . . . . . . . . . 158vi Presheaf models 160 C has finite coproducts then C T has finite coproducts . . . . . . . . 1777.8 The functor H : D → ˆ C T . . . . . . . . . . . . . . . . . . . . . . . . . 1787.8.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 1787.8.2 Definition of H . . . . . . . . . . . . . . . . . . . . . . . . . . . 1807.8.3 C : ˆ C T → ˆ D Γ is a strong monoidal functor . . . . . . . . . . . 1827.8.4 H is a strong monoidal functor . . . . . . . . . . . . . . . . . 1847.8.5 H preserves coproducts . . . . . . . . . . . . . . . . . . . . . . 1887.9 F T ⊣ G T is a monoidal adjunction . . . . . . . . . . . . . . . . . . . . 1897.10 Abstract model of the quantum lambda calculus . . . . . . . . . . . . 190 Srel fn . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1918.2 The category Q ′′ and the functors Φ and Ψ . . . . . . . . . . . . . . . 1938.3 A concrete model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 vii hapter 1Introduction Quantum computers are computing devices which are based on the laws of quantumphysics. While no actual general-purpose quantum computer has yet been built,research in the last two decades indicates that quantum computers would be vastlymore powerful than classical computers. For instance, Shor proved in 1994 that theinteger factoring problem can be solved in polynomial time on a quantum computer,while no efficient classical algorithm is known.The goal of this research is to extend existing connections between logic andcomputation, and to apply them to the field of quantum computation. Logic hasbeen applied to the study of classical computation in many ways. For instance, thelambda calculus, a prototypical programming language invented by Church and Curryin the 1930’s, can be simultaneously regarded as a programming language and as aformalism for writing mathematical proofs. This observation has become the basis forthe development of several modern programming languages, including ML, Haskell,and Lisp.Recent research by Selinger, Valiron, and others has shown that the logical sys-tem which corresponds most closely to quantum computation is the so-called “linearlogic” of Girard. Linear logic, a resource sensitive logic, formalizes one of the cen-tral principles of quantum physics, the so-called “no-cloning property”, which assertsthat a given quantum state cannot be replicated. This property is reflected on thelogical side by the requirement that a given logical assumption (or “resource”) can1
HAPTER 1. INTRODUCTION
HAPTER 1. INTRODUCTION
B → C → D , and on a family Γ of cones in D . We use this data to obtain a pairof adjunctions [ B op , Set ] L / / [ C op , Set ] F / / Φ ∗ ⊥ o o [ D op , Set ] Γ G ⊥ o o and give sufficient conditions on B → C → D and Γ so that the resulting structure isa model of the quantum lambda calculus.This provides a general framework in which one can describe various classes ofmodels that depend on the concrete choice of the parameters B , C , D , and Γ. hapter 2Some mathematical background The aim of this chapter is to review some basic categorical background material thatis needed to understand this thesis. For a more detailed discussion, see [54], [15], and[52]. The reader who is already familiar with category theory can skip this chapterinitially, and refer back to it when needed.
In what follows, Id C is the identity functor on a category C and 1 G is the identitynatural transformation on a functor G . Given a category C , the symbol C ( A, B )denotes the set of morphisms from A to B . Definition 2.1.1 (Adjunction) . Let A and B be categories. An adjunction from A to B is a quadruple ( F, G, η, ε ) where F : A → B and G : B → A are functorsand η : Id A ⇒ GF and ε : F G ⇒ Id B are natural transformations such that:( Gε ) ◦ ( ηG ) = 1 G and ( εF ) ◦ ( F η ) = 1 F . The functor F is said to be a left adjoint for G or G a right adjoint for F and we use the following notation: F ⊣ G or( F, G, η, ε ) : A ⇀ B or even more graphically A F / / B . G ⊥ o o . 4 HAPTER 2. SOME MATHEMATICAL BACKGROUND Definition 2.1.2 (Monads) . A monad or a triple on a category C is a 3-tuple ( T, η, µ )where T : C → C is an endofunctor and η : Id C ⇒ T (unit law), µ : T ⇒ T (multi-plication law) are two natural transformations, satisfying the following conditions: T ◦ Id C T η + ❍❍❍❍❍❍❍❍❍❍ ❍❍❍❍❍❍❍❍❍❍ T µ (cid:11) (cid:19) Id C ◦ T ηT k s ✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈✈ T T µ + µT (cid:11) (cid:19) T µ (cid:11) (cid:19) T T µ + T Theorem 2.1.3 (Huber) . If F ⊣ G with unit η : Id A ⇒ GF and co-unit ε : F G ⇒ Id B , then ( GF, η, GεF ) is a monad on A .Proof. See Lambek and Scott [52].Suppose we have two adjunctions: (
F, G, η, ε ) : A ⇀ B and ( F ′ , G ′ , η ′ , ε ′ ) : B ⇀ CA F / / B F ′ / / G ⊥ o o C G ′ ⊥ o o We can consider the composite: ( F ′ F, GG ′ , Gη ′ F ◦ η, ε ′ ◦ F ′ εG ′ ) : A ⇀ C yielding anadjunction from A to C . Therefore, by Theorem 2.1.3 ( T, ˜ η, ˜ µ ) with T = GG ′ F ′ F ,˜ η = Gη ′ F ◦ η and ˜ µ = GG ′ ( ε ′ ◦ F ′ εG ′ ) F ′ F is a monad defined by this new adjunction.Next we recall the comparison theorem for the Kleisli category. Definition 2.1.4.
Given a monad (
T, η, µ ) on a category C , the Kleisli category C T is determined by the following conditions:- Obj ( C T ) = Obj ( C )- C T ( A, B ) = C ( A, T B )- f K ◦ K g K = µ C ◦ T ( g ) ◦ f when A f K → B and B g K → C are arrows in C T . Theidentity is given by 1 KC = η C : C → T C .There is an adjunction between the category C and the Kleisli category C T , givenby the following: • F T ( A ) = A and F T ( f ) = η B ◦ f if A f → B is an arrow in C . HAPTER 2. SOME MATHEMATICAL BACKGROUND • G T ( B ) = T ( B ) and G T ( f K ) = µ B ◦ T ( f ) if A f K → B is an arrow in C T .The adjunction F T ⊣ G T has the following universal property: given any otheradjunction F ⊣ G such that G ◦ F = T , there exists a unique functor C : C T → D ,called the comparison functor , with the following properties C ◦ F T = F and G ◦ C = G T . • C ( A ) = F ( A ) on objects and F T ( f ) = η B ◦ f if A f → B is an arrow in C . • C ( f ) = ε F B ◦ F ( f ) when A f K → B is an arrow in C T . C F T (cid:15) (cid:15) F / / D G ⊥ o o C TG T ⊢ O O C ? ? ⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦⑦ First we evaluate the identity: C (1 KA ) = C ( η A ) = ε F A ◦ F ( η A ) = 1 F A by definitionof the adjoint pair.Now, suppose we have A f K → B and B g K → C a pair of arrows in C T i.e. a pair A f → GF B and B g → GF C in C . We want to prove that C ( g ◦ K f ) = C ( g ) ◦ C ( f ).We have that: F A F ( g ◦ K f )= F ( µ C T ( g ) f ) " " ❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋ F f / / F GF B
F T g = F GF g (cid:15) (cid:15) ε F B / / F B
F g (cid:15) (cid:15)
F GF GF C F ( µ C )= F Gε
F C (cid:15) (cid:15) ε F GF C / / F GF C ε F C (cid:15) (cid:15)
F GF C ε F C / / F C
Where the top square commutes by naturality of ε with F ( g ) and the bottom squareby naturality of ε with ε F C . The top leg of the diagram is C ( g ) ◦ C ( f ) since C ( f ) = ε F B ◦ F f and C ( g ) = ε F C ◦ F g . The bottom leg is C ( g ◦ K f ) = ε F C ◦ F ( g ◦ K f ). HAPTER 2. SOME MATHEMATICAL BACKGROUND C : C T ( A, B ) → D ( C ( A ) , C ( B ))( A f → T B ) ( F A F ( f ) → F GF B ε F B → F B )Therefore define a function C − by C − : D ( F A, F B ) → C T ( A, B )( F A g → F B ) ( A η A → GF A G ( g ) → GF B )i.e., C − ( g ) = G ( g ) ◦ η A . Definition 2.2.1. A monoidal category, also often called tensor category, is a cate-gory V with a unit object I ∈ V together with a bifunctor ⊗ : V × V → V and naturalisomorphisms ρ : A ⊗ I ∼ = → A , λ : I ⊗ A ∼ = → A , α : A ⊗ ( B ⊗ C ) ∼ = → ( A ⊗ B ) ⊗ C ,satisfying the following coherence axioms: A ⊗ ( I ⊗ B ) ⊗ λ ( ( ◗◗◗◗◗◗◗ α / / ( A ⊗ I ) ⊗ B ρ ⊗ v v ♠♠♠♠♠♠♠ A ⊗ B and A ⊗ ( B ⊗ ( C ⊗ D )) α (cid:15) (cid:15) α / / ( A ⊗ B ) ⊗ ( C ⊗ D ) α / / (( A ⊗ B ) ⊗ C ) ⊗ D α (cid:15) (cid:15) ( A ⊗ (( B ⊗ C ) ⊗ D ) α / / ( A ⊗ ( B ⊗ C )) ⊗ D. Definition 2.2.2. A symmetric monoidal category consists of a monoidal category( V , ⊗ , I, α, ρ, λ ) with a chosen natural isomorphism σ : A ⊗ B ∼ = → B ⊗ A , called symmetry , which satisfies the following coherence axioms: A ⊗ B σ / / id & & ▼▼▼▼▼▼ B ⊗ A σ (cid:15) (cid:15) A ⊗ B A ⊗ I σ / / ρ & & ▲▲▲▲▲▲▲▲ I ⊗ A λ (cid:15) (cid:15) A HAPTER 2. SOME MATHEMATICAL BACKGROUND A ⊗ ( B ⊗ C ) ⊗ σ (cid:15) (cid:15) α / / ( A ⊗ B ) ⊗ C σ / / C ⊗ ( A ⊗ B ) α (cid:15) (cid:15) A ⊗ ( C ⊗ B ) α / / ( A ⊗ C ) ⊗ B σ ⊗ / / ( C ⊗ A ) ⊗ B. Definition 2.2.3. A symmetric monoidal closed category category is a symmetricmonoidal category V for which each functor − ⊗ B : V → V has a right adjoint[ B, − ] : V → V , i.e. : V ( A ⊗ B, C ) ∼ = V ( A, [ B, C ]) . Definition 2.2.4. A monoidal functor ( F, m
A,B , m I ) between monoidal categories( V , ⊗ , I, α, ρ, λ ) and ( W , ⊗ ′ , I ′ , α ′ , ρ ′ , λ ′ ) is a functor F : V → W equipped with:- morphisms m A,B : F ( A ) ⊗ ′ F ( B ) → F ( A ⊗ B ) natural in A and B ,- a morphism m I : I ′ → F ( I ),which satisfy the following coherence axioms: F A ⊗ ′ ( F B ⊗ ′ F C ) α ′ (cid:15) (cid:15) ⊗ ′ m / / F A ⊗ ′ F ( B ⊗ C ) m / / F ( A ⊗ ( B ⊗ C )) F α (cid:15) (cid:15) ( F A ⊗ ′ F B ) ⊗ ′ F C m ⊗ ′ / / F ( A ⊗ B ) ⊗ ′ F C m / / F (( A ⊗ B ) ⊗ C ) F A ⊗ ′ I ′ ρ ′ / / ⊗ ′ m (cid:15) (cid:15) F AF A ⊗ ′ F I m / / F ( A ⊗ I ) F ρ O O I ′ ⊗ ′ F A m ⊗ ′ (cid:15) (cid:15) λ ′ / / F AF I ⊗ ′ F A m / / F ( I ⊗ A ) . F ( λ ) O O A monoidal functor is strong when m I and for every A and B m
A,B are isomor-phisms. It is said to be strict when all the m A,B and m I are identities. Remark 2.2.5.
Throughout the remainder of this exposition whenever we write(
F, m ) we symbolize a monoidal functor where m not only represents the naturaltransformation m A,B : F A ⊗ F B → F ( A ⊗ B ) but also m I : I → F I relating theunits of the two monoidal categories.
HAPTER 2. SOME MATHEMATICAL BACKGROUND Definition 2.2.6. If V and W are symmetric monoidal categories with naturalsymmetry maps σ and σ ′ , a symmetric monoidal functor is a monoidal functor( F, m
A,B , m I ) satisfying the following axiom: F A ⊗ ′ F B σ ′ / / m (cid:15) (cid:15) F B ⊗ ′ F A m (cid:15) (cid:15) F ( A ⊗ B ) F ( σ ) / / F ( B ⊗ A ) Definition 2.2.7. A monoidal natural transformation θ : ( F, m ) → ( G, n ) betweenmonoidal functors is a natural transformation θ A : F A → GA such that the followingaxioms hold: F A ⊗ ′ F B m / / θ A ⊗ ′ θ B (cid:15) (cid:15) F ( A ⊗ B ) θ A ⊗ B (cid:15) (cid:15) GA ⊗ ′ GB n / / G ( A ⊗ B ) I ′ m I / / n I ! ! ❈❈❈❈❈❈❈❈ F I θ I (cid:15) (cid:15) GI.
Definition 2.3.1. A monoidal adjunction ( V , ⊗ , I ) ( F,m ) / / ( W , ⊗ ′ , I ′ ) ( G,n ) ⊥ o o between two monoidal categories V and W consists of an adjunction ( F, G, η, ε ) inwhich (
F, m ) and (
G, n ) are monoidal functors and the unit η : Id ⇒ G ◦ F andthe counit ε : F ◦ G ⇒ Id are monoidal natural transformations, as defined inDefinition 2.2.7. Definition 2.3.2.
Let ( V , ⊗ , I ) be a monoidal category. A monoidal monad ( T, η, µ, m ) on V is a monad ( T, η, µ ) such that the endofunctor T : V → V is amonoidal functor (
T, m ) with m A,B : T A ⊗ T B → T ( A ⊗ B ) and m : I → T I ascoherence maps, and the natural transformations η : Id ⇒ T and µ : T ⇒ T aremonoidal natural transformations. HAPTER 2. SOME MATHEMATICAL BACKGROUND Lemma 2.3.3.
Let T be a monoidal monad. Consider the Kleisli adjunction C F T / / C TG T ⊥ o o as in Definition 2.1.4. Then C T is a monoidal category and F T ⊣ G T is a monoidal adjunction, where- m TA,B : F T ( A ) ⊗ F T ( B ) → F T ( A ⊗ B ) is given by η : A ⊗ B → T ( A ⊗ B ) ,- m TI = η I : I → T ( I ) ,- n TA,B : G T ( A ) ⊗ G T ( B ) → G T ( A ⊗ B ) is given by m A,B : T ( A ) ⊗ T ( B ) → T ( A ⊗ B ) , and- n TI = η I : I → T ( I ) . Definition 2.3.4. A strong monad ( T, η, µ, t ) is a monad (
T, η, µ ) and a naturaltransformation t A,B : A ⊗ T B → T ( A ⊗ B ) called a strength satisfying the followingaxioms: I ⊗ T A λ & & ▼▼▼▼▼▼▼▼▼▼▼ t I,A / / T ( I ⊗ A ) T ( λ ) (cid:15) (cid:15) T ( A ) A ⊗ B η A ⊗ B & & ▼▼▼▼▼▼▼▼▼▼ ⊗ η B / / A ⊗ T B t A,B (cid:15) (cid:15) T ( A ⊗ B )( A ⊗ B ) ⊗ T C t A ⊗ B,C / / α A,B,TC (cid:15) (cid:15) T (( A ⊗ B ) ⊗ C ) T ( α A,B,C ) (cid:15) (cid:15) A ⊗ ( B ⊗ T C ) ⊗ t B,C / / A ⊗ T ( B ⊗ C ) t A,B ⊗ C / / T ( A ⊗ ( B ⊗ C )) A ⊗ T B ⊗ µ B (cid:15) (cid:15) t A,TB / / T ( A ⊗ T B ) T ( t A,B ) / / T ( A ⊗ B ) µ A ⊗ B (cid:15) (cid:15) A ⊗ T B t A,B / / T ( A ⊗ B ) . Remark 2.3.5.
Let (
T, η, µ, m ) be a symmetric monoidal monad. A strong monadcan be defined in which the strength t A,B is given by the following formula: A ⊗ T B η ⊗ −→ T A ⊗ T B m A,B −→ T ( A ⊗ B )see Theorem 2.1 in [49]. HAPTER 2. SOME MATHEMATICAL BACKGROUND
Proposition 2.3.6 (Kelly) . Let ( F, m ) :
C → C ′ be a monoidal functor. Then F hasa right adjoint G for which the adjunction ( F, m ) ⊣ ( G, n ) is monoidal if and only if F has a right adjoint F ⊣ G and F is strong monoidal.Proof. Here we give a sketch; see [42], [44] or [55] for a detailed proof. Since we havethat C ′ ( F A, B ) ∼ = C ( A, GB ) then there is a unique n A,B and n I such that: F ( GA ⊗ GB ) F ( n A,B ) / / m − GA,GB (cid:15) (cid:15)
F G ( A ⊗ ′ B ) ǫ A ⊗ B (cid:15) (cid:15) F GA ⊗ ′ F GB ǫ A ⊗ ǫ B / / A ⊗ ′ B F I F ( n I ) / / m − I ●●●●●●●●●● F GI ′ ǫ I ′ (cid:15) (cid:15) I ′ Then using the adjunction we check that this candidates satisfy the definition.
We recall some properties of the finite coproduct completion of a category. A referencecan be found in [17].
Definition 2.4.1.
Let us consider the category
FinSet whose objects are finite sets A = { a , . . . , a n } and whose arrows are functions. To avoid any problem about thesize of this category, we assume without loss of generality that all objects of FinSet are subsets of a given fixed infinite set; thus
FinSet can be regarded as a smallcategory.Note that
FinSet has finite coproducts and products.
Definition 2.4.2.
Let C be a category. The category C + has as its objects finitefamilies of objects of C : V = { V a } a ∈ A , with A a finite set. A morphism from V = { V a } a ∈ A to W = { W b } b ∈ B consists of the following two items:- a function φ : A → B - a family f = { f a } a ∈ A of morphisms of C f a : V a → W φ ( a ) . HAPTER 2. SOME MATHEMATICAL BACKGROUND Notation : We shall denote a morphism of C + as a pair F = ( φ, f ). Moreover,sometimes we write V ab instead of ( V a ) b to emphasize some particular set index sub-script, and in the same way for arrows.Before we study any possible structure in C + we observe that this is really acategory. The identity map is given by taking φ = id A the identity function on A and f a = 1 V a , the identity map in C , for every a ∈ A .Composition is defined by the following rule: if F = ( φ, f ) and G = ( ψ, g ) then G ◦ C + F = ( ψ ◦ φ, { g φ ( a ) ◦ f a } a ∈ A ).To verify the associative law for the composition we have that if F = ( φ, f ), G = ( ψ, g ) and H = ( λ, h ) then: H ◦ ( G ◦ F ) = H ◦ ( ψ ◦ φ, { g φ ( a ) ◦ f a } a ∈ A ) = ( λ ◦ ( ψ ◦ φ ) , { h ψ ◦ φ ( a ) ◦ ( g φ ( a ) ◦ f a ) } a ∈ A ) =(( λ ◦ ψ ) ◦ φ, { ( h ψ ( φ ( a )) ◦ g φ ( a ) ) ◦ f a } a ∈ A ) = ( λ ◦ ψ, { h ψ ( b ) ◦ g b } b ∈ B ) ◦ F = ( H ◦ G ) ◦ F. Lemma 2.4.3. C + has finite coproducts.Proof. On objects we have that if V = { V a } a ∈ A , W = { W b } b ∈ B then V ⊕ W = { Z c } c ∈ C where C = A + B is the coproduct in FinSet . We take Z in ( a ) = V a and Z in ( b ) = W b for every a ∈ A , b ∈ B . Thus, V ⊕ W is just a concatenation of families of objects of C . Injections maps are defined in the following way: { V a } a ∈ A i −→ { Z c } c ∈ C and { W b } b ∈ B i −→ { Z c } c ∈ C where i = ( in , Id VA ), i = ( in , Id WB ) are given by: A in −→ A + B B in −→ A + B injections in FinSet and Id VA = { Va } a ∈ A , Id WB = { Wb } b ∈ B where V a Va −→ V a and W b Wb −→ W b are identitiesin C . Notation : Sometimes we shall use V ⊕ W for Z , so we have the following notation V ⊕ W = { ( V ⊕ W ) c } c ∈ A + B .There is also an initial object that we shall denote by ǫ . It is the empty family ofobjects. The unique morphism ǫ ǫ W −→ { W b } b ∈ B is given by ǫ W = ( ∅ , ∅ ). HAPTER 2. SOME MATHEMATICAL BACKGROUND C , we associate a functor I : C → C + as follows, I ( V ) = { V ∗ } ∗∈ , V ∗ = V and when there is a f : V → W in C then I ( f ) = ( id , { f ∗ } ∗∈ ) with f ∗ = f . Proposition 2.4.4.
Given any category A with finite coproducts ` and any functor F : C → A , there is a unique finite coproduct preserving functor G : C + → A , up tonatural isomorphism, such that G ◦ I = F . C I (cid:15) (cid:15) F / / AC + G > > ⑥⑥⑥⑥⑥⑥⑥⑥ Proof.
We shall begin by considering the definition of the functor G : C + → A thatassigns to each object V = { V a } a ∈ A the coproduct G ( { V a } a ∈ A ) = ` a ∈ A F ( V a ) in thecategory A . For any arrow { V a } a ∈ A ( φ,f ) −→ { W b } b ∈ B we define G ( φ, f ) = [ i F ( W φ ( a ) ) ◦ F ( f a )] a ∈ A as the unique arrow in A such that the following diagram commutes: F ( V a ) i F ( Va ) (cid:15) (cid:15) F ( f a ) / / F ( W φ ( a ) ) i F ( Wφ ( a )) (cid:15) (cid:15) ` a ∈ A F ( V a ) G ( φ,f ) / / ❴❴❴ ` b ∈ B F ( W b )We must show that G is a functor. To see this, suppose we have { V a } a ∈ A ( φ,f ) −→ { W b } b ∈ B ( ψ,g ) −→ { Z c } c ∈ C then by hypothesis F ( W b ) i F ( Wb ) (cid:15) (cid:15) F ( g b ) / / F ( Z ψ ( b ) ) i F ( Zψ ( b )) (cid:15) (cid:15) ` b ∈ B F ( W b ) G ( ψ,g ) / / ` c ∈ C F ( Z c )therefore using the case b = φ ( a ) we obtain F ( V a ) i F ( Va ) (cid:15) (cid:15) F ( f a ) / / F ( W φ ( a ) ) i F ( Wφ ( a )) (cid:15) (cid:15) F ( g φ ( a ) ) / / F ( Z ψ ( φ ( a )) ) i F ( Zψ ( φ ( a ))) (cid:15) (cid:15) ` a ∈ A F ( V a ) G ( φ,f ) / / ` b ∈ B F ( W b ) G ( ψ,g ) / / ` c ∈ C F ( Z c ) HAPTER 2. SOME MATHEMATICAL BACKGROUND G (( ψ, g ) ◦ ( φ, f )) = G ( ψ, g ) ◦ G ( φ, f ). Also by uniqueness it is easily to check that G ( id A , id VA ) = id ` a ∈ A F ( V a ) .The functor G preserves coproducts. To see this let us consider V i = { V ia } a ∈ A i , i ∈ I then G ( ⊕ i ∈ I V i ) = G ( ⊕ i ∈ I { V ia } a ∈ A i ) = G ( { Z c } c ∈⊕ i ∈ I A i ) = a c ∈⊕ i ∈ I A i F ( Z c ) ∼ = a i ∈ I ( a a ∈ A i F ( V ia )) == a i ∈ I G ( { V ia } a ∈ A i ) = a i ∈ I G ( V i )with Z c = V ia if in A i ( a ) = c . It remains to verify that G is unique up to naturalisomorphism. Suppose there is another H preserving coproducts such that H ◦ I = F .Therefore, using the definitions given above of coproduct in C + , the functor G andthe fact that by hypothesis H preserves coproducts, we calculate on objects H ( { V a } a ∈ A ) ∼ = H ( ⊕ a ∈ A { V a ∗ } ∗∈ ) ∼ = a a ∈ A H ( { V a ∗ } ∗∈ ) == a a ∈ A H ( I ( V a )) = a a ∈ A F ( V a ) = G ( { V a } a ∈ A )Suppose we have a morphism { V a } a ∈ A ( φ,f ) −→ { W b } b ∈ B with φ : A → B and f = { f a } a ∈ A then using the coproduct in C + we consider a decomposition of it, up toisomorphism, in the following way ⊕ a ∈ A { V a ∗ } ∗∈ i Wφ ( a ) ◦ I ( f a )] a ∈ A −→ ⊕ b ∈ B { W b ∗ } ∗∈ these morphisms are explicitly given by { V a ∗ } ∗∈ I ( f a ) −→ { W φ ( a ) ∗ } ∗∈ i Wφ ( a ) −→ ⊕ b ∈ B { W b ∗ } ∗∈ where I ( f a ) = ( id , { f a ∗ } ∗∈ ), i W φ ( a ) = ( in φ ( a ) , { W φ ( a ) ∗ } ∗∈ ) with 1 in φ ( a ) −→ ⊕ B W φ ( a ) 1 Wφ ( a ) ∗ =1 −→ W φ ( a ) and ⊕ b ∈ B { V b ∗ } ∗∈ = { Z c } c ∈⊕ B with Z in b ( ∗ ) = W b ∗ = W b .Since H preserves coproducts H ([ i W φ ( a ) ◦ I ( f a )] a ∈ A ) ∼ = [ H ( i W φ ( a ) ) ◦ H ( I ( f a ))] a ∈ A = [ i F ( W φ ( a ) ) ◦ F ( f a )] a ∈ A = G ( φ, f ) HAPTER 2. SOME MATHEMATICAL BACKGROUND H ( { W φ ( a ) ∗ } ∗∈ ) H ( i Wφ ( a ) ) −→ H ( ⊕ b ∈ B { W b ∗ } ∗∈ )hence using again that H preserves coproducts, up to isomorphism, we have H ( { W φ ( a ) ∗ } ∗∈ ) H ( i Wφ ( a ) ) −→ a b ∈ B H ( { W b ∗ } ∗∈ )this means by definition of the functor I , H ( I ( W φ ( a ) )) H ( i Wφ ( a ) ) −→ a b ∈ B H ( I ( W b ))but, by hypothesis we know that H ◦ I = F , F ( W φ ( a ) ) i F ( Wφ ( a )) −→ a b ∈ B F ( W b ) Corollary 2.4.5. C + is the free finite coproduct completion generated by C . Proposition 2.4.6. If C is a symmetric monoidal category then C + is also a sym-metric monoidal category.Proof. Assume that V = { V a } a ∈ A and W = { W b } b ∈ B are objects in C + then we take V ⊗ C + W = { V a ⊗ W b } ( a,b ) ∈ A × B where A × B is the finite product of sets.The tensor extends to morphisms, if V F −→ X , W G −→ Y , with X = { X c } c ∈ C , Y = { Y d } d ∈ D , F = ( φ, f ), G = ( ψ, g ) then F ⊗ G = ( φ × ψ, f ¯ ⊗ g ) is given by thefollowing data:- φ × ψ : A × B → C × D , ( φ × ψ )( a, b ) = ( φ ( a ) , ψ ( b ))- f ¯ ⊗ g = { ( f ¯ ⊗ g ) ( a,b ) } ( a,b ) ∈ A × B where we have that( f ¯ ⊗ g ) ( a,b ) : ( V ⊗ W ) ( a,b ) −→ ( X ⊗ Y ) ( φ × ψ )( a,b ) is defined by: f a ⊗ g b : V a ⊗ W b −→ X φ ( a ) ⊗ Y ψ ( b ) HAPTER 2. SOME MATHEMATICAL BACKGROUND − ⊗ C + − : C + × C + → C + is a bifunctor one first calculates the definitionby using that 1 A × B = 1 A × B and 1 V a ⊗ W a = 1 V a ⊗ W b .Next, we shall prove that ( F ◦ F ′ ) ⊗ ( G ◦ G ′ ) = ( F ⊗ G ) ◦ ( F ′ ⊗ G ′ ). Suppose: F ′ = ( φ, f ), F = ( η, h ), G ′ = ( ψ, g ), G = ( ξ, k ) where { V a } a ∈ A ( φ,f ) −→ { X c } c ∈ C ( η,h ) −→ { Z e } e ∈ E and { W b } b ∈ B ( ψ,g ) −→ { Y d } d ∈ D ( ξ,k ) −→ { H f } f ∈ F Therefore, ( F ◦ F ′ ) ⊗ ( G ◦ G ′ ) = (( η ◦ φ ) × ( ξ ◦ ψ ) , { ( h φ ( a ) ◦ f a ) ⊗ ( k ψ ( b ) ◦ g b ) } ( a,b ) ∈ A × B ) =(( η × ξ ) ◦ ( φ × ψ ) , { ( h φ ( a ) ⊗ k ψ ( b ) ) ◦ ( f a ⊗ g b ) } ( a,b ) ∈ A × B ) = ( F ⊗ G ) ◦ ( F ′ ⊗ G ′ ) wherewe simplify the notation of the tensor symbol. The unit of the tensor is given by I = { I ∗ } ∗∈{∗} . The tensor functor is equipped with the following set of isomorphisms:- V ⊗ I ¯ ρ −→ V , and I ⊗ V ¯ λ −→ V where V = { V a } a ∈ A , I = { I ∗ } ∗∈{∗} then V ⊗ I = { V a ⊗ I ∗ } ( a, ∗ ) ∈ A × .These maps are given by: ¯ ρ = ( ρ, r ) with ρ : A × {∗} → A , ρ ( a, ∗ ) = a and with r = { r ( a, ∗ ) } ( a, ∗ ) ∈ A × where r ( a, ∗ ) = r V a , V a ⊗ I r Va −→ V a . In an analogous way isdefined ¯ λ = ( λ, l ).- If V = { V a } a ∈ A and W = { W b } b ∈ B then ¯ σ = ( σ, s ) with σ : A × B → B × A , σ ( x, y ) = ( y, x ) and s = { s ( x,y ) } ( x,y ) ∈ A × B , where s ( x,y ) = s i.e, V x ⊗ W y s −→ W y ⊗ V x - If V = { V a } a ∈ A , W = { W b } b ∈ B , Z = { Z c } c ∈ C , then ¯ α = ( α, a ) with α : A × ( B × C ) → ( A × B ) × C , α ( x, ( y, z )) = (( x, y ) , z ) and a = { a ( x, ( y,z )) } ( x, ( y,z )) ∈ A × ( B × C ) ,where a ( x, ( y,z )) = a i.e., V x ⊗ ( W y ⊗ Z z ) a −→ ( V x ⊗ W y ) ⊗ Z z Coherence follows by definition, coherence in
FinSet and coherence in the symmetricmonoidal category C . HAPTER 2. SOME MATHEMATICAL BACKGROUND Remark 2.4.7.
Notice that the distributivity condition V ⊗ ( W ⊕ Z ) ∼ = ( V ⊗ W ) ⊕ ( V ⊗ Z ) is satisfied with the map: D : V ⊗ ( W ⊕ Z ) → ( V ⊗ W ) ⊕ ( V ⊗ Z )where V = { V a } a ∈ A , W = { W b } b ∈ B , Z = { Z c } c ∈ C , D = ( δ, Id ) in which δ is thebijective function δ : ( A + B ) × C → ( A × C ) + ( B × C ) and Id = { d } d ∈ ( A + B ) × C . Example . If is the one object, one arrow strict symmetric monoidal categorywith the evident monoidal structure then + ∼ = FinSet and ⊗ + = × and I = 1. Proposition 2.4.9.
Under the hypotheses of Proposition 2.4.4, assume that the cate-gories C and A are symmetric monoidal. Then I is a symmetric monoidal functor. Ifmoreover F is a symmetric monoidal functor and tensor distributes over coproductsin A , then G is a symmetric monoidal functor. Moreover, if F is strong monoidalthen so is G .Proof. We first show that I is a monoidal functor by considering: I ( V ) ⊗ I ( W ) u −→ I ( V ⊗ W )where V = { V ∗ } ∗∈ , W = { W ∗ } ∗∈ and u = ( µ, { V ∗ ⊗ W ∗ } ( ∗ , ∗ ) ∈ × ) with µ : 1 × → V ∗ ⊗ W ∗ = 1 V ⊗ W . It is easy to check that all the axioms of the definition aresatisfied. As an example we have that by routine calculations the following axiom issatisfied: { V ∗ ⊗ I ∗ } ( ∗ , ∗ ) ∈ × ⊗ (cid:15) (cid:15) ( ρ, { r ∗ , ∗ } ( ∗ , ∗ ) ∈ × ) / / { V ∗ } ∗∈ { V ∗ ⊗ I ∗ } ( ∗ , ∗ ) ∈ × u / / { ( V ⊗ I ) ∗ } ∗∈ I ( r ) O O since ρ I = µ and ( r V ) µ ( ∗ , ∗ ) = ( r V ) ∗ = r V = r ∗ , ∗ .Next assuming that ( F, m ) is monoidal we wish to show that G is also a monoidalfunctor.Since we also assumed that the category ( A , ρ ⊗ , λ ⊗ , α ⊗ , ρ ⊕ , λ ⊕ , α ⊕ , δ, σ ⊗ , σ ⊕ , λ , ρ , I, )is symmetric distributive then there exists isomorphisms of type: φ : ( a a ∈ A F ( V a )) ⊗ ( a b ∈ B F ( W b )) ∼ = → a a ∈ A,b ∈ B F ( V a ) ⊗ F ( W b ) HAPTER 2. SOME MATHEMATICAL BACKGROUND ξ given by the universal property of the coproduct: F ( V a ) ⊗ F ( W b ) m Va,Wb (cid:15) (cid:15) i a,b / / ` a ∈ A,b ∈ B F ( V a ) ⊗ F ( W b ) ξ =[ m Va,Wb ◦ j a,b ] a ∈ A,b ∈ B (cid:15) (cid:15) F ( V a ⊗ W b ) j a,b / / ` a ∈ A,b ∈ B F ( V a ⊗ W b ) (1)Using these maps we define the mediating arrow ϑ : G ( V ) ⊗ G ( W ) → G ( V ⊗ W ) asthe composition ϑ V,W = ξ ◦ φ . We also have that ϑ I : I → G ( I ) is given by m I .To show that ϑ satisfies the axioms of a symmetric monoidal functor we shallonly provide the proof of one of the diagrams. This is justified by obvious coproductproperties: the exterior diagram commutes for every a ∈ A and this implies that theinterior diagram commutes by pre-composing with injections i and using the universalproperty of coproducts: F ( V a ) ⊗ I ρ / / ⊗ m I (cid:15) (cid:15) i ) ) ❘❘❘❘❘❘❘❘❘❘❘❘❘❘ F ( V a ) i v v ♠♠♠♠♠♠♠♠♠♠♠♠♠ ` a ∈ A ( F ( V a ) ⊗ I ) [ i ◦ (1 ⊗ m I )] a ∈ A (cid:15) (cid:15) [ i F ( Va ) ◦ ρ ] a ∈ A / / ` a ∈ A F ( V a ) ` a ∈ A ( F ( V a ) ⊗ F I ) ξ / / ` a ∈ A F ( V a ⊗ I ) [ i F ( Va ) ◦ F ( ρ )] a ∈ A O O F ( V a ) ⊗ F I i ❧❧❧❧❧❧❧❧❧❧❧❧❧❧ m Va,I / / F ( V a ⊗ I ) i h h ◗◗◗◗◗◗◗◗◗◗◗◗ F ( ρ ) O O Then by coherence [53], distributivity of the tensor through coproduct:( A ` B ) ⊗ I ρ * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ δ / / ( A ⊗ I ) ` ( B ⊗ I ) ρ ` ρ (cid:15) (cid:15) A ` B HAPTER 2. SOME MATHEMATICAL BACKGROUND ϑ we may infer that:( ` F ( V a )) ⊗ I ρ , , ❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩ ( ` a ∈ A ⊗ m I (cid:15) (cid:15) ¯ δ ) ) ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ ` a ∈ A ( F ( V a ) ⊗ I ) ` a ∈ A (1 ⊗ m I ) (cid:15) (cid:15) ` a ∈ A ρ F ( Va ) / / ` a ∈ A F ( V a ) ` a ∈ A ( F ( V a ) ⊗ F I ) ξ / / ` a ∈ A F ( V a ⊗ I ) ` a ∈ A F ( ρ Va ) O O ( ` F ( V a )) ⊗ F I φ ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ ϑ V,I ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ commutes, which turns to be: G ( V ) ⊗ I ⊗ ϑ I (cid:15) (cid:15) ρ / / G ( V ) G ( V ) ⊗ G ( I ) ϑ V,I / / G ( V ⊗ I ) G ( ρ ) O O Similarly one could prove the rest of the axioms.Notice that if the mediating arrows m V a ,W b are isomorphisms in diagram (1) abovethen ξ is an isomorphism. Therefore this implies that ϑ V,W is an isomorphism for every V and W i.e., G is a strong functor. Φ :
FinSet → C + . Now we turn to prove that when C is affine, there exists a functor Φ : FinSet → C + which is fully faithful and preserves tensor and coproduct. Definition 2.5.1.
A monoidal category C is called affine if the tensor unit I is aterminal object. Lemma 2.5.2.
Let C be an affine category. Then there exists a fully-faithful strongmonoidal functor Φ : (
FinSet , × , → ( C + , ⊗ C + , I ) that preserves coproducts.Proof. We shall begin by considering the functor Φ which assigns to each finite set A a family Φ( A ) = { C a } a ∈ A , such that for every a ∈ A , C a = I is the unit of the HAPTER 2. SOME MATHEMATICAL BACKGROUND C .Now let A φ −→ B be a function in FinSet , thenΦ( A ) Φ( φ ) −→ Φ( B ) with Φ( φ ) = ( φ, Id A ) and Id A = { a } a ∈ A , I a = id I −→ I. The kind of functor obtained in this way has been motivated in order to satisfy thefollowing properties which are essential for the model.Φ is faithful:
The way we define morphisms in C + allows us to infer that if Φ( φ ) =Φ( ψ ) then φ = ψ .Φ is full: Suppose we have a pair ( φ, f ) ∈ C + (Φ( A ) , Φ( B )) then f = { f a } a ∈ A with I f a −→ I ; since I is a terminal object this implies that f a = 1 a = ! for every a ∈ A .Therefore Φ( φ ) = ( φ, f ).Φ preserves coproducts: Take objects A and B ; then by definition we have thatΦ( A ⊕ B ) = { C c } c ∈ A ⊕ B = { C a } a ∈ A ⊕ { C b } b ∈ B = Φ( A ) ⊕ Φ( B ) . Suppose we have two arrows A φ −→ C , B ψ −→ D then:Φ( φ ⊕ ψ ) = ( φ ⊕ ψ, Id A ⊕ B ) = ( φ ⊕ ψ, Id A ⊕ Id B ) def. = ( φ, Id A ) ⊕ ( ψ, Id B ) = Φ( φ ) ⊕ Φ( ψ )Φ preserves tensor product: Assuming A and B are finite sets thenΦ( A × B ) = { C ( a,b ) } ( a,b ) ∈ A × B = { C a ⊗ C b } ( a,b ) ∈ A × B = { C a } a ∈ A ⊗{ C b } b ∈ B = Φ( A ) ⊗ Φ( B )at the level of objects. If A φ −→ C , B ψ −→ D then we have that naturality is satisfied:Φ( φ × ψ ) = ( φ × ψ, Id A × B ) = ( φ × ψ, Id A ¯ ⊗ Id B ) = ( φ, Id A ) ⊗ ( ψ, Id B ) = Φ( φ ) ⊗ Φ( ψ )since Id A ¯ ⊗ Id B = { (1 ⊗ ( a,b ) } ( a,b ) ∈ A × B = { a ⊗ b } ( a,b ) ∈ A × B = { ( a,b ) } ( a,b ) ∈ A × B = Id A × B .Also, Φ(1) = Φ( {∗} ) = { C ∗ } ∗∈ = I C + .This implies that Φ is a monoidal functor with identity id : Φ( A ) ⊗ Φ( B ) → Φ( A × B ), id : I → Φ(1) as mediating natural transformations. It is a routine
HAPTER 2. SOME MATHEMATICAL BACKGROUND α , ρ and λ , are satisfied.For example, the diagram Φ( B ) ⊗ I ⊗ (cid:15) (cid:15) ¯ ρ / / Φ( B )Φ( B ) ⊗ Φ(1) / / Φ( B × Φ( ρ ) O O is satisfied. To see this, we calculate Φ( ρ ) = ( ρ, { ( a, ∗ ) } ( a, ∗ ) ∈ A × ). On the other handby definition we have that ¯ ρ = ( ρ, r ) with ρ : A × {∗} → A , ρ ( a, ∗ ) = a and with r = { r ( a, ∗ ) } ( a, ∗ ) ∈ A × where r ( a, ∗ ) = r V a , V a ⊗ I r Va −→ V a but since V a = I this implies I r Va =1 I −→ I . Hence, these two arrows are equal. Recall from Definition 2.5.1 that a monoidal category is affine when the tensor unit I is a terminal object. The following construction is well-known. Definition 2.6.1 (Free affine symmetric monoidal category) . Let K be a category.The free affine symmetric monoidal category F wm ( K ) is the category defined asfollows:(a) objects are finite sequences of objects of K : { V i } i ∈ [ n ] = { V , . . . , V n } (b) maps ( φ, { f i } i ∈ [ m ] ) : { V i } i ∈ [ n ] −→ { W i } i ∈ [ m ] are determined by:- an injective function φ : [ m ] → [ n ]- a family of morphism f i : V φ ( i ) → W i in the category K (c) composition ( φ, { f i } i ∈ [ m ] ) ◦ ( ψ, { g i } i ∈ [ s ] ) = ( ψ ◦ φ, { f i ◦ g φ ( i ) } i ∈ [ s ] )(d) the unit is given by the empty sequence. HAPTER 2. SOME MATHEMATICAL BACKGROUND ⊗ is given by concatenation of sequences of objects and arrows: { V i } i ∈ [ n ] ⊗ { W i } i ∈ [ m ] = { Z i } i ∈ [ n + m ] where Z i = V i if 1 ≤ i ≤ n and Z i = W i − n if n + 1 ≤ i ≤ n + m { V i } i ∈ [ n ] ⊗ { W i } i ∈ [ m ] = { P i } i ∈ [ n + m ] ( φ,f ) ⊗ ( ψ,g ) −→ { Q i } i ∈ [ s + t ] = { X i } i ∈ [ s ] ⊗ { Y i } i ∈ [ t ] given by ( φ, f ) ⊗ ( ψ, g ) = ( φ + ψ, f + g ) where φ + ψ : [ s + t ] → [ n + m ] isdefined by ( φ + ψ )( i ) = φ ( i ) if 1 ≤ i ≤ s and ( φ + ψ )( i ) = ψ ( i − s ) + n if s + 1 ≤ i ≤ s + t and f + g = { ( f + g ) j } j ∈ [ s + t ] where ( f + g ) j : P ( φ + ψ )( j ) → Q j is defined by ( f + g ) j = f j if 1 ≤ j ≤ s and ( f + g ) j = g j − s if s + 1 ≤ j ≤ s + t (f) the canonical isomorphisms are strict given by l = r = 1, a = 1 and symmetriesby s = ( σ,
1) with σ : [ n + m ] → [ n + m ] such that σ ( i ) = i + n if 1 ≤ i ≤ m and σ ( i ) = i − m if m + 1 ≤ i ≤ n + m . Remark 2.6.2.
The tensor unit of F wm ( K ) is a terminal object: { V i } i ∈ [ n ] ( ∅ , ∅ ) −→ {} for every { V i } i ∈ [ n ] object in K . In addition, notice that F wm ( K )( {} , { V i } i ∈ [ n ] ) = ∅ if { V i } i ∈ [ n ] = {} . Proposition 2.6.3.
Given any symmetric monoidal category A whose tensor unitis terminal and any functor F : K → A , there is a unique strong monoidal functor G : F wm ( K ) → A , up to isomorphism, such that G ◦ I = F . K I (cid:15) (cid:15) F / / AF wm ( K ) G : : ✉✉✉✉✉✉✉✉✉✉ Proof. (sketch) The functor G is defined on objects by: G ( {} ) = I and G ( { V i } i ∈ [ n ] ) =( . . . ( F ( V ) ⊗ F ( V ) ⊗ F ( V )) . . . ⊗ F ( V n )).Let ( φ, { f i } i ∈ [ m ] ) : { V i } i ∈ [ n ] → { W i } i ∈ [ m ] be a map in F wm ( K ) then G ( φ, { f i } i ∈ [ m ] ) : G ( { V i } i ∈ [ n ] ) → G ( { W i } i ∈ [ m ] ) HAPTER 2. SOME MATHEMATICAL BACKGROUND F ( V ) ⊗ F ( V )) ⊗ . . . ⊗ F ( V n ) = G ( { V i } i ∈ [ n ] ) ( x ⊗ x ) ... ⊗ x n (cid:15) (cid:15) / / ( F ( W ) ⊗ F ( W )) . . . F ( W m ) = G ( { W i } i ∈ [ m ] )( X ⊗ X ) ⊗ . . . ⊗ X n ∼ = / / ( F ( V φ (1) ) ⊗ F ( V φ (2) )) . . . F ( V φ ( m ) ) ( F ( f ) ⊗ F ( f )) ... ⊗ F ( f m ) O O where x i = 1 F ( V i ) : F ( V i ) → F ( V i ) if i ∈ φ ([ m ]) and x i = ! : F ( V i ) → I if i ∈ [ n ] − φ ([ m ]).Using coherence of the category A we prove that G is a strong functor: themediating isomorphism is given by the unique morphism that shifts all the parenthesisto the left: G ( { V i } i ∈ [ n ] ) ⊗ G ( { W i } i ∈ [ m ] ) m −→ G ( { V i } i ∈ [ n ] ⊗ { W i } i ∈ [ m ] )and I m =1 −→ G ( {} ) . To prove uniqueness we use the fact that x i = ! : F ( V i ) → I transforms into x i = ! : G ◦ I ( V i ) → G {} if i ∈ [ n ] − φ ([ m ]) and also that the coherence structure ispreserved, up to isomorphism, for any functor satisfying these conditions. Corollary 2.6.4.
F wm ( K ) is the free affine symmetric monoidal category generatedby K .Example . To illustrate the definition of the functor G in the proof of Proposi-tion 2.6.3, let us consider ( φ, { f i } i ∈ [2] ) : { V , V , V } → { W , W } with φ : [2] → [3], φ (1) = 3 , φ (2) = 1 then G ( φ, { f i } i ∈ [2] ) : G ( { V , V , V } ) → G ( { W , W } )is given by( F ( V ) ⊗ F ( V )) ⊗ F ( V ) = G ( { V , V , V } ) F ( f ) ⊗ ! ⊗ F ( f ) (cid:15) (cid:15) G ( φ, { f i } i ∈ [2] ) / / F ( W ) ⊗ F ( W ) = G ( { W , W } )( F ( W ) ⊗ I ) ⊗ F ( W ) ρ ⊗ / / F ( W ) ⊗ F ( W ) . σ O O HAPTER 2. SOME MATHEMATICAL BACKGROUND We recall the definition of a trace from [41].
Definition 2.7.1. A trace for a symmetric monoidal category ( C , ⊗ , I, ρ, λ, s ) consistsof a family of functions Tr UA,B : C ( A ⊗ U, B ⊗ U ) → C ( A, B )natural in A , B , and dinatural in U , satisfying the following axioms: Vanishing I: Tr IX,Y ( f ) = f , Vanishing II: Tr U ⊗ VX,Y ( g ) = Tr UX,Y (Tr VX ⊗ U,Y ⊗ U ( g )), Superposing: Tr UA ⊗ C,B ⊗ D ((1 B ⊗ σ − D,U ) ◦ ( f ⊗ g ) ◦ (1 A ⊗ σ C,U )) = Tr
UA,B ( f ) ⊗ g =Tr UA ⊗ C,B ⊗ D ((1 B ⊗ σ U,D ) ◦ ( f ⊗ g ) ◦ (1 A ⊗ σ − U,C )),
Yanking:
For every U , we have Tr UU,U ( σ U,U ) = 1 U .Explicitly, naturality and dinaturality mean the following Naturality in A and B : For any g : X ′ → X and h : Y → Y ′ we have thatTr UX ′ ,Y ′ (( h ⊗ U ) ◦ f ◦ ( g ⊗ U ) = h ◦ Tr UX,Y ( f ) ◦ g. Dinaturality in U : For any f : X ⊗ U → Y ⊗ U ′ , g : U ′ → U we have thatTr UX,Y ((1 Y ⊗ g ) ◦ f ) = Tr U ′ X,Y ( f ◦ (1 X ⊗ g )) . HAPTER 2. SOME MATHEMATICAL BACKGROUND Definition 2.7.2.
Suppose we have two traced monoidal categories ( V , Tr) and( W , c Tr). We say that a strong monoidal functor (
F, m ) :
V → W is traced monoidal when it preserves the trace operator in the following way: for f : A ⊗ U → B ⊗ U c Tr F UF A,F B ( m − A,U ◦ F ( f ) ◦ m A,U ) = F (Tr UA,B ( f )) : F A → F B.
Graphical calculi are an important tool for reasoning about monoidal categories,dating back at least to the work of Penrose [59]. There are various graphical lan-guages which are provably complete for reasoning about diagrams in different kindsof monoidal categories. They allow efficient geometrical and topological insights tobe used in a kind of calculus of “wirings”, which simplifies diagrammatic reasoning.See [66] for a detailed survey of such graphical languages.In particular, there is a graphical language for traced monoidal categories, whichwas already used in the original paper of Joyal, Street, and Verity [41]. The axiomsof traced monoidal categories are represented in the following way.Naturality: f g h = f g h Dinaturality: f g = f g Vanishing I: f I = f Vanishing II: f XY = f X Y HAPTER 2. SOME MATHEMATICAL BACKGROUND f g = f g Yanking:Strength (equivalent formulation of superposing): fg = fg The following theorem shows the validity of such diagrammatic reasoning in com-pact closed categories:
Theorem 2.8.1 (Coherence, see [66]) . A well-formed equation between morphismsin the language of symmetric traced categories follows from the axioms of symmetrictraced categories if and only if it holds, up to isomorphism of diagrams, in the graphicallanguage.
Here by isomorphism of diagrams we mean a bijective correspondence betweenwires and boxes in which the structure of the graph is preserved.
Definition 2.9.1. A compact closed category is a symmetric monoidal category V for which every object A has assigned another object A ∗ , called the dual, and a pairof arrows η : I → A ∗ ⊗ A (unit), ε : A ⊗ A ∗ → I (counit) such that the followingdiagrams commute: A (cid:15) (cid:15) ρ / / A ⊗ I ⊗ η / / A ⊗ ( A ∗ ⊗ A ) α (cid:15) (cid:15) A I ⊗ A λ − o o ( A ⊗ A ∗ ) ⊗ A ε ⊗ o o HAPTER 2. SOME MATHEMATICAL BACKGROUND A ∗ (cid:15) (cid:15) λ / / I ⊗ A ∗ η ⊗ / / ( A ∗ ⊗ A ) ⊗ A ∗ α − (cid:15) (cid:15) A ∗ A ∗ ⊗ I ρ − o o A ∗ ⊗ ( A ⊗ A ∗ ) . ⊗ ε o o In a compact closed category we can define a functor ( − ) ∗ : V op → V where if f : A → B then f ∗ : B ∗ → A ∗ is given by: B ∗ λ −→ I ⊗ B ∗ η ⊗ −→ A ∗ ⊗ A ⊗ B ⊗ f ⊗ −→ A ∗ ⊗ B ⊗ B ∗ ⊗ ε −→ A ∗ ⊗ I ρ − −→ A ∗ . Proposition 2.9.2.
Let ( V , ⊗ , η, ε ) be a compact closed category. There exists atrace, which we call the canonical trace, defined by: Tr UA,B ( f ) = (1 ⊗ εσ )( f ⊗ ⊗ η ) . Moreover every symmetric strong monoidal functor between compact categories istraced monoidal with respect to the canonical trace.Proof.
See [41].
Proposition 2.9.3.
Let C be a compact closed category. Then C has a unique trace,i.e., the canonical trace Tr UA,B ( f ) = (1 ⊗ εσ )( f ⊗ ⊗ η ) . Proof.
Appendix B of [38]. hapter 3Categories of completely positivemaps
Definition 3.1.1.
Let H be a finite dimensional Hilbert space, i.e., a finite dimen-sional complex inner product space. Let us write L ( H ) for the space of linear functions ρ : H → H . Equivalently, we can write L ( H ) = H ∗ ⊗ H .Recall that the adjoint of a linear function F : H → K is defined to be the uniquefunction F † : K → H such that h F † v, w i = h v, F w i , for all v ∈ K and w ∈ H . Definition 3.1.2.
Let
H, K be finite dimensional Hilbert spaces. A linear function F : L ( H ) → L ( K ) is said to be completely positive if it can be written in the form F ( ρ ) = m X i =1 F i ρF † i , where F i : H → K is a linear function for i = 1 , . . . , m . Definition 3.1.3.
The category
CPM s of simple completely positive maps has finitedimensional Hilbert spaces as objects, and the morphisms F : H → K are completelypositive maps F : L ( H ) → L ( K ). 28 HAPTER 3. CATEGORIES OF COMPLETELY POSITIVE MAPS Definition 3.1.4.
The category
CPM of completely positive maps is defined as CPM = CPM ⊕ s , the biproduct completion of CPM s . Specifically, the objectsof CPM are finite sequences ( H , . . . , H n ) of finite-dimensional Hilbert spaces, anda morphism F : ( H , . . . , H n ) → ( K , . . . , K m ) is a matrix ( F ij ), where each F ij : H j → K i is a completely positive map. Composition is defined by matrixmultiplication. Remark 3.1.5.
In quantum mechanics, completely positive maps correspond to gen-eral transformations between quantum systems. Two special cases are of note: first, F ( ρ ) = U ρU † , where U is a unitary transformation. This represents the unitaryevolution of an isolated quantum system. Second, F ( ρ ) = ( P ρP † , . . . , P m ρP † m ) , where P , . . . , P m is a system of commuting self-adjoint projections. This correspondsto measurement with possible outcomes 1 , . . . , m . For more details on the physicalinterpretation, see e.g. [58] or [63]. Remark 3.1.6.
Note that the category
CPM is the same (up to equivalence) as thecategory W of [63] and the category CPM ( FdHilb ) ⊕ of [65].Note that for any two finite dimensional Hilbert spaces V and W , there is acanonical isomorphism φ V,W : L ( V ⊗ W ) → L ( V ) ⊗ L ( W ) . Remark 3.1.7.
The categories
CPM s and CPM are symmetric monoidal. For
CPM s , the tensor product is given on objects by the tensor product defined onHilbert spaces V ¯ ⊗ W = V ⊗ W , and on morphisms by the following map f ¯ ⊗ g : L ( V ⊗ W ) f ¯ ⊗ g / / φ V,W (cid:15) (cid:15) L ( X ⊗ Y ) φ X,Y (cid:15) (cid:15) L ( V ) ⊗ L ( W ) f ⊗ g / / L ( X ) ⊗ L ( Y ) . The left and right unit, associativity, and symmetry maps are inherited fromthe symmetric monoidal structure of Hilbert spaces. For the symmetric monoidalstructure on
CPM , define( V i ) i ∈ I ⊗ ( W j ) j ∈ J = ( V i ⊗ W j ) i ∈ I,j ∈ J . HAPTER 3. CATEGORIES OF COMPLETELY POSITIVE MAPS
Definition 3.2.1.
We say that a linear map F : L ( V ) → L ( W ) is a trace preserving linear function when it satisfies tr W ( F ( ρ )) = tr V ( ρ ) (2)for all positive ρ ∈ L ( V ) . F is called trace non-increasing when it satisfiestr W ( F ( ρ )) ≤ tr V ( ρ ) (3)for all positive ρ ∈ L ( V ) . Definition 3.2.2.
A linear function F : L ( V ) → L ( W ) is called a trace preservingsuperoperator if it is completely positive and trace preserving, and it is called a tracenon-increasing superoperator if it is completely positive and trace non-increasing. Definition 3.2.3.
A completely positive map F : ( H , . . . , H n ) → ( K , . . . , K m ) inthe category CPM is called a trace preserving superoperator if for all j and all positive ρ ∈ L ( H j ), X i tr( F ij ( ρ )) = tr( ρ ) , and a trace non-increasing superoperator if for all j and all positive ρ ∈ L ( H j ), X i tr( F ij ( ρ )) ≤ tr( ρ ) . Definition 3.2.4.
We define four symmetric monoidal categories of superoperators.All of them are symmetric monoidal subcategories of
CPM .- Q and Q ′ have the same objects as CPM , and Q s and Q ′ s have the same objectsas CPM s .- The morphisms of Q and Q s are trace non-increasing superoperators, and themorphisms of Q ′ and Q ′ s are trace preserving superoperators. HAPTER 3. CATEGORIES OF COMPLETELY POSITIVE MAPS
CPM s CPM trace non-increasing Q s Q trace preserving Q ′ s Q ′ Remark 3.2.5.
The categories Q , Q s , Q ′ , and Q ′ s are all symmetric monoidal. Thesymmetric monoidal structure is as in CPM and
CPM s , and it is easy to check thatall the structural maps are trace preserving. Lemma 3.2.6. Q and Q ′ have finite coproducts.Proof. The injection and copairing maps are as in
CPM ; we only need to show thatthey are trace preserving. But this is trivially true. hapter 4Partially traced categories
Traced monoidal categories were introduced by Joyal, Street and Verity [41] as anattempt to organize properties from different fields of mathematics, such as algebraictopology and computer science. This abstraction has been useful in formulating newinsights in concrete topics of theoretical computer science such as feedback, fixed-pointoperators, the execution formula in Girard’s Geometry of Interaction (GoI) [27], etc.In this spirit, an axiomatization for partially traced symmetric monoidal categorieswas introduced by Haghverdi and Scott [34] providing an appropriate framework fora typed version of the Geometry of Interaction.An important part of the treatment of the dynamics of proofs in the Geometryof Interaction relies on the expressiveness of its model: proofs are interpreted aslinear operators in Hilbert spaces and an invariant for the cut-elimination process ismodelled by a convergent sum in some linear space. Haghverdi and Scott [34] havedemonstrated that the categorical notion of partially traced category is a useful toolfor capturing the dynamic behavior of all of these conceptual ideas as described byGirard. The word “partial” here refers to the fact that the trace operator is definedon a subset of the set of morphisms
Hom ( A ⊗ U, B ⊗ U ) called the trace class .A large portion of Haghverdi and Scott’s work is concerned with constructing theappropriate abstract notion of a typed GoI aided by the idea of orthogonality in thesense of Hyland and Schalk. Partial traces play a central role in Haghverdi and Scott’swork. For example, their analysis of the idea of an abstract algorithm concerns the32 HAPTER 4. PARTIALLY TRACED CATEGORIES
We recall the definition of a monoidal partially traced category from [34].
Definition 4.1.1.
Let f and g be partially defined operations. We write f ( x ) ↓ if f ( x ) is defined, and f ( x ) ↑ if it is undefined. Following Freyd and Scedrov [25], wealso write f ( x ) ✄ (cid:0)✂ ✁ g ( x ) if f ( x ) and g ( x ) are either both undefined, or else they areboth defined and equal. The relation “ ✄ (cid:0)✂ ✁ ” is known as Kleene equality . We also write f ( x ) ✄✂ g ( x ) if either f ( x ) is undefined, or else f ( x ) and g ( x ) are both defined andequal. The relation “ ✄✂ ” is known as directed Kleene equality . Definition 4.1.2.
Suppose ( C , ⊗ , I, ρ, λ, s ) is a symmetric monoidal category. A partial trace is given by a family of partial functions Tr UX,Y : C ( X ⊗ U, Y ⊗ U ) ⇀ C ( X, Y ), satisfying the following axioms:
Naturality:
For any f : X ⊗ U → Y ⊗ U , g : X ′ → X and h : Y → Y ′ we have that h Tr UX,Y ( f ) g ✄✂ Tr UX ′ ,Y ′ (( h ⊗ U ) f ( g ⊗ U )) . Dinaturality:
For any f : X ⊗ U → Y ⊗ U ′ , g : U → U ′ we haveTr UX,Y ((1 Y ⊗ g ) f ) ✄ (cid:0)✂ ✁ Tr U ′ X,Y ( f (1 X ⊗ g )) . HAPTER 4. PARTIALLY TRACED CATEGORIES Vanishing I:
For every f : X ⊗ I → Y ⊗ I we haveTr IX,Y ( f ) ✄ (cid:0)✂ ✁ ρ Y f ρ − X . Vanishing II:
For every g : X ⊗ U ⊗ V → Y ⊗ U ⊗ V , ifTr VX ⊗ U,Y ⊗ U ( g ) ↓ , then Tr U ⊗ VX,Y ( g ) ✄ (cid:0)✂ ✁ Tr UX,Y (Tr VX ⊗ U,Y ⊗ U ( g )) . Superposing:
For any f : X ⊗ U → Y ⊗ U and g : W → Z , g ⊗ Tr UX,Y ( f ) ✄✂ Tr UW ⊗ X,Z ⊗ Y ( g ⊗ f ) . Yanking:
For any U , Tr UU,U ( σ U,U ) ✄ (cid:0)✂ ✁ U . Definition 4.1.3. A partially traced category is a symmetric monoidal category witha partial trace. Remark 4.1.4.
Comparing this to the definition of a traced monoidal category inSection 2.7, we see that a traced monoidal category is exactly the same as a partiallytraced category where the trace operation happens to be total. We sometimes referto traced monoidal categories as totally traced monoidal categories, when we want toemphasize that they are not partial.
Definition 4.1.5.
The subset of C ( X ⊗ U, Y ⊗ U ) where Tr UX,Y is defined is sometimescalled the trace class , and is written T UX,Y = { f : X ⊗ U → Y ⊗ U | Tr UX,Y ( f ) ↓} . HAPTER 4. PARTIALLY TRACED CATEGORIES Lemma 4.1.6.
Let ( C , ⊗ , I, Tr , s ) be a partially traced category. The superpositionaxioms is equivalent to the following axiom (called strength ):For f : A ⊗ U → B ⊗ U and g : C → D , Tr UA,B ( f ) ⊗ g ✄✂ Tr UA ⊗ C,B ⊗ D ((1 B ⊗ s U,D ) ◦ ( f ⊗ g ) ◦ (1 A ⊗ s C,U )) . Proof. ( ⇒ ) First, from the original version we shall prove this second version.By hypothesis and by naturality of the symmetries we have: g ⊗ f ∈ T UC ⊗ A,D ⊗ B and s DB ◦ Tr UC ⊗ A,D ⊗ B ( g ⊗ f ) ◦ s AC = s DB ◦ ( g ⊗ Tr UA,B ( f )) ◦ s AC = Tr UAB ( f ) ⊗ g .Thus by the naturality axiom we have that since g ⊗ f ∈ T UC ⊗ A,D ⊗ B :( s DB ⊗ U ) ◦ ( g ⊗ f ) ◦ ( s AC ⊗ U ) ∈ T UA ⊗ C,B ⊗ D andTr UA ⊗ C,B ⊗ D ( s DB ⊗ U ) ◦ ( g ⊗ f ) ◦ ( s AC ⊗ U ) = s DB ◦ Tr UC ⊗ A,D ⊗ B ( g ⊗ f ) ◦ s AC .Finally by coherence we obtain:( s DB ⊗ U ) ◦ ( g ⊗ f ) ◦ ( s AC ⊗ U ) = (1 B ⊗ s UD ) ◦ ( f ⊗ g ) ◦ (1 A ⊗ s CU )( ⇐ ) Conversely by hypothesis and composing with symmetries we get:(1 B ⊗ s UD ) ◦ ( f ⊗ g ) ◦ (1 A ⊗ s CU ) ∈ T UA ⊗ C,B ⊗ D and s BD ◦ Tr UA ⊗ C,B ⊗ D ((1 B ⊗ s U,D ) ◦ ( f ⊗ g ) ◦ (1 A ⊗ s C,U )) ◦ s CA = s BD ◦ (Tr UA,B ( f ) ⊗ g ) ◦ s BD .Which implies by the naturality axiom that: α = ( s BD ⊗ U ) ◦ (1 B ⊗ s U,D ) ◦ ( f ⊗ g ) ◦ (1 A ⊗ s C,U ) ◦ ( s CA ⊗ U ) ∈ T UC ⊗ A,D ⊗ B andTr UC ⊗ A,D ⊗ B ( α ) = g ⊗ Tr UA,B ( f ) . But by coherence α = g ⊗ f . Among the examples that motivated this notion of partially traced category in Def-inition 4.1.3 a particularly important one [34], [36] is the category (
Vect fn , ⊕ , ) of HAPTER 4. PARTIALLY TRACED CATEGORIES ⊕ as thetensor product.We recall that in an additive category a morphism f : X ⊕ U → Y ⊕ V is charac-terized by compositions with injections and projections: f ij = π i ◦ f ◦ in j , 1 ≤ i, j ≤ f by a matrix of morphisms of type " f f f f where composition cor-responds to multiplication of matrices. Definition 4.2.1.
The trace class in ( Vect fn , ⊕ , ) is defined as follows: we say that f : X ⊕ U → Y ⊕ U ∈ T UX,Y iff I − f is invertible, where I = id on U .When this is the case we define Tr UX,Y ( f ) = f + f ( I − f ) − f . Proposition 4.2.2.
With the operation defined in Definition 4.2.1, the category offinite dimensional vector spaces is partially traced.Proof. [34], [36].
In order to capture classical probabilistic computation (as a stepping stone towardsquantum computation), we now describe a trace class in the category
Srel of stochas-tic relations. In fact, this partial trace arises from the canonical total trace on(
Vect fn , ⊗ ) by a general construction that we will examine in detail in Section 4.3.Note that it differs from the trace on Srel given by Abramsky [2], [31]. Abramsky’strace is with respect to the coproduct structure ⊕ and is total; here we discuss apartial trace with respect to the tensor structure ⊗ .The category of stochastic relations attempts to model the probability of a bitbeing in states 0 or 1, or more generally, of a variable taking a specific value in afinite set of possible values. Morphisms in this category correspond to the behavioursof finitary probabilistic systems. The general category of stochastic relations, Srel ,is described in [2] and [4]. It arises as the Kleisli category of the Giry Monad [30].We look at the special case where the objects are finite sets.
Definition 4.2.3.
The category
Srel fn of finite stochastic relations consists of thefollowing data: HAPTER 4. PARTIALLY TRACED CATEGORIES
37- objects are finite sets: A , B . . .- morphisms: A f / / B are finite matrices f : B × A → [0 ,
1] such that ∀ a ∈ A X b ∈ B f ( b, a ) ≤ . The composite of two morphisms is defined by matrix multiplication:If A f / / B and B g / / C then g ◦ f : C × A → [0 ,
1] is:( g ◦ f )( c, a ) = X b ∈ B g ( c, b ) .f ( b, a ) . It is immediate that composition as defined above is associative, with identities 1 A : A × A → [0 , A ( x, y ) = ( x = y x = y . Remark 4.2.4.
Note that we allow P b ∈ B f ( b, a ) ≤
1, rather than requiring equality.This is also called a “partial” stochastic relation. A probability that is less than 1corresponds to a computational process that may not terminate.One obtains a symmetric monoidal category (
Srel fn , ⊗ , I ) where the tensor prod-uct on objects is given by the set product A ⊗ B = A × B . For arrows f : A → B and g : C → D , i.e., f : B × A → [0 ,
1] and g : D × C → [0 ,
1] then we have f ⊗ g : A ⊗ C → B ⊗ D is given by a map of type f ⊗ g : B × D × A × C → [0 , f ⊗ g )( b, d, a, c ) = f ( b, a ) · g ( d, c ) . Let
A, B be finite sets. There is a canonical way to encode a function f : A → B as a stochastic map: we write ˆ f : B × A → [0 ,
1] where ˆ f ( b, a ) = 1 if f ( a ) = b andˆ f ( b, a ) = 0 otherwise. We define the symmetric monoidal coherence isomorphisms byapplying this codification to the coherence structure of the cartesian category FinSet of finite sets.
Definition 4.2.5.
Let f : X ⊗ U → Y ⊗ U be a stochastic map. We define thefollowing trace class T UX,Y ⊆ Srel fn ( X ⊗ U, Y ⊗ U ) for all X and Y : f ∈ T UX,Y iff P y ∈ Y P u ∈ U f ( y, u, x, u ) ≤ , ∀ x ∈ X HAPTER 4. PARTIALLY TRACED CATEGORIES
UX,Y : T UX,Y → Srel fn ( X, Y ) with Tr
UX,Y ( f )( y, x ) = P u ∈ U f ( y, u, x, u ) . Proposition 4.2.6.
The formula given in Definition 4.2.5 defines a partial trace on
Srel fn .Proof. We check the axioms of partial trace.
Naturality:
Let f ∈ T UX,Y and g : X ′ → X and h : Y → Y ′ be stochastic maps, first we want toprove that( h ⊗ U ) f ( g ⊗ U ) ∈ T UX ′ ,Y ′ with ( h ⊗ U ) f ( g ⊗ U ) : X ′ ⊗ U → Y ′ ⊗ U. Since we have a map of type ( h ⊗ U ) f : X ⊗ U → Y ′ ⊗ U we evaluate:( h ⊗ U ) f ( y ′ , u, x, v ) = P y ∈ Y,u ′ ∈ U ( h ⊗ U )( y ′ , u, y, u ′ ) f ( y, u ′ , x, v ) = P y ∈ Y,u ′ ∈ U h ( y ′ , y )1( u, u ′ ) f ( y, u ′ , x, v ) = P y ∈ Y h ( y ′ , y ) f ( y, u, x, v ) . Now we compose again:( h ⊗ U ) f ( g ⊗ U )( y ′ , u, x ′ , v ) = P x ∈ X,u ′ ∈ U ( h ⊗ U ) f ( y ′ , u, x, u ′ )( g ⊗ U )( x, u ′ , x ′ , v ) = P x ∈ X,u ′ ∈ U ( P y ∈ Y h ( y ′ , y ) f ( y, u, x, u ′ )) .g ( x, x ′ ) . U ( u ′ , v ) = P x ∈ X,y ∈ Y h ( y ′ , y ) f ( y, u, x, v ) g ( x, x ′ ) . Thus ( h ⊗ U ) f ( g ⊗ U ) ∈ T UX ′ ,Y ′ iff P y ′ ∈ Y ′ ,u ∈ U ( h ⊗ U ) f ( g ⊗ U )( y ′ , u, x ′ , u ) ≤ , ∀ x ′ ∈ X ′ . We know by hypothesis that f ∈ T UX,Y which implies that P y ∈ Y,u ∈ U f ( y, u, x, u ) ≤ , ∀ x ∈ X. On the other hand we also know that P x ∈ X g ( x, x ′ ) ≤ ∀ x ′ ∈ X and P y ′ ∈ Y ′ h ( y ′ , y ) ≤ ∀ y ∈ Y since g : X ′ → X and h : Y → Y ′ are stochastic maps.Thus, P x ∈ X ( P y ∈ Y,u ∈ U f ( y, u, x, u )) g ( x, x ′ ) ≤ P x ∈ X .g ( x, x ′ ) ≤ ∀ x ′ ∈ X ′ . Therefore, P x ∈ X,y ∈ Y,u ∈ U f ( y, u, x, u )) g ( x, x ′ ) ≤ ∀ x ′ ∈ X ′ . Now using this and the fact that P y ′ ∈ Y ′ h ( y ′ , y ) ≤ HAPTER 4. PARTIALLY TRACED CATEGORIES P x ∈ X,y ∈ Y,u ∈ U ( P y ′ ∈ Y ′ h ( y ′ , y )) .f ( y, u, x, u )) g ( x, x ′ ) ≤ P x ∈ X,y ∈ Y,u ∈ U .f ( y, u, x, u )) g ( x, x ′ ) ≤ ∀ x ′ ∈ X ′ . This implies the following: P x ∈ X,y ∈ Y,u ∈ U,y ′ ∈ Y ′ h ( y ′ , y )) f ( y, u, x, u )) g ( x, x ′ ) ≤ ∀ x ′ ∈ X ′ . Therefore, P y ′ ∈ Y ′ ,u ∈ U ( P x ∈ X,y ∈ Y h ( y ′ , y )) f ( y, u, x, u )) g ( x, x ′ ) = P y ′ ∈ Y ′ ,u ∈ U ( h ⊗ U ) f ( g ⊗ U )( y ′ , u, x ′ , u ) ≤ ∀ x ′ ∈ X ′ which implies that the following assertion holds:( h ⊗ U ) f ( g ⊗ U ) ∈ T UX ′ ,Y ′ . Next, we preliminary compute the partial trace. For that purpose, we first need someprevious calculations:Tr UX ′ ,Y ′ (( h ⊗ U ) f ( g ⊗ U ))( y ′ , x ′ ) = P u ∈ U ( P x ∈ X,y ∈ Y h ( y ′ , y ) f ( y, u, x, u )) g ( x, x ′ )) . If we apply the definition of partial trace to f and compose with h then this comesdown to h ◦ Tr UX,Y ( f )( y ′ , x ) = P y ∈ Y h ( y ′ , y ) . ( P u ∈ U f ( y, u, x, u )) = P y ∈ Y,u ∈ U h ( y ′ , y ) f ( y, u, x, u ) . Similarly, we compose with g (( h Tr UX,Y ( f )) g )( y ′ , x ′ ) = P x ∈ X ( h Tr UX,Y ( f ))( y ′ , x ) .g ( x, x ′ ) = P x ∈ X ( P y ∈ Y,u ∈ U h ( y ′ , y ) f ( y, u, x, u )) .g ( x, x ′ ) = P x ∈ X,y ∈ Y,u ∈ U h ( y ′ , y ) f ( y, u, x, u ) g ( x, x ′ )which proves that both previous calculations are equal. Yanking:
Let σ : A ⊗ B → B ⊗ A be defined as the matrix σ : B × A × A × B → [0 ,
1] with σ ( b, a, a ′ , b ′ ) = 1 iff b = b ′ and a = a ′ otherwise is 0.It may be seen immediately that if σ : U ⊗ U → U ⊗ U Tr UU,U ( σ )( u, v ) = P x ∈ U σ ( u, x, v, x ) = 1 if and only if u = x = v otherwise is 0. HAPTER 4. PARTIALLY TRACED CATEGORIES U ( u, v ) = 1 if and only if u = v , otherwise it is 0 we obtain thatTr UU,U ( σ )( u, v ) = 1 U ( u, v ) for every u and v . Dinaturality:
Consider the stochastic maps f : X ⊗ U → Y ⊗ U ′ and g : U ′ → U . First we want toprove that (1 Y ⊗ g ) f ∈ T UX,Y if and only if f (1 X ⊗ g ) ∈ T U ′ X,Y . By definition of trace class we know that(1 Y ⊗ g ) f ∈ T UX,Y if and only if P y ∈ Y,u ∈ U (1 Y ⊗ g ) f ( y, u, x, u ) ≤ ∀ x ∈ X. Also, by definition of composition in the category
Srel fn :(1 Y ⊗ g ) f ( y, u, x, v ) = P y ′ ∈ Y,u ′ ∈ U ′ (1 Y ⊗ g )( y, u, y ′ , u ′ ) f ( y ′ , u ′ , x.v ) = P y ′ ∈ Y,u ′ ∈ U ′ Y ( y, y ′ ) g ( u, u ′ ) f ( y ′ , u ′ , x, v ) = P u ′ ∈ U ′ g ( u, u ′ ) f ( y, u ′ , x, v ) . Thus, we have seen that(1 Y ⊗ g ) f ∈ T UX,Y if and only if P y ∈ Y,u ∈ U ( P u ′ ∈ U ′ g ( u, u ′ ) f ( y, u ′ , x, u )) ≤ ∀ x ∈ X. Following a similar argument we have that f (1 X ⊗ g ) ∈ T U ′ X,Y if and only if P y ∈ Y,u ′ ∈ U ′ f (1 X ⊗ g )( y, u ′ , x, u ′ ) ≤ ∀ x ∈ X. But, again by definition of composition f (1 X ⊗ g )( y, u ′ , x, v ′ ) = P x ′ ∈ X,u ∈ U f ( y, u ′ , x ′ , u )(1 X ⊗ g )( x ′ , u, x, v ′ ) = P x ′ ∈ X,u ∈ U f ( y, u ′ , x ′ , u )1 X ( x ′ , x ) g ( u, v ′ ) = P u ∈ U f ( y, u ′ , x, u ) g ( u, v ′ ) . This means that f (1 X ⊗ g ) ∈ T U ′ X,Y if and only if P y ∈ Y,u ′ ∈ U ′ ( P u ∈ U f ( y, u ′ , x, u ) g ( u, u ′ )) ≤ ∀ x ∈ X. This implies that the condition on the trace class is satisfied. Next, it remains tocalculate the corresponding partial traces.Tr
UX,Y ((1 Y ⊗ g ) f )( y, x ) = P u ∈ U (1 Y ⊗ g ) f ( y, u, x, u ) = P u ∈ U ( P u ′ ∈ U ′ g ( u, u ′ ) f ( y, u ′ , x, u )) = P u ∈ U,u ′ ∈ U ′ g ( u, u ′ ) f ( y, u ′ , x, u ) = P u ′ ∈ U ′ ,u ∈ U f ( y, u ′ , x, u )) g ( u, u ′ ) = P u ′ ∈ U ′ f (1 X ⊗ g )( y, u ′ , x, u ′ ) =Tr U ′ X,Y ( f (1 X ⊗ g ))( y, x ) . HAPTER 4. PARTIALLY TRACED CATEGORIES Vanishing I:
Let f : X ⊗ I → Y ⊗ I be a stochastic map. Therefore, this implies by definition P y ∈ Y,u ∈{∗} f ( y, u, x, ∗ ) = P y ∈ Y f ( y, ∗ , x, ∗ ) ≤ x ∈ X. Thus, this is equivalent to P y ∈ Y,u ∈{∗} f ( y, u, x, u ) ≤ x ∈ X which is the condition f ∈ T IX,Y . Now, we compute the partial traces. Let us consider the following composition X ρ − X → X ⊗ I f → Y ⊗ I ρ Y → Y. We have f ρ − X ( y, ∗ , x ) = X x ′ ∈ X,u ∈ I f ( y, ∗ , x ′ , u ) ρ − X ( x ′ , u, x )= f ( y, ∗ , x, ∗ ) f ρ − X ( y, ∗ , x ) = P x ′ ∈ X,u ∈ I f ( y, ∗ , x ′ , u ) ρ − X ( x ′ , u, x ) = f ( y, ∗ , x, ∗ ) . Now, we compose with ρ Y to get: ρ Y ( f ρ − X )( y, x ) = X y ′ ∈ Y,u ∈ I ρ Y ( y, y ′ , u )( f ρ − X )( y ′ , u, , x )= f ρ − X ( y, ∗ , x ) = f ( y, ∗ , x, ∗ ) ρ Y ( f ρ − X )( y, x ) = P y ′ ∈ Y,u ∈ I ρ Y ( y, y ′ , u )( f ρ − X )( y ′ , u, , x ) = f ρ − X ( y, ∗ , x ) = f ( y, ∗ , x, ∗ )which clearly means thatTr IX,Y ( f )( y, x ) = f ( y, ∗ , x, ∗ ) = ρ Y f ρ − X ( y, x ) for every x ∈ X and y ∈ Y. Thus, we proved that Tr
IX,Y ( f ) = ρ Y f ρ − X . Vanishing II:
Suppose we have a stochastic map g : X ⊗ U ⊗ V → Y ⊗ U ⊗ V such that g ∈ T VX ⊗ U,Y ⊗ U . We need to check that
HAPTER 4. PARTIALLY TRACED CATEGORIES g ∈ T U ⊗ VX,Y if and only if Tr VX ⊗ U,Y ⊗ U ( g ) ∈ T UX,Y . By definition, it follows that g ∈ T U ⊗ VX,Y if and only if P y ∈ Y, ( u,v ) ∈ U × V g ( y, u, v, x, u, v ) ≤ . On the other hand we haveTr VX ⊗ U,Y ⊗ U ( g )( y, u, x, u ′ ) = P v ∈ V g ( y, u, v, x, u ′ , v ) . We obtain Tr VX ⊗ U,Y ⊗ U ( g ) ∈ T UX,Y if and only if P y ∈ Y ( P u ∈ U Tr VX ⊗ U,Y ⊗ U ( g )( y, u, x, u ) = P y ∈ Y,u ∈ U,v ∈ V g ( y, u, v, x, u, v ) ≤ . Thus, we have shown that both conditions are equivalent. Now we move to thecalculation of the partial traces.Tr U ⊗ VX,Y ( g )( y, x ) = X ( u,v ) ∈ U × V g ( y, u, v, x, u, v )= X u ∈ U X v ∈ V g ( y, u, v, x, u, v )= X u ∈ U Tr VX ⊗ U,Y ⊗ U ( g )( y, u, x, u )= Tr UX,Y (Tr VX ⊗ U,Y ⊗ U ( g ))( y, x ) . In conclusion we obtain thatTr U ⊗ VX,Y ( g ) = Tr UX,Y (Tr VX ⊗ U,Y ⊗ U ( g )) . Superposing:
Consider the stochastic maps f : X ⊗ U → Y ⊗ U with f ∈ T UX,Y and g : W → Z .First, we want to prove that g ⊗ f ∈ T UW ⊗ X,Z ⊗ Y . In order to prove this we have that g ⊗ f ∈ T UW ⊗ X,Z ⊗ Y if and only if P ( z,y ) ∈ Z × Y,u ∈ U g ⊗ f ( z, y, u, w, x, u ) ≤ ∀ w ∈ W, ∀ x ∈ X if and only if P z ∈ Z,y ∈ Y,u ∈ U g ( z, w ) f ( y, u, x, u ) ≤ ∀ w ∈ W, ∀ x ∈ X if and only if P z ∈ Z g ( z, w ) P y ∈ Y,u ∈ U f ( y, u, x, u ) ≤ , ∀ w ∈ W, ∀ x ∈ X. HAPTER 4. PARTIALLY TRACED CATEGORIES g is stochastic i.e., P z ∈ Z g ( z, w ) ≤ ∀ w ∈ W .Since we have that f ∈ T UX,Y this implies P y ∈ Y,u ∈ U f ( y, u, x, u ) ≤ ∀ x ∈ X. Weshow now that the partial traces are equal.Tr UW ⊗ X,Z ⊗ Y ( g ⊗ f )( z, y, w, x ) = P u ∈ U ( g ⊗ f )( z, y, u, w, x, u ) = P u ∈ U g ( z, w ) .f ( y, u, x, u ) = g ( z, w ) . P u ∈ U f ( y, u, x, u ) = g ( z, w ) . Tr UX,Y ( f )( y, x ) = g ⊗ Tr UX,Y ( f )( z, y, w, x ) . This means that Tr UW ⊗ X,Z ⊗ Y ( g ⊗ f ) = g ⊗ Tr UX,Y ( f ) . ⊗ In this section, we define a total trace on the category
CPM s of simple completelypositive maps (see Section 3.1). As a matter of fact, this category is compact closed,and therefore is has a unique total trace. Here, we describe it explicitly via a Krausoperator-sum representation.Recall that the category fHilb of finite dimensional Hilbert spaces and linearmaps is compact closed, and therefore (totally) traced. Let, H A , H B and H C be finitedimensional Hilbert spaces with orthonormal bases { e i } , { f i } and { w i } , respectively,and let F : H A ⊗ H B → H C ⊗ H B be a linear function, i.e., F = X j,l,k,m F j,l,k,m | w j , f k ih e l , f m | . Then tr B ( F ) = P j,l,k F j,l,k,k | w j ih e l | defines a total trace on fHilb . Proposition 4.2.7.
Let F : L ( H A ) ⊗L ( H B ) → L ( H C ) ⊗L ( H B ) be a complete positivemap with representation F = P nj =1 F j ρF † j . Then Tr A,CB ( F )( ρ ) = P nj =1 tr B F j ρ tr B F † j defines a (total) trace on the category CPM s .Proof. Suppose we take two representations of F ( ρ ) = P ni =1 E i ρE † i = P nj =1 F j ρF † j . HAPTER 4. PARTIALLY TRACED CATEGORIES
A,CB ( F )( ρ )= P ni =1 tr B F i ρ tr B F † i = P ni =1 tr B ( P j U i,j F j ) ρ tr B ( P j U i,j F j ) † = P ni =1 ( P j U i,j tr B F j ) ρ ( P j U ∗ i,j tr B F † j ) = P i,j,k U i,j U ∗ i,k tr B F j ρ tr B F † k = P j,k ( P i U i,j U ∗ i,k ) tr B F j ρ tr B F † k = P j,k ( P i U † k,i U i,j ) tr B F j ρ tr B F † k = P j,k δ k,j tr B F j ρ tr B F † k = P j tr B F j ρ tr B F † j since U is unitary.Now we check all the axioms. Naturality:
Let us consider f = P i U i − U † i and g = P j V j − V † j where f : L ( H A ) ⊗ L ( H B ) →L ( H C ) ⊗ L ( H B ) and g : L ( H A ′ ) → L ( H A ).Since f ( g ⊗ id ) = ( λρ P i U i ρU † i )( λρ P j V j ρV † j ⊗ id ) = λρ P i,j U i ( V j ⊗ I ) ρ ( V † j ⊗ I ) U † i therefore, we have:Tr BA ′ C ( f ( g ⊗ id )) = λρ P i,j tr B ( U i ( V j ⊗ I )) ρ tr B (( V † j ⊗ I ) U † i ) = λρ P i,j (tr B U i ) V j ρV † j (tr B U † i ) = λρ P i,j (tr B U i ) ρ (tr B U † i ) ◦ λρ P j V j ρV † j = Tr BAC ( f ) ◦ g . Dinaturality:
Suppose we have f = P i U i − U † i and g = P j V j − V † j where f : L ( H A ) ⊗ L ( H B ) →L ( H C ) ⊗ L ( H B ′ ) and g : L ( H B ′ ) → L ( H B ).Then Tr BAC ((1 ⊗ g ) f ) = Tr BAC (( λρ P j ( I ⊗ V j ) ρ ( I ⊗ V † j )) ◦ ( λρ P i U i ρU † i )) =Tr BAC ( λρ P i,j ( I ⊗ V j )( U i ρU † i )( I ⊗ V † j )) = P i,j tr B (( I ⊗ V j ) U i ) ρ tr B ( U † i ( I ⊗ V † j )) = P i,j tr B ′ ( U i ( I ⊗ V j )) ρ tr B ′ (( I ⊗ V † j ) U † i ) = Tr B ′ AC ( λρ P i,j U i ( I ⊗ V j ) ρ ( I ⊗ V † j ) U † i ) =Tr B ′ AC (( λρ P i U i ρU † i ) ◦ ( λρ P j ( I ⊗ V j ) ρ ( I ⊗ V † j ))) = Tr B ′ AC ( f (1 ⊗ g )) . Vanishing I:
Consider the map f : L ( H A ) ⊗ L ( H I ) → L ( H B ) ⊗ L ( H I ) with the following represen-tation f = P i U i − U † i , soTr IA,B ( f ) = λρ P i tr I U i ρ tr I U † i = P i U i ρ U † i . HAPTER 4. PARTIALLY TRACED CATEGORIES Vanishing II:
Let us consider g : L ( H X ) ⊗ L ( H U ) ⊗ L ( H V ) → L ( H Y ) ⊗ L ( H U ) ⊗ L ( H V ) withrepresentation g = P i E i − E † i then: T UX,Y ( T VX ⊗ U,Y ⊗ U ( g )) = T UX,Y ( λρ P i tr V E i ρ tr V E † i ) = λρ P i tr U (tr V ( E i )) ρ tr U (tr V ( E † i )) = λρ P i tr U ⊗ V ( E i )) ρ tr U ⊗ V ( E † i ) = T U ⊗ VX,Y ( g ) . Yanking:
Before we study the proof of this axiom we consider a representation of the symmetricisomorphism: σ N,M : L ( H N ) ⊗ L ( H M ) → L ( H M ) ⊗ L ( H N ) . Let { e ni } , { e mj } be an orthonormal basis for H N and H M respectively. Then { E ni,j } and { E mk,l } are orthonormal basis for L ( H N ) and L ( H N ) respectively with E ni,j = e ni e n † j , E mk,l = e mk e n † l and h A, B i = tr( A † B ) as a inner product.Thus we have: σ ( E ni,j ⊗ E mk,l ) = σ ( | e ni ih e nj | ⊗ | e mk ih e ml | ) = σ ( | e ni i| e mk i ⊗ h e nj |h e ml | ) = U ( | e ni i| e mk i ⊗ h e nj |h e ml | ) U † = U | e ni , e mk i ⊗ ( U | e nj , e ml i ) † = | e mk , e ni i ⊗ ( | e ml , e nj i ) † = | e mk , e ni i ⊗ h e ml , e nj | = | e mk ih e ml | ⊗ | e ni ih e nj | = E mk,l ⊗ E ni,j for every vector basis where the action U is defined by U | e ni ⊗ e mj i = | e mj ⊗ e ni i on the basis of the tensor space. This implies that σ ( A ) = U AU † for every A ∈L ( H N ) ⊗ L ( H M ).Now, let σ N,N : L ( H N ) ⊗ L ( H N ) → L ( H N ) ⊗ L ( H N ) be the symmetric naturalisomorphism with the representation σ N,N = U − U † , σ N,N : H N ⊗ H N → H N ⊗ H N where U = P k,l | e ml ih e nk | ⊗ | e nk ih e ml | and U † = P k,l | e nk ih e nl | ⊗ | e nl ih e nk | . Thus we havethat tr N U = P k,l | e nl ih e nk | ⊗ tr( | e nk ih e nl | ) = P k,l | e nl ih e nk | ⊗ h e nk | e nl i = P k = l | e nl ih e nk | = P l | e nl ih e nl | = Id N . In an analogous way we trace U † obtaining the identity. HenceTr NN,N ( σ N,N )( ρ ) = tr N U ρ tr N U † = Id N ρId N = ρ . Remark 4.2.8.
The category
CPM s is compact closed, due to the existence of amonoidal functor F : fHilb → CPM s which is onto objects. (This functor takes eachobject to itself, and each linear map f to F ( ρ ) = f ρf † ). This already implies that HAPTER 4. PARTIALLY TRACED CATEGORIES
In Definition 4.2.1, we considered a partial trace on the category of finite dimensionalvector spaces with ⊕ as a tensor product. Now, we relax conditions on the definition ofthe trace class and we define another partial trace on vector spaces for not necessarilyfinite dimensions. Definition 4.2.9.
Let (
Vect , ⊕ ,
0) be the symmetric monoidal category of vectorspaces and linear transformations with the monoidal tensor taken to be the directsum. We define a trace class in the following way. Given a map f : V ⊕ U → W ⊕ U we say f ∈ T UV,W iff • imf ⊆ im ( I − f ) and • ker ( I − f ) ⊆ kerf ,where I is the identity map. Whenever these conditions are satisfied we defineTr UV,W ( f ) : V → W :Tr UV,W ( f )( v ) = f ( v ) + f ( u ) for some u ∈ U such that ( I − f )( u ) = f ( v ) . To show that this is well-defined, suppose u ′ is another candidate satisfying( I − f )( u ′ ) = f ( v ) . Then ( I − f )( u − u ′ ) = 0 which implies by the second condition of Definition 4.2.9that f ( u ) = f ( u ′ ). This shows that the value of the trace does not depend on thechoice of the pre-image, but on its existence. Remark 4.2.10.
Notice that the partial trace of Definition 4.2.9 generalizes that ofDefinition 4.2.1. Indeed, if I − f is invertible, then im ( I − f ) = U and ker ( I − f ) =0, which implies that Definition 4.2.9 is trivially satisfied and in this case, Tr UV,W ( f ) = HAPTER 4. PARTIALLY TRACED CATEGORIES f + f ( I − f ) − f (where u = ( I − f ) − f ( v )). Moreover, Definition 4.2.9 isstrictly more general than Definition 4.2.1, because the identity maps are traceablein Definition 4.2.9, but not in Definition 4.2.1. Theorem 4.2.11.
The formula given in Definition 4.2.9 is a partial trace.Proof.
Naturality:
Let f ∈ T UX,Y , g : X ′ → X and h : Y → Y ′ be linear maps. First, we want to provethat ( h ⊕ U ) f ( g ⊕ U ) ∈ T UX ′ ,Y ′ with ( h ⊕ U ) f ( g ⊕ U ) : X ′ ⊕ U → Y ′ ⊕ U. The following equations are satisfied by naturality on injections and projections: • (( h ⊕ U ) f ( g ⊕ U )) = hf g • (( h ⊕ U ) f ( g ⊕ U )) = hf • (( h ⊕ U ) f ( g ⊕ U )) = f g • (( h ⊕ U ) f ( g ⊕ U )) = f . Thus, we have im (( h ⊕ U ) f ( g ⊕ U )) = imf g ⊆ imf ⊆ im ( I − f ) = im ( I − (( h ⊕ U ) f ( g ⊕ U )) )by the hypotheses, properties of the image, and the equations above.Also, ker ( I − (( h ⊕ U ) f ( g ⊕ U )) ) = ker ( I − f ) ⊆ kerf ⊆ ker hf = ker (( h ⊕ U ) f ( g ⊕ U )) by the equations above, by the hypothesis, the properties of the kernel and the equa-tions above again.Now, we want to check the value of the trace. In view of the definition, we may write:Tr UX ′ ,Y ′ (( h ⊕ U ) f ( g ⊕ U ))( x ) = (( h ⊕ U ) f ( g ⊕ U )) ( x ) + (( h ⊕ U ) f ( g ⊕ U )) ( u ) HAPTER 4. PARTIALLY TRACED CATEGORIES u ∈ U such that ( I − (( h ⊕ U ) f ( g ⊕ U )) )( u ) = ( h ⊕ U ) f ( g ⊕ U )) ( x ) . But, this implies using the equations above that:Tr UX ′ ,Y ′ (( h ⊕ U ) f ( g ⊕ U ))( x ) = hf g ( x ) + hf ( u ) = h ( f ( g ( x )) + f ( u )) = h Tr UX,Y ( f )( g ( x )) = h Tr UX,Y ( f ) g ( x )for some u ∈ U such that ( I − f )( u ) = f ( g ( x )). Dinaturality :For any f : X ⊗ U → Y ⊗ U ′ , g : U ′ → U we must prove that(1 Y ⊗ g ) f ∈ T UX,Y iff f (1 X ⊗ g ) ∈ T U ′ X,Y and also we need to check: Tr
UX,Y ((1 Y ⊗ g ) f ) = Tr U ′ X,Y ( f (1 X ⊗ g )) . On the one hand, we know by naturality on injections and projections that we havethe following equations: • ((1 Y ⊕ g ) f ) = f • ((1 Y ⊕ g ) f ) = f • ((1 Y ⊕ g ) f ) = gf • ((1 Y ⊕ g ) f ) = gf . On the other hand we know: • ( f (1 X ⊕ g )) = f • ( f (1 X ⊕ g )) = f g • ( f (1 X ⊕ g )) = f • ( f (1 X ⊕ g )) = f g. First, let us now prove the following equivalence: im ((1 Y ⊕ g ) f ) ⊆ im ( I − ((1 Y ⊕ g ) f ) ) iff im ( f (1 X ⊕ g )) ⊆ im ( I − ( f (1 X ⊕ g )) ) . By the equations above, it corresponds to the following equivalence:
HAPTER 4. PARTIALLY TRACED CATEGORIES im gf ⊆ im ( I − gf ) iff imf ⊆ im ( I − f g ) . ( ⇒ ) Given x = f ( z ) for some z we want to prove that x ∈ im ( I − f g ) . Since, by hypothesis g ( x ) = g ( f ( z )) ∈ im ( I − gf ) then g ( f ( z )) = z ′ − g ( f ( z ′ ))for some z ′ , which implies that g ( f ( z ) + f ( z ′ )) = z ′ . Thus, now choose v = f ( z ) + f ( z ′ ) allowing us to obtain: v − f ( g ( v )) = f ( z ) + f ( z ′ ) − f ( g ( v )) = f ( z ) + f ( z ′ ) − f ( z ′ ) = f ( z ) = x. ( ⇐ ) Given y = g ( f ( u )) for some u we want to prove y ∈ im ( I − gf ) . Since by hypothesis there is a z such that f ( u ) = z − f ( g ( z )) consider v = g ( z );then we get the following:( I − gf )( v ) = g ( z ) − g ( f ( g ( z ))) = g ( z − f ( g ( z ))) = g ( f ( u )) = y. Next, we want to check the following: ker ( I − ((1 Y ⊕ g ) f ) ) ⊆ ker ((1 Y ⊕ g ) f ) iff ker ( I − ( f (1 X ⊕ g )) ) ⊆ ker ( f (1 X ⊕ g )) which by the equations above is equivalent to: ker ( I − gf ) ⊆ kerf iff ker ( I − f g ) ⊆ kerf g. ( ⇒ ) If z = f g ( z ) then g ( z ) = g ( f ( g ( z ))) which implies that g ( z ) ∈ ker ( I − gf )and by hypothesis that f ( g ( z )) = 0 i.e., z ∈ kerf g .( ⇐ ) If v − gf ( v ) = 0 then choosing z = f ( v ) there is a z such that g ( z ) = v . But,clearly z ∈ ker ( I − f g ) since v = gf ( v ) implies:( I − f g )( z ) = f ( v ) − f g ( f ( v )) = f ( v ) − f ( g ( f ( v ))) = f ( v ) − f ( v ) = 0 . Then by hypothesis z ∈ kerf g , which means that f g ( z ) = 0 i.e., f ( v ) = 0.Hence, we proved that if v − gf ( v ) = 0 then f ( v ) = 0 . Now we are ready to check the values of the traces.Tr
UX,Y ((1 Y ⊕ g ) f )( u ) = ((1 Y ⊕ g ) f ) ( u ) + ((1 Y ⊕ g ) f ) ( v ) for some v with( I − ((1 Y ⊕ g ) f ) )( v ) = ((1 Y ⊕ g ) f ) ( u )which by the equations above we get: HAPTER 4. PARTIALLY TRACED CATEGORIES
UX,Y ((1 Y ⊕ g ) f )( u ) = f ( u ) + f ( v ) for some v such that ( I − gf )( v ) = gf ( u ) . On the other hand we have that:Tr U ′ X,Y ( f (1 X ⊕ g ))( u ) = ( f (1 X ⊕ g )) ( u ) + ( f (1 X ⊕ g )) ( v ′ ) for some v ′ such that( I − ( f (1 X ⊕ g )) )( v ′ ) = ( f (1 X ⊕ g )) ( u )and again by the equations above:Tr U ′ X,Y ( f (1 X ⊕ g ))( u ) = f + f g ( v ′ ) for some v ′ such that ( I − f g )( v ′ ) = f ( u ) . ( ⇒ ) Given v as above there is a v ′ such that g ( v ′ ) = v since we have ( I − gf )( v ) = gf ( u ) then v = g ( f ( v ) + f ( u )) so choose v ′ = f ( v ) + f ( u ) and this vectorsatisfies the condition required since ( I − f g )( v ′ ) = v ′ − f g ( v ′ ) = f ( v ) + f ( u ) − f g ( v ′ ) = f ( u ) . ( ⇐ ) Choose v = g ( v ′ ) and then we get ( I − gf )( v ) = ( I − gf )( g ( v ′ )) = g ( v ′ ) − gf g ( v ′ ) = g ( v ′ − f g ( v ′ )) = g ( I − f g )( v ′ )) = g ( f ( u )) = gf ( u ) . Vanishing I :Now, we want to check that: b T IX,Y = C ( X ⊗ I, Y ⊗ I ) and Tr IX,Y ( f ) = ρ Y f ρ − X . Let us consider f : X ⊕ I → Y ⊕ I , we notice first that imf = im ⊆ im ( I − f )and ker ( I − f ) = kerI = 0 ⊆ kerf since f , f , f are constant 0 functions.Next, we move to the value of the trace:Tr X,Y ( f ) = f ( u ) + f ( v ) for some v such that ( I − f )( v ) = f ( u ) . Therefore, since f = 0 we choose v = 0 as a representative and we obtain:Tr X,Y ( f ) = f ( u ) = π f in ( u ) = ρ Y f ρ − X ( u ).since injection, projection and ρ isomorphism coincide in this case. Vanishing II :For any g : X ⊗ U ⊗ V → Y ⊗ U ⊗ V , with g ∈ T VX ⊗ U,Y ⊗ U we want to prove thefollowing equivalence: g ∈ T U ⊗ VX,Y iff Tr VX ⊗ U,Y ⊗ U ( g ) ∈ T UX,Y . HAPTER 4. PARTIALLY TRACED CATEGORIES g using matrix notation: g = g g g g g g g g g . First, we translate the general hypothesis g ∈ T VX ⊗ U,Y ⊗ U in terms of this matrixrepresentation. • c g = g g g g ! : X ⊕ U → Y ⊕ U • c g = (cid:16) g g (cid:17) : X ⊕ U → V • c g = g g ! : V → Y ⊕ U • c g = (cid:16) g (cid:17) : V → V. Thus the condition im c g ⊆ im ( I − c g ) is actually im (cid:16) g g (cid:17) ⊆ im ( I − g )which implies that: ∀ x ∈ X, ∀ u ∈ U, ∃ v ∈ V : g ( x ) + g ( u ) + g ( v ) = v. On theother hand, the condition ker ( I − c g ) ⊆ ker c g is ker ( I − g ) ⊆ ker g g ! whichimplies that: ∀ v ∈ V such that g ( v ) = v then g ( v ) + g ( v ) = 0.We are now ready to translate the condition g ∈ T U ⊗ VX,Y in terms of the matrix repre-sentation of g . • ˜ g = (cid:16) g (cid:17) : X → Y • ˜ g = g g ! : X → U ⊕ V • ˜ g = (cid:16) g g (cid:17) : U ⊕ V → Y • ˜ g = g g g g ! : U ⊕ V → U ⊕ V. HAPTER 4. PARTIALLY TRACED CATEGORIES im ˜ g ⊆ im ( I − ˜ g ) is actually im g g ! ⊆ im ( I − g g g g ! ) which implies that: ∀ x ∈ X, ∃ u ∈ U, ∃ v ∈ V : ( g ( x ) + g ( u ) + g ( v ) = ug ( x ) + g ( u ) + g ( v ) = v ! On the other hand, the condition ker ( I − ˜ g ) ⊆ ker ˜ g is ker ( I − g g g g ! ) ⊆ ker (cid:16) g g (cid:17) which implies that: ∀ u ∈ U, ∀ v ∈ V such that u = g ( u ) + g ( v )and v = g ( u ) + g ( v ) then g ( u ) + g ( v ) = 0 . Now we express Tr VX ⊕ U,Y ⊕ U ( g ) ∈ T UX,Y in terms of the components of g Tr VX ⊕ U,Y ⊕ U ( g )( x, u ) = c g ( x, u ) + c g ( v ) for some v ∈ V such that ( I − c g )( v ) = c g ( x, u ) which implies:Tr VX ⊕ U,Y ⊕ U ( g )( x, u ) = ( g ( x ) + g ( u ) , g ( x ) + g ( u )) + ( g ( v ) , g ( v )) for some v ∈ V such that v − g ( v ) = g ( x ) + g ( u ) . Now we renamed ¯ g = Tr VX ⊕ U,Y ⊕ U ( g ) and compose with injections and projections. • ¯ g = π ¯ g in : X → Y , ¯ g ( x ) = g ( x )+ g ( v ) with v such that v − g ( v ) = g ( x ) • ¯ g = π ¯ g in : X → U , ¯ g ( x ) = g ( x )+ g ( v ) with v such that v − g ( v ) = g ( x ) • ¯ g = π ¯ g in : U → Y , ¯ g ( u ) = g ( u ) + g ( v ) with v such that v − g ( v ) = g ( u ) • ¯ g = π ¯ g in : U → U , ¯ g ( u ) = g ( u ) + g ( v ) with v such that v − g ( v ) = g ( u ) . Thus we have that:¯ g ∈ T UX,Y iff im ¯ g ⊆ im ( I − ¯ g ) and ker ( I − ¯ g ) ⊆ ker ¯ g . HAPTER 4. PARTIALLY TRACED CATEGORIES im ¯ g ⊆ im ( I − ¯ g ) implies that ∀ x ∈ X, ∀ v ∈ V such that v − g ( v ) = g ( x ) , ∃ u ∈ U, ∃ v ∈ V such that v − g ( v ) = g ( u ) and g ( x ) + g ( v ) + g ( u ) + g ( v ) = u. On the other hand, the condition ker ( I − ¯ g ) ⊆ ker ¯ g implies by the equationsabove that ∀ u ∈ U, ∀ v ∈ V such that v − g ( v ) = g ( u ), if g ( u ) + g ( v ) = u then g ( u ) + g ( v ) = 0 . Now since we have all the conditions in term of g we can prove the equivalence.( ⇒ ) We have by general hypothesis that the condition im c g ⊆ im ( I − c g ) is actually ∀ x ∈ X, ∀ u ∈ U, ∃ v ∈ V : g ( x ) + g ( u ) + g ( v ) = v . We also have now ashypothesis that the condition im ˜ g ⊆ im ( I − ˜ g ) is ∀ x ∈ X, ∃ u ∈ U, ∃ v ∈ V : g ( x ) + g ( u ) + g ( v ) = u and g ( x ) + g ( u ) + g ( v ) = v .By the equations above we want to prove that: ∀ x ∈ X, ∀ v ∈ V such that v − g ( v ) = g ( x ) ( α )then ∃ u ∈ U, ∃ v ∈ V such that the following two equations hold: v − g ( v ) = g ( u ) ( β ) g ( x ) + g ( v ) + g ( u ) + g ( v ) = u ( γ ) . By hypothesis given x ∈ X , let us consider u, v such that g ( x ) + g ( u ) + g ( v ) = u and g ( x ) + g ( u ) + g ( v ) = v . Now choose v = v − v ; then we have that g ( x ) + g ( v ) + g ( u ) + g ( v − v ) = g ( x ) + g ( u ) + g ( v ) = u which provesequation ( γ ) using the first of the equations above. It can be seen that: v = v − v = g ( x )+ g ( u )+ g ( v ) − v = g ( x )+ g ( u )+ g ( v ) − ( g ( v )+ g ( x )) = g ( u ) + g ( v ) − g ( v ) = g ( u ) + g ( v − v ) = g ( u ) + g ( v ) which proves equation( β ) using the equations above.( ⇐ ) Now assume the same general hypothesis as before: ∀ x ∈ X, ∀ u ∈ U, ∃ v ∈ V : g ( x ) + g ( u ) + g ( v ) = v . We know by hypothesis that: ∀ x ∈ X, ∀ v ∈ V such that v − g ( v ) = g ( x ) , ∃ u ∈ U, ∃ v ∈ V such that HAPTER 4. PARTIALLY TRACED CATEGORIES v − g ( v ) = g ( u ) and g ( x ) + g ( v ) + g ( u ) + g ( v ) = u .We want to prove that: ∀ x ∈ X, ∃ u ∈ U, ∃ v ∈ V such that g ( x ) + g ( u ) + g ( v ) = u ( ⋆ ) g ( x ) + g ( u ) + g ( v ) = v ( ⋆⋆ )Using the general hypothesis with u = 0 we obtain: ∀ x ∈ X, ∃ v ∈ V : g ( x ) + g (0) + g ( v ) = v . ( ⋆ ⋆ ⋆ )Now by hypothesis we have: given x ∈ X, v ∈ V , since v = g ( x ) + g ( v ) we havethat ∃ u ∈ U, ∃ v ∈ V such that v − g ( v ) = g ( u ) ( ) g ( x ) + g ( v ) + g ( u ) + g ( v ) = u ( ) . Now consider v = v + v we have by the equation above ( ) that: g ( x )+ g ( v + v )+ g ( u ) = u which proves ( ⋆ ). We also have that v + v = g ( x ) + g ( v ) + g ( v ) + g ( u ) by adding equations ( ) and ( ⋆ ⋆ ⋆ ). Thus v + v = g ( x ) + g ( v + v ) + g ( u )which proves ( ⋆⋆ ).Now we move to checking that the condition on kernels is also satisfied. It followsfrom the general hypothesis that: ∀ v ∈ V such that g ( v ) = v then g ( v )+ g ( v ) = 0.( ⇒ ) By hypothesis we know that the two equations ∀ u ∈ U, ∀ v ∈ V u = g ( u ) + g ( v ) ( ⋆ ) v = g ( u ) + g ( v ) ( ⋆ ) imply g ( u ) + g ( v ) = 0. We want to prove that: ∀ u ∈ U if ∃ v ∈ V : v = g ( v ) + g ( u ) with g ( u ) + g ( v ) = u then g ( u ) + g ( v ) = 0 ( ⋆⋆ ) . So, given u ∈ U, v ∈ V : v = g ( v ) + g ( u ) with g ( u ) + g ( v ) = u then byhypothesis since u = g ( u ) + g ( v ) and v = g ( u ) + g ( v ) so ( ⋆ ) and ( ⋆ ) aresatisfied with v = v which implies g ( u ) + g ( v ) = 0 ( ⋆⋆ ).( ⇐ )By a similar argument with v = v . HAPTER 4. PARTIALLY TRACED CATEGORIES VX ⊕ U,Y ⊕ V ( g )( x, u ) = c g ( x, u ) + c g ( v ) for some v such( I − c g )( v ) = c g ( x, u ). If we apply Tr UX,Y to this function, it is equivalent in termsof the g to Tr UX,Y (Tr VX ⊕ U,Y ⊕ V ( g ))( x ) = g ( x ) + g ( v + v ) + g ( u ) with u ∈ U , v ∈ V , v ∈ V such that: u = g ( x ) + g ( v + v ) + g ( u ), v = g ( x ) + g ( v )and v = g ( u ) + g ( v ).On the other hand, we may also calculate Tr U ⊗ VX,Y ( g )( x ) = ˜ g ( x ) + ˜ g ( u, v ) forsome u ∈ U , v ∈ V such that ( I − ˜ g ( u, v )) = ˜ g ( x ) and we get by the equationsabove: Tr U ⊗ VX,Y ( g )( x ) = g ( x ) + g ( v ) + g ( u ) with u = g ( x ) + g ( v ) + g ( u ), v = g ( x ) + g ( v ) + g ( u ). In both implications we obtain the same value of thetrace. Notice that the value is independent of the choice of the vectors that satisfythe auxiliary conditions. When we chose v = v + v we have:Tr UX,Y (Tr VX ⊕ U,Y ⊕ V ( g ))( x ) = g ( x ) + g ( v + v ) + g ( u ) = Tr U ⊗ VX,Y ( g )( x )and when we chose v = v − v we haveTr UX,Y (Tr VX ⊕ U,Y ⊕ V ( g ))( x ) = g ( x ) + g ( v + v ) + g ( u ) = g ( x ) + g ( v + v − v ) + g ( u ) = g ( x ) + g ( v ) + g ( u ) = Tr U ⊗ VX,Y ( g )( x ). Superposing :Suppose now that f ∈ b T UX,Y and g : W → Z ; we want to prove that g ⊕ f ∈ b T UW ⊕ X,Z ⊕ Y .First, we start writing the matrix representation of g ⊕ f in terms of g . • ( g ⊕ f ) = g ⊕ f : W ⊕ X → Z ⊕ Y • ( g ⊕ f ) = (cid:16) f (cid:17) : W ⊕ X → U • ( g ⊕ f ) = f ! : U → Z ⊕ Y • ( g ⊕ f ) = f : U → U. If z ∈ im ( g ⊕ f ) then z = 0 w + f ( x ) for some w ∈ W , x ∈ X which by hypothesisand the equation above implies that f ( x ) ∈ im ( I − f ) = im ( I − ( g ⊕ f ) ).On the other hand, we have ker ( I − ( g ⊕ f ) ) = ker ( I − f ) ⊆ kerf ⊆ HAPTER 4. PARTIALLY TRACED CATEGORIES ker f ! = ker ( g ⊕ f ) by hypothesis and properties of kernels.Now we evaluate the traces:Tr UW ⊕ X,Z ⊕ Y ( g ⊕ f )( w, x ) = ( g ⊕ f ) ( w, x ) + ( g ⊕ f ) ( u ) = g ⊕ f ( w, x ) + f ! ( u ) = ( g ( w ) , f ( x )) + (0 , f ( u )) = ( g ( w ) , f ( x ) + f ( u )) =( g ( w ) , Tr UX,Y ( f )( x ) = ( g ⊕ Tr UX,Y ( f ))( w, x )with u − f ( u ) = (cid:16) f (cid:17) ( w, x ) which by the equations above is equivalent to u − f ( u ) = f ( x ) . ThusTr UW ⊕ X,Z ⊕ Y ( g ⊕ f ) = g ⊕ Tr UX,Y ( f ) . Yanking:
We want to prove that σ U,U ∈ T UU,U , and also Tr
UU,U ( σ U,U ) = 1 U where σ U,U : U ⊕ U → U ⊕ U is the coherent isomorphism. • σ = π σ UU in : U → U , with σ = 0 • σ = π σ UU in : U → U , with σ = id U • σ = π σ UU in : U → U , with σ = id U • σ = π σ UU in : U → U , with σ = 0 . Thus, we have σ ( u ) = u = ( I − σ )( u ) which means that im σ ⊆ im ( I − σ ).On the other hand we have that if u = σ ( u ) then u = 0. This means that ker ( I − σ ) ⊆ ker σ .The value of the trace is the following:Tr UU,U ( σ UU )( u ) = σ ( u ) + σ ( v ) = 0 + v = v with the condition: ( I − σ )( v ) = σ ( u ) for some v ∈ U . But this implies by theequations above that v = u . Thus Tr UU,U ( σ UU )( u ) = u , i.e., Tr UU,U ( σ UU ) = id U . HAPTER 4. PARTIALLY TRACED CATEGORIES ⊕ Definition 4.2.12.
On the category
CPM with monoidal structure ⊕ , we define apartial trace as follows. We say that f ∈ e T UX,Y for some objects X , Y , U iff(a) ( I − f ) is invertible as linear function and(b) the inverse map ( I − f ) − is a completely positive map.We define f Tr UX,Y ( f ) = f + f ( I − f ) − f where I is the identity map.Thus, we are demanding that ( I − f ) − should be regarded as an inverse in thecategory CPM . Lemma 4.2.13.
Let M = " A BC D be a partitioned matrix with sub-block A ∈ M at m × m , B ∈ M at m × n , C ∈ M at n × m and D ∈ M at n × n . Assume D is invertible.Then M is invertible if and only if A − BD − C is invertible. Lemma 4.2.14.
Let us consider A ∈ M at m × n and B ∈ M at n × m . Then ( I m − AB ) isinvertible if and only if ( I n − BA ) is invertible and ( I m − AB ) − A = A ( I n − BA ) − . Proposition 4.2.15. ( CPM , ⊕ , is a partially traced category with respect to Def-inition 4.2.12.Proof. The partial trace axioms, restricted to condition (a) of Definition 4.2.12, arebasically proved in [34]. This picture is completed by adding the proof of the traceaxioms for the positiveness condition (b) of Definition 4.2.12.
Vanishing I:
This follows from the definition of the unit I as the empty list and the fact that theidentity map is an invertible map where its inverse is a completely positive map.Thus e T IX,Y = CPM ( X, Y ) and f Tr IX,Y ( f ) = f for every f ∈ e T IX,Y . Superposing:
Let us consider f ∈ e T UX,Y and g : W → Z then g ⊕ f ∈ e T UW ⊕ X,Z ⊕ Y since ( g ⊕ f ) = f . HAPTER 4. PARTIALLY TRACED CATEGORIES f Tr UW ⊕ X,Z ⊕ Y ( g ⊕ f ) = " g f + f ( I − f ) − f = g ⊕ f Tr UX,Y ( f ). Naturality: If f ∈ e T UX,Y and we have two arrows g : X ′ → X , h : Y → Y ′ then since(( h ⊕ id U ) f ( g ⊕ id U )) = f always is satisfied for linear maps since composition computes as matrix product i.e., " h
00 1 u . " f f f f . " g
00 1 u = " hf g hf f g f Thus then the conditions remain exactly the same, meaning that(( h ⊕ id u ) f ( g ⊕ id u )) = f ∈ e T UX ′ ,Y ′ . Moreover f Tr UX ′ ,Y ′ (( h ⊕ u ) f ( g ⊕ u ) = hf g +( hf )( f )( f g ) = h ( f + f f f ) g = h f Tr UX,Y ( f ) g. Yanking :Note that s U,U ∈ e T UU,U since ( s U,U ) , = 0 which implies that I − ( s U,U ) , is invertibleand ( I − − is a completely positive map. Moreover f Tr UU,U ( σ U,U ) = 1 u since( s U,U ) , = ( s U,U ) , = 0 and ( s U,U ) , = ( s U,U ) , = 1 u . Vanishing II :Let us consider g : X ⊕ U ⊕ V → Y ⊕ U ⊕ V , we write using matrix notation g = a b cd e fm n p .Now, assuming by hypothesis that g ∈ e T VX ⊕ U,Y ⊕ U , i.e., I − p is invertible and( I − p ) − is a completely positive map we must show that g ∈ e T U ⊕ VX,Y iff f Tr VX ⊕ U,Y ⊕ U ( g ) ∈ e T UX,Y .First, we analyze the conditions of definition 4.2.12 in terms of its matrix termcomponents. If we represent functions using matrix notation we have:
HAPTER 4. PARTIALLY TRACED CATEGORIES f Tr VX ⊕ U,Y ⊕ U ( g ) xu ! = " a bd e ! + cf ! ◦ (1 − p ) − ◦ (cid:16) m n (cid:17) xu ! andwe obtain ( f Tr VX ⊕ U,Y ⊕ U ( g )) ( u ) = ( e + f (1 − p ) − n )( u )by composing with the second injection and the second projection.Thus we know by definition: f Tr VX ⊕ U,Y ⊕ U ( g ) ∈ e T UX,Y i.e., I − e − f (1 − p ) − n is invertible and ( I − e − f (1 − p ) − n ) − is a complete positive map.On the other hand, g ∈ e T U ⊕ VX,Y i.e., I − e fn p ! is invertible and ( I − e fn p ! ) − is a complete positive map. Also we obtain the explicit inverse by( I − e fn p ! ) − = " ( I − e − f qn ) − ( I − e − f qn ) − ( f ) qqn ( I − e − f qn ) − q + qn ( I − e − f qn ) − f q where q = ( I − p ) − . Now we prove the equivalence on the trace class:First of all, by Lemma 4.2.13 above we get: if we know that ( I − p ) is invertiblethen ( I − e fn p ! ) is invertible iff I − e − f (1 − p ) − n is invertible, which meansthat the first part of the definition is satisfied. Also from Lemma 4.2.13 we have thatthe equation on traces f Tr U ⊕ VX,Y ( g ) = f Tr UX,Y ( f Tr VX ⊕ U,Y ⊕ U ( g )) is satisfied by using matrixmultiplication and the explicit inverse ( I − e fn p ! ) − written above.Positiveness condition (b):( ⇐ ) Injections and projections are completely positive maps and by the fact that g is a completely positive map this implies by definition that n and f are completepositive maps. Also, ( I − e − f (1 − p ) − n ) − = ( I − ( f Tr VX ⊕ U,Y ⊕ U ( g )) ) − is a completelypositive map by conditional hypothesis and ( I − p ) − is also a completely positivemap by the general hypothesis. This implies that ( I − e fn p ! ) − is a completelypositive since each component of the matrix is obtained by sum and composition ofcompletely positive maps. HAPTER 4. PARTIALLY TRACED CATEGORIES ⇒ ) If ( I − e fn p ! ) − is a completely positive map then π ◦ ( I − e fn p ! ) − ◦ i = ( I − e − f ( I − p ) − n ) − is a completely positive map where π and i are the first projection and first injection. Therefore, we showed that( I − e − f ( I − p ) − n ) − = ( I − ( e + f (1 − p ) − n )) − = ( I − ( f Tr vX ⊕ U,Y ⊕ U ( g )) ) − is acompletely positive map. Dinaturality:
Now, suppose f : X ⊕ U → Y ⊕ U ′ and g : U ′ → U are completely positive mapsthen we want to prove that ( id y ⊕ g ) ◦ f ∈ e T UX,Y iff f ◦ ( id x ⊕ g ) ∈ e T U ′ X,Y which meansthat (( id y ⊕ g ) ◦ f ) = g ◦ f satisfies conditions (a) and (b) of Definition 4.2.12 if andonly if ( f ◦ ( id x ⊕ g )) = f ◦ g does.Given f : X ⊕ U → Y ⊕ U ′ and g : U ′ → U by Lemma 4.2.14 above, I − g ◦ f isinvertible if and only if I − f ◦ g is invertible and we have that f Tr UX,Y ((1 y ⊕ g ) f ) = f + f ( I − gf ) − gf = f + f g ( I − f g ) − f = f Tr U ′ X,Y ( f (1 x ⊕ g )) . Therefore, it suffices to prove the following: if I u − g ◦ f is invertible and ( I u − g ◦ f ) − is a completely positive map then ( I u ′ − f ◦ g ) − is a completely positive map, where f : U → U ′ and g : U ′ → U .We know by hypothesis that ∀ τ, ∀ A ′ ∈ V τ ⊗ V u , if A ′ ≥ Id τ ⊗ ( I u − g ◦ f ) − )( A ′ ) ≥ ∀ τ, ∀ A ∈ V τ ⊗ V u ′ , if A ≥ Id τ ⊗ ( I u ′ − f ◦ g )) − )( A ) ≥ . Suppose we name ( Id τ ⊗ ( I u ′ − f ◦ g ) − )( A ) = B then by hypothesis A = ( Id τ ⊗ ( I u ′ − f ◦ g ))( B ) ≥ . (4)Since g is a completely positive map this implies that: if A ≥ Id τ ⊗ g )( A ) ≥ HAPTER 4. PARTIALLY TRACED CATEGORIES ≤ ( Id τ ⊗ g ) ◦ ( Id τ ⊗ ( I u ′ − f ◦ g ))( B ) = ( Id τ ⊗ g ( I u ′ − f ◦ g ))( B ) =( Id τ ⊗ ( g − g ◦ f ◦ g ))( B ) = ( Id τ ⊗ ( I u − g ◦ f ) g )( B ) = ( Id τ ⊗ ( I u − g ◦ f )) ◦ ( Id τ ⊗ g )( B )which implies (rename it C )( Id τ ⊗ ( I u − g ◦ f ))(( Id τ ⊗ g )( B )) = C ≥ . Thus we have( Id τ ⊗ g )( B ) = ( Id τ ⊗ ( I u − g ◦ f )) − ( C ) = ( Id τ ⊗ ( I u − g ◦ f ) − )( C ) ≥ I u − g ◦ f ) − is a completely positive map by hypothesis. Therefore, ( Id τ ⊗ g )( B ) ≥ f is a completely positive map which implies( Id τ ⊗ f )(( Id τ ⊗ g )( B )) ≥
0, which means ( Id τ ⊗ f ◦ g )( B ) ≥ A ≥ Id τ ⊗ I u ′ )( B ) − ( Id τ ⊗ f ◦ g )( B ) ≥ B ≥ ( Id τ ⊗ f ◦ g )( B ) hence B = ( Id τ ⊗ ( I u ′ − f ◦ g ) − )( A ) ≥ τ . For the converse implication we repeat this argumentinterchanging f and g . The aim of this section is to provide a general construction of partially traced cate-gories as subcategories of other partially (or totally) traced categories.Suppose ( D , ⊗ , I, s, Tr) is a partially traced category with traceTr
UX,Y : D ( X ⊗ U, Y ⊗ U ) ⇀ D ( X, Y ) . Given a monoidal subcategory
C ⊆ D , we get a partial trace on C , defined by c Tr UX,Y ( f ) = Tr UX,Y ( f ) if Tr UX,Y ( f ) ↓ and Tr UX,Y ( f ) ∈ C ( X, Y ), and undefined other-wise.More generally, we shall show a method of constructing one partially traced cate-gory from another in such a way that the first one is faithfully embedded in the second.
HAPTER 4. PARTIALLY TRACED CATEGORIES Proposition 4.3.1.
Let F : C → D be a faithful strong symmetric monoidal functorwith ( D , ⊗ , I, s, Tr) a partially traced category and ( C , ⊗ , I, s ) a symmetric monoidalcategory. Then we obtain a partial trace c Tr on C as follows. For f : X ⊗ U → Y ⊗ U ,we define c Tr UX,Y ( f ) = g if there exists some (necessarily unique) g : X → Y such that F ( g ) = Tr F UF X,XY ( m − Y,U ◦ F ( f ) ◦ m X,U ) is defined, and c Tr UX,Y ( f ) undefined otherwise.Proof. To clarify the notation used here we recall that there are two partial functions: c Tr UX,Y : C ( X ⊗ U, Y ⊗ U ) ⇀ C ( X, Y )and Tr
UX,Y : D ( X ⊗ U, Y ⊗ U ) ⇀ D ( X, Y ) . Then we have two maps c Tr UX,Y : b T UX,Y → C ( X, Y )where b T UX,Y ⊆ C ( X ⊗ U, Y ⊗ U ) and we also haveTr UX,Y : T UX,Y → D ( X, Y )where T UX,Y ⊆ D ( X ⊗ U, Y ⊗ U ) . Naturality :For any X , Y , U objects in C , f ∈ b T UX,Y and g : X ′ → X , h : Y → Y ′ arrows in C . We want to prove that the two conditions given above hold: (1) we must prove that m − Y ′ U ◦ F ( h ⊗ U ) F ( f ) F ( g ⊗ U ) ◦ m X ′ ,U ∈ T F UF X ′ ,F Y ′ By naturality of the map m − with h , g and identities we have: m − Y ′ U ◦ F ( h ⊗ U ) = ( F ( h ) ⊗ F U ) ◦ m − Y U and also F ( g ⊗ U ) ◦ m X ′ ,U = m X,U ◦ ( F ( g ) ⊗ F U ) . (5) HAPTER 4. PARTIALLY TRACED CATEGORIES F ( h ) ⊗ F U ) ◦ m − Y U ◦ F ( f ) ◦ m X,U ◦ ( F ( g ) ⊗ F U ) ∈ T F UF X ′ ,F Y ′ . Notice that by hypothesis m − Y U ◦ F ( f ) ◦ m X,U ∈ T F UF X,F Y . Then by the naturality axiom in the category D we have that( F ( h ) ⊗ F U ) ◦ m − Y U ◦ F ( f ) ◦ m X,U ◦ ( F ( g ) ⊗ F U ) ∈ T F UF X ′ ,F Y ′ and alsoTr F UF X ′ ,F Y ′ (( F ( h ) ⊗ F U ) ◦ m − Y U ◦ F ( f ) ◦ m X,U ◦ ( F ( g ) ⊗ F U )) == F ( h ) ◦ Tr F UF X,F Y ( m − Y U ◦ F ( f ) ◦ m X,U ) ◦ F ( g ) . (2) Since by hypothesis there exists an arrow p : X → Y such that F ( p ) = Tr F UF X,F Y ( m − Y U ◦ F ( f ) ◦ m X,U )thenTr
F UF X ′ ,F Y ′ ( m − Y ′ U ◦ F (( h ⊗ U ) ◦ f ◦ ( g ⊗ U )) ◦ m X ′ ,U ) = (equation (5) above)= Tr F UF X ′ ,F Y ′ (( F ( h ) ⊗ F U ) ◦ m − Y U ◦ F ( f ) ◦ m X,U ◦ ( F ( g ) ⊗ F U )) =(naturality axiom in D and hyp.)= F ( h ) ◦ F ( p ) ◦ F ( g ) = F ( h ◦ p ◦ g ) . This means that we can choose p = h ◦ p ◦ g. Now we are able to compute the trace: c Tr UX ′ ,Y ′ (( h ⊗ U ) f ( g ⊗ U )) = p = h ◦ p ◦ g = h ◦ c Tr UX,Y ( f ) ◦ g. Dinaturality :For any f : X ⊗ U → Y ⊗ U ′ , g : U ′ → U where f and g are in C we must prove that(1 Y ⊗ g ) f ∈ b T UX,Y iff f (1 X ⊗ g ) ∈ b T U ′ X,Y . We must check condition (1) and (2). (1)
By definition we have(1 Y ⊗ g ) f ∈ b T UX,Y implies m − Y,U ◦ F ((1 Y ⊗ g ) f ) ◦ m X,U ∈ T F UF X,F Y . (6) HAPTER 4. PARTIALLY TRACED CATEGORIES m it follows that m − Y,U ◦ F (1 Y ⊗ g ) = ( F (1) ⊗ F ( g )) ◦ m − Y,U ′ . Then we can replace it in (6) obtaining: m − Y,U ◦ F ((1 Y ⊗ g ) f ) ◦ m X,U = m − Y,U ◦ F (1 Y ⊗ g ) ◦ F f ◦ m X,U =(1
F Y ⊗ F ( g )) ◦ m − Y,U ′ ◦ F f ◦ m X,U ∈ T F UF X,F Y . It now follows by the dinaturality axiom of the category D that this condition isequivalent to proving: m − Y,U ′ ◦ F ( f ) ◦ m X,U ◦ (1 F X ⊗ F ( g )) ∈ T F U ′ F X,F Y and again by naturality of m we have that m X,U ◦ (1 F X ⊗ F ( g )) = ( F (1 ⊗ g )) ◦ m X,U ′ and we replace it: m − Y,U ′ ◦ F ( f ) ◦ F ((1 ⊗ g ) ◦ m X,U ′ = m − Y,U ′ ◦ F ( f (1 ⊗ g )) ◦ m X,U ′ ∈ T F U ′ F X,F Y which is condition (1) in the definition f ◦ (1 X ⊗ g ) ∈ b T U ′ X,Y . In the same way weprove the converse. (2)
Also there is an arrow p such that F ( p ) = Tr F UF X,F Y ( m − Y,U ◦ F ((1 Y ⊗ g ) f ) ◦ m X,U ) if and only if there is an arrow F ( p ) = Tr F U ′ F X,F Y ( m − Y,U ′ ◦ F ( f (1 X ⊗ g )) ◦ m X,U ′ ).Since the value of the trace remains invariant under the dinaturality axiom and allthe transformations made in part (1) then it is enough to take p = p . Vanishing I :Now we want to check that: b T IX,Y = C ( X ⊗ I, Y ⊗ I ). Given any f : X ⊗ I → Y ⊗ I we want to prove that f ∈ b T IX,Y by verifying conditions (1) and (2). (1)
Let us consider g = (1 F Y ⊗ m − I ) ◦ m − Y,I ◦ F ( f ) ◦ m X,I ◦ (1 F X ⊗ m I ). By the vanishingI axiom in the category D we know that g ∈ b T IF X,F Y . Then, since (1
F Y ⊗ m − I ) ◦ (1 F Y ⊗ m I ) ◦ g = g ∈ b T IF X,F Y we can apply the dinaturality axiom in D to conclude that(1 F Y ⊗ m I ) ◦ g ◦ (1 F X ⊗ m − I ) ∈ b T F IF X,F Y but we have that (1
F Y ⊗ m I ) ◦ g ◦ (1 F X ⊗ m − I ) = m − Y,I ◦ F ( f ) ◦ m X,I . So we proved that m − Y,I ◦ F ( f ) ◦ m X,I ∈ b T F IF X,F Y . (2) Since g ∈ D ( F X ⊗ I, F Y ⊗ I ) we can say also, by the dinaturality axiom, thatTr IF X,F Y ( g ) = Tr F IF X,F Y ( m − Y,I ◦ F ( f ) ◦ m X,I ) HAPTER 4. PARTIALLY TRACED CATEGORIES
IF X,F Y ( g ) = ρ F Y ◦ g ◦ ρ − F X by vanishing I in D which implies thatTr IF X,F Y ( g ) = ρ F Y ◦ (1 F Y ⊗ m − I ) ◦ m − Y,I ◦ F ( f ) ◦ m X,I ◦ (1 F X ⊗ m I ) ◦ ρ − F X =(since F is monoidal ) F ( ρ Y ) ◦ F ( f ) ◦ F ( ρ − X ) = F ( ρ Y ◦ f ◦ ρ − X ) . Thus there exists a p = ρ Y ◦ f ◦ ρ − X such that F ( p ) = Tr F IF X,F Y ( m − Y,I ◦ F ( f ) ◦ m X,I ).Also notice that we prove that c Tr IX,Y ( f ) = p = ρ Y ◦ f ◦ ρ − X , which is the equation ofthe trace value in the category C . Vanishing II :Let g : X ⊗ U ⊗ V → Y ⊗ U ⊗ V be an arrow in the category C . By hypothesis,we are given g ∈ b T VX ⊗ U,Y ⊗ U (general hypothesis) and we want to prove the followingequivalence: g ∈ b T U ⊗ VX,Y iff c Tr VX ⊗ U,Y ⊗ U ( g ) ∈ b T UX,Y . According to the general hypothesis there is a map: F ( X ⊗ U ) ⊗ F V m X ⊗ U,V −→ F ( X ⊗ U ⊗ V ) F g −→ F ( Y ⊗ U ⊗ V ) m − Y ⊗ U,V −→ F ( Y ⊗ U ) ⊗ F V ∈ b T F VF ( X ⊗ U ) ,F ( Y ⊗ U ) and also there exists p : X ⊗ U → Y ⊗ U such that F ( p ) = Tr F VF ( X ⊗ U ) ,F ( Y ⊗ U ) ( m − Y ⊗ U,V ◦ F ( g ) ◦ m X ⊗ U,V ) i.e., by definition p = c Tr VX ⊗ U,Y ⊗ U ( g ) . ( ⇒ ) We have a conditional hypothesis g ∈ b T U ⊗ VX,Y which asserts that the map:
F X ⊗ F ( U ⊗ V ) m X,U ⊗ V −→ F ( X ⊗ U ⊗ V ) F g −→ F ( Y ⊗ U ⊗ V ) m − Y,U ⊗ V −→ F Y ⊗ F ( U ⊗ V ) ∈ b T F ( U ⊗ V ) F X,F Y and also that there exists an p : X → Y such that F ( p ) = Tr F ( U ⊗ V ) F X,F Y ( m − Y,U ⊗ V ◦ F ( g ) ◦ m X,U ⊗ V ) i.e., by definition p = c Tr U ⊗ VX,Y ( g ) . Recalling that p = c Tr VX ⊗ U,Y ⊗ U ( g ), we want to prove that p ∈ b T UX,Y . For thatpurpose, we shall prove the two conditions that characterize the trace class definitionwhich are the following:
HAPTER 4. PARTIALLY TRACED CATEGORIES
F X ⊗ F U m X,U −→ F ( X ⊗ U ) F ( p ) −→ F ( Y ⊗ U ) m − Y,U −→ F Y ⊗ F U ) ∈ T F UF X,F Y (2) there exists an p : X → Y such that F ( p ) = Tr F UF X,F Y ( m − Y,U ◦ F ( p ) ◦ m X,U ) i.e., by definition p = c Tr UX,Y ( p ) = c Tr UX,Y ( c Tr VX ⊗ U,Y ⊗ U ( g )) . (1) To prove condition (1) we notice that since by definition F ( p ) = Tr F VF ( X ⊗ U ) ,F ( Y ⊗ U ) ( m − Y ⊗ U,V ◦ F ( g ) ◦ m X ⊗ U,V )then we must prove that
F X ⊗ F U m X,U −→ F ( X ⊗ U ) Tr F VF ( X ⊗ U ) ,F ( Y ⊗ U ) ( m − ◦ F ( g ) ◦ m ) −→ F ( Y ⊗ U ) m − Y,U −→ F Y ⊗ F U ) ∈ b T F UF X,F Y . But since m − Y ⊗ U,V ◦ F ( g ) ◦ m X ⊗ U,V ∈ T F VF ( X ⊗ U ) ,F ( Y ⊗ U ) this condition allows us to apply the naturality axiom in the category D : F X ⊗ F U ⊗ F V m X,U ⊗ F V −→ F ( X ⊗ U ) ⊗ F V m − Y ⊗ U,V ◦ F ( g ) ◦ m X ⊗ U,V −→ F ( Y ⊗ U ) F V m − Y,U ⊗ F V −→ F Y ⊗ F U ⊗ F V ∈ b T F VF X ⊗ F U,F Y ⊗ F U . And also by the same axiom we have that:Tr
F VF X ⊗ F U,F Y ⊗ F U (( m − Y,U ⊗ F V ) ◦ m − Y ⊗ U,V ◦ F ( g ) ◦ m X ⊗ U,V ◦ ( m X,U ⊗ F V )) == m − ◦ Tr F VF ( X ⊗ U ) ,F ( Y ⊗ U ) ( m − Y ⊗ U,V ◦ F ( g ) ◦ m X ⊗ U,V ) ◦ m. Hence, this is equivalent to proving that:Tr
F VF X ⊗ F U,F Y ⊗ F U (( m − Y,U ⊗ F V ) ◦ m − Y ⊗ U,V ◦ F ( g ) ◦ m X ⊗ U,V ◦ ( m X,U ⊗ F V )) ∈ Tr F UF X,F Y . Consequently, by vanishing II in the category D , it would be enough that the map λ = ( m − Y,U ⊗ F V ) ◦ m − Y ⊗ U,V ◦ F ( g ) ◦ m X ⊗ U,V ◦ ( m X,U ⊗ F V ) ∈ T F U ⊗ F VF X,F Y
HAPTER 4. PARTIALLY TRACED CATEGORIES λ ∈ T F VF X ⊗ F U,F Y ⊗ F U . But by coherence of monoidal functors wehave:( m − Y,U ⊗ F V ) ◦ m − Y ⊗ U,V ◦ F ( g ) ◦ m X ⊗ U,V ◦ ( m X,U ⊗ F V ) == (1
F X ⊗ m − U,V ) ◦ m − Y,U ⊗ V ◦ F ( g ) ◦ m X,U ⊗ V ◦ (1 F X ⊗ m U,V ) . Therefore, by the dinaturality axiom in the category D :(1 F X ⊗ m − U,V ) ◦ m − Y,U ⊗ V ◦ F ( g ) ◦ m X,U ⊗ V ◦ (1 F X ⊗ m U,V ) ∈ T F U ⊗ F VF X,F Y if and only if( m − Y,U ⊗ V ◦ F ( g ) ◦ m X,U ⊗ V ) ◦ (1 F X ⊗ m U,V ) ◦ (1 F X ⊗ m − U,V ) ∈ T F ( U ⊗ V ) F X,F Y which is valid since this is the conditional hypothesis. (2)
We shall prove that there exists an arrow p : X → Y in C such that F ( p ) = Tr F UF X,F Y ( m − Y,U ◦ F ( p ) ◦ m X,U ) . For that purpose, take p = p . Hence by the conditional hypothesis if g ∈ b T U ⊗ VX,Y holds then there is p with F ( p ) = Tr F ( U ⊗ V ) F X,F Y ( m − Y,U ⊗ V ◦ F ( g ) ◦ m X,U ⊗ V ) . Therefore, this is equal to,Tr F ( U ⊗ V ) F X,F Y (( m − Y,U ⊗ V ◦ F ( g ) ◦ m X,U ⊗ V ) ◦ (1 F X ⊗ m U,V ) ◦ (1 F X ⊗ m − U,V )) =Tr
F U ⊗ F VF X,F Y ((1
F X ⊗ m − U,V ) ◦ m − Y,U ⊗ V ◦ F ( g ) ◦ m X,U ⊗ V ◦ (1 F X ⊗ m U,V )) = (dinaturality)Tr
F UF X,F Y (Tr
F VF X ⊗ F U,F Y ⊗ F U ((1
F X ⊗ m − U,V ) ◦ m − Y,U ⊗ V ◦ F ( g ) ◦ m X,U ⊗ V ◦ (1 F X ⊗ m U,V ))) =(vanishing II)
HAPTER 4. PARTIALLY TRACED CATEGORIES
F UF X,F Y (Tr
F VF X ⊗ F U,F Y ⊗ F U (( m − Y,U ⊗ F V ) ◦ m − Y ⊗ U,V ◦ F ( g ) ◦ m X ⊗ U,V ◦ ( m X,U ⊗ F V ))) =(coherence)Tr
F UF X,F Y ( m − Y,U ◦ Tr F VF ( X ⊗ U ) ,F ( Y ⊗ U ) ( m − Y ⊗ U,V ◦ F ( g ) ◦ m X ⊗ U,V ) ◦ m X,U ) =(naturality axiom)Tr
F UF X,F Y ( m − Y,U ◦ F ( p ) ◦ m X,U ) = (definition of general hypothesis) . So we have proved that: F ( p ) = Tr F UF X,F Y ( m − Y,U ◦ F ( p ) ◦ m X,U )which means that c Tr UX,Y ( c Tr VX ⊗ U,Y ⊗ U ( g )) = c Tr UX,Y ( p ) = p = p = c Tr U ⊗ VX,Y ( g ) . ( ⇐ ) Similarly, we prove the converse. The proof is just a matter of using the conversehypothesis of vanishing II in the category D . Superposing:
Suppose f ∈ b T UX,Y and g : W → Z with g ∈ C we want to prove that g ⊗ f ∈ b T UW ⊗ X,Z ⊗ Y by checking conditions (1) and (2). Also we want to show that c Tr UW ⊗ X,Z ⊗ Y ( g ⊗ f ) = g ⊗ c Tr UX,Y ( f ) . (1) By hypothesis we know that
F X ⊗ F U m X,U −→ F ( X ⊗ U ) F ( f ) −→ F ( Y ⊗ U ) m − Y,U −→ F Y ⊗ F U ) ∈ b T F UF X,F Y and also there exists an arrow p : X → Y such that F ( p ) = Tr F UF X,F Y ( m − Y,U ◦ F ( f ) ◦ m X,U ) . Then by the superposing axiom in the category D it follows that F ( g ) ⊗ ( m − Y,U ◦ F ( f ) ◦ m X,U ) ∈ T F UF W ⊗ F X,F W ⊗ F Y
HAPTER 4. PARTIALLY TRACED CATEGORIES
F UF W ⊗ F X,F Z ⊗ F Y ( F ( g ) ⊗ ( m − Y,U ◦ F ( f ) ◦ m X,U )) = F ( g ) ⊗ Tr F UF X,F Y ( m − Y,U ◦ F ( f ) ◦ m X,U ) . But by functoriality of the tensor we obtain F ( g ) ⊗ ( m − Y,U ◦ F ( f ) ◦ m X,U ) = (1
F Z ⊗ m − Y,U ) ◦ ( F ( g ) ⊗ F ( f )) ◦ (1 F W ⊗ m X,U ) = β (To simplify notation, we name this equation β ).We can apply the naturality axiom in the category D and we obtain:( m Z,Y ⊗ F U ) ◦ β ◦ ( m − W,X ⊗ F U ) ∈ T F UF ( W ⊗ X ) ,F ( Z ⊗ Y ) andTr F UF ( W ⊗ X ) ,F ( Z ⊗ Y ) (( m Z,Y ⊗ F U ) ◦ β ◦ ( m − W,X ⊗ F U )) = m Z,Y ◦ Tr F UF W ⊗ F X,F Z ⊗ F Y ( β ) ◦ m − W,X but by naturality and monoidal functor axioms we have that( m Z,Y ⊗ F U ) ◦ β ◦ ( m − W,X ⊗ F U ) = m Z ⊗ Y,U ◦ F ( g ⊗ f ) ◦ m W ⊗ X,U . Therefore, we proved that m − ◦ F ( g ⊗ f ) ◦ m ∈ T F UF ( W ⊗ X ) ,F ( Z ⊗ Y . (2) Let us consider β = (1 F Z ⊗ m − Y,U ) ◦ ( F g ⊗ F f ) ◦ (1 F W ⊗ m X,U ). It follows thatTr
F UF ( W ⊗ X ) ,F ( Z ⊗ Y ) (( m Z,Y ⊗ F U ) ◦ β ◦ ( m − W,X ⊗ F U )) = (naturality axiom)= m Z,Y ◦ Tr F UF W ⊗ F X,F Z ⊗ F Y ( β ) ◦ m − W,X = (functoriality of the tensor)= m Z,Y ◦ Tr F UF W ⊗ F X,F Z ⊗ F Y ( F ( g ) ⊗ ( m − Y,U ◦ F g ◦ m X,U )) ◦ m − W,X = (superposing)= m Z,Y ◦ ( F ( g ) ⊗ Tr F UF W ⊗ F X,F Z ⊗ F Y ( m − Y,U ◦ F g ◦ m X,U )) ◦ m − W,X = (by hypothesis)= m Z,Y ◦ ( F ( g ) ⊗ F ( p )) ◦ m − W,X = (by naturality of m )= F ( g ⊗ p ).Thus, we proved that there exists an arrow p = g ⊗ p such that F ( p ) = Tr F UF ( W ⊗ X ) ,F ( Z ⊗ Y ) (( m Z,Y ⊗ F U ) ◦ β ◦ ( m − W,X ⊗ F U )) . HAPTER 4. PARTIALLY TRACED CATEGORIES F is a monoidalfunctor: ( m Z,Y ⊗ F U ) ◦ β ◦ ( m − W,X ⊗ F U ) = m − Z ⊗ Y,U ◦ F ( g ⊗ f ) ◦ m W ⊗ X,U which means, according to our definition, that c Tr UW ⊗ X,Z ⊗ Y ( g ⊗ f ) = p = g ⊗ p = g ⊗ c Tr UX,Y ( f ) . Yanking:
Let us consider σ : U ⊗ U → U ⊗ U ; we want to prove that σ U,U ∈ b T UU,U and c Tr UU,U ( σ U,U ) = 1 U . To show that σ U,U ∈ b T UU,U we recall from the trace class definitionthat we must check two conditions: (1)
First, we notice that since F is a symmetric monoidal functor and by theyanking axiom in the category D : m − U,U ◦ F ( σ U,U ) ◦ m U,U = σ F U,F U ∈ T UU,U . From which it follows that Tr
F UF U,F U ( σ F U,F U ) = 1
F U = F (1 U ). (2) Therefore there exists an arrow p = 1 U such that F (1 U ) = Tr F UF U,F U ( m − U,U ◦ F ( σ U,U ) ◦ m U,U ) . Hence, we are saying that c Tr UU,U ( σ U,U ) = p = 1 U . ⊕ Definition 4.4.1.
Consider the forgetful functor F : ( CPM , ⊕ ) → ( Vect fn , ⊕ ),where ( Vect fn , ⊕ , , T ) is partially traced by Definition 4.2.1, i.e., CPM is a monoidalsubcategory of
Vect fn . We define a partial trace c Tr with trace class given by b T on CPM by the method of Section 4.3.
HAPTER 4. PARTIALLY TRACED CATEGORIES Remark 4.4.2.
Comparing this with the partial trace (on the same category) definedin Section 4.2.5, we note that if f and ( I − f ) − are completely positive then f + f ( I − f ) − f is a completely positive map. This implies that e T UX,Y as in Definition 4.2.12 satisfies: e T UX,Y ⊆ b T UX,Y . However, consider the
CPM -map f : U ⊕ U → U ⊕ U given by thefollowing matrix: I
00 2 I ! We have f = I , f = f = 0 and f = 2 I . Then I − f = I − I = ( − I is an invertible map with inverse ( − I but is not a positive map. On the otherhand, f + f ( I − f ) − f = I + 0(( − I )0 = I is a CPM -map, i.e., f ∈ b T UU,U but f / ∈ e T UU,U . ⊕ and ⊗ As an application of the construction of Section 4.3, we now focus on the category Q which is not a compact closed category. We discuss examples of partial traces inconnection with its two monoidal structures. Example . ( Q , ⊕ ) has a total trace operator Tr ux,y : Q ( x ⊕ u, y ⊕ u ) → Q ( x, y )defined by Tr ux,y ( f ) = f + P ∞ i =0 f f i f , see [63] for details. Example . By Proposition 4.3.1, ( Q , ⊕ ) has a partial trace Tr ux,y : Q ( x ⊕ u, y ⊕ u ) ⇀ Q ( x, y ) , given by Tr ux,y ( f ) ✄ (cid:0)✂ ✁ f + f ( I − f ) − f . Example . Another partial trace on ( Q , ⊕ ) is given by considering the forgetfulfunctor from Q to the category of vector spaces ( Vect , ⊕ ) with the kernel-imagepartial trace of Definition 4.2.9 given in Section 4.2.4. Notice that the identity is asuperoperator satisfying Definition 4.2.9 which implies that these two partial tracesstill remain different on Q . Example . We can consider the category ( Q s , ⊗ ) of simple superoperators as asubcategory of the compact closed category ( CPM s , ⊗ ), see Definition 3.2.4. It has HAPTER 4. PARTIALLY TRACED CATEGORIES
72a partial trace Tr given by Proposition 4.3.1 where Tr
UX,Y : Q s ( X ⊗ U, Y ⊗ U ) ⇀ Q s ( X, Y ) is the canonical trace on
CPM s . Since linear maps f in the category offinite dimensional vector spaces are continuous functions we can prove that for everycompletely positive map there exists a 0 < λ ≤ λf is a superoperator.Then, for every unit map η U : I → U ∗ ⊗ U in CPM there exists a λ U such that λ U .η U is a superoperator. Therefore, if λ − U .f is a superoperator then f ∈ T UX,Y . hapter 5A representation theorem forpartially traced categories The goal of this chapter is to prove a strong converse to Proposition 4.3, i.e.: everypartially traced category arises as a monoidal subcategory of a totally traced cate-gory. More precisely, we show that every partially traced category can be faithfullyembedded in a compact closed category in such a way that the trace is preserved.Our construction uses a partial version of the “
Int ” construction of Joyal, Street,and Verity [41]. When we try to apply the
Int construction to a partially traced cat-egory C , we find that the composition operation in Int( C ) is a well-defined operationonly if the trace is total. We therefore consider a notion of “categories” with partiallydefined composition, namely, Freyd’s paracategories [39]. Specifically, we introducethe notion of a strict symmetric compact closed paracategory.We first show that every partially traced category can be fully and faithfullyembedded in a compact closed paracategory, by an analogue of the Int construction.We then show that every compact closed paracategory can be embedded (faithfully,but not necessarily fully) in a compact closed (total) category, using a constructionsimilar to Freyd’s. Finally, every compact closed category is (totally) traced, yieldingthe desired result. 73
HAPTER 5. A REPRESENTATION THEOREM The aim of this section is to recall Freyd’s notion of paracategory. A reference onthis subject is [39]. Informally, a paracategory is a category with partially definedcomposition.
Definition 5.1.1. A (directed) graph C consists of: • a class of elements called objects obj ( C ) • for every pair of objects A, B a set C ( A, B ) called arrows from A to B . Let Arrow ( C )be the class of all the arrows in C . Definition 5.1.2.
Let C be a graph. We define P ( C ), the path category of C , by obj ( P ( C )) = obj ( C ) and arrows from A to A n are finite sequences( A , f , A , f , . . . , A n ) of alternating objects and arrows of the graph C , where n ≥ n is the length of the path. Two arrows are equal when the sequencescoincide. Composition is defined by concatenation and the identity arrow at A is thepath of zero length ( A ) with an object A . We write ǫ A = ( A ) for the identity arrow. Notation:
For the sake of simplification, we often write ~f = f , f , . . . , f n for apath and the symbol “; ” or “ , ” for concatenation.Recall the definition of Kleene equality “ ✄ (cid:0)✂ ✁ ” and directed Kleene equality “ ✄✂ ”from Definition 4.1.1.We write φ ( f ) ↓ to say a partial function φ is defined on input f . Definition 5.1.3. A paracategory ( C , [ − ]) consists of a directed graph C and a partialoperation [ − ] : Arrow ( P ( C )) ⇀ Arrow ( C ) called composition , which satisfies thefollowing axioms:(a) for all A , [ ǫ A ] ↓ , i.e., [ − ] is a total operation on empty paths(b) for paths of length one, [ f ] ↓ and [ f ] = f (c) for all paths ~r : A → B , ~f : B → C , and ~s : C → D , if [ ~f ] ↓ then[ ~r, [ ~f ] , ~s ] ✄ (cid:0)✂ ✁ [ ~r, ~f , ~s ] . HAPTER 5. A REPRESENTATION THEOREM A = [( A )] = [ ǫ A ] for every object A in C .- for a path ~f = f , f , . . . , f n and an operation ⊗ , defined on C (see Defini-tion 5.2.1), we extend it to the category of paths using the following notation:1 ⊗ p ~f = 1 ⊗ f , ⊗ f , . . . , ⊗ f n and in the same way: ~f ⊗ p
1. We drop the symbol p when it is clear from thecontext. Definition 5.1.4.
Let ( C , [ − ]) and ( D , [ − ] ′ ) be two paracategories. A func-tor between paracategories is a graph morphism F : Obj ( C ) → Obj ( D ), F : C ( A, B ) → D ( F A, F B ) such that when [ ~p ] ↓ then F [ ~p ] = [ ~F p ] ′ . Let PCat be the category of (small) paracategories and functors.We say that such a functor is faithful if it is faithful as a morphism of graphs.
Remark 5.1.5.
Every category C can be regarded as a paracategory with[ f , . . . , f n ] = f n ◦ . . . ◦ f . In this case, composition is a totally defined operation.This yields a forgetful functor Cat → PCat . Definition 5.2.1. A strict symmetric monoidal paracategory ( C , [ − ] , ⊗ , I, σ ), alsocalled an ssmpc , consists of: • a paracategory ( C , [ − ]) • a total operation ⊗ : C × C → C which satisfies:( A ⊗ B ) ⊗ C = A ⊗ ( B ⊗ C ) on objects, ( f ⊗ g ) ⊗ h = f ⊗ ( g ⊗ h ) on arrows(associative); there is an object I such that A ⊗ I = I ⊗ A = A and f ⊗ I =1 I ⊗ f = f for every object A and arrow f (unit). Subject to the followingconditions: HAPTER 5. A REPRESENTATION THEOREM A ⊗ B = 1 A ⊗ B .(b) [ f, f ′ ] ⊗ [ g, g ′ ] ✄✂ [ f ⊗ g, f ′ ⊗ g ′ ] where f, g, f ′ , g ′ are arrows of C and ✄✂ denotes Kleene directed equality.(c) 1 ⊗ [ ~p ] ✄✂ [ 1 ⊗ ~p ] and [ ~p ] ⊗ ✄✂ [ ~p ⊗ • for all objects A and B there is an arrow σ A,B : A ⊗ B → B ⊗ A such that:- for every f : B ⊗ A → X , g : Y → A ⊗ B , [ σ A,B , f ] ↓ and [ g, σ A,B ] ↓ - for every f : A → A ′ and g : B → B ′ : [ f ⊗ B , σ ] = [ σ, B ⊗ f ] and[1 A ⊗ g, σ ] = [ σ, g ⊗ A ]- for every A and B : [ σ A,B , σ
B,A ] = 1 A ⊗ B - for every A , B , and C : [ σ A,B ⊗ C , σ B,A ⊗ C ] = 1 A ⊗ σ B,C . Remark 5.2.2.
Conditions (b) and (c) are equivalent to the condition [ f , . . . , f n ] ⊗ [ g . . . , g n ] ✄✂ [ f ⊗ g , . . . , f n ⊗ g n ] for all natural numbers n . Proposition 5.2.3.
Let ( C , [ − ] , ⊗ , I, σ ) be a ssmpc. Then for paths p , q of lengthone we have that [ p ⊗ C , B ⊗ q ] ↓ , [1 A ⊗ q, p ⊗ D ] ↓ and are equal to p ⊗ q . Moreover,for paths ~p and ~q [ 1 A ⊗ ~q, ~p ⊗ D ] ✄ (cid:0)✂ ✁ [ ~p ⊗ C , B ⊗ ~q ] .Proof. Let us first prove the result for paths of length 1, say p : A → B , q : C → D .Observe that [ p, B ] = [ p, [( B )]] = [ p, ( B )] = [ p ] = p since the last equation isdefined and by the axioms. In the same way p = [1 A , p ], [1 C , q ] = q = [ q, D ]. Then p ⊗ q = [ p, B ] ⊗ [1 C , q ] ✄✂ [ p ⊗ C , B ⊗ q ] and p ⊗ q = [1 A , p ] ⊗ [ q, D ] ✄✂ [1 A ⊗ q, p ⊗ D ], bycondition (b) of Definition 5.2.1, which implies that [1 A ⊗ q, p ⊗ D ] ↓ , [ p ⊗ C , B ⊗ q ] ↓ and [1 A ⊗ q, p ⊗ D ] = [ p ⊗ C , B ⊗ q ] = p ⊗ q .Now since we have already proved that [1 A ⊗ q, p ⊗ D ] ↓ , [ p ⊗ C , B ⊗ q ] ↓ andthat they are equal we can use the axioms of paracategories and extend this to[1 A ⊗ ~q, ~p ⊗ D ] ✄ (cid:0)✂ ✁ [ ~p ⊗ C , B ⊗ ~q ] by iterating this procedure in the following way: HAPTER 5. A REPRESENTATION THEOREM A ⊗ ~q, ~p ⊗ D ] = [1 A ⊗ q , . . . , A ⊗ q n , p ⊗ D , . . . , p m ⊗ D ] ✄ (cid:0)✂ ✁ [1 A ⊗ q , . . . , [1 A ⊗ q n , p ⊗ D ] , . . . , p m ⊗ D ] ✄ (cid:0)✂ ✁ [1 A ⊗ q , . . . , [ p ⊗ C , B ⊗ q n ] , . . . , p m ⊗ D ] ✄ (cid:0)✂ ✁ [1 A ⊗ q , . . . , p ⊗ C , B ⊗ q n , . . . , p m ⊗ D ] ✄ (cid:0)✂ ✁ . . . we move p ⊗ C to the first position ✄ (cid:0)✂ ✁ [ p ⊗ C , B ⊗ q , . . . , A ⊗ q n , p ⊗ C , . . . , p m ⊗ D ] ✄ (cid:0)✂ ✁ . . . we iterate this procedure m − ✄ (cid:0)✂ ✁ [ ~p ⊗ C , B ⊗ ~q ] . Definition 5.2.4.
Let ( C , [ − ] , ⊗ , I, σ ) and ( D , [ − ] ′ , ⊗ ′ , I ′ , σ ′ ) be two ssmpcs. A func-tor between them is strict monoidal when F ( A ) ⊗ ′ F ( B ) = F ( A ⊗ B ), F ( I ) = I ′ onobjects, F ( f ) ⊗ ′ F ( g ) = F ( f ⊗ g ) and F ( σ ) = σ ′ on arrows. From now on C denotes a ssmpc. We wish to prove the following theorem: Theorem 5.3.1.
Every strict symmetric monoidal paracategory can be faithfully em-bedded in a strict symmetric monoidal category.
Definition 5.3.2. A congruence relation S on P ( C ) is given as follows: for every pairof objects A , B , an equivalence relation ∼ A,B S on the hom-set P ( C )( A, B ), satisfyingthe following axioms. We usually omit the superscripts when they are clear from thecontext.(1) If ~p ∼ S ~p ′ and ~q ∼ S ~q ′ , then ~p ; ~q ∼ S ~p ′ ; ~q ′ .(2) Whenever [ ~p ] ↓ , then ~p ∼ S [ ~p ]. HAPTER 5. A REPRESENTATION THEOREM ~p ∼ S ~q , then ~p ⊗ ∼ S ~q ⊗ ⊗ ~p ∼ S ⊗ ~q . Remark 5.3.3.
Technically Definition 5.3.2 can be regarded as a “congruence sub-category” on P ( C ), i.e., S is a subcategory of P ( C ) × P ( C ) satisfying axioms (2) and(3). Definition 5.3.4.
We define a particular congruence relation ˆ S as follows: ~p ∼ ˆ S ~q ifand only if ∀ ~r, ~s, ∀ A, B ∈ Obj ( C ) [ ~r, A ⊗ ~p ⊗ B , ~s ] ✄ (cid:0)✂ ✁ [ ~r, A ⊗ ~q ⊗ B , ~s ]. Remark 5.3.5.
It should be observed that ~p ∼ ˆ S ~q implies [ ~p ] ✄ (cid:0)✂ ✁ [ ~q ] by letting ~r, ~s be empty lists and A = B = I .Let us check that ˆ S is a congruence relation. Lemma 5.3.6. ˆ S is a congruence relation.Proof. We need to show axioms (1), (2), and (3). To show (1), assume ~p ∼ ˆ S ~q and ~u ∼ ˆ S ~t we have to check that ~p ; ~u ∼ ˆ S ~q ; ~t . Consider arbitrary ~r , ~s , A , B . We have:[ ~r, A ⊗ ( ~p ; ~u ) ⊗ B , ~s ] ✄ (cid:0)✂ ✁ [ ~r, A ⊗ ~p ⊗ B , A ⊗ ~u ⊗ B , ~s ] ✄ (cid:0)✂ ✁ [ ~r, A ⊗ ~q ⊗ B , A ⊗ ~u ⊗ B , ~s ] . The first equation is by definition of the tensor ⊗ p on paths, the second equation isbecause by hypothesis we have that: ~p ∼ ˆ S ~q implies [ ~r ′ , A ⊗ ~p ⊗ B , ~s ′ ] ✄ (cid:0)✂ ✁ [ ~r ′ , A ⊗ ~q ⊗ B , ~s ′ ] with ~r = ~r ′ and ~s ′ = 1 A ⊗ ~u ⊗ B , ~s . In a similar way we have that:[ ~r, A ⊗ ( ~q ; ~t ) ⊗ B , ~s ] ✄ (cid:0)✂ ✁ [ ~r, A ⊗ ~q ⊗ B , A ⊗ ~t ⊗ B , ~s ] ✄ (cid:0)✂ ✁ [ ~r, A ⊗ ~q ⊗ B , A ⊗ ~u ⊗ B , ~s ] . It follows that ~p ; ~u ∼ ˆ S ~q ; ~t . To prove (2), assume [ ~p ] ↓ , and let ~r, ~s, A, B be given. Weobserve first that 1 A ⊗ [ ~p ] ⊗ B ✄✂ [ 1 A ⊗ ~p ⊗ B ] by ( c ) in the definition of a ssmpc.Then [ ~p ] ↓ implies that 1 A ⊗ [ ~p ] ⊗ B ↓ and then [ 1 A ⊗ ~p ⊗ B ] ↓ and and they areequal. Thus we have by one of the axioms of paracategory that:[ ~r, A ⊗ [ ~p ] ⊗ B , ~s ] ✄ (cid:0)✂ ✁ [ ~r, [ 1 A ⊗ ~p ⊗ B ] , ~s ] ✄ (cid:0)✂ ✁ [ ~r, A ⊗ ~p ⊗ B , ~s ] . To prove (3), assume ~p ∼ ˆ S ~p ′ . We observe that this implies for every C ∈ Obj ( C ),[ ~r, A ⊗ ~p ⊗ C ⊗ B , ~s ] ✄ (cid:0)✂ ✁ [ ~r, A ⊗ ~p ′ ⊗ C ⊗ B , ~s ], ∀ ~r, ~s, ∀ A, B ∈ Obj ( C ), therefore ~p ⊗ ∼ ˆ S ~p ′ ⊗
1. In a similar way we get the other equation.
HAPTER 5. A REPRESENTATION THEOREM Definition 5.3.7.
Let ∼ be the smallest congruence relation on P ( C ), i.e., the inter-section of all congruence relations. Proposition 5.3.8. ~p ∼ ~q implies [ ~p ] ✄ (cid:0)✂ ✁ [ ~q ] .Proof. Since ( ~p, ~q ) is in the intersection of all congruence relations then in particular ~p ∼ ˆ S ~q which implies that [ ~p ] ✄ (cid:0)✂ ✁ [ ~q ] by Remark 5.3.5. Corollary 5.3.9.
For paths p, q : A → B of length 1, p ∼ q iff p = q .Proof. Obvious from Proposition 5.3.8 and axiom (b) of paracategories.We now introduce the following notation: ~f ⊗ p ~g = ( ~f ⊗ p , (1 ⊗ p ~g ) . Note that, as a path, this is not equal to (1 ⊗ p ~g ) , ( ~f ⊗ p p . Lemma 5.3.10.
Let S be a congruence relation of P ( C ) . Then if ~f ∼ S ~f ′ and ~g ∼ S ~g ′ then ~f ⊗ ~g ∼ S ~f ′ ⊗ ~g ′ .Proof. By assumption ~f ∼ S ~f ′ therefore by (3) we have ~f ⊗ ∼ S ~f ′ ⊗
1. Similarly1 ⊗ ~q ∼ S ⊗ ~q ′ . Therefore by (1), we have: ~f ⊗ , ⊗ ~q ∼ S ~f ′ ⊗ , ⊗ ~q ′ . Lemma 5.3.11.
Let S be a congruence relation of P ( C ) . Then ~f ⊗ , ⊗ ~g ∼ S ⊗ ~g, ~f ⊗ . Proof.
Given ~f = f , . . . , f n and ~g = g , . . . , g m we have that by Proposition 5.2.3above [ f n ⊗ , ⊗ g ] ↓ , [1 ⊗ g , f n ⊗ ↓ and are equal to f n ⊗ g . This yields, byDefinition 5.3.2 of congruence relation, the following sequence of equivalences: f n ⊗ , ⊗ g ∼ S [ f n ⊗ , ⊗ g ] = [1 ⊗ g , f n ⊗ ∼ S ⊗ g , f n ⊗ HAPTER 5. A REPRESENTATION THEOREM f ⊗ , . . . , f n − ⊗ , z }| { ( f n ⊗ , ⊗ g ) , ⊗ g , . . . ⊗ g m ∼ S f ⊗ , . . . , f n − ⊗ , (1 ⊗ g , f n ⊗ | {z } , ⊗ g , . . . ⊗ g m .By iterating this procedure we end up moving 1 ⊗ g into the first place. We finishthe proof by repeating this m − Lemma 5.3.12.
Let S be a congruence relation defined on a strict symmetricmonoidal paracategory ( C , [ − ] , ⊗ , I, σ ) . Then the quotient ( P ( C ) / S , ˆ ⊗ , I, s ) is a strictsymmetric monoidal category, where ˆ ⊗ is the obvious tensor and s = σ .Proof. Let ( C , [ − ] , ⊗ , I, σ ) be a strict symmetric monoidal paracategory. It induces astrict symmetric monoidal category ( P ( C ) / S , ˆ ⊗ , I, s ) in the following way:The objects of P ( C ) / S are the same as the objects of the graph C and the arrows ~f = f . . . , f n are S -equivalence classes of paths. Composition on classes is inducedby composition on paths by axiom (1) of congruences. The identity is the class of theidentity of the path category.A bifunctor ˆ ⊗ : P ( C ) / S × P ( C ) / S → P ( C ) / S is defined by ~f ˆ ⊗ ~g = ~f ⊗ p , ⊗ p ~g .The tensor is well-defined by the Lemma 5.3.10 above.We must check the interchange law:( ~f ˆ ⊗ ~g ); ( ~f ′ ˆ ⊗ ~g ′ ) = ( ~f ; ~f ′ ) ˆ ⊗ ( ~g ; ~g ′ )We have:( ~f ˆ ⊗ ~g ); ( ~f ′ ˆ ⊗ ~g ′ ) = ( ~f ⊗ , ⊗ ~g ); ( ~f ′ ⊗ , ⊗ ~g ′ ) = ~f ⊗ , ⊗ ~g, ~f ′ ⊗ , ⊗ ~g ′ = ~f ⊗
1; 1 ⊗ ~g, ~f ′ ⊗
1; 1 ⊗ ~g ′ ( ∗ ) = ~f ⊗ ~f ′ ⊗ , ⊗ ~g ; 1 ⊗ ~g ′ = ~f ⊗ , ~f ′ ⊗ , ⊗ ~g, ⊗ ~g ′ = ( ~f , ~f ′ ) ⊗ , ⊗ ( ~g, ~g ′ ) = ( ~f , ~f ′ ) ⊗ p ( ~g, ~g ′ ) =( ~f , ~f ′ ) ˆ ⊗ ( ~g, ~g ′ ) = ( ~f ; ~f ′ ) ˆ ⊗ ( ~g ; ~g ′ ).Where in ( ∗ ) we used the property of the Lemma 5.3.11 above: 1 ⊗ ~g, ~f ′ ⊗ ~f ′ ⊗ , ⊗ ~g .Also we want to check that ǫ A ⊗ B = ǫ A ˆ ⊗ ǫ B . HAPTER 5. A REPRESENTATION THEOREM ǫ A ˆ ⊗ ǫ B = ǫ A ⊗ p B , A ⊗ p ǫ B = ǫ A ⊗ B , ǫ A ⊗ B = ǫ A ⊗ B since [ 1 A ⊗ B , A ⊗ B ] ↓ .Given paths ~f : A → B , ~g : C → D and ~h : E → F we check the associative property:( ~f ˆ ⊗ ~g ) ˆ ⊗ ~h = ( ~f ⊗ p ~g ) ˆ ⊗ ~h = ( ~f ⊗ C , B ⊗ ~g ) ˆ ⊗ ~h = ( ~f ⊗ C , B ⊗ ~g ) ⊗ p ~h =( ~f ⊗ C , B ⊗ ~g ) ⊗ E , B ⊗ D ⊗ ~h = ~f ⊗ C ⊗ E , B ⊗ ~g ⊗ E , B ⊗ D ⊗ ~h = ~f ⊗ C ⊗ E , B ⊗ ~g ⊗ E , B ⊗ D ⊗ ~h = ~f ⊗ C ⊗ E , B ⊗ ( ~g ⊗ E , D ⊗ ~h ) = ~f ⊗ p ( ~g ⊗ E , D ⊗ ~h ) = ~f ⊗ p ( ~g ⊗ p ~h ) = ~f ˆ ⊗ ( ~g ⊗ p ~h ) = ~f ˆ ⊗ ( ~g ˆ ⊗ ~h ) . Also if ~f : A → B and 1 I : I → I then: ~f ˆ ⊗ I = ~f ⊗ p I = ~f ⊗ I , B ⊗ I = ~f , B = ~f . Since ~f ⊗ I = −−−−→ f ⊗ I = ~f and 1 B ⊗ I = 1 B . In the same way we get 1 I ˆ ⊗ ~f = ~f .The symmetry is defined as s A,B : A ˆ ⊗ B → B ˆ ⊗ A , s A,B = σ A,B . This arrow is anisomorphism since [ σ A,B , σ
B,A ] ↓ implies σ A,B , σ
B,A ∼ S [ σ A,B , σ
B,A ] and then: s A,B ; s B,A = σ A,B ; σ B,A = σ A,B , σ
B,A = [ σ A,B , σ
B,A ] = 1 A ⊗ B . Similarly s B,A ; s A,B = 1 B ⊗ A .Next, we check the following coherence diagram: ( s A,B ⊗ C ); s B,A ⊗ C = 1 A ⊗ s B,C .( s A,B ⊗ C ); s B,A ⊗ C = ( σ A,B ˆ ⊗ C ); σ B,A ⊗ C = ( σ A,B ⊗ C ); σ B,A ⊗ C =( σ A,B ⊗ C ) , σ B,A ⊗ C = [( σ A,B ⊗ C ) , σ B,A ⊗ C ] = 1 A ⊗ σ B,C = 1 A ˆ ⊗ σ B,C = 1 A ˆ ⊗ s B,C . Next we prove naturality of the map s A,B : A ˆ ⊗ B → B ˆ ⊗ A . To see this, it is enoughto prove it on simple path of length one and then extend it by composition. Let usconsider f : A → A ′ and since [ σ A,B , ⊗ f ] ↓ s A,B ; (1 ˆ ⊗ f ) = σ A,B ; (1 ˆ ⊗ f ) = σ A,B ; 1 ⊗ f = σ A,B , ⊗ f = [ σ A,B , ⊗ f ] =[ f ⊗ , σ A ′ ,B ] = f ⊗ , σ A ′ ,B = f ⊗ σ A ′ ,B = ( f ˆ ⊗ s A ′ ,B . HAPTER 5. A REPRESENTATION THEOREM
Proof of Theorem 5.3.1
Proof.
A functor between paracategories F : ( C , [ − ] , ⊗ , I, σ ) → ( P ( C ) / S , ˆ ⊗ , I, s ),where the category P ( C ) / S is taken as a (total) paracategory, is defined in the fol-lowing way:- on objects as the identity and- on arrows F ( f ) = f as the projection on classes.Observe that F preserves identities and composition when [ ~f ] is defined: F [ ~f ] = [ ~f ] = ~f = f , . . . , f n = f ; . . . ; f n = F f ; . . . ; F f n . Following the definition, we have that F preserves symmetries: F ( σ ) = σ = s .In addition, if f : A → C and g : B → D then F ( f ⊗ g ) = f ⊗ g = [ f ⊗ B , C ⊗ g ] = f ⊗ B , C ⊗ g = f ⊗ p g = F f ˆ ⊗ F g where the last sequence of equations is justified by Proposition 5.2.3, the propertyabove, axioms and by definition of congruence relation.Moreover, if S is the smallest congruence relation, or indeed any congruence relationsatisfying S ⊆ ˆ S , then F is faithful by Corollary 5.3.9. Definition 5.4.1. A (strict symmetric) compact closed paracategory ( C , [ − ] , ⊗ , I, σ, η, ǫ ) is a strict symmetric monoidal paracategory such that forevery object A there is an object A ∗ and arrows η A : I → A ⊗ A ∗ , ǫ A : A ∗ ⊗ A → I such that [ η A ⊗ A , A ⊗ ǫ A ] ↓ , [1 A ∗ ⊗ η A , ǫ A ⊗ A ∗ ] ↓ and [ η A ⊗ A , A ⊗ ǫ A ] = 1 A ,[1 A ∗ ⊗ η A , ǫ A ⊗ A ∗ ] = 1 A ∗ . HAPTER 5. A REPRESENTATION THEOREM Theorem 5.4.2.
Every compact closed paracategory can be faithfully embedded in acompact closed category.Proof.
Let us consider the paracategory ( C , [ − ] , ⊗ , I, σ, η, ǫ ).As a result of the proof of Theorem 5.3.1 above, it suffices to show that( P ( C ) / S , ˆ ⊗ , I, s, η ′ , ǫ ′ ) is compact closed, where η ′ = η and ǫ ′ = ǫ . Noticethat by definition the functor F preserves η and ǫ . Consequently, the compactnessdiagrams are satisfied, since the condition [ η ⊗ , ⊗ ǫ ] ↓ implies: η ˆ ⊗ A ; 1 A ˆ ⊗ ǫ = η ⊗ A ; 1 A ⊗ ǫ = η ⊗ A , A ⊗ ǫ = [ η ⊗ A , A ⊗ ǫ ]= 1 A . In the same way, 1 A ∗ ˆ ⊗ η ; ǫ ˆ ⊗ A ∗ = 1 A ∗ We can strengthen Theorem 5.3.1 by noting that the faithful embedding satisfies auniversal property.
Theorem 5.5.1.
The category ( P ( C ) / ∼ , ˆ ⊗ , I, s ) satisfies the following property: forany strict symmetric monoidal category D and any strict symmetric monoidal functor G : C → D between paracategories, there exists a unique strict symmetric monoidalfunctor L : P ( C ) / ∼ → D such that L ◦ F = G , where F is the inclusion map definedin Theorem 5.3.1 above. C F / / G ' ' PPPPPPPPPPPPPPPP P ( C ) / ∼ L (cid:15) (cid:15) D HAPTER 5. A REPRESENTATION THEOREM Proof.
Consider the set S : { ( ~f , ~g ) ∈ P ( C ) × P ( C ) : G ( f ) ◦ · · · ◦ G ( f n ) = G ( g ) ◦ · · · ◦ G ( g m ) } where ~f = f , . . . , f n and ~g = g , . . . , g m . We claim that S is a congruence relation in the sense of Definition 5.3.2 stipulatedabove. Clearly, it is an equivalence relation. To show that it satisfies axiom (1),assume p , . . . p n ∼ S q , . . . , q m and f , . . . f s ∼ S q , . . . , g t , then by hypothesis G ( p ) ◦ · · · ◦ G ( p n ) = G ( q ) ◦ · · · ◦ G ( q m ) and G ( f ) ◦ · · · ◦ G ( f s ) = G ( g ) ◦ · · · ◦ G ( g t ).Then by composing the left hand side and the right hand side we get the condition p , . . . , p n , f , . . . , f s ∼ S q , . . . , q m , g , . . . , g t . To show (3), assume p , . . . , p n ∼ ˆ S q , . . . , q m , then G ( p ) ◦ · · · ◦ G ( p n ) = G ( q ) ◦ · · · ◦ G ( q m ) which in C implies that ( G ( p ) ◦ · · · ◦ G ( p n )) ⊗ G G ( q ) ◦ · · · ◦ G ( q m )) ⊗ G G and functoriality we obtain G ( p ⊗ ◦· · ·◦ G ( p n ⊗
1) = G ( q ⊗ ◦ · · · ◦ G ( q m ⊗ p ⊗ , . . . , p n ⊗ ∼ S q ⊗ , . . . , q m ⊗ . Inthe same way ~p ∼ S ~q implies 1 ⊗ ~p ∼ S ⊗ ~q .To show (2), since G is a functor between paracategories, we have G ( p ) ◦ . . . G ( p n ) = G ([ ~p ]) when [ ~p ] ↓ hence ~p ∼ S [ ~p ].Now we define the functor L in the following way: L ( A ) = G ( A ) on objects and L ( ~p ) = G ( p ) ◦ · · · ◦ G ( p n ), where ~p = p , . . . , p n .It should be apparent that F is well-defined since when ~p ∼ ~q then in particular it istrue that ~p ∼ S ~q and this implies L ( ~p ) = L ( ~q ).We check functoriality: L ( p , . . . , p n ; q , . . . , q m ) = L ( p , . . . , p n , q , . . . , q m )= G ( p ) ◦ · · · ◦ G ( p n ) ◦ G ( q ) ◦ · · · ◦ G ( q m )= L ( p , . . . , p n ) ◦ L ( q , . . . , q m ) HAPTER 5. A REPRESENTATION THEOREM L (( A )) = L (1 A ) = G (1 A ) = 1 GA . Furthermore L is strict symmetric monoidal: L ( ~p ˆ ⊗ ~q ) = L ( ~p ⊗ , ⊗ ~q )= G ( p ⊗ ◦ · · · ◦ G ( p n ⊗ ◦ G (1 ⊗ q ) ◦ · · · ◦ G (1 ⊗ q m )= ( G ( p ) ⊗ G ◦ · · · ◦ ( G ( p n ) ⊗ G ◦ ( G ⊗ G ( q )) ◦ · · · ◦ ( G ⊗ G ( q m ))= ( G ( p ) ⊗ ◦ · · · ◦ ( G ( p n ) ⊗ ◦ (1 ⊗ G ( q )) ◦ · · · ◦ (1 ⊗ G ( q m ))= ( G ( p ) ◦ · · · ◦ G ( p n )) ⊗ ◦ (1 ⊗ ( G ( q ) ◦ · · · ◦ G ( q m ))= ( G ( p ) ◦ · · · ◦ G ( p n )) ⊗ ( G ( q ) ◦ · · · ◦ G ( q m ))= L ( ~p ) ⊗ L ( ~q )Finally, since G is strict symmetric L ( s ) = L ( σ ) = G ( σ ) = σ ′ where σ ′ is thesymmetry of the category D . Joyal, Street, and Verity proved in [41] that every (totally) traced monoidal category C can be faithfully embedded in a compact closed category Int ( C ). Here, we give asimilar construction for partially traced categories. We call the corresponding con-struction the partial Int construction , or the Int p construction for short. When C isa partially traced category, Int p ( C ) will be a compact closed para category. Definition 5.6.1.
Let ( C , ⊗ , I, σ ) be a symmetric monoidal category. There is agraph Int p ( C ) associated to this category defined in the following way: • objects: are a pair of object ( A + , A − ) of the category C . HAPTER 5. A REPRESENTATION THEOREM • arrows: f Int p : ( A + , A − ) → ( B + , B − ) are arrows of type f : A + ⊗ B − → B + ⊗ A − in the category C .When it is clear from the context we drop the symbol Int p on the arrows of Int p ( C ).We want to define a partial composition on this graph. For that purpose, considerthe following natural transformation, uniquely induced by the symmetric monoidalstructure, for n ≥ γ n : A ⊗ A ⊗ . . . A n − ⊗ A n → A n ⊗ A n − . . . A ⊗ A . Also, given a path ~p = p , . . . , p m ∈ P ( Int p ( C )), using graphical language of symmet-ric monoidal categories, we shall define an arrow ǫ ( ~p ) ∈ C in the following way: if ~p = p , . . . , p m then ǫ ( ~p ) pictorially is equal to:For m = 1 arrow: ❄❄❄❄❄❄❄ ❄❄❄❄❄❄❄⑧⑧⑧⑧⑧⑧⑧ p ⑧⑧⑧⑧⑧⑧⑧ For m = 2 arrows: ❄❄❄❄❄❄❄ ❄❄❄❄❄❄❄ ❄❄❄❄❄❄❄⑧⑧⑧⑧⑧⑧⑧ p ⑧⑧⑧⑧⑧⑧⑧ p ⑧⑧⑧⑧⑧⑧⑧ For m = 3 arrows: ❄❄❄❄❄❄❄ ❄❄❄❄❄❄❄ ❄❄❄❄❄❄❄❄❄❄❄❄❄❄ ⑧⑧⑧⑧⑧⑧⑧ ❄❄❄❄❄❄❄ ⑧⑧⑧⑧⑧⑧⑧ ❄❄❄❄❄❄❄ ⑧⑧⑧⑧⑧⑧⑧ ❄❄❄❄❄❄❄⑧⑧⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧⑧⑧ p p p HAPTER 5. A REPRESENTATION THEOREM ǫ ( p , . . . , p m ) we form a pyramid of m − Definition 5.6.2.
Let ( C , ⊗ , I, Tr , s ) be a symmetric monoidal partially traced cate-gory. We turn the graph Int p ( C ) into a paracategory by defining a partial compositionoperation [ ~p ]. First of all, when it is applied to an empty path it will be defined asthe identity arrow i.e., [( A + , A − )] = 1 A + ⊗ A − . On path of length one it will be bydefinition the same arrow, i.e., [ f ] = Tr U ( ǫ ( f )(1 X +1 ⊗ σ X − ,X − )) = f with U = X − . Suppose that we have a family of arrows f Int p i : ( X + i , X − i ) → ( X + i +1 , X − i +1 ) with1 ≤ i ≤ n ( n ≥
2) in the graph
Int p ( C ) such that dom ( f i +1 ) = cod ( f i ) and1 ≤ i ≤ n −
1. Let U = X − n ⊗ X − n − ⊗ · · · ⊗ X − ⊗ X − and the permutation γX − n ⊗ X − n − ⊗ · · · ⊗ X − ⊗ X − γ −→ X − ⊗ X − ⊗ · · · ⊗ X − n − ⊗ X − n . We define the following operation for n ≥ f , . . . , f n ] ✄ (cid:0)✂ ✁ Tr U ( ǫ ( f , . . . , f n )(1 X +1 ⊗ X − n +1 ⊗ γ n − )) . Note that therefore, [ f , . . . , f n ] is defined if and only if ǫ ( f , . . . , f n )(1 X +1 ⊗ X − n +1 ⊗ γ ) ∈ T U . We show now that the operation [ − ] satisfies the axioms required in order to bea paracategory. Lemma 5.6.3.
Let ( C , ⊗ , I, Tr , s ) be a strict symmetric monoidal partially tracedcategory. The operation defined in Definition 5.6.2 determines a paracategory ( Int p ( C ) , [ − ]) .Proof. Properties ( a ) and ( b ) of Definition 5.1.3 hold by definition. The goal is toprove ( c ), i.e., if [ ~g ] ↓ then [ ~f , [ ~g ] , ~h ] ✄ (cid:0)✂ ✁ [ ~f , ~g, ~h ] for every ~f and ~h . The value of thetrace remains always invariant or follows the variations that the axioms trace dictate.Without loss of generality we are going to represent these paths using graphical lan-guage in a concrete situation. Therefore, suppose we have ~f = f , f , ~g = g , g , g , g and ~h = h , h , h . The most general case follows the same pattern. HAPTER 5. A REPRESENTATION THEOREM ~g ] ↓ means that the map: V ✶✶✶✶✶✶✶✶ ❈❈❈❈❈ ❈❈❈❈❈ ❈❈❈❈❈ V ❈❈❈❈❈ ④④④④④ ❈❈❈❈❈ ④④④④④ ❈❈❈❈❈ ④④④④④ ❈❈❈❈❈✌✌✌✌✌✌✌✌ ❈❈❈❈❈ ④④④④④ ❈❈❈❈❈ ④④④④④ ❈❈❈❈❈ ④④④④④ ❈❈❈❈❈ ④④④④④ ❈❈❈❈❈④④④④④ ④④④④④ ④④④④④ ④④④④④ ④④④④④ g g g g (7)(without the dotted lines) is in the trace class T V . We symbolize that it is in thetrace class of this type with these dotted lines. Moreover, [ ~f , [ ~g ] , ~h ] ↓ means that: U ✯✯✯✯✯✯✯✯✯✯✯✯✯ ❈❈❈❈❈ ❈❈❈❈❈ ❈❈❈❈❈ U ✹✹✹✹✹✹✹ ❈❈❈❈❈ ④④④④④ ❈❈❈❈❈ ④④④④④ ❈❈❈❈❈ ④④④④④ ❈❈❈❈❈❈❈❈❈❈ ④④④④④ ❈❈❈❈❈ ④④④④④ ❈❈❈❈❈ ④④④④④ ❈❈❈❈❈ ④④④④④ ❈❈❈❈❈✡✡✡✡✡✡✡✡ ❈❈❈❈❈ ④④④④④ ❈❈❈❈❈ ④④④④④ ❈❈❈❈❈ ④④④④④ ❈❈❈❈❈ ④④④④④ ❈❈❈❈❈ ④④④④④ ❈❈❈❈❈✕✕✕✕✕✕✕✕✕✕✕✕✕✕ ❈❈❈❈❈ ④④④④④ ❈❈❈❈❈ ④④④④④ ❈❈❈❈❈ ④④④④④ ❈❈❈❈❈ ④④④④④ ❈❈❈❈❈ ④④④④④ ❈❈❈❈❈ ④④④④④ ❈❈❈❈❈④④④④④ ④④④④④ ④④④④④ ④④④④④ ④④④④④ ④④④④④ ④④④④④ f f [ ~g ] h h h (8)without the dotted lines is in trace class T U . We want to obtain [ ~f , ~g, ~h ]. So, forthat purpose, we start by replacing the first diagram (7) traced on V into the seconddiagram (8). Then we apply superposition, and the naturality axiom and we get thefollowing diagram: V ✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪ ✯✯✯✯✯✯✯✯✯✯✯ V ✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪ ❄❄❄❄❄ ✯✯✯✯✯✯✯✯✯✯✯✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪ ⑧⑧⑧⑧⑧ ✯✯✯✯✯✯✯✯✯✯✯ U ✮✮✮✮✮✮✮✮✮✮✮✮✮ ❄❄❄❄❄ ✔✔✔✔✔✔✔✔✔✔✔ ❄❄❄❄❄ ✯✯✯✯✯✯✯✯✯✯✯ ❄❄❄❄❄ ❄❄❄❄❄ U ✷✷✷✷✷✷✷ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ✔✔✔✔✔✔✔✔✔✔✔ ⑧⑧⑧⑧⑧ ✯✯✯✯✯✯✯✯✯✯✯ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ✔✔✔✔✔✔✔✔✔✔✔ ✴✴✴✴✴✴✴✴ ❄❄❄❄❄ ❄❄❄❄❄ ❄❄❄❄❄ ✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄☞☞☞☞☞☞☞☞ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ✔✔✔✔✔✔✔✔✔✔✔ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄✖✖✖✖✖✖✖✖✖✖✖✖✖ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ✔✔✔✔✔✔✔✔✔✔✔ ✎✎✎✎✎✎✎✎ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ f f g g g g h h h (9) HAPTER 5. A REPRESENTATION THEOREM α (without the dotted lines). Notice that since [ ~g ] ↓ and afterapplying superposing and the naturality axioms we have that α ∈ T V . This turnsout to be the general condition that we need in order to use the Vanishing II axiom,i.e., if we consider α ∈ T V as a general hypothesis then the equivalence α ∈ T U ⊗ V ⇔ Tr V ( α ) ∈ T U is precisely the condition required to apply the Vanishing II axiom in which thecondition [ ~f , [ ~g ] , ~h ] translates into Tr V ( α ) ∈ T U and [ ~f , ~g, ~h ] into α ∈ T U ⊗ V . Thus wecan replace the previous diagram by the next one: U ⊗ V ✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪ ✯✯✯✯✯✯✯✯✯✯✯ U ⊗ V ✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪ ❄❄❄❄❄ ✯✯✯✯✯✯✯✯✯✯✯✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪ ⑧⑧⑧⑧⑧ ✯✯✯✯✯✯✯✯✯✯✯✮✮✮✮✮✮✮✮✮✮✮✮✮ ❄❄❄❄❄ ✔✔✔✔✔✔✔✔✔✔✔ ❄❄❄❄❄ ✯✯✯✯✯✯✯✯✯✯✯ ❄❄❄❄❄ ❄❄❄❄❄✷✷✷✷✷✷✷ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ✔✔✔✔✔✔✔✔✔✔✔ ⑧⑧⑧⑧⑧ ✯✯✯✯✯✯✯✯✯✯✯ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ✔✔✔✔✔✔✔✔✔✔✔ ✴✴✴✴✴✴✴✴ ❄❄❄❄❄ ❄❄❄❄❄ ❄❄❄❄❄ ✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄☞☞☞☞☞☞☞☞ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ✔✔✔✔✔✔✔✔✔✔✔ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄✖✖✖✖✖✖✖✖✖✖✖✖✖ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ✔✔✔✔✔✔✔✔✔✔✔ ✎✎✎✎✎✎✎✎ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ f f g g g g h h h By coherence we can replace this part of the diagram: ✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬ ✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬ ✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬ ✒✒✒✒✒✒✒✒✒✒✒ ✒✒✒✒✒✒✒✒✒✒✒ ✒✒✒✒✒✒✒✒✒✒✒ ✴✴✴✴✴✴✴✴✒✒✒✒✒✒✒✒✒✒✒ ✒✒✒✒✒✒✒✒✒✒✒ ✎✎✎✎✎✎✎✎ by this one
HAPTER 5. A REPRESENTATION THEOREM ✷✷✷✷✷✷✷ ✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪ ✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪☞☞☞☞☞☞☞☞ ✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪ ✔✔✔✔✔✔✔✔✔✔✔ ✔✔✔✔✔✔✔✔✔✔✔ ✔✔✔✔✔✔✔✔✔✔✔ ✔✔✔✔✔✔✔✔✔✔✔ ✔✔✔✔✔✔✔✔✔✔✔ So, by this substitution and functoriality we get: ✷✷✷✷✷✷✷ ✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪ ✯✯✯✯✯✯✯✯✯✯✯✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪ ❄❄❄❄❄ ✯✯✯✯✯✯✯✯✯✯✯☞☞☞☞☞☞☞☞ ✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪ ⑧⑧⑧⑧⑧ ✯✯✯✯✯✯✯✯✯✯✯✮✮✮✮✮✮✮✮✮✮✮✮✮ ❄❄❄❄❄ ✔✔✔✔✔✔✔✔✔✔✔ ❄❄❄❄❄ ✯✯✯✯✯✯✯✯✯✯✯ ❄❄❄❄❄ ❄❄❄❄❄✷✷✷✷✷✷✷ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ✔✔✔✔✔✔✔✔✔✔✔ ⑧⑧⑧⑧⑧ ✯✯✯✯✯✯✯✯✯✯✯ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ✔✔✔✔✔✔✔✔✔✔✔ ❄❄❄❄❄ ❄❄❄❄❄ ❄❄❄❄❄ ✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄☞☞☞☞☞☞☞☞ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ✔✔✔✔✔✔✔✔✔✔✔ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄✖✖✖✖✖✖✖✖✖✖✖✖✖ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ✔✔✔✔✔✔✔✔✔✔✔ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ f f g g g g h h h From now, we are going to permute the objects that are traced in order to get theformula [ ~f , ~g, ~h ]. The dinaturality axiom allows us to commute the objects that aretraced by composing with a permutation and pre-composing with its inverse. For thatpurpose we define a permutation which will impose an order at the level of objects insuch a way that creates a sequence where the objects that are connected to ~g followthe objects connected to ~f and the objects of ~h follow the objects of ~g : τ : A ⊗ A · · · ⊗ A n ⊗ B m ⊗ C ⊗ · · · ⊗ C s − ⊗ B ⊗ · · · ⊗ B m − → A ⊗ A · · · ⊗ A n ⊗ B ⊗ · · · ⊗ B m − B m ⊗ C ⊗ · · · ⊗ C s − . HAPTER 5. A REPRESENTATION THEOREM ~f , [ ~g ] , ~h ]: γ ′ : C s − ⊗ C s − ⊗ . . . C ⊗ C ⊗ B m ⊗ A n . . . A ⊗ A → A ⊗ A . . . A n ⊗ B m ⊗ C ⊗ C . . . C s − and with [ ~g ]: γ ′′ : B m − ⊗ B m − ⊗ · · · ⊗ B ⊗ B → B ⊗ B ⊗ · · · ⊗ B m − ⊗ B m − and with [ ~f , ~g, ~h ]: C s − ⊗ C s − ⊗ . . . C ⊗ B m ⊗ B m − ⊗ . . . B ⊗ A n ⊗ A n − · · ·⊗ A γ → A ⊗ . . . A n ⊗ B · · ·⊗ B m ⊗ C . . . C s − As we said, we want to compose and pre-compose with a permutation, let us callit y , for our purpose this permutation should satisfy: y ; ( γ ′ ⊗ γ ′′ ) = γ ; τ − . Thus, since all this map are invertible we define: y = γ ; τ − ; ( γ ′− ⊗ γ ′′− ) . In our concrete graphical description after applying dinaturality we get: y γ ′ ⊗ γ ′′ ✴✴✴✴✴✴✴✴ ✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪ ✯✯✯✯✯✯✯✯✯✯✯ y − ✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪ ❄❄❄❄❄ ✯✯✯✯✯✯✯✯✯✯✯✎✎✎✎✎✎✎✎ ✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪ ⑧⑧⑧⑧⑧ ✯✯✯✯✯✯✯✯✯✯✯✬✬✬✬✬✬✬✬✬✬✬✬✬✬ ❄❄❄❄❄ ✔✔✔✔✔✔✔✔✔✔✔ ❄❄❄❄❄ ✯✯✯✯✯✯✯✯✯✯✯ ❄❄❄❄❄ ❄❄❄❄❄✴✴✴✴✴✴✴✴ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ✔✔✔✔✔✔✔✔✔✔✔ ⑧⑧⑧⑧⑧ ✯✯✯✯✯✯✯✯✯✯✯ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ✔✔✔✔✔✔✔✔✔✔✔ ❄❄❄❄❄ ❄❄❄❄❄ ❄❄❄❄❄ ✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄✎✎✎✎✎✎✎✎ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ✔✔✔✔✔✔✔✔✔✔✔ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄✗✗✗✗✗✗✗✗✗✗✗✗✗✗ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ✔✔✔✔✔✔✔✔✔✔✔ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ f f g g g g h h h and using the equation that we defined above y ; ( γ ′ ⊗ γ ′′ ) = γ ; τ − we replace it andwe obtain: HAPTER 5. A REPRESENTATION THEOREM γ ✩✩✩✩✩✩✩✩✩✩✩✩✩✩✩✩✩✩✩✩✩✩ τ − ✯✯✯✯✯✯✯✯✯✯✯ ✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪ ✯✯✯✯✯✯✯✯✯✯✯ ✴✴✴✴✴✴✴✴✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬ ✯✯✯✯✯✯✯✯✯✯✯ ✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪ ❄❄❄❄❄ ✯✯✯✯✯✯✯✯✯✯✯ ✴✴✴✴✴✴✴✴✱✱✱✱✱✱✱✱✱✱ ✯✯✯✯✯✯✯✯✯✯✯ ✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪ ⑧⑧⑧⑧⑧ ✯✯✯✯✯✯✯✯✯✯✯ ✴✴✴✴✴✴✴✴❉❉❉❉❉ ✔✔✔✔✔✔✔✔✔✔✔ ❄❄❄❄❄ ✔✔✔✔✔✔✔✔✔✔✔ ❄❄❄❄❄ ✯✯✯✯✯✯✯✯✯✯✯ ❄❄❄❄❄ ❄❄❄❄❄ ✔✔✔✔✔✔✔✔✔✔✔③③③③③ ✔✔✔✔✔✔✔✔✔✔✔ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ✔✔✔✔✔✔✔✔✔✔✔ ⑧⑧⑧⑧⑧ ✯✯✯✯✯✯✯✯✯✯✯ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ✔✔✔✔✔✔✔✔✔✔✔✒✒✒✒✒✒✒✒✒✒✒ ✔✔✔✔✔✔✔✔✔✔✔ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ✔✔✔✔✔✔✔✔✔✔✔ ❄❄❄❄❄ ❄❄❄❄❄ ❄❄❄❄❄ ✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ✔✔✔✔✔✔✔✔✔✔✔ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ✔✔✔✔✔✔✔✔✔✔✔ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ f f g g g g h h h Now we split the diagram in two sets of different types of symmetries, those whichare functorially free from the set of arrows { f i , g j , h k : i, j, k } and those that are not.Here, in the next diagram, the dotted boxes contain part of the free ones: ✩✩✩✩✩✩✩✩✩✩✩✩✩✩✩✩✩✩✩✩✩✩ ✯✯✯✯✯✯✯✯✯✯✯ ✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪ ✯✯✯✯✯✯✯✯✯✯✯ ✴✴✴✴✴✴✴✴✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬ ✯✯✯✯✯✯✯✯✯✯✯ ✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪ ❄❄❄❄❄ ✯✯✯✯✯✯✯✯✯✯✯ ✴✴✴✴✴✴✴✴✱✱✱✱✱✱✱✱✱✱ ✯✯✯✯✯✯✯✯✯✯✯ ✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪ ⑧⑧⑧⑧⑧ ✯✯✯✯✯✯✯✯✯✯✯ ✴✴✴✴✴✴✴✴❉❉❉❉❉ ✔✔✔✔✔✔✔✔✔✔✔ ❄❄❄❄❄ ✔✔✔✔✔✔✔✔✔✔✔ ❄❄❄❄❄ ✯✯✯✯✯✯✯✯✯✯✯ ❄❄❄❄❄ ❄❄❄❄❄ ✔✔✔✔✔✔✔✔✔✔✔③③③③③ ✔✔✔✔✔✔✔✔✔✔✔ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ✔✔✔✔✔✔✔✔✔✔✔ ⑧⑧⑧⑧⑧ ✯✯✯✯✯✯✯✯✯✯✯ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ✔✔✔✔✔✔✔✔✔✔✔✒✒✒✒✒✒✒✒✒✒✒ ✔✔✔✔✔✔✔✔✔✔✔ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ✔✔✔✔✔✔✔✔✔✔✔ ❄❄❄❄❄ ❄❄❄❄❄ ❄❄❄❄❄ ✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ✔✔✔✔✔✔✔✔✔✔✔ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ✔✔✔✔✔✔✔✔✔✔✔ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ f f g g g g h h h So, we replace this box:
HAPTER 5. A REPRESENTATION THEOREM ✱✱✱✱✱✱✱✱✱✱ ✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✱✱✱✱✱✱✱✱✱✱ ✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✱✱✱✱✱✱✱✱✱✱ ✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✪✒✒✒✒✒✒✒✒✒✒✒ ❄❄❄❄❄ ✔✔✔✔✔✔✔✔✔✔✔✒✒✒✒✒✒✒✒✒✒✒ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ✔✔✔✔✔✔✔✔✔✔✔✒✒✒✒✒✒✒✒✒✒✒ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ✔✔✔✔✔✔✔✔✔✔✔ ❄❄❄❄❄⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ✔✔✔✔✔✔✔✔✔✔✔ ❄❄❄❄❄ ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ ❄❄❄❄❄ ✔✔✔✔✔✔✔✔✔✔✔ ❄❄❄❄❄ ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ By this one: ❄❄❄❄❄❄❄❄❄❄ ⑧⑧⑧⑧⑧❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄❉❉❉❉❉ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧③③③③③ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ and this one: ✯✯✯✯✯✯✯✯✯✯✯ ✴✴✴✴✴✴✴✴✯✯✯✯✯✯✯✯✯✯✯ ✴✴✴✴✴✴✴✴✯✯✯✯✯✯✯✯✯✯✯ ✴✴✴✴✴✴✴✴✯✯✯✯✯✯✯✯✯✯✯ ❄❄❄❄❄ ❄❄❄❄❄ ✔✔✔✔✔✔✔✔✔✔✔✯✯✯✯✯✯✯✯✯✯✯ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ✔✔✔✔✔✔✔✔✔✔✔❄❄❄❄❄ ✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ ❄❄❄❄❄ ✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ ❄❄❄❄❄ ✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙✙ ❄❄❄❄❄ ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧
HAPTER 5. A REPRESENTATION THEOREM ❉❉❉❉❉ ❄❄❄❄❄③③③③③ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄❉❉❉❉❉ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄③③③③③ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄❉❉❉❉❉ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄③③③③③ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧
Finally, we get the desired diagram: ✩✩✩✩✩✩✩✩✩✩✩✩✩✩✩✩✩✩✩✩✩✩ ❄❄❄❄❄ ❄❄❄❄❄ ❄❄❄❄❄✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬✬ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄✱✱✱✱✱✱✱✱✱✱ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄❉❉❉❉❉ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄③③③③③ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄✒✒✒✒✒✒✒✒✒✒✒ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗✗ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚✚ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄ ⑧⑧⑧⑧⑧ ❄❄❄❄❄⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ ⑧⑧⑧⑧⑧ f f g g g g h h h To go from [ ~f , ~g, ~h ] to [ ~f , [ ~g ] , ~h ] we use the same arguments in the reverse ordersince [ ~g ] ↓ .Next, we wish to show that the paracategory Int p ( C ) is strict symmetric monoidal. Definition 5.6.4.
Let ( C , ⊗ , I, Tr , s ) be a symmetric monoidal partially traced cat-egory, the tensor in the graph Int p ( C ) is defined as follows: • The unit is (
I, I ) • on objects: ( A + , A − ) ⊗ ( B + , B − ) = ( A + ⊗ B + , B − ⊗ A − ) HAPTER 5. A REPRESENTATION THEOREM • on arrows: given f Int p : ( A + , A − ) → ( C + , C − ) and g Int p : ( B + , B − ) → ( D + , D − ) then ( f ⊗ g ) Int p : ( A + , A − ) ⊗ ( B + , B − ) → ( C + , C − ) ⊗ ( D + , D − )is defined by A + ⊗ B + ⊗ D − ⊗ C − s ⊗ s → B + ⊗ A + ⊗ C − ⊗ D − ⊗ f ⊗ → B + ⊗ C + ⊗ A − ⊗ D − s ⊗ s → C + ⊗ B + ⊗ D − ⊗ A − ⊗ g ⊗ → C + ⊗ D + ⊗ B − ⊗ A − . Let us derive some immediate consequences of this definition:(i) 1 ( A + ,A − ) ⊗ ( B + ,B − ) = = = 1 ( A + ⊗ B + ,B − ⊗ A − ) (ii) ( A + , A − ) ⊗ ( I, I ) = ( A + ⊗ I, I ⊗ A − ) = ( A + , A − ) and ( I, I ) ⊗ ( A + , A − ) =( A + , A − ).(iii) ( A + , A − ) ⊗ (( B + , B − ) ⊗ ( C + , C − )) = ( A + ⊗ B + ⊗ C + , C − ⊗ B − ⊗ A − ) =(( A + , A − ) ⊗ ( B + , B − )) ⊗ ( C + , C − ) . Definition 5.6.5.
The symmetry ( A + , A − ) ⊗ ( B + , B − ) σ → ( B + , B − ) ⊗ ( A + , A − ) isdefined in Int p ( C ) by the following formula: σ = s A + B + ⊗ s A − B − . Lemma 5.6.6.
Let ( C , ⊗ , I, Tr , s ) be a symmetric monoidal partial traced category.Given f Int p : ( Y + , Y − ) → ( C + , C − ) ⊗ ( D + , D − ) , and g Int p : ( A + , A − ) ⊗ ( B + , B − ) → ( X + , X − ) then [ f, σ ] ↓ and [ σ, g ] ↓ .Proof. To simplify the notation we use the symbol “;” for the composition in thecategory C with the order given by graphical concatenation.We first consider the composition of f : Y + ⊗ D − ⊗ C − → C + ⊗ D + ⊗ Y − with thefollowing symmetries and identities in the category C : (1 Y + ⊗ s C − D − ); f ; ( s C + D + ⊗ Y − ).Next, since by the yanking axiom s D − ⊗ C − ,D − ⊗ C − ∈ T D − ⊗ C − D − ⊗ C − ,D − ⊗ C − andTr D − ⊗ C − D − ⊗ C − ,D − ⊗ C − ( s D − ⊗ C − ,D − ⊗ C − ) = 1 D − ⊗ C − then by superposing axiom we have that1 Y + ⊗ s D − ⊗ C − ,D − ⊗ C − ∈ T D − ⊗ C − Y + ⊗ D − ⊗ C − ,Y + ⊗ D − ⊗ C − and1 Y + ⊗ Tr D − ⊗ C − D − ⊗ C − ,D − ⊗ C − ( s D − ⊗ C − ,D − ⊗ C − ) =Tr D − ⊗ C − Y + ⊗ D − ⊗ C − ,Y + ⊗ D − ⊗ C − (1 Y + ⊗ ( s D − ⊗ C − ,D − ⊗ C − )) . HAPTER 5. A REPRESENTATION THEOREM Y + ⊗ s C − D − ); Tr D − ⊗ C − Y + ⊗ D − ⊗ C − ,Y + ⊗ D − ⊗ C − (1 Y + ⊗ ( s D − ⊗ C − ,D − ⊗ C − )); f ; ( s C + D + ⊗ Y − ) =Tr D − ⊗ C − Y + ⊗ C − ⊗ D − ,Y + ⊗ D − ⊗ C − ((1 Y + ⊗ s C − D − ⊗ D − ⊗ C − ); (1 Y + ⊗ s D − ⊗ C − D − ⊗ C − )); f ; ( s C + D + ⊗ Y − ) =by coherence:Tr D − ⊗ C − Y + ⊗ C − ⊗ D − ,Y + ⊗ D − ⊗ C − ((1 Y + ⊗ s C − ⊗ D − ,D − ⊗ C − ); (1 Y + ⊗ D − ⊗ C − ⊗ s C − D − )); f ; ( s C + D + ⊗ Y − ) =by the naturality axiom:Tr D − ⊗ C − Y + ⊗ C − ⊗ D − ,D + ⊗ C + ⊗ Y − ((1 Y + ⊗ s C − ⊗ D − ,D − ⊗ C − ); (1 Y + ⊗ D − ⊗ C − ⊗ s C − D − ); f ⊗ D − ⊗ C − ; ( s C + D + ⊗ Y − ⊗ D − ⊗ C − )) =and by functoriality:Tr D − ⊗ C − Y + ⊗ C − ⊗ D − ,D + ⊗ C + ⊗ Y − ((1 Y + ⊗ s C − ⊗ D − ,D − ⊗ C − ); ( f ⊗ C − ⊗ D − ); ( s C + D + ⊗ Y − ⊗ s C − D − )) =Now by coherence, we can replace: s C + D + ⊗ Y − ⊗ s C − D − by the following(1 C + ⊗ D + ⊗ s Y − ,C − ⊗ D − ); ( s C + D + ⊗ s C − D − ⊗ Y − ); (1 D + ⊗ C + ⊗ s D − ⊗ C − ,Y − ) . Which, by definition, is [ f, s C + D + ⊗ s C − D − ], i.e., we proved that [ f, σ ( C + ,C − ) , ( D + ,D − ) ] ↓ .After repeating a similar argument as above, we have: [ s A + B + ⊗ s A − B − , f ] ↓ . Now we repeat the proof using graphical language. The purpose of this is topersuade the reader of the advantages of using this methodology. We start with thefollowing diagram f HAPTER 5. A REPRESENTATION THEOREM f by the superposition axiom f naturality axiom f naturality of the symmetry σ f functoriality f HAPTER 5. A REPRESENTATION THEOREM σ and coherence axiom in C f From now we use the graphical language systematically.
Lemma 5.6.7. σ is a natural transformation.Proof. We want to prove that σ ◦ ( f ⊗ g ) = ( f ⊗ g ) ◦ σ . Notice that we have alreadyproved that ( f ⊗ g ) ◦ σ ↓ and σ ◦ ( f ⊗ g ) ↓ by Lemma 5.6.6. We have by assumptionthat σ ◦ ( f ⊗ g ) is defined. In the graphical language this means that h ∈ T U ⊗ V ,where h is the following diagram: f g the trace given by Tr U ⊗ V ( h ) = f g UV Here the issue is to justify the use of the Vanishing II axiom. Putting the matterschematically without to much emphasis on the name of the objects, we want to splitthe trace over U ⊗ V by using a general hypothesis of type h ∈ T V and a conditionalhypothesis of type h ∈ T U ⊗ V and we must prove that Tr V ( h ) ∈ T U . This is the kindof back and forward process of proof that we have repeatedly used before where thejustification of the use of the axiom is also the proof that we need. Let us start byconsidering the following diagram: f g . (10)Then by the yanking axiom, which is totally defined: σ ∈ T V and we can replace theformer graph by this one f g , HAPTER 5. A REPRESENTATION THEOREM T V and the graph, after tracing it out isgiven by f g . Then the naturality axiom allows us to include the full diagram in the trace class T V and we are allowed also to trace it: f g . Finally, by coherence f g ∈ T V and the trace is represented by f g . Now we are in a position where we can use the vanishing II axiom and to conclude thatTr V ( h ) ∈ T U and of course the value of the trace is given by Tr U (Tr V ( h )) = Tr U ⊗ V ( h ).After justifying the use of the vanishing II axiom we move to ensure that bothdiagram are equal. First notice that the following diagrams are equivalent : f g by coherence g f . Starting with the last diagram and applying the axioms, where the existence ofthe trace is justified by the axiom that we are mentioning, we obtain: g f coh. g f yank. g f nat. g f coh. g f yank., sup. g f coh.
HAPTER 5. A REPRESENTATION THEOREM g f nat. g f coh. g f where the last diagram is of type Tr U (Tr V ( h ′ )). Therefore, by the same reasoningas given at the beginning of the proof we find that Tr U ⊗ V ( h ′ ) = Tr U (Tr V ( h ′ )) also forthis diagram. As before we repeat our arguments to justify the existence and valueof the trace for the case when we start with the graph f g obtaining the following diagram: f g UV . Lemma 5.6.8. [ σ ⊗ , σ ] = 1 ⊗ σ .Proof. Here again, as in Lemma 5.6.7, the key point is to justify the use of theVanishing II axiom. We will apply this strategy twice. Since by Lemma 5.6.6 [ σ ⊗ , σ ] ↓ we want to be able to use an scheme proof of type: g ∈ T U ⊗ V ⊗ W iff Tr W ( g ) ∈ T U ⊗ V , but for this we need an hypothesis of type g ∈ T W . To justify this, we wantto split the trace over U ⊗ V by using a general hypothesis of type h ∈ T V and aconditional hypothesis of type h ∈ T U ⊗ V and we must prove that Tr V ( h ) ∈ T U .We start with the following diagram that represents 1 ⊗ σ : V Id which by coherence is equivalent to the following HAPTER 5. A REPRESENTATION THEOREM V Therefore, by yanking, (let us name the variable U ) we have V Then by the naturality and the superposition axioms we obtain that it is equal tothe trace represented by: V in which the diagram below the trace, let us call it h , satisfies h ∈ T U . Notice thatthis is true because our axioms of partially traced category allow us to entail this laststatement.Now, by coherence we have that is equal to V Let us still call h the new graph below the trace. By yanking with respect to a variable HAPTER 5. A REPRESENTATION THEOREM V V Now again by naturality, superposition and coherence we conclude that the graphbelow the trace, name it h ′ , is in the trace class T V . Moreover, the value of the tracealong V is equal to h , i.e., Tr V ( h ′ ) = h , which implies that is in the trace class T U ,this means that we are allowed to use vanishing II and to conclude that h ′ ∈ T U ⊗ V : V and coherence V Now we repeat the idea with a new parameter W . V and V Hence, this yields after applying vanishing II again U V W Id V which represents [ σ ⊗ , σ ]. Lemma 5.6.9. ⊗ [ ~p ] ✄✂ [ 1 ⊗ ~p ] and [ ~p ] ⊗ ✄✂ [ ~p ⊗ .Proof. Without loss of generality we consider the case when ~p = p , p , p , p . Bydefinition [ ~p ] ⊗ HAPTER 5. A REPRESENTATION THEOREM Id p p p p Then using superposing axiom we obtain:and since by the yanking axiom Tr( σ ) = 1, we have that:Now by the fact that the trace is defined on symmetries this is the hypothesis thatI need in order to apply superposing (equivalent version) axiom, thus by the samereason we can apply also the naturality axiom: V U We name g the diagram without being traced, i.e., g is VU Then g ∈ T V by the reasons given above and if we reverse this procedure infact we are showing that Tr V ( g ) ∈ T U (after applying superposition, yanking and HAPTER 5. A REPRESENTATION THEOREM g ∈ T U ⊗ V .Now we are allowed to apply the dinaturality axiom in order to permute the order ofthe objects that are going to be traced out:Thus by coherence we have that:Again by coherence:Now by coherence and the yanking axiom:Again since the trace is total on symmetries, and after applying superposing (equiv-alent version), the naturality axiom shows that: HAPTER 5. A REPRESENTATION THEOREM
HAPTER 5. A REPRESENTATION THEOREM
HAPTER 5. A REPRESENTATION THEOREM σ ◦ σ − = 1:Finally by coherence we get:Which is by definition [ ~p ⊗ Theorem 5.6.10.
Let ( C , ⊗ , I, Tr , s ) be a symmetric monoidal partially traced cate-gory. The operation defined above [ − ] determines a ssmpc ( Int p ( C ) , [ − ] , ⊗ , I, σ ) .Proof. It follows from the previous lemmas.Next, we wish to show that
Int p ( C ) is a compact closed paracategory. Let( I, I ) η −→ ( A, B ) ⊗ ( A, B ) ∗ and ( A, B ) ∗ ⊗ ( A, B ) ε −→ ( I, I ) be the unit and counitassociated to the paracategory
Int p ( C ). Actually, since C is a strict category, we canregard these morphisms as id : I ⊗ A ⊗ B → A ⊗ B ⊗ I and id : B ⊗ A ⊗ B → I ⊗ B ⊗ A respectively. HAPTER 5. A REPRESENTATION THEOREM
Lemma 5.6.11. [ η ⊗ , ⊗ ε ] ↓ , [1 ⊗ η, ε ⊗ ↓ and [ η ⊗ , ⊗ ε ] = 1 , [1 ⊗ η, ε ⊗
1] = 1 .Proof.
Notice that σ A,I = id A for every object A ∈ C . We start with the identity map( A, B ) −→ ( A, B ) which is the map 1 : A ⊗ B → A ⊗ B in C . Since,1 A ⊗ B = A B holds by coherence, using the yanking axiomand naturality . Notice that, all along this proof, we implicitly claim that the graph below the trace isin the corresponding trace class. For instance, in the last diagram from the naturalityaxiom it follows that ∈ T B . Then by superposing axiom and coherence . Therefore, by applying yanking and naturality again we obtainwhere the graph below this new trace is in the trace class T A and this, of course, willbe preserved by any coherent modification of the graph. We have from superpositionand coherence axioms that . HAPTER 5. A REPRESENTATION THEOREM T B . Since C is a strict categorythen is equal to I II I . Finally, since the trace class conditions for applying vanishing II are satisfied, weapply the vanishing II axiom twice and we obtain that is equal to id id I A I B BAB AB B I II A ABAB ABI = [ η ⊗ , ⊗ ε ] . In the same way as before we prove that [1 ⊗ η, ε ⊗
1] = 1.We sketch schematically the rest of the proof leaving details to the reader. Westart with the identity 1 B ⊗ A .coherence: yanking: naturality:coh.: yanking: superposing:nat.: coh.: dinaturality:yanking: naturality: coherence: HAPTER 5. A REPRESENTATION THEOREM id id I A I B BAB AB B I II A ABA A BIA
Corollary 5.6.12.
Let C be partially traced. Then Int p ( C ) is a compact closed para-category.Proof. This is a consequence of Lemma 5.6.11.Our final result for this section is that there exists a full and faithful, trace pre-serving functor from C to Int p ( C ). Definition 5.6.13.
In a similar way as done in [41], we define a fully faithful functorbetween paracategories N : C →
Int p ( C ) defined by N ( A ) = ( A, I ) and N ( f ) = f bystrictness of the category C . Lemma 5.6.14. N is a well-defined, full and faithful functor of paracategories.Proof. To prove well-definedness, note that we are considering the category C as aparacategory with composition [ f , . . . , f n ] = f n ◦ . . . ◦ f as its partial operation, and[ − ] ′ the partial composition defined in Int p ( C ). Thus, N ([ ~f ]) = ~ [ N ( f )] ′ , since by theVanishing I axiom, the trace operator is totally defined when we restrict it to thistype of arrows i.e., T IA,B = C ( A ⊗ I, B ⊗ I ) and Tr IA,B ( f ) = f .By definition, N ( f ) = N ( g ) implies f = g , which proves faithfulness. If we takeand arrow in Int p (( A, I ) , ( B, I )), let us say for example f : ( A, I ) → ( B, I ), whichreally means in C an arrow of type f : A ⊗ I → B ⊗ I , then we just choose the same f obtaining N f = f . This proves fullness. Lemma 5.6.15.
The functor N : C →
Int p ( C ) preserves the trace, i.e., if f : A ⊗ U → B ⊗ U is in T UA,B then N (Tr UA,B ( f )) = Tr NUNA,NB ( N f ) : (
A, I ) → ( B, I ) which means N (Tr UA,B ( f )) = [1 ⊗ η ; f ⊗
1; 1 ⊗ σ ; 1 ⊗ ε ] . HAPTER 5. A REPRESENTATION THEOREM
Proof.
Let us start with N (Tr UA,B ( f )) : A ⊗ I → B ⊗ I in C which is represented by f A BUU
Notice that by hypothesis we have f ∈ T U . Let us call this hypothesis: condition( A ).By the yanking axiom σ U,U ∈ T U where the trace is locally represented by f A BUU and by applying superposing axiom σ ⊗ A ∈ T U and then by applying the natu-rality axiom we obtain that the full diagram below this trace is in T U (let us call itcondition ( B )), i.e., f A BUU ∈ T U . The trace of this graph is equal to f which implies by condition ( A ) that is in T U i.e., f A BUU ∈ T U . From condition ( A ) and ( B ) and the vanishing II axiom we conclude that f A BUU ∈ T U ⊗ U HAPTER 5. A REPRESENTATION THEOREM A + B ) and the trace is represented by: f A BUU . We repeat this operation, by yanking: f A BUU and naturality we obtain that the diagram in the dotted box: f A BUU is in T U .Hence, after any further coherent change we made in the graph, it will remain inthe trace class T U . Let us call it condition ( C ); where the trace will be representedby f AA A B BU UUU U
HAPTER 5. A REPRESENTATION THEOREM f AA A B BU UU UUU U and f IAA A B BB BBU U UU UUUU UUU U . In the same way as above: by condition A + B , C and the vanishing II axiom weobtain that the graph is in the trace class T U ⊗ U ⊗ U and the trace given by f II II II II II II IIII II II III IAA A B BB BBU U UU UUUU UUU U
Now, since C is a strict category we can represent the last diagram in the followingway: f I I II II II II II III I II II I I I I A A A B BB B BU U U U UUUU UU U U which is equal to [1 ⊗ η ; f ⊗
1; 1 ⊗ σ ; 1 ⊗ ε ] . HAPTER 5. A REPRESENTATION THEOREM
Theorem 5.7.1.
Every (strict) symmetric partially traced category can be faithfullyembedded in a totally traced category.Proof.
This follows from the various lemmas. Let C be a strict symmetric partiallytraced category. By Lemmas 5.6.14 and 5.6.15, C can be faithfully embedded in acompact closed paracategory Int p ( C ), and the embedding is trace preserving. ByLemma 5.4.2, Int p ( C ) can be faithfully embedded in a compact closed category P ( Int p ( C )) / ∼ (and the embedding preserves the compact closed structure, hencethe trace). Since P ( Int p ( C )) / ∼ is compact closed, it is totally traced, which provesthe theorem. Remark 5.7.2.
Notice that by the Lemma 5.6.15 above if f : A ⊗ U → B ⊗ U is in T UA,B then [1 ⊗ η ; f ⊗
1; 1 ⊗ σ ; 1 ⊗ ε ] ↓ ; therefore the projection functor F : Int p ( C ) → P ( Int p ( C )) / ∼ also preserves the trace F (Tr UA,B ( f )) = Tr F UF A,F B ( F f ) since we have that F (Tr UA,B ( f )) = F [1 ⊗ η ; f ⊗
1; 1 ⊗ σ ; 1 ⊗ ε ] = [1 ⊗ η ; f ⊗
1; 1 ⊗ σ ; 1 ⊗ ε ] =1 ⊗ η ; f ⊗
1; 1 ⊗ σ ; 1 ⊗ ε = 1 ⊗ η ◦ f ⊗ ◦ ⊗ σ ◦ ⊗ ε = 1 ˆ ⊗ η ◦ f ˆ ⊗ ◦ ⊗ σ ◦ ⊗ ε =Tr F UF A,F B ( F f ) . The category ( P ( Int p ( C )) / ∼ , ˆ ⊗ , I, s ) satisfies the following universal property. Proposition 5.8.1.
Let C be a partially traced category and D a compact closedcategory. If F : C → D is a strict monoidal traced functor then there exists a uniquemonoidal functor L : P ( Int p ( C )) / ∼ → D such that C ˆ N / / F * * ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ P ( Int p ( C )) / ∼ L (cid:15) (cid:15) D HAPTER 5. A REPRESENTATION THEOREM where ˆ N is C N −→ Int p ( C ) π −→ P ( Int p ( C )) / ∼ Proof.
We first construct a monoidal functor K : Int p ( C ) → D such that K ◦ N = F .This functor is defined in the same way as in [41], and is in fact unique.On objects K ( A, U ) =
F A ⊗ ( F U ) ∗ and given ( A, U ) f −→ ( B, V ) we define K ( f )as F A ⊗ F U ∗ ⊗ η ⊗ −→ F A ⊗ F V ⊗ F V ∗ ⊗ F U ∗ F f ⊗ −→ F B ⊗ F U ⊗ F V ∗ ⊗ F U ∗ (1 ⊗ σ ⊗ ◦ (1 ⊗ εσ ) −→ F B ⊗ F V ∗ Graphically this is represented by the following diagram f F . We need to prove that K is a functor between paracategories, i.e., if [ f , . . . f n ] ↓ then K [ f , . . . f n ] = [ Kf , . . . Kf n ]. The remaining properties of K are proved asin [41].Without loss of generality we take n = 4. Therefore we have K [ f , . . . f ] = f F . (11)where f = f f f f Since F preserves the trace, composition and symmetries we have that equa-tion (11) is equal to the following diagram HAPTER 5. A REPRESENTATION THEOREM F f f F F ff F . Notice that the category D is compact closed and its trace is totally defined andgiven by composition of unit η , counit ε , symmetries σ and arrows F f i in D , i = 1 . . . D , we transform the previous diagram into F f fff F FF i.e., [ Kf , . . . Kf ].Given K , we use Theorem 5.5.1 to obtain a unique L such that: C N / / F * * ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ Int p ( C ) π / / K ' ' PPPPPPPPPPPPPP P ( Int p ( C )) / ∼ L (cid:15) (cid:15) D Uniqueness: Suppose L ′ : P ( Int p ( C )) / ∼ → D is another monoidal functor suchthat L ′ ◦ π ◦ N = F . Then K ′ = L ′ ◦ π satisfies K ′ ◦ N = F so by uniqueness of K , itfollows that K = K ′ . But then L ′ ◦ π = K , and by uniqueness of L , we have L = L ′ . hapter 6Background material on presheafcategories Here we review some of the basic and advanced concepts of functor categories thatwill be used in Chapters 7 and 8. For additional details, see [54], [15], [51], [46].
Definition 6.1.1.
Let F : A → B be a functor and B ∈ B . A pair ( A, f ) where A ∈ A and f : B → F ( A ) is said to be a universal arrow from B to F when for everyarrow f ′ : B → F ( A ′ ) there is a unique arrow g : A → A ′ in the category A such that B f / / f ′ ! ! ❉❉❉❉❉❉❉❉❉ F ( A ) F ( g ) (cid:15) (cid:15) F ( A ′ )is a commutative diagram. Definition 6.1.2. A universal element of the functor F : A →
Set is an object A ∈ A and an element x ∈ F ( A ) such that for any other pair A ′ ∈ A and x ′ ∈ F ( A ′ )there exists a unique f : A → A ′ that satisfies F ( f )( x ) = x ′ .117 HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES
Definition 6.1.3.
Let F : C →
Set be a functor. The category El ( F ) of elements isgiven by the following data:(a) objects of El ( F ) are pairs ( C, x ) where x ∈ F C and C ∈ |C| .(b) morphisms f : ( C, x ) → ( D, y ) are arrows f : C → D in the category C suchthat F f ( x ) = y . Definition 6.1.4.
An object A ∈ A is said to be the representing object of a functor F : A →
Set when there is a natural isomorphism φ : A ( A, − ) φ → F. When this occurs we said that F is a representable functor . There is a distinguishedelement of this isomorphism φ A (1 A ) ∈ F ( A ), which is called the unit of the represen-tation. Theorem 6.1.5 (The Yoneda Lemma) . Let F : A →
Set be a functor, A ∈ A .There exists a bijection ϑ F,A : [ A , Set ]( A ( A, − ) , F ) → F ( A ) which is natural in A and if A is a small category ϑ is natural in F .Proof. [15] Theorem 6.1.6.
Let F : A →
Set be a functor, F is representable iff it has auniversal element.Proof. [54] Let A and B be categories. For every object A ∈ A the constant functor isdefined to be ∆ A : B → A with ∆ A ( B ) = A and ∆ A ( f ) = 1 A when B f → B ′ .If A g → A ′ is an arrow in A there is a natural transformation ∆( g ) : ∆ A ⇒ ∆ A ′ HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES g ))( B ) = g . These functors and natural transformations define a functor∆ : A → [ B , A ].Let F : J → A be a functor. The definition of limits and colimits can becharacterized by objects that represent the following functors: A ( − , lim F ) ∼ = [ J , A ](∆ − , F ) : A op → Set (12)and A (colim F, − ) ∼ = [ J , A ]( F, ∆ − ) : A →
Set . (13)To see this, suppose we have A ( − , lim F ) φ → [ J , A ](∆ − , F ). Then φ limF (1 limF ) :∆ limF ⇒ F is a cone determined by the universal element. If ∆ α ⇒ F is another conethen φ − A ( α ) : A → lim F is an arrow on the category A such that by naturality wehave: A (lim F, lim F ) φ limF / / A ( φ − A ( α ) , lim F ) (cid:15) (cid:15) [ J , A ](∆lim F, F ) [ J , A ](∆( φ − A ( α )) ,F ) (cid:15) (cid:15) A ( A, lim F ) φ A / / [ J , A ](∆ A, F )which implies by evaluating at 1lim F that: φ limF (1 limF ) ◦ ∆( φ − A ( α )) = φ A ( φ − A ( α )) : ∆ A ⇒ F. Graphically: ∆ A α + ∆( φ − A ( α )) ( ❏❏❏❏❏❏❏❏❏ ❏❏❏❏❏❏❏❏❏ F ∆lim F φ limF (1 limF ) K S Therefore, evaluating at i ∈ J : A α i / / φ − A ( α ) " " ❉❉❉❉❉❉❉❉❉ F ( i )lim F ( φ limF (1 limF ))( i ) O O HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES
Next, we recall the notion of dinatural transformation. The case which interests usthe most is when one of the functors involved is a constant functor.
Definition 6.3.1 (Dinatural transformation) . Suppose we have two functors
F, G : A op × A → B , a family of maps α : F .. −→ G = { α A : F ( A, A ) → G ( A, A ) } A ∈|A| is called a dinatural transformation when for every arrow f : A → B the followingholds: F ( A, A ) α A / / G ( A, A ) G (1 ,f ) & & ▲▲▲▲▲▲▲▲▲▲ F ( B, A ) F ( f, rrrrrrrrrr F (1 ,f ) & & ▲▲▲▲▲▲▲▲▲▲ G ( A, B ) F ( B, B ) α B / / G ( B, B ) G ( f, rrrrrrrrrr Example . Let S : A op → Set be a functor, and let B ∈ |A| . There are twofunctors F, G : A op × A → Set defined by F ( A ′ , A ) = S ( A ′ ) × A ( B, A ), and G =∆( S ( B )), the constant functor. Let us consider maps of type λ A : S ( A ) × A ( B, A ) → S ( B ) with λ A ( x, f ) = S ( f )( x ). Then λ : F → G is a dinatural transformation: forall f : A ′ → A , S ( A ) × A ( B, A ) λ A ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖ S ( A ) × A ( B, A ′ ) ×A ( B,f ) ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ S ( f ) × ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ S ( B ) S ( A ′ ) × A ( B, A ′ ) λ A ′ ♦♦♦♦♦♦♦♦♦♦♦♦♦ . Definition 6.3.3 (Wedge) . Given a functor F : A op × A → B , a wedge is a dinaturaltransformation from a constant functor to F , λ : ∆( E ) .. −→ F. Definition 6.3.4 (End) . Given a functor F : A op × A → B , an end is a wedge λ : ∆( E ) .. −→ F HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES α : ∆( A ) .. −→ F then thereis a unique g : A → E with λ A ◦ g = α A for every A ∈ A .In an analogous way we define the notion of co-end. Example . In the example above we have that S ( B ) with component λ is a co-end for the functor F . Given a dinatural transformation α A : S ( A ) × A ( B, A ) −→ X there is a unique g : S ( B ) → X given by g ( y ) = α B ( y, B ) that satisfies the definition.From the uniqueness of the universal property we conclude that, up to isomor-phism, all the ends are equal. This justifies the following notation to indicate an end E with components λ A : Z A F ( A, A ) λ A −→ F ( A, A )and in the same way the co-end: F ( A, A ) λ A −→ Z A F ( A, A ) . Theorem 6.3.6.
Let α : F ⇒ G : A op ×A → B be a natural transformation. Supposealso that there exists the ends induced by F and G : Z A F ( A, A ) λ A −→ F ( A, A ) and Z A G ( A, A ) µ A −→ G ( A, A ) (14) then there is a unique map R A α A,A in the category B such that: R A F ( A, A ) λ A / / R A α A,A (cid:15) (cid:15) F ( A, A ) α A,A (cid:15) (cid:15) R A G ( A, A ) µ A / / G ( A, A ) Proof. [54]
Theorem 6.3.7.
Let F : A × B op × B → C be a functor such that for each A ∈ |A| there exists an end Z B,B F ( A, B, B ) λ AB −→ F ( A, B, B ) . Then there is a unique functor U : A → C with U ( A ) = R B F ( A, B, B ) making λ AB natural in A ∈ |A| .Proof. [54] HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES
Definition 6.4.1.
Let A be a small category and G : A → B be functors. We definea functor ˆ G : B op → [ A , Set ] whose values on objects are functorsˆ G ( B ) = B ( B, G − ) : A →
Set and whose value on a morphism B f → B ′ is a natural transformation B ( f, G − ) : B ( B ′ , G − ) → B ( B, G − ) . Let F : A →
Set be a functor. Thus we have a composition of functors: B op ˆ G → [ A , Set ] [ A , Set ]( F, − ) −→ Set . Suppose now that this composition admits a representation: φ : B ( − , C ) ∼ = [ A , Set ]( F, ˆ G ( − )) . Definition 6.4.2 (Indexed limit) . Let us denote C = { F, G } , so we have that B ( B, { F, G } ) ∼ = [ A , Set ]( F, B ( B, G − ))natural in B with counit µ = φ { F,G } (1 { F,G } ) : F → B ( { F, G } , G − ) which has theproperty of being a universal element. Following Kelly’s definition [46], we name thispair ( { F, G } , µ ) the limit of G indexed by F .Thus µ ∈ [ A , Set ]( F, B ( { F, G } , G − )) and if there is another λ ∈ [ A , Set ]( F, B ( B ′ , G − )) then there exists a unique { F, G } g op → B ′ in the category B op such that ([ A , Set ]( F, B ( g, G − )))( µ ) = λ which means that B ( g, G − ) ◦ µ = λ .Therefore, F ( A ) µ A / / λ A ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ B ( { F, G } , G ( A )) B ( g,G ( A )) (cid:15) (cid:15) B ( B ′ , G ( A )) . Thus, after evaluating at x ∈ F ( A ) we obtain: HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES B ′ g / / λ A ( x ) " " ❋❋❋❋❋❋❋❋❋ { F, G } µ A ( x )) (cid:15) (cid:15) G ( A )There is a bijection:[ A , Set ]( F, B ( B, G − )) ∼ = [ El ( F ) , B ](∆ B, G ◦ π F )natural in B , with the projection π F : El ( F ) → A .By equation (12): B ( B, lim G ◦ π F ) ∼ = [ El ( F ) , B ](∆ B, G ◦ π F )we conclude that: Proposition 6.4.3. lim G ◦ π F = { F, G } Proof.
To see this bijection we have that every natural transformation α ∈ [ A , Set ]( F, B ( B, G − )) and for every f : A → A ′ there is a diagram: F ( A ) α A / / F ( f ) (cid:15) (cid:15) B ( B, G ( A )) B ( B,G ( f )) (cid:15) (cid:15) F ( A ′ ) α A ′ / / B ( B, G ( A ′ ))which translates into a diagram: B α A ( a ) (cid:15) (cid:15) α A ′ ( F ( f )( a )) ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ G ( A ) G ( f ) / / G ( A ′ )for every a ∈ F ( A ). HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES
Remark 6.4.4.
When we choose F = ∆1 B ( B, lim G ) ∼ = [ A , B ](∆ B, G ) ∼ = [ A , Set ](∆1 , B ( B, G − ))we obtain by definition that lim G = { ∆1 , G } Definition 6.4.5 (Indexed colimit) . In the same way as above by duality we definethe colimit of G : A → B indexed by F : A op → Set as the representing pair (
F ⋆G, λ )of the functor: [ A op , Set ]( F, ˜ G ( − )) : B →
Set where ˜ G : B → [ A op , Set ] whose values on objects are functors˜ G ( B ) = B ( G − , B ) : A op → Set and whose value on a morphism B f → B ′ is a natural transformation B ( G − , f ) : B ( G − , B ) → B ( G − , B ′ ) . Therefore, we have that B ( F ⋆ G, B ) ∼ = [ A op , Set ]( F, B ( G − , B )) (15)and after evaluating the representation isomorphism on the identity with B = F ⋆ G we obtain a unit λ : F → B ( G − , F ⋆ G ). Remark 6.4.6.
With enough conditions, for example when B in cocomplete, there isa functor • ⋆G : [ A op , Set ] → B . Also, from equation (15) we conclude that • ⋆G is leftadjoint of the functor B ( G − , • ) : B → [ A op , Set ] where B ( G − , • )( B ) = B ( G − , B ) : A op → Set . We write • ⋆ G ⊣ B ( G − , • ).The functor • ⋆ G is the unique, up to isomorphism, colimit preserving functorsuch that the following diagram commutes: A Y / / G * * ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ [ A op , Set ] • ⋆G (cid:15) (cid:15) B In the next section we shall discuss this construction in more detail in the context ofa coproduct preserving Yoneda embedding.
HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES
Proposition 6.4.7. If F : A op → Set and G : A → B thencolim G ◦ π opF ∼ = F ⋆ G
Proof.
Analogously, there is a bijection:[ A op , Set ]( F, B ( G − , B )) ∼ = [ El ( F ) op , B ]( G ◦ π opF , ∆ B )natural in B , with projection π opF : El ( F ) op → A .From this since by equation (13): B (colim G ◦ π opF , B ) ∼ = [ El ( F ) op , B ]( G ◦ π opF , ∆ B )we conclude that: colim G ◦ π opF ∼ = F ⋆ G.
Remark 6.4.8.
Since all colimits may be expressed in terms of coproducts andcoequalizers we have the following explicit formula: ` x ∈ F ( A ) ,f : A ′ → A G ( A ′ ) θ / / τ / / ` A,x ∈ F ( A ) G ( A ) λ / / F ⋆ G where λ is a coequalizer of the unique maps τ and θ : G ( A ′ ) id / / i ( x,f ) (cid:15) (cid:15) G ( A ′ ) i ( A ′ ,F ( f )( x )) (cid:15) (cid:15) ` x ∈ F ( A ) ,A ′ f → A G ( A ′ ) θ / / ` A,x ∈ F ( A ) G ( A ) G ( A ′ ) G ( f ) / / i ( x,f ) (cid:15) (cid:15) G ( A ) i ( A,x ) (cid:15) (cid:15) ` x ∈ F ( A ) ,A ′ f → A G ( A ′ ) τ / / ` A,x ∈ F ( A ) G ( A )obtained by the coproduct definition.Now, suppose we take F = A ( − , A ) : A op → Set , then for every B we have that: B ( A ( − , A ) ⋆ G, B ) ∼ = [ A op , Set ]( A ( − , A ) , B ( G − , B )) = B ( G ( A ) , B )by the Yoneda Lemma. Therefore A ( − , A ) ⋆ G ∼ = G ( A ). In the same way we obtainthat {A ( A, − ) , G } ∼ = G ( A ). HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES
Proposition 6.4.9. Z A F ( A ) ⊗ G ( A ) ∼ = F ⋆ G
Proof.
Let F : A →
Set and G : A → B be functors and suppose now that thecategory B has copowers . We denote by X ⊗ A = ` X A ∈ |B| where X is a set and A ∈ |B| . Then we have B ( R A F ( A ) ⊗ G ( A ) , B ) ∼ = R A B ( F ( A ) ⊗ G ( A ) , B ) ∼ = R A [ F ( A ) , B ( G ( A ) , B )] ∼ =[ A op , Set ]( F, B ( G ( − ) , B ))by properties of ends, copowers, hom as end in the functor category.Thus, by definition this implies that Z A F ( A ) ⊗ G ( A ) ∼ = F ⋆ G.
In particular when G = Y : A → [ A op , Set ] we have that: Z A F ( A ) ⊗ A ( − , A ) ∼ = F ⋆ Y ∼ = F as we already have proved (Example 6.3.2). Proposition 6.5.1.
Let A F / / B G ⊥ o o be an adjunction with unit η : 1 A ⇒ GF andcounit ε : F G ⇒ B . Then (i) F is full and faithful if and only if (ii) η is anisomorphism. When these conditions are satisfied, ε ∗ G and F ∗ ε are isomorphisms.Dually, G is full and faithful iff and only if ε is an isomorphism. When this happens η ∗ G and F ∗ η are isomorphism as well. If X is a set and B an object, the copower X × B is defined to be a coproduct of X copies of B , i.e., ` x ∈ X B . HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES
Proof. (i) ⇒ (ii): We have that φ : B ( F A, B ) → A ( A, GB ), where φ − ( g ) = ε B ◦ F ( g ).Since F is full there is an f such that F ( f ) = ε F A . Hence since F is faithful, F ( f ◦ η A ) = F ( f ) F ( η A ) = ε F A F ( η A ) = 1 F A = F (1 A ) implies f ◦ η A = 1 A has aleft inverse.Therefore we have: φ − ( η A ◦ f ) = ε F B ◦ F ( η A ◦ f ) = ε F B ◦ F ( η A ) ◦ F ( f ) = ε F B ◦ F ( η A ) ◦ ε F B = 1
F A ◦ ε F B = ε F B ◦ F (1 GF A ) = φ − (1 GF A ). This implies that η A ◦ f = 1 GF A isalso a right inverse.(ii) ⇒ (i): Consider the following isomorphism A ( A, A ′ ) A ( A,η A ′ ) −→ A ( A, GF A ′ ) φ − −→ B ( F A, F A ′ ) . When we evaluate at g : A → A ′ we obtain that: φ − ( A ( A, η A ′ )( g )) = φ − ( η A ′ ◦ g )) = ε F A ′ ◦ F ( η A ◦ g ) = ε F A ′ ◦ F ( η A ) ◦ F ( g ) = F ( g )by definition of adjunction. Thus φ − ◦ A ( A, η A ′ ) = F , is an isomorphism. In this section, we review some material from [51] relevant to the following question:how to embed a small category as a full subcategory of a complete and cocompletecategory in which the embedding preserves existing limits and colimits.
Definition 6.6.1.
Let G : A → B be a functor, A a small category. Recall thefunctor ˜ G defined in Definition 6.4.1 by ˜ G ( B ) = B ( G ( − ) , B ) on objects and ˜ G ( f ) = B ( G ( − ) , f ) on arrows. We say that G is left adequate for the category B if the functor˜ G : B → [ A op , Set ] is fully faithful.
Proposition 6.6.2.
Suppose we have a functor G : A → B , A a small category, B a co-complete category. If G is a left adequate functor then for every B ∈ B thereexists a small category I and a functor H : I → A such that colim GH = B . HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES
Proof.
For every B ∈ B let us consider F = B ( G ( − ) , B ) : A op → Set . Also considerthe category El ( F ) op of elements of F , defined in Definition 6.1.3. We claim that H = π op : El ( F ) op → A , i.e.,colim ( El ( F ) op π op → A G → B ) ∼ = B. If ( A ′ , x ′ ) f op → ( A, x ) then (
A, x ) f → ( A ′ , x ′ ) with G ( A ′ ) x ′ / / G ( f ) (cid:15) (cid:15) BG ( A ) x = = ③③③③③③③③③ since x ′ = F ( f op )( x ).We define the following set of arrows Gπ op ( A, x ) β ( A,x ) → B with β ( A,x ) = x . Naturalityfollows from the previous diagram: Gπ op ( A, x ) β ( A,x ) / / Gπ op ( f op ) (cid:15) (cid:15) ∆ B ( A ′ , x ′ ) ∆ B ( f op ) (cid:15) (cid:15) Gπ op ( A, x ) β ( A,x ) / / ∆ B ( A, x )for every ( A ′ , x ′ ) f op → ( A, x ). Now since B is co-complete we have that there existsa co-cone ( C, u ( A,x ) : Gπ op ( A, x ) → C ) such that colim Gπ op = C . This implies, bydefinition of colimit, that there exists a unique p : C → B such that the followingdiagram commutes: Gπ op ( A, x ) u ( A,x ) / / β ( A,x ) ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ C p (cid:15) (cid:15) B Actually p is an epimorphism. If f p = gp with f : B → B ′ and g : B → B ′ then we have that f pu ( A,x ) = gpu ( A,x ) for every g ( A ) x → B . This implies f x = f β ( A,x ) = gβ ( A,x ) = gx for every g ( A ) x → B . Now we use the fact that by hypothesis˜ G is faithful. By definition we have ˜ G ( f ) = ˜ G ( g ) : B ( G − , B ) → B ( G − , B ′ ) since˜ G ( f )( A )( x ) = f x = gx = ˜ G ( g )( A )( x ), which implies f = g . HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES α A : B ( G ( A ) , B ) → B ( G ( A ) , C ) with α A ( x ) = u ( A,x ) for every A ∈ A and g ( A ) x → B . We check that α is a natural transformation: B ( G ( A ′ ) , B ) α A ′ / / B ( G ( f ) ,B ) (cid:15) (cid:15) B ( G ( A ′ ) , C ) B ( G ( f ) ,C ) (cid:15) (cid:15) B ( G ( A ) , B ) α A / / B ( G ( A ) , C )for every A f → A ′ . B ( G ( f ) , C )( α A ′ ( x ′ )) = B ( G ( f ) , C )( u ( A ′ ,x ′ ) ) = u ( A ′ ,x ′ ) G ( f ) = ( ∗ ) u ( A,x ) = α A ( x ′ G ( f )) = α A ( B ( G ( f ) , C )( x ′ )) . This equality ( ∗ ) is justified because u is a co-cone, i.e., for every ( A, x ) f op → ( A ′ , x ′ ) Gπ op ( A, x ) u ( A,x ) % % ❏❏❏❏❏❏❏❏❏❏ Gπ op ( f op ) / / Gπ op ( A ′ , x ′ ) u ( A ′ ,x ′ ) y y sssssssssss C since we have that G ( A ) u ( A,x ) " " ❉❉❉❉❉❉❉❉❉ G ( f ) / / G ( A ′ ) u ( A ′ ,x ′ ) | | ②②②②②②②②② C The rest of the proof follows now from the fact that ˜ G is a full functor. Hence thereexists a morphism b : B → C such that α = B ( G − , b ) : B ( G − , B ) → B ( G − , C ).Therefore using this representation we get that u ( A,x ) = α A ( x ) = B ( G ( A ) , b )( x ) = bx for every ( A, x ) ∈ El( F ) op . Thus by definition of colimits we get that bpu ( A,x ) = bx = u ( A,x ) for every ( A, x ) ∈ El( F ) op implies that bp = 1 C . But p is an epimorphism, so wecancel to obtain pbp = p p and thus pb = 1 B , which means it is an isomorphism.Therefore colim Gπ op = ( C, u ( A,x ) ) ( A,x ) ∈ El ( F ) ∼ = ( B, β ( A,x ) ) ( A,x ) ∈ El ( F ) . Corollary 6.6.3.
For every F ∈ [ A op , Set ] F = colim ( El ( F ) op π op → A Y → [ A op , Set ]) . HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES
Proof.
The Yoneda functor A Y → [ A op , Set ] is left adequate since we have that:˜ Y : [ A op , Set ] → [ A op , Set ] is defined ˜ Y ( F ) = [ A op , Set ]( Y − , F ) = F on objects and˜ Y ( α ) = [ A op , Set ]( Y − , F ) → [ A op , Set ]( Y − , F ′ ) = Id ( α ) on arrows by the YonedaLemma. Definition 6.6.4.
A functor F : A → B reflects limits when for each functor G : I → A with I small and given a cone ( A, u i ) i ∈I , u i : A → G ( i ), if ( F ( A ) , F ( u i )) i ∈I isa limit of F G then (
A, u i ) i ∈I is a limit of G . Proposition 6.6.5.
Let F : A → B be a functor. F preserves colimits if and only if B ( F − , B ) : A op → Set preserves limits for every B ∈ B .Proof. ( ⇒ ) Let us first observe that we have a composition of functors B ( F − , B ) = B ( − , B ) ◦ F op where F op : A op → B op preserves limits since F preserves colimits and B ( − , B ) : B op → Set preserves limits [15].( ⇐ ) Now consider the functor G : I → A with colim G = ( A, u i ) i ∈I , u i : G ( i ) → A .Thus lim G op = ( A, u opi ) i ∈I op where G op : I op → A op . By hypothesis we know that B ( F − , B ) : A op → Set preserves limits, hence for every B ∈ B the limit takes theform lim B ( F − , B ) ◦ G op = ( B ( F ( A ) , B ) , B ( F ( u opi ) , B )) i ∈I op , so we have: I op G op → A op F op → B op Y → [ B , Set ] i G ( i ) F ( G ( i ))
7→ B ( F ( G ( i )) , − ),where Y ( B ′ ) = B ( B ′ , − ) : B →
Set .Therefore for any B ∈ B it may be verified that Y ◦ F op ◦ G op ( − )( B ) : I op → Set hasa limit by hypothesis, since ∀ B ∈ B :lim Y ◦ F op ◦ G op ( − )( B ) = ( B ( F ( A ) , B ) , B ( F ( u opi ) , B )) i ∈I op .Then, by proposition 2.15.1 of [15] we have Y ◦ F op ◦ G op : I op → [ B , Set ] has a limitbeing compute pointwise. Which means we have:lim Y ◦ F op ◦ G op = ( B ( F ( A ) , − ) , B ( F ( u opi ) , − )) i ∈I op .But Y is a full and faithful functor, it reflects limits (see proposition 2.9.9 [15]) whichimplies that (see definition 2.9.6 [15]) since ( Y ( F ( A )) , Y (( F ( u opi )) i ∈I op is the limit of Y ◦ F op ◦ G op then ( F ( A ) , F ( u opi )) i ∈I op is the limit of F op ◦ G op in B op . Equivalently, HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES F ( A ) , F ( u i )) i ∈I is the colimit of F ◦ G in B .Summarizing, we started with colim G = ( A, u i ) i ∈I and we end with colim F G =( F A, F u i ) i ∈I , i.e., F preserves colimits. Proposition 6.6.6.
Let F : A → B be a functor. F preserves coproducts if and onlyif B ( F − , B ) : A op → Set preserves products for every B ∈ B . Proposition 6.6.7.
Let F : A → B be a functor. F preserves limits if and only if B ( B, F − ) : A →
Set preserves limits for every B ∈ B . Let F : A → C be a fully faithful functor. Consider the full subcategories B of C such that | F ( A ) | ⊆ |B| ⊆ |C| and define: A F B → B j B → C with F = j B F B , F B ( A ) = F ( A ), F B ( f ) = F ( f ) and j the inclusion functor. Define B a full subcategory of C in the following way: |B | = { B ∈ |C| : C ( F ( − ) , B ) : A op → Set preserves limits } . Remark 6.6.8. If F : A → C is a fully faithful functor then | F ( A ) | ⊆ |B | . To seethis we have that C ( F − , F ( A )) ∼ = A ( − , A ) are naturally isomorphic which impliesthat C ( F − , F ( A )) preserves limits. Proposition 6.6.9.
Let F : A → C be a fully faithful functor. Then:(a) if j B F B preserves colimits then B ⊆ B (b) let J be a small category, and consider the following composition of functors: J ∆ → B j B → C if lim j B ∆ = ( C, v j ) then C ∈ |B | .Proof. (a) Take B ∈ |B| , since j B F B preserves colimits then by Proposition 6.6.5 C ( j B ( F B ( − )) , B ) = C ( F ( − ) , B ) preserves limits, which by definition means that B ∈|B | . HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES |B ′ | = |B | ∪ { C } also satisfies property of part (a)above. This implies that B ′ ⊆ B i.e., C ∈ |B | . We have that A F B′ → B ′ j B′ → C and we want to show that if I Γ → A with colim Γ = ( A, u ), with Γ( i ) u i → A , i ∈ I thencolim F B ′ Γ = ( F B ′ ( A ) , F B ′ ( u i )) i ∈I = ( F ( A ) , F ( u i )) i ∈I .Let t : F B ′ Γ ⇒ C be a co-cone. Without loss of generality, we assume that F B ′ (Γ( i )) = C for every i ∈ I . If there exists a i with F B ′ (Γ( i )) = C then since | F ( A ) | ⊆ |B| this implies that C ∈ |B| .We fix an object j ∈ |J | . Therefore since t is a co-cone we consider the followingco-cone: F (Γ( i )) t i → C v j → ∆( j ) . These arrows are contained in the category B because F (Γ( i )) and ∆( j ) are objectof B . We know by part (a) that F B has the property of preserving colimits:colim F B Γ = ( F B ( A ) , F B ( u i )) i ∈I = ( F ( A ) , F ( u i )) i ∈I . For that reason there exists a unique x j : F ( A ) → ∆( j ) such that F (Γ( i )) t i / / F ( u i ) (cid:15) (cid:15) C v j (cid:15) (cid:15) F ( A ) x j / / ∆( j )for every i ∈ |I| . We will show that x j is a cone in order to use the universal propertyof the limit. Let f : j → j ′ be an arrow in J . We want to prove that ∆( f ) x j = x ′ j .This follows from the fact that x j is defined using colim F B Γ. We must check that v ′ j t i = ∆( f ) x j F ( u i ) for every i ∈ |I| . Then by uniqueness of the colimit definition weget that ∆( f ) x j = x ′ j .But we know by definition of x j that: x j F ( u i ) = v j t i for every i ∈ |I| , then composingwith ∆( f ) we obtain ∆( f ) x j F ( u i ) = ∆( f ) v j t i for every i ∈ |I| . Therefore, it will beenough to prove that ∆( f ) v j = v ′ j , but this follows from the naturality of the cone C ⇒ ∆. HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES F ( A ) x ⇒ ∆ is a cone in B . Then by definition of lim∆ = ( C, v )there exists a unique y : F ( A ) → C such that: F ( A ) y / / x j ●●●●●●●●● C v j (cid:15) (cid:15) ∆( j )We therefore put all the equations together: v j t i = x j F ( u i ) = v j yF ( u i ) for every j ∈ |J | . Thus since this is true for every j ∈ |J | , by definition of limit we have that t i = yF ( u i ).So now suppose there exists another y ′ satisfying the same property as above: t i = y ′ F ( u i ). We want to prove that y = y ′ . It will be enough to prove that: v ( j ) y ′ = x j for every j ∈ |J | . For that purpose, we know by hypothesis that t i = y ′ F ( u i ) forevery i ∈ I . Then by composing we get v ( j ) y ′ F ( u i ) = v ( j ) t i for every i ∈ I , andsince v j t i = x j F ( u i ) we replace it: v j y ′ F ( u i ) = x j F ( u i ) for every i ∈ I . This impliesby uniqueness of the colimit that v j y ′ = x j .We proved that colim F B ′ Γ = ( F B ′ ( A ) , F B ′ ( u i )) i ∈I where |B ′ | = |B | ∪ { C } , i.e., for anarbitrary co-cone in B ′ , ( F ( A ) , F ( u i )) i ∈I is still a limit co-cone and this implicationis the the property that characterizes the set |B | . Corollary 6.6.10.
Let F : A → C be a fully faithful functor such that for every C ∈ C there exists a functor G : I → A with lim
F G = C . Then F preserves colimits.Proof. We consider B as above. Since lim F G = C for some G , then by part (b) ofthe Proposition 6.6.9 above we have that C ∈ B , therefore F = F B and it preservescolimits by Proposition 6.6.5. Remark 6.6.11.
To prove that Y : A → [ A op , Set ] preserves limits is equivalent toproving that Y op : A op → [ A op , Set ] op preserves colimits and since by Corollary 6.6.3: F = colim ( El ( F ) op π op → A Y → [ A op , Set ])for every F ∈ [ A op , Set ] this implies that: F = lim ( El ( F ) π → A op Y op → [ A op , Set ] op ) HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES F ∈ [ A op , Set ] op . But we know by the Corollary 6.6.10 above that thisimplies that A op Y op → [ A op , Set ] op preserves colimits. Definition 6.6.12.
Let [ A , Set ] inf ⊆ [ A , Set ] be the full subcategory of limit pre-serving functors. Since the representable functors A ( − , A ) : A op → Set preservelimits, we can define a functor A Y inf → [ A op , Set ] inf by co-restriction induced by theYoneda embedding. Remark 6.6.13.
Let A be a small category. The functor A Y inf → [ A op , Set ] inf is leftadequate since the induced functor [ A , Set ] inf ˜ Y inf → [ A op , Set ] is fully faithful. To seethis, we check that we have on objects:˜ Y inf ( F ) = [ A op , Set ] inf ( Y inf − , F ) = [ A op , Set ]( Y − , F ) ∼ = F since is a full subcategory and Y inf − = Y − ∈ [ A op , Set ] inf . Thus we have that[ A op , Set ] inf ( F, G ) = [ A op , Set ]( F, G ) ∼ = [ A op , Set ]( ˜ Y inf ( F ) , ˜ Y inf ( G ))which means that ˜ Y inf is fully faithful, i.e., Y inf left adequate. Therefore, using thesame argument we get that A Y inf → [ A op , Set ] inf preserves limits. Proposition 6.6.14.
Let B be a full subcategory of C such that for every C ∈ C there exists functor G : I → B with colim j B G = C . If B is a co-complete categorythen B is a left reflective subcategory of C . Conversely, suppose B is a left reflectivesubcategory of C . If C is co-complete then B is co-complete.Proof. We want to prove that the inclusion functor B j ֒ → C has a left adjoint C R → B . Itis enough to prove that for every C ∈ C there is an object R ( C ) ∈ B , a map C η C → R ( C )such that for every f : C → B ′ with B ′ ∈ B there is a unique g : R ( C ) → B ′ suchthat the following diagram commutes: C η C / / f ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖❖ R ( C ) g (cid:15) (cid:15) B ′ . HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES C ∈ C . By hypothesis we have that there exists a functor G : I → B with colim j B G = C . But since B is a co-complete category then there is an object B ∈ B and a co-cone { G ( i ) u i → B } i ∈ I with colim G = ( B, u ).We define R ( C ) = B , and since { jG ( i ) = G ( i ) u i → B } i ∈ I is a co-cone of jG in thecategory C therefore there exists a unique C η C → R ( C ), such that: G ( i ) v i / / u i ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ C η C (cid:15) (cid:15) B = R ( C )commutes for every i ∈ I . Now suppose we have a map f : C → B ′ with B ′ ∈ B .Then since G ( i ) is an object of B for every i ∈ I and { v i } i ∈I is a co-cone in C thisimplies that { jG ( i ) = G ( i ) v i → C f → B ′ } i ∈ I is a co-cone in the category B . Thereforeby definition of colim G = ( B, u ) there is a unique g : R ( C ) → B ′ , g ∈ B with f v i = gu i for every i ∈ I . Hence f v i = gu i = gη C v i for every i ∈ I , and this impliesby definition (uniqueness) of colimit that f = gη C .If there is a morphism ˜ g : R ( C ) → B ′ such that f = ˜ gη C then by composing with v i we get f v i = gη C v i for every i ∈ I which means that f v i = ˜ gu i for every i ∈ I therefore g = ˜ g . If C f → C ′ a morphism in C then R ( f ) is defined as the unique arrowsuch that: C f (cid:15) (cid:15) η C / / R ( C ) R ( f ) (cid:15) (cid:15) C ′ η C ′ / / R ( C ′ )commutes. By uniqueness we obtain that R is a functor and naturality of Id η ⇒ j ◦ R follows from the diagram.Conversely, let G : I → B be a functor. Since C is co-complete there exists colim jG =( C, v ) with j : B → C the inclusion functor and G ( i ) v i → C . By hypothesis weknow that R is a reflection of j , which means B ( R ( A ) , B ) ∼ = C ( A, j ( B )) for every A ∈ C , and B ∈ B . When A ∈ B then since B is a full subcategory we have that B ( R ( A ) , B ) ∼ = B ( A, B ) for every B ∈ B . By the Yoneda Lemma this implies that R ( A ) ∼ = A . On the other hand R preserves colimits because is a left adjoint. Thus G ( i ) ∼ = R ( G ( i )) = RjG ( i ) R ( v i ) → R ( C ) is a colimit of G with R ( C ) ∈ B . HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES
Remark 6.6.15.
Notice that, from the proof above, colimits in B are induced by thereflection, i.e., if G : I → B is a functor with I small then:colim B G ∼ = R (colim C j ◦ G ) . Remark 6.6.16.
For every F ∈ [ A op , Set ] there exists a functor I G → [ A op , Set ] inf such that colim jG = F : F = colim ( El ( F ) op π op → A Y inf → [ A op , Set ] inf j ֒ → [ A op , Set ]) . Proposition 6.6.17.
Let A be a small category. Then [ A , Set ] inf is a reflectivesubcategory of [ A , Set ] .Proof. [47]. Remark 6.6.18.
This implies that [ A , Set ] inf is a co-complete category. Proposition 6.6.19.
Let A Y inf → [ A op , Set ] inf be the restricted Yoneda embedding fromDefinition 6.6.12 above. Then Y inf is a full and faithful, limit and colimit preservingfunctor such that for every F ∈ [ A op , Set ] inf there exists a functor G : I → A withlim Y inf G = F . Moreover, [ A op , Set ] inf is a complete and co-complete category.Proof. First, [ A op , Set ] inf is a co-complete category by Remark 6.6.18 above. In viewof the Remark 6.6.13 above A Y inf → [ A op , Set ] inf preserves limits.Using Proposition 6.6.5: A Y inf → [ A op , Set ] inf preserves co-limits if and only if[ A op , Set ] inf ( Y inf − , F ) : A op → Set preserves limits for all F ∈ [ A op , Set ] inf .But by the Yoneda Lemma we have that[ A op , Set ] inf ( Y inf − , F ) = [ A op , Set ]( Y − , F ) ∼ = F which is the condition that defines the subcategory. Notice that we used the fact that[ A op , Set ] inf is a full subcategory.Now, in view of Proposition 6.6.9, consider the fully faithful functor F : A → C , with F = Y , B = [ A op , Set ] inf and C = [ A op , Set ]. By part (b) when there is a functor J ∆ → B j B → C HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES A op , Set ] is a complete category then lim j B ∆ = ( C, v j ) exists. But this impliesthat C ∈ |B | which means that B = [ A op , Set ] inf is complete.To see why B = [ A op , Set ] inf , consider B = [ A op , Set ] inf and j B F B = Y with F B = Y inf . Since it preserves colimits then B ⊆ B . On the other hand if B ∈ B suchthat Y B : A → B preserves colimits then by Proposition 6.6.5 this implies that: B ( Y B − , B ) : A op → Set preserves limits. But B ( Y B − , B ) = [ A op , Set ]( Y − , B ) ∼ = B. Thus it means that B preserves limits, i.e., B ∈ [ A op , Set ] inf .It remains to show that if F ∈ [ A op , Set ] inf then there exists a functor G : I → A with lim Y inf G = F . For this, it is enough to prove that Y inf is left adequate, whichwas done on Remark 6.6.13. Remark 6.6.20.
This amounts to proving that for every F ∈ [ A op , Set ] there is anobject R ( F ) ∈ [ A op , Set ] inf , a co-cone Y inf π F u ⇒ ∆ R ( F ), and a co-cone jY inf π F v ⇒ ∆ F such that colim Y inf π F = ( R ( F ) , u ) and colim jY inf π F = ( F, v ). Therefore thereis a unique F η F → R ( F ) such that F η F / / R ( F ) jY inf π F ( A, a ) = Y inf π F ( A, a ) = A ( − , A ) v ( A,a ) j j ❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯❯ u ( A,a ) ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ commutes for every i ∈ I .To conclude this section, we briefly comment on the reflective adjoint pair i ⊢ R of Proposition 6.6.17. Since [ A op , Set ] inf is a co-complete category, all small colimitsexists and we are in a position to consider co-powers A ⊗ inf B where A ∈ Set and B ∈ [ A op , Set ] inf . On the other hand, co-powers in the category [ A op , Set ] inf areinduced by copowers in [ A op , Set ] using the reflection above: A ⊗ inf B = R ( A ⊗ i ( B )) . Therefore, since R preserves coends we have that we can express R ( F ) = F ⋆ Y inf as an indexed colimit where the definition of the operation ⋆ , taken from [46] (see HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES
F ⋆Y inf = Z x F ( x ) ⊗ int Y inf ( x ) = Z x R ( F ( x ) ⊗ A ( − , x )) ∼ = R ( Z x F ( x ) ⊗ A ( − , x )) ∼ = R ( F )Notice that we are using the fact that every representable functor is included inthe category [ A op , Set ] inf . Thus, in terms of left Kan extension (see Section 6.7) orindexed colimits we have the following diagram: A Y / / Y inf ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗◗ [ A op , Set ] ˜ Y inf (cid:15) (cid:15) [ A op , Set ] inf ˆ Y inf ⊢ O O where ˜ Y inf = R = − ⋆ Y inf = Lan Y ( Y inf ) since Lan Y ( Y inf )( F ) = Z A [ A op , Set ]( Y ( A ) , F ) ⊗ inf Y inf ( A ) ∼ = Z A F ( A ) ⊗ inf Y inf ( A )and ˆ Y inf ∼ = i the inclusion functor sinceˆ Y inf ( F ) = [ A op , Set ]( Y ( − ) , F ) ∼ = F. This section provides a brief overview of the left Kan extension. A large portion ofChapter 7 depends on this central notion. To mention two examples: the definitionof a left adjoint of a certain functor and the monoidal enrichment of the functorcategory.
Definition 6.7.1.
Let F : A → B and G : A → C be two functors. The left Kanextension of the functor G along F , if it exists, is a functor K : B → C together witha natural transformation α : G ⇒ KF satisfying the following universal property:if H : B → C and β : G ⇒ HF then there is a unique natural transformation γ : K ⇒ H satisfying ( γ ∗ F ) ◦ α = β . HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES
Notation:
We denote the functor K by Lan F ( G ).Let F : A → B , and consider the functor between functor categories[ B , C ] F ∗ → [ A , C ] (16)defined by precomposition with F , i.e., F ∗ ( G ) = G ◦ F for any functor G : B → C . Corollary 6.7.2. If Lan F ( G ) exists for all G , then Lan F ⊣ F ∗ .Proof. The definition above turns out to be the following: for every β : G ⇒ F ∗ ( K )there exists a unique γ : Lan F ( G ) → H such that: G β ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ α (cid:15) (cid:15) F ∗ ( Lan F ( G )) F ∗ ( γ ) / / F ∗ ( H )which means that: [ B , C ]( Lan F ( G ) , H ] ∼ = [ A , C ]( G, F ∗ ( H ))with unit α = η G : G ⇒ F ∗ ( Lan F ( G )). Proposition 6.7.3. If A is a small category and C is co-complete then the left Kanextension of G along F exists. Remark 6.7.4.
We can also formulate the left Kan extension as a coend. If ∀ a, a ′ ∈ A and b ∈ B the copowers B ( F ( a ′ ) , b ) × G ( a ) exist in C ; and the following coend exists ∀ b ∈ B then: Lan F ( G )( b ) = Z a B ( F ( a ) , b ) × G ( a ) . Notation:
For the sake of brevity we sometimes write
Lan F instead of Lan F op when the extension is along the opposite functor F op : A op → B op . Remark 6.7.5.
Notice that for a functor Φ :
A → B we can express the adjunction
Lan Φ ⊣ Φ ∗ as a left Kan extension of Y ◦ Φ :
A → [ B op , Set ] along Y : B → [ B op , Set ]in the following way: for some F : A op → Set we have
HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES
Lan Y ( Y ◦ Φ)( F ) = ( R a [ A op Set ]( Y ( a ) , − ) × Y ◦ Φ( a ))( F ) = R a [ A op Set ]( Y ( a ) , F ) × Y ◦ Φ( a ) ∼ = R a F ( a ) × B ( − , Φ( a )) = Lan Φ op ( F )and also for some G : B op → Set :[ B op , Set ]( Y (Φ( − )) , G ) ∼ = G (Φ( − )) = Φ ∗ ( G ) . A symmetric monoidal category can be fully and faithfully embedded in a symmetricmonoidal closed category in such a way that the tensor is preserved. This constructionis a particular instance of a more general notion called promonoidal categories definedby Day [18]. In fact there is a correspondence between promonoidal categories andbiclosed monoidal structures defined on the functor categories.
Proposition 6.8.1.
Let A be a symmetric monoidal category. Then [ A op , Set ] can beequipped with a symmetric monoidal structure (called the Day tensor [18]), such thatthe Yoneda embedding Y : A → [ A op , Set ] is a strong monoidal functor. Moreover, [ A op , Set ] is monoidal closed.Proof. (sketch)We consider the monoidal closed case on functor categories([ A op , Set ] , ⊗ D , I D , ⊸ ) . This structure is obtained by using the Kan extension to closed functor categories:
A × A Y × Y / / ⊗ (cid:15) (cid:15) [ A op , Set ] × [ A op , Set ] Lan Y × Y ( −⊗− ) (cid:15) (cid:15) A Y / / [ A op , Set ]In more detail the following data is obtained: • − ⊗ D − : [ A op , Set ] × [ A op , Set ] → [ A op , Set ] is defined by S ⊗ D T = Z a S ( a ) × Z b T ( b ) × A ( − , a ⊗ b )This operation is also called the convolution of S and T . HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES • I D = A ( − , I ) • l : I D ⊗ D T → T is given by: R x I D ( x ) × ( R a T ( a ) × A ( − , x ⊗ a )) ∼ = R a ( R x I D ( x ) × A ( − , x ⊗ a )) × T ( a ) R λ ∗ × → R a A ( − , a ) × T ( a ) ∼ = T where λ ∗ : R x A ( x, I ) × A ( − , x ⊗ a )) ∼ = A ( − , I ⊗ a ) A ( − ,λ ) → A ( − , a ) • r : T ⊗ D I D → T : analogous. • a : ( R ⊗ D S ) ⊗ D T → R ⊗ D ( S ⊗ D T )( R ⊗ D S ) ⊗ D T = R x ( R a R ( a ) × ( R b S ( b ) × A ( x, a ⊗ b )) × ( R c T ( c ) × A ( − , x ⊗ c ))) ∼ = ◦ ( R × R × R (1 × α )) ◦∼ = → R a R ( a ) × R x ( R b S ( b ) × ( R c T ( c ) × A ( x, b ⊗ c ))) × A ( − , a ⊗ x )) = R ⊗ D ( S ⊗ D T ) • c : S ⊗ D T → T ⊗ D S is S ⊗ D T = Z a S ( a ) × ( Z b T ( b ) ×A ( − , a ⊗ b )) ∼ = Z b T ( b ) × ( Z a S ( a ) ×A ( − , a ⊗ b )) R × ( R × σ ) → Z b T ( b ) × ( Z a S ( a ) × A ( − , b ⊗ a )) = T ⊗ D S • the internal hom is: [ S, T ] D ∼ = Z b [ S ( b ) , T ( − ⊗ b )]For more details on this construction we refer the reader to [18]. [ C , A ] ΓIn this section we give a brief overview the methodology of Freyd and Kelly [24] inorder to build reflections in a more general way using the notion of orthogonality. Inparticular, we are interested in some full subcategories of presheaves. This construc-tion generalizes Lambek’s presentation in Section 6.6 by regarding the condition of
HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES A ∈ A , we define a preorder among the class of monomorphismswith codomain A : if f : B → A , g : C → A are two monomorphisms f is said to besmaller than g ( f ≤ g ) when f factors through g i.e., f = gk for some k : B → C .Note that k is unique and also a monomorphism.We have an equivalence relation f ≡ g iff f ≤ g and g ≤ f . Definition 6.9.1. A subobject of A is an equivalence class of these monomorphisms.The class of subobjects is partially ordered by the order induced by the represen-tatives. Definition 6.9.2.
We say that a category A is well-powered when for every A ∈ A the class of subobjects of A is a set.The dual notions applied to epimorphisms are called quotient for an equivalenceclass of epimorphisms, and co-well-powered . Definition 6.9.3.
Let A be an object. The intersection of a family of subobjectsof A , if it exists, is the greatest lower bound defined in the partially ordered classof subobjects of A . Analogously, by the union we mean the least upper bound, if itexists.Concretely, we mean the following: if { A i f i → A } i ∈ I are subobjects of A then thereexists an arrow ∩ i ∈ I A i f → A satisfying the following properties:- f ≤ f i ∀ i ∈ I , i.e., for every i ∈ I there exists an arrow ∩ i ∈ I A i t i → A i such that f i ◦ t i = f .- if there exists a p such that p ≤ f i ∀ i ∈ I then p ≤ f , i.e., if there are maps B p → A and B p i → A i with the property f i ◦ p i = p ∀ i ∈ I then there exists aunique h : B → ∩ i ∈ I A i such that p = f ◦ h . Definition 6.9.4.
An infinite limit cardinal α is regular when it is equal to itscofinality: cf( α ) = α . Here cf( α ) is the least limit ordinal β such that there exists anincreasing sequence { α η } η<β with lim η → β α η = α. HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES α is regular means cannot be written as a sum of a lesser numberof cardinals less than α . Definition 6.9.5.
Let α be a regular cardinal. An ordered set J is α -directed whenfor every subset I ⊆ J with | I | ≤ α there exists an upper bound in J . Definition 6.9.6.
Let S = { f ξ : C ξ → B } be a family of subobjects of B withthe monotonic property: f ξ ≤ f ζ whenever ξ ≤ ζ . The family S is called α -directed provided that the set J is α -directed . Definition 6.9.7.
We say that an object A ∈ A is bounded by a regular cardinal α when for every morphism from A to a α -directed union ∪ ξ ∈ J C ξ factors through aunion ∪ ξ ∈ K C ξ for some K ⊆ J with | K | < α . We call A bounded if each A ∈ A isbounded. Definition 6.9.8.
Let
E, M ⊆ Mor( A ) be two classes of morphisms. A factorizationsystem ( E, M ) on a category A consists of the following data:- Isos( A ) ⊆ E ∩ M , isomorphisms belong to the intersection of the two classes- E and M are closed under composition- for every morphism f there is a factorization f = m ◦ e with e ∈ E and m ∈ M - for every f and g if m ′ ◦ e ′ ◦ f = g ◦ m ◦ e with e, e ′ ∈ E and m, m ′ ∈ M thenthere exists a unique w making the whole diagram • f (cid:15) (cid:15) e / / • w (cid:15) (cid:15) ✤✤✤ m / / • g (cid:15) (cid:15) • e ′ / / • m ′ / / • commutative. A factorization system ( E, M ) is called a proper factorization when E ⊆ Epis( A ), M ⊆ Monos( A ) where Epis( A ) is the class of all epimor-phisms of A and Monos( A ) is the class of all monomorphisms of A . HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES
Definition 6.9.9.
An epimorphism p is called extremal provided that whenever wehave p = m ◦ g , where m is a monomorphism then m is also an isomorphism. Duallywe define the notion of extremal monomorphism . Epi ∗ denotes the class of extremalepimorphism and Mon ∗ the class of extremal monomorphism. Proposition 6.9.10.
If one of these two conditions below are satisfied- the category A is finitely complete and has arbitrary intersections of monomor-phisms- the category A is finitely co-complete and has arbitrary co-intersections of ex-tremal epimorphismsthen (Epi ∗ , Mon) is a proper factorization system.Proof. [24]In the case of the category of sets a direct calculation shows that Epi ∗ = Epi andMon ∗ = Mon since we have: if p ∈ Epi with p = m ◦ g then a ◦ m = b ◦ m implies a ◦ m ◦ g = b ◦ m ◦ g which is a ◦ p = b ◦ p and this a = b . Definition 6.9.11.
Given factorization system (
E, M ) a generator of the category A is a small full subcategory G such that for each A ∈ A , ∪ G ∈G A ( G, A ) ⊆ E .When a factorization system ( E, M ) is proper and G a generator then given anypair of morphisms f, g : A → B then for every p : G → A with G ∈ G we have that f ◦ p = g ◦ p ⇒ f = g .If A has coproducts then G is a generator if and only if for every A ∈ A the map: k A : a G ∈G ( a A ( G,A ) ) → A is in E ; where k A is defined by the universal property of the coproduct, i.e., k A ◦ i G,f = f : G → A and i G,f : G → ` G ∈G ( ` A ( G,A ) ) is the coproduct injection. Definition 6.9.12.
Let
P, Q : K → C be functors with K a small category. A cylinder in C is just a natural transformation α : P → Q . HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES
Definition 6.9.13.
A functor T : C → A is continuous with respect to the cylinder α when:- there exists lim T P and lim
T Q as cones in A .- the unique morphism lim T α : lim
T P → lim T Q determined by the definitionof limit (lim
T Q, π Q K ): T P K
T α K / / T QK lim
T P π P K O O lim T α / / ❴❴❴❴❴❴ lim T Q π Q K O O is an isomorphism. Remark 6.9.14.
In the case when P = ∆ C is a constant functor, C ∈ C , then α isjust a cone in the usual sense and continuity is the standard definition of continuityof functors. Definition 6.9.15.
Let Γ be a class of cylinders in the category C . Then [ C , A ] Γ is thefull subcategory of [ C , A ] of functors T that are continuous w.r.t. each ( P, Q, α ) ∈ Γ. Definition 6.9.16.
Consider an arrow f : A → B and an object C ∈ A . We saythat f is orthogonal to C , and we write C ⊥ f , if for every morphism y : A → C there exists a unique x : B → C such that x ◦ f = y .This definition is basically the definition of a bijective function since is equivalentto the fact that the representables A ( B, C ) A ( f,C ) −→ A ( A, C ) are isos in the category ofsets.Dually we consider f ⊥ C . Definition 6.9.17.
Given a class ∆ of morphisms in a category A , let us consider thefull subcategory of A defined by the following object: ∆ ⊥ = { B ∈ A : B ⊥ f, ∀ f ∈ ∆ } . Definition 6.9.18.
Let us consider X ∈ Set , where A ∈ A . The tensor product X ⊗ A ∈ A is the co-power, i.e., the coproduct of | X | copies of the object A in thecategory A characterized by the following natural isomorphism: A ( X ⊗ A, B ) ∼ = Set ( X, A ( A, B )) . HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES α : P → Q : K → C we associate an arrow ˜ α : ˜ Q → ˜ P inthe presheaf category [ C , A ] in the following way.First we consider the functor ˆ P : K op → [ C , Set ] defined by: K op P op −→ C op Y −→ [ C , Set ]thus ˆ P ( K ) = C ( P ( K ) , − ), and C ( f, − ) if f op ∈ K op .Then, we take ˜ P = colim ˆ P the pointwise colimit in the category [ C , Set ], i.e.,˜ P ∈ [ C , Set ].In the same way, at the level of arrows we get:ˆ α K = C ( α K , − ) : C ( QK, − ) −→ C ( P K, − )and then we obtain: ˆ Q ( K ) π Q K (cid:15) (cid:15) ˆ α K / / ˆ P ( K ) π P K (cid:15) (cid:15) ˜ Q ˜ α / / ❴❴❴❴❴❴❴❴ ˜ P by definition of colimit ( ˜ Q, π Q K ), since π P K ◦ ˆ α K is natural in K . So, ˜ α is given as theunique arrow in [ C , Set ] making the previous diagram commute. Now we considerthe class of morphisms ∆ ⊆ [ C , A ] depending on a choice of a class of cylinders Γ:∆ = { ˜ α ⊗ A : ˜ Q ⊗ A → ˜ P ⊗ A, with A ∈ A , α ∈ Γ } where ˜ Q ⊗ A : C → A and ˜ α ⊗ A are defined using the pointwise co-power as( ˜ Q ⊗ A )( C ) = ˜ Q ( C ) ⊗ A . Proposition 6.9.19.
Let A be a complete and co-complete category and let Γ be aclass of cylinders in the small category C . Then [ C , A ] Γ = ∆ ⊥ .Proof. Since both categories are full it is enough to check that they contain the sameobjects. We want to prove that T ∈ ∆ ⊥ if and only if T ∈ [ C , A ] Γ .By definition of the orthogonal class, T ∈ ∆ ⊥ if and only if for every ˜ α ⊗ A ∈ ∆we have that [ C , A ]( ˜ α ⊗ A, T ) is a bijective map, i.e., for every µ : ˜ Q ⊗ A → T there HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES ν such that, ˜ Q ⊗ A ˜ α ⊗ A (cid:15) (cid:15) µ / / T ˜ P ⊗ A ν ♣♣♣♣♣♣♣♣♣♣♣♣♣ But since when A ( F ( X ) , B ) ∼ = Set ( X, G ( B )) is natural with F : Set → A , F ( X ) = X ⊗ A and G : A →
Set , G ( B ) = A ( A, B ). Then we have that: A ( F ( X ) , B ) A ( F ( f ) ,g ) (cid:15) (cid:15) φ X,B / / Set ( X, G ( B )) Set ( f,G ( g )) (cid:15) (cid:15) A ( F ( X ′ ) , B ′ ) φ X ′ ,B ′ / / Set ( X ′ , G ( B ′ ))with X ′ f → X and B g → B ′ .This implies that G ( g ) ◦ φ X,B ( x ) ◦ f = φ X ′ ,B ′ ( g ◦ x ◦ F ( f )) for every x : F ( X ) → B .Therefore choosing g = 1, x = ν , X = ˜ P , X ′ = ˜ Q , f = ˜ α , B = T we have that since ν ◦ F ( ˜ α ) = µ then φ X ′ ,B ′ ( ν ◦ F ( ˜ α )) = φ X ′ ,B ′ ( µ ) and then G (1) ◦ φ X,B ( ν ) ◦ ˜ α = φ X ′ ,B ′ ( µ ).Using the natural isomorphism let us call ν ′ = φ X,B ( ν ) : ˜ P → A ( A, T − ) where ν : F ( ˜ P ) → T and µ ′ = φ X ′ ,B ′ ( µ ) : ˜ Q → A ( A, T − ) where µ : F ( ˜ Q ) → T . So thisturns out to be ν ′ ◦ ˜ α = µ , ˜ Q ˜ α (cid:15) (cid:15) µ ′ / / A ( A, T − )˜ P ν ′ ♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥ Then by definition of ˜ Q = colim ˆ Q with injection ˆ QK i QK → ˜ Q and ˜ P = colim ˆ P withinjection ˆ P K i PK → ˜ P we define µ ′′ and ν ′′ by the following compositions: µ ′′ = µ ′ ◦ i QK where C ( QK, − ) = ˆ QK i QK −→ ˜ Q µ ′ −→ A ( A, T − ) and ν ′′ = ν ′ ◦ i PK where C ( P K, − ) =ˆ P K i PK −→ ˜ P µ ′ −→ A ( A, T − ). Therefore we have C ( QK, − ) C ( α K , − ) (cid:15) (cid:15) i QK / / ˜ Q ˜ α (cid:15) (cid:15) µ ′ / / A ( A, T − ) C ( P K, − ) i PK / / ˜ P ν ′ ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES F = A ( A, T − ), then by naturality of the Yoneda Lemma with respect to α K we have that: [ C , Set ]( C ( P K, − ) , F ) [ C , Set ]( C ( α K , − ) ,F ) (cid:15) (cid:15) θ P / / F ( P K ) F ( α K ) (cid:15) (cid:15) [ C , Set ]( C ( QK, − ) , F ) θ Q / / F ( QK )Thus if we evaluate ν : C ( P K, − ) → F we obtain: θ Q ( ν ◦ C ( α K , − )) = F ( α K )( θ P ( ν ))and since F = A ( A, T − ) then we get θ Q ( ν ◦ C ( α K , − )) = T ( α K ) ◦ θ P ( ν )Therefore since µ ′′ = ν ′′ ◦ C ( α K , − ) we have by choosing ν = ν ′′ : θ Q ( µ ′′ ) = T ( α K ) ◦ θ P ( ν ′′ )where θ Q ( µ ′′ ) ∈ F ( QK ) = A ( A, T QK ), θ Q ( µ ′′ ) : A → T QK and θ Q ( ν ′′ ) ∈ F ( P K ) = A ( A, T P K ), θ P ( ν ′′ ) : A → T P K .So by naturality of K and the definition of limit we obtain the following diagram:lim T Q π Q z z tttttttttt A ¯ µ ❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣❣ θ Q ( µ ′′ ) / / T QKA ¯ ν + + ❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳ θ P ( ν ′′ ) / / T P K T ( α K ) O O lim T P lim
T α O O π P e e ❑❑❑❑❑❑❑❑❑❑ Thus the condition of T ∈ [ C , A ] Γ (continuity) is by definition that lim T α is anisomorphism and T ∈ ∆ ⊥ (orthogonality) iff [ C , A ]( ˜ α ⊗ A, T ) is an isomorphism.
Theorem 6.9.20.
Let A be a complete and co-complete category with a given properfactorization system ( E, M ) . Let A be bounded and co-well-powered. Let us considerthe class ∆ = Φ ∪ Ψ where Φ is small and where Ψ ⊆ E . Then ∆ ⊥ is a reflectivesubcategory of A . HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES
Proof. [24]
Theorem 6.9.21.
Let A be a complete and co-complete category with a given properfactorization system ( E , M ) . Let A be bounded with a generator, and co-well-powered.Let Γ be a class of cylinders in the small category C , and let all but a set of thesecylinders be cones. Then [ C , A ] Γ is a reflective subcategory of [ C , A ] .Proof. [24] Let B be a symmetric monoidal closed category. Day’s so-called reflection theo-rem [19] can be used to derive a monoidal closed structure in a reflective subcategoryof [ B op , Set ]. In Chapter 7, we shall utilize this to determine a strong monoidalfunctor which, in turns, determines a monoidal adjunction. Here, we review Day’sreflection theorem.
Definition 6.10.1.
A class of objects
A ⊆ |B| is strongly generating when B (1 , f ) : B ( A, B ) → B ( A, B ′ ) is an isomorphism for every A ∈ A implies that f : B → B ′ isan isomorphism in B .Dually we define the notion of strongly cogenerating class of object by consideringthe maps B ( f, Example . The class
A ⊆ [ B op , Set ], where A = {B ( − , B ) : B ∈ |B|} arerepresentables, is strongly generating. To see this we must prove that if (1 , α ) :[ B op , Set ]( B ( − , B ) , F ) → [ B op , Set ]( B ( − , B ) , G ) is an isomorphism for every B ∈ B ,where (1 , α ) = [ B op , Set ]( B ( − , B ) , α ) acts on natural transformations as (1 , α )( β ) = α ◦ β , then α : F ⇒ G is an isomorphism. To prove this, consider the followingdiagram: [ B op , Set ]( B ( − , B ) , F ) (1 ,α ) / / [ B op , Set ]( B ( − , B ) , G ) φ G (cid:15) (cid:15) F ( B ) φ − F O O α B / / G ( B ) HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES φ − F : F ( B ) → [ B op , Set ]( B ( − , B ) , F ) is defined φ − F ( x ) : B ( − , B ) ⇒ F as( φ − F ( x )) C ( g ) = F ( g )( x ) and φ G : [ B op , Set ]( B ( − , B ) , G ) → G ( B ) is defined as φ G ( β ) = β B (1 B ).Therefore, we have( φ G ◦ (1 , α ) ◦ φ − F )( x ) = φ G ((1 , α )(( φ − F ( x )))) == φ G ( α ◦ φ − F ( x )) = ( α ◦ φ − F ( x )) B (1 B ) = α B ◦ ( φ − F ( x )) B (1 B ) == α B (( φ − F ( x )) B (1 B )) = α B ( F (1 B )( x )) = α B (1 F B )( x )) = α B ( x ) , which means φ G ◦ (1 , α ) ◦ φ − F = α B . Theorem 6.10.3. (Day’s reflection theorem) Let ( B , ⊗ , I, [ − ]) be a symmetricmonoidal closed category, and let B F / / C G ⊥ o o be an adjunction from B to C , where G is full and faithful. Let A ⊆ |B| be a strongly generating class in B and D ⊆ |C| bea strongly cogenerating class in C . Then the following are equivalent:(0) there exists a monoidal closed structure on C for which F is a monoidal strongfunctor.(a) η : [ B, GC ] → GF [ B, GC ] , is an isomorphism for all C ∈ C , B ∈ B .(b) η : [ A, GD ] → GF [ A, GD ] , is an isomorphism for all A ∈ A , D ∈ D .(c) [ η,
1] : [
GF B, GC ] → [ B, GC ] , is an isomorphism for all C ∈ C , B ∈ B .(d) F ( η ⊗
1) : F ( B ⊗ B ′ ) → F ( GF B ⊗ B ′ ) , is an isomorphism for all B, B ′ ∈ B .(e) F ( η ⊗
1) : F ( B ⊗ A ) → F ( GF B ⊗ A ) , is an isomorphism for all A ∈ A , B ∈ B .(f ) F ( η ⊗ η ) : F ( B ⊗ B ′ ) → F ( GF B ⊗ GF B ′ ) , is an isomorphism for all B, B ′ ∈ B . HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES
Proof. ( a ) ⇒ ( b ) Since A ⊆ B and
D ⊆ C .( b ) ⇒ ( e ) C ( F ( GF B ⊗ A ) , D ) adjunction (cid:15) (cid:15) C ( F ( η ⊗ , / / C ( F ( B ⊗ A ) , D ) adjunction (cid:15) (cid:15) B ( GF B ⊗ A, GD ) (4) closed (cid:15) (cid:15) B ( η ⊗ , / / B ( B ⊗ A, GD ) closed (cid:15) (cid:15) B ( GF B, [ A, GD ]) (3) η iso by hypothesis (cid:15) (cid:15) B (1 ⊗ η ) B ( η, / / B ( B, [ A, GD ]) η iso by hypothesis (cid:15) (cid:15) B (1 ⊗ η ) B ( GF B, GF [ A, GD ]) (2) G fully faithful (cid:15) (cid:15) B ( η, / / B ( B, GF [ A, GD ]) C ( F B, F [ A, GD ]) (1) adjunction ❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢ (1) commutes since we have θ ( f ) = G ( f ) ◦ η B = ( B ( η, ◦ G F B,F [ A,GD ] )( f ) B ( GF B, GF [ A, GD ]) B ( η, / / B ( B, GF [ A, GD ]) C ( F B, F [ A, GD ]) G O O θ ❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢ (2) by functoriality; (3) and (4) by naturality. The vertical and bottom arrows areisos then the top is an isomorphism. Hence since D ⊆ C is strongly cogenerating wehave that F ( η ⊗
1) : F ( B ⊗ A ) → F ( GF B ⊗ A ) is an isomorphism for every A ∈ A and B ∈ B .( e ) ⇒ ( c ) C ( F ( GF B ⊗ A ) , C ) adjunction (cid:15) (cid:15) C ( F ( η ⊗ , / / C ( F ( B ⊗ A ) , C ) adjunction (cid:15) (cid:15) B ( GF B ⊗ A, GC ) (2) closed (cid:15) (cid:15) B ( η ⊗ , / / B ( B ⊗ A, GC ) closed (cid:15) (cid:15) B ( A, [ GF B, GC ]) (1) B (1 , [ η, / / B ( A, [ B, GC ]) HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES A is strongly generating then [ η,
1] : [
GF B, GC ] → [ B, GC ]is an isomorphism as well.( c ) ⇒ ( d ) We use the same diagram with A ∈ B .( d ) ⇒ ( f )By functoriality F ( B ⊗ B ′ ) F ( η ⊗ η ) / / F ( η ⊗ ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ F ( GF B ⊗ GF B ′ ) F ( GF B ⊗ B ′ ) F (1 ⊗ η ) ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ ( f ) ⇒ ( a )We want to find an arrow ν : GF [ B, GC ] → [ B, GC ] such that η ◦ ν = ν ◦ η = 1.From naturality of the following diagram B ( GF [ B, GC ] ⊗ B, GC ) φ / / B ( η ⊗ , (cid:15) (cid:15) B ( GF [ B, GC ] , [ B, GC ]) B ( η, (cid:15) (cid:15) B ([ B, GC ] ⊗ B, GC ) φ / / B ([ B, GC ] , [ B, GC ])we obtain B ( η ⊗ , φ − ( ν )) = φ − ( B ( η, ν )) which implies that φ − ( ν ) ◦ ( η ⊗
1) = φ − ( ν ◦ η ). On the other hand we have that1 = ν ◦ η if and only if φ − (1) = φ − ( ν ◦ η ) if and only if ev = φ − ( ν ) ◦ ( η ⊗ . Therefore by uniqueness it is enough to find an arrow x of the correct type which isa solution of the following equation ev = x ◦ ( η ⊗ x = φ − ( ν ), i.e., φ ( x ) = ν . We choose x = G ( θ − ( ev )) GF ( η ⊗ η ) η (1 ⊗ η )satisfying the following diagram HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES
B, GC ] ⊗ B η ⊗ (cid:15) (cid:15) η ⊗ η $ $ ■■■■■■■■■■■■■■■■■■■■■■■■ η , , ❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩ ev / / GC (1) GF ([ B, GC ] ⊗ B ) G ( θ − ( ev )) O O GF [ B, GC ] ⊗ B ⊗ η / / GF [ B, GC ] ⊗ GF B η / / GF ( GF [ B, GC ] ⊗ GF B ) GF ( η ⊗ η ) O O To justify (1), let φ : B ([ B, GC ] ⊗ B, GC ) → B ([ B, GC ] , [ B, GC ]) be the tensoradjunction. By definition we have ev = φ − (1 [ B,GC ] ). Now consider the adjunctionbetween functors F and G , θ : C ( F ([ B, GC ] ⊗ B ) , C ) → B ([ B, GC ] ⊗ B, GC )and take e ′ = θ − ( ev ). Then we have that G ( e ′ ) ◦ η [ B,GC ] ⊗ B = θ ( e ′ ) = θ ( θ − ( ev )) = ev .It remains to prove that η ◦ ν = 1. Since G : C → B is a fully faithful functor, thereis a unique f such that G ( f ) = η ◦ ν . Also we know that ν ◦ η = 1 Hence, we have G ( f ) ◦ η = ( η ◦ ν ) ◦ η = η ◦ ( ν ◦ η ) = η ◦ (1) = η = G (1) ◦ η. Finally, from the adjunction θ : C ( F [ B, GC ] , F [ B, GC ] → B ([ B, GC ] , GF [ B, GC ]) weobtain θ ( f ) = G ( f ) ◦ η [ B,GC ] , which implies that θ ( f ) = θ (1), i.e., f = 1. Therefore η ◦ ν = G (1) = 1.(0) ⇒ ( f ) See [43].( f ) ⇒ (0) The monoidal closed structure induced on C : Now using Theorem 6.10.3 we are able to induce a monoidal structure on the category C . Define C ˜ ⊗ C ′ = F ( GC ⊗ GC ′ ) and f ˜ ⊗ g = F ( Gf ⊗ Gg ). Also define ˜ I = F I and (
F, m ) is monoidal functor, where m A,B : F ( A ) ˜ ⊗ F ( B ) → F ( A ⊗ B ) is given by: m A,B = ( F ( η ⊗ η )) − with ( F ( η ⊗ η )) − : F ( GF A ⊗ GF B ) → F ( A ⊗ B ).The tensor has right adjoint given by the following formula [ C, E ] C = F [ GC, GE ], C, E ∈ |C| C ( D ˜ ⊗ C, E ) = C ( F ( GD ⊗ GC ) , E ) ∼ = B ( GD ⊗ GC, GE ) HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES ∼ = B ( GD, [ GC, GE ]) ∼ = B ( GD, GF [ GC, GE ]) ∼ = C ( D, F [ GC, GE ]) ∼ = C ( D, [ C, E ] C ) . In order to obtain a monoidal structure on the category C we define natural isomor-phisms ˜ λ , ˜ ρ and ˜ α determined by the following diagrams:˜ I ˜ ⊗ C = F ( GF I ⊗ GC ) ˜ λ / / F ( η I ⊗ − (cid:15) (cid:15) CF ( I ⊗ GC ) F ( λ ) / / F GC ǫ C O O C ˜ ⊗ ˜ I = F ( GC ⊗ GF I ) ˜ ρ / / F (1 ⊗ η I ) − (cid:15) (cid:15) CF ( GC ⊗ I ) F ( ρ ) / / F GC ǫ C O O ( C ˜ ⊗ C ′ ) ˜ ⊗ C ′′ = F ( GF ( GC ⊗ GC ′ ) ⊗ GC ′′ ) ˜ α O O F ( η ⊗ − / / C ˜ ⊗ ( C ′ ˜ ⊗ C ′′ ) = F (( GC ⊗ GF ( GC ′ ⊗ GC ′ )) F (( GC ⊗ GC ′ ) ⊗ GC ′′ ) F ( α ) O O F (( GC ⊗ ( GC ′ ⊗ GC ′′ )) F (1 ⊗ η ) o o For example we want to check that:( C ˜ ⊗ ˜ I ) ˜ ⊗ C ′ ˜ α / / ˜ ρ ˜ ⊗ & & ▲▲▲▲▲▲▲▲▲▲ C ˜ ⊗ ( ˜ I ˜ ⊗ C ′ ) ⊗ ˜ λ x x rrrrrrrrrr C ˜ ⊗ C ′ HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES F (( GC ⊗ I ) ⊗ GC ′ ) ABF ( α ) A A F ( ηGC ⊗ I ⊗ / / F ( ρ ⊗ - - ❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩ F (( GC ⊗ GF I ) ⊗ GC ′ ) CF ((1 ⊗ η ) ⊗ − (cid:15) (cid:15) F ( α ) O O F ( GC ⊗ ( GF I ⊗ GC ′ )) EF (1 ⊗ η ) / / F ( GC ⊗ ( I ⊗ GC ′ )) F (1 ⊗ ( η ⊗ (cid:15) (cid:15) F (1 ⊗ ηI ⊗ GC ′ ) / / F (1 ⊗ λ ) ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ F ( G ( F ( GC ⊗ I )) ⊗ GC ′ ) DFF ( G ( F ( ρ )) ⊗ ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ F ( G ( F ( GC ⊗ GF I )) ⊗ GC ′ ) H ˜ ρ ˜ ⊗ ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ F ( G ( F (1 ⊗ ηI ) − ⊗ (cid:15) (cid:15) ˜ α O O F ( η ⊗ − o o F ( GC ⊗ GF ( GF I ⊗ GC ′ )) I ⊗ ˜ λ * * ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ F (1 ⊗ G ( F ( η ⊗ − O O F ( GC ⊗ GF ( I ⊗ GC ′ )) F (1 ⊗ GF ( λ )) * * ❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱❱ G F ( GC ⊗ GC ′ ) ^ ^ F ( ηGC ⊗ O O F ( G ( F ( GC )) ⊗ GC ′ ) F ( G ( εC ) ⊗ O O F ( GC ⊗ GC ′ ) F ( GC ⊗ GF ( GC ′ )) F (1 ⊗ G ( εC ′ )) (cid:15) (cid:15) F ( GC ⊗ GC ′ ) F (1 ⊗ ηGC ′ ) (cid:15) (cid:15) (cid:0) (cid:0) Diagram A: By naturality of η with 1 ⊗ η I : GC ⊗ I → GC ⊗ GF I , then byfunctoriality of − ⊗ GC ′ and F we obtain: F (( GC ⊗ I ) ⊗ GC ′ ) F ( η GC ⊗ I ⊗ / / F ((1 ⊗ η I ) ⊗ (cid:15) (cid:15) F ( GF ( GC ⊗ I ) ⊗ GC ′ ) F ( GF (1 ⊗ η I ) ⊗ (cid:15) (cid:15) F (( GC ⊗ GF I ) ⊗ GC ′ ) F ( η GC ⊗ GF I ⊗ / / F ( GF ( GC ⊗ GF I ) ⊗ GC ′ )Since F (1 GC ⊗ η I ), F ( η GC ⊗ I ⊗ GC ′ ) and F ( η GC ⊗ GF I ⊗ GC ′ ) are invertible mapthis implies that F ((1 GC ⊗ η I ) ⊗ GC ′ ) is invertible as well.Diagram D: by naturality of η with ρ : GC ⊗ I → GC we have that GF ( ρ ) ◦ η GC ⊗ I = η GC ◦ ρ then by functoriality of − ⊗ GC ′ and F .Diagram H: by definition we have ˜ ρ = F (1 ⊗ η ) − ; F ( ρ ); ε , then we apply functor − ˜ ⊗− = F ( G ( − ) ⊗ G ( − )) to the pair of arrows ( ˜ ρ, C ′ ). HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES α .Diagram B: by considering the diagram A , the map F ((1 GC ⊗ η I ) ⊗ GC ′ ) − makessense, also by naturality of α with 1 GC , η I and 1 GC ′ , and then compose with F .Diagram E: this is analogous to diagram A. We consider naturality of η with themap η I ⊗ I ⊗ GC ′ → GF I ⊗ GC ′ , then compose with the functor GC ⊗ − and F .Since F ( η I ⊗
1) is invertible then F (1 ⊗ GF ( η I ⊗ F (1 ⊗ η I ⊗ GC ′ ) = F (1 ⊗ G (( F ( η I ⊗ − )) ◦ F (1 ⊗ η GF I ⊗ GC ′ ) ◦ F (1 ⊗ ( η I ⊗ η with λ : I ⊗ GC ′ → GC ′ then compose with GC ⊗ − and F .At the bottom of the diagram we have an adjoint equation: η G ◦ G ( ε ) = 1.We can also define ρ on the image of F in the following way: F B ˜ ⊗ ˜ I = F ( GF B ⊗ GF I ) ˜ ρ / / F ( η B ⊗ η I ) − (cid:15) (cid:15) F BF ( B ⊗ I ) F ( ρ ) ✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ This coincides with the above definition: F ( GF B ⊗ GF I ) F ( η B ⊗ η I ) − (cid:15) (cid:15) F (1 GF B ⊗ η I ) − / / F ( GF B ⊗ I ) F ( ρ ) (cid:15) (cid:15) F ( G ( F ( B ))) ε F B (cid:15) (cid:15) F ( B ⊗ I ) F ( ρ ) / / F B
To see this we have that: F ( ρ ) ◦ F ( η B ⊗ η I ) − = ε F B ◦ F ( ρ ) ◦ F (1 GF B ⊗ η I ) − iff F ( ρ ) = ε F B ◦ F ( ρ ) ◦ F (1 GF B ⊗ η I ) − ◦ F ( η B ⊗ η I ) iff F ( ρ ) = ε F B ◦ F ( ρ ) ◦ F (1 GF B ⊗ η I ) − ◦ F (1 GF B ⊗ η I ) ◦ F ( η B ⊗ I ) iff F ( ρ ) = ε F B ◦ F ( ρ ) ◦ F ( η B ⊗ I ) iff ε − F B ◦ F ( ρ ) = F ( ρ ) ◦ F ( η B ⊗ I ) iff F ( η B ) ◦ F ( ρ ) = F ( ρ ) ◦ F ( η B ⊗ I ) iff HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES F ( η B ◦ ρ ) = F ( ρ ◦ ( η B ⊗ I ))where the last two equations are justified by naturality of ρ with η B : B → GF B and, since G is full and faithful, we have that ε is an isomorphism and ε − F B = F ( η B ).We can also define an associativity isomorphism on the image of G ˜ α : ( GC ˜ ⊗ GC ′ ) ˜ ⊗ GC ′′ → GC ˜ ⊗ ( GC ′ ˜ ⊗ GC ′′ )in the following way: F ( G ( F ( G ( F B ) ⊗ G ( F B ′ ))) ⊗ G ( F B ′′ )) ˜ α / / F ( G ( F ( η ⊗ η ) − ) ⊗ (cid:15) (cid:15) F ( GF B ⊗ GF ( GF B ′ ⊗ GF B ′′ )) F ( GF ( B ⊗ B ′ ) ⊗ GF B ′′ ) F ( η ⊗ η ) − (cid:15) (cid:15) F ( GF B ⊗ GF ( B ′ ⊗ B ′′ )) F (1 ⊗ GF ( η ⊗ η )) O O F (( B ⊗ B ′ ) ⊗ B ′′ ) F ( α ) / / F ( B ⊗ ( B ′ ⊗ B ′′ )) F ( η ⊗ η ) O O F ( G ( F ( G ( F B ) ⊗ G ( F B ′ ))) ⊗ G ( F B ′′ )) F ( G ( F ( η ⊗ η ) − ) ⊗ (cid:15) (cid:15) F ( η ⊗ − / / F (( GF B ⊗ GF B ′ ) ⊗ GF B ′′ ) F ( α ) (cid:15) (cid:15) F ( GF ( B ⊗ B ′ ) ⊗ GF B ′′ ) F ( η ⊗ η ) − (cid:15) (cid:15) A F ( GF B ⊗ ( GF B ′ ⊗ GF B ′′ )) F (1 ⊗ η ) (cid:15) (cid:15) F (( B ⊗ B ′ ) ⊗ B ′′ ) F ( α ) (cid:15) (cid:15) F (( η ⊗ η ) ⊗ η ) ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ F ( GF B ⊗ GF ( GF B ′ ⊗ GF B ′′ )) F ( B ⊗ ( B ′ ⊗ B ′′ )) B F ( η ⊗ η ) / / F ( η ⊗ ( η ⊗ η )) ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ F ( GF B ⊗ GF ( B ′ ⊗ B ′′ )) F (1 ⊗ GF ( η ⊗ η )) O O C Diagram A commutes by naturality of η with η ⊗ η : B ⊗ B ′ → GF B ⊗ GF B ′ : weapply − ⊗ η B ′′ ( B ⊗ B ′ ) ⊗ B ′′ η ⊗ η / / ( η ⊗ η ) ⊗ η (cid:15) (cid:15) GF ( B ⊗ B ′ ) ⊗ GF B ′′ GF ( η ⊗ η ) ⊗ (cid:15) (cid:15) ( GF B ⊗ GF B ′ ) ⊗ GF B ′′ η ⊗ / / GF ( GF B ⊗ GF B ′ ) ⊗ GF B ′′ HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES F .Diagram B commutes by naturality of the isomorphism α Diagram C is analogous to diagram A: it commutes by naturality of η with η ⊗ η : B ′ ⊗ B ′′ → GF B ′ ⊗ GF B ′′ , then we apply η ⊗ − and finally we evaluate the functor F on this diagram. Now we consider a particular case of Theorem 6.10.3 studied in [20]. Let us con-sider [ B op , Set ] F / / C G ⊥ o o with G fully faithful and where ([ B op , Set ] , ⊗ , I ) has themonoidal structure induced by the convolution product (defined in Proposition 6.8.1).When A = B ( − , B ) is a representable functor, by the Yoneda Lemma we have that:[ A, G ( C )] = Z B ′ [ B ( B ′ , B ) , G ( C )( − ⊗ B ′ )] ∼ = G ( C )( − ⊗ B ) (17)Now suppose there exists C ′ ∈ C such that G ( C )( − ⊗ B ) ∼ = G ( C ′ ) (18)is a natural isomorphism between functors. Let us explicitly call φ the compositionof these two isomorphisms (17) and (18) above: φ : [ A, G ( C )] → G ( C ′ ). Then wehave: [ A, G ( C )] φ / / η [ A,G ( C )] (cid:15) (cid:15) G ( C ′ ) η G ( C ′ ) (cid:15) (cid:15) GF [ A, G ( C )] GF ( φ ) / / GF ( G ( C ′ ))From this diagram we conclude that the condition of η [ A,G ( C )] being an isomorphismis equivalent to the condition of η G ( C ′ ) of being an isomorphism. Thus, since G isfully faithful we have by Proposition 6.5.1 that η ∗ G is always an isomorphismwhich implies that η [ A,G ( C )] is an isomorphism as well. Therefore, the adjunction ismonoidal if and only if condition (18) is satisfied. HAPTER 6. BACKGROUND MATERIAL ON PRESHEAF CATEGORIES G is an inclusion this translates to the condition thatthere exists an isomorphism C ( − ⊗ B ) ∼ = C ′ where C ∈ C ⊆ [ B op , Set ], B ∈ B forsome C ′ ∈ C . Remark 6.11.1.
Consider C = [ B op , Set ] inf . Suppose we have two functors F and H isomorphic in [ B op , Set ]. Then F preserves limits if and only if H preserves limits.Therefore the condition C ( − ⊗ B ) ∼ = C ′ ∈ C implies that C ( − ⊗ B ) preserves limits,i.e., C ( − ⊗ B ) ∈ [ B op , Set ] inf . We have by hypothesis that C ∈ C and hence itdepends on whether the functor − ⊗ B : B op → B op preserves limits. The same isvalid if we consider not all but some specific limits: a certain class Γ. hapter 7Presheaf models of a quantumlambda calculus In this chapter we study a categorical model for the quantum lambda calculus ofSelinger and Valiron [67]. We focus on exploring the existence of such a model usingpresheaf categories.In [63], Selinger defined an elementary quantum flow chart language and gave adenotational model in terms of superoperators. This axiomatic framework capturesthe behavior and interconnection between the basic quantum computation conceptssuch as the manipulation of quantum bits by considering two basic operations: mea-surement and unitary transformation in a lower-level language. In particular, thesemantics of this framework is very well understood: each program corresponds to aconcrete superoperator.Higher-order functions are functions that can input or output other functions.In order to deal with higher-order functions, Selinger and Valiron introduced, inseveral papers [68], [69], [70] a typed lambda calculus for quantum computation andinvestigated several aspects of its semantics. In this context, they combined two verywell-established languages in the literature of computer science: the intuitionisticfragment of Girard’s linear logic [26] and the computational monads introduced byMoggi in [56].The type system of Selinger and Valiron’s quantum lambda calculus is based on160
HAPTER 7. PRESHEAF MODELS A can only be used once, whereas a value of type ! A can becopied and used multiple times. The impossibility of copying quantum informationis one of the fundamental differences between quantum information and classicalinformation, and is known as the no-cloning property . From the logical perspective,it therefore seems natural to relate quantum computation and linear logic. Note thatthe operator “!” satisfies the properties of a comonad .Since we have higher-order functions, as well as probabilistic operations (namelyquantum measurement), the language needs to address the question of evaluationstrategies. Otherwise, in some concrete situation, it would be impossible to give acoherent outcome every time for identical circumstances. In order to deal with thisissue, Selinger and Valiron chose to incorporate a methodology `a la Moggi by makingthe distinction between values and computations. Moggi [56] proposed the notion ofa monad as an appropriate tool for interpreting computational behavior. At the levelof the denotational model, this will be reflected by a strong monad.To summarize, let us say that the exponential operator ! will be modelled bya monoidal comonad arising from an adjunction between a cartesian category (ac-counting for classical duplicability) and a symmetric monoidal category (accountingfor quantum non-duplicability) while the manipulation of the probabilistic aspect ofthe quantum computation is handled by a monoidal monad. The result of combin-ing these two methodologies is what Selinger and Valiron call a linear category forduplication .This is not the first time that this interaction between a monad and a comonadhas been invoked in order to express denotational aspects of a system in computerscience (see [10] for example). But what is new in Selinger and Valiron’s work, isputting this interaction in the context of quantum computation.In this thesis, we will focus exclusively on the categorical aspects of the modelconstruction. Thus, we will not review the syntax of the quantum lambda calculusitself. Instead, we will take as our starting point Selinger and Valiron’s definition of
HAPTER 7. PRESHEAF MODELS categorical model of the quantum lambda calculus [70]. It was already proven in [70]that the quantum lambda calculus forms an internal language for the class of suchmodels. This is similar to the well-known interplay between typed lambda calculusand cartesian closed categories [52]. What was left open in [70] was the constructionof a concrete such model (other than that given by the syntax itself). This is thequestion we answer here.The use of category theory to model and to explain formal languages has anestablished tradition in logic, but in quantum computation it constitutes a relativelyrecent trend. We finish this introduction by stressing that the field of quantumcomputation in connection with category theory is fast-growing. The ability to createbridges among these different branches of mathematics that are apparently far fromone another is one of the motivating goals of this thesis and we hope to contribute inthis direction.
In the introduction we informally described the main ideas and motivation of whatshould be a categorical model for quantum lambda calculus. Here we shall take theformal definition in [70] as our starting point. However, before presenting it, we willgive some preliminary definitions and we shall make some remarks about how tosimplify its presentation. Several of the definitions sketched here will be made moreprecise in Section 7.3 and beyond.Let ( C , ⊗ , I, α, ρ, λ, σ ) be a symmetric monoidal category. Definition 7.1.1.
A symmetric monoidal comonad (! , δ, ε, m A,B , m I ) is a comonad(! , δ, ε ) where the functor ! is a monoidal functor (! , m A,B , m I ), i.e., with naturaltransformations m A,B : ! A ⊗ ! B → !( A ⊗ B ) and m I : I → ! I satisfying the coher-ence axioms of Definition 2.2.4, such that δ and ε are symmetric monoidal naturaltransformations. HAPTER 7. PRESHEAF MODELS
Definition 7.1.2. A linear exponential comonad is a symmetric monoidal comonad(! , δ, ε, m A,B , m I ) in which the following conditions hold:- for every A ∈ C there exists a commutative comonoid, with d A :! A → ! A ⊗ ! A and e A :! A → I as associated maps,- d A and e A are monoidal natural transformation with respect to the naturaltransformations m ,- d A and e A are coalgebra morphisms when we consider (! A, δ A ), (! A ⊗ ! A, m ! A, ! A ◦ ( δ A ⊗ δ A )), and ( I, m I ) as coalgebras,- the maps δ A : (! A, e A , d A ) → (!! A, e ! A , d ! A ) are comonoid morphisms. Definition 7.1.3.
Let (
T, η, µ ) be a strong monad. We say that C has Kleisli expo-nentials if there exists a functor [ − , − ] k : C op × C → C and a natural isomorphism: C ( A ⊗ B, T C ) ∼ = C ( A, [ B, C ] k )) Remark 7.1.4.
When the category ( C , ⊗ , [ − , − ]) is a monoidal closed category thenit certainly has Kleisli exponentials just by putting [ B, C ] k = [ B, T C ]. Definition 7.1.5 (Linear category for duplication [70]) . A linear category for du-plication consists of a symmetric monoidal category ( C , ⊗ , I ) satisfying the followingdata:- an idempotent, strongly monoidal, linear exponential comonad (! , δ, ε, d, e ),- a strong monad ( T, µ, η, t ),- C has Kleisli exponentials.Further, if the unit I is a terminal object we shall speak of an affine linear categoryfor duplication , cf. Definition 2.5.1. Remark 7.1.6.
The definition of a linear category for duplication (Definition 7.1.5)is equivalent to the existence of a pair of monoidal adjunctions ([9], [55] and [49]):( B , × , ( L,l ) / / ( C , ⊗ , I ) ( F,m ) / / ( I,i ) ⊥ o o ( D , ⊗ , I ) ( G,n ) ⊥ o o HAPTER 7. PRESHEAF MODELS B has finite products and C and D are symmetric monoidal closedcategories. The monoidal adjoint pair of functors on the left represents a linear-non-linear model in the sense of Benton [9] in which we obtain a monoidal comonad by! = L ◦ I . The monoidal adjoint on the right gives rise to T = G ◦ F a strong monadin the sense of Kock [48], [49] which is also a computational monad in the sense ofMoggi [56].We now state the main definition of a model of the quantum lambda calculus. Definition 7.1.7 (Model of the quantum lambda calculus [70]) . An abstract model ofthe quantum lambda calculus is an affine linear category for duplication C with finitecoproducts, preserved by the comonad !. Moreover, a concrete model of the quantumlambda calculus is an abstract model of the quantum lambda calculus such that thereexists a full and faithful embedding Q ֒ → C T , preserving tensor ⊗ and coproduct ⊕ ,from the category Q of norm non-increasing superoperators (see Definition 3.2.4) intothe Kleisli category generated by the monad T . Remark 7.1.8.
To make the connection to quantum lambda calculus: the category C ,the Kleisli category C T , and the co-Kleisli category C ! all have the same objects, whichcorrespond to types of the quantum lambda calculus. The morphism f : A → B of C correspond to values of type B (parameterized by variables of type A ). A morphism f : A → B in C T , which is really a morphism f : A → T B in C , corresponds toa computation of type B (roughly, a probability distribution of values). Finally, amorphism f : A → B in C ! , which is really a morphism f : ! A → B in C , correspondsto a classical value of type B , i.e., one which only depends on classical variables. Theidempotence of “!” implies that morphisms ! A → B are in one-to-one correspondencewith morphisms ! A → ! B , i.e., classical values are duplicable. For details, see [70]. Our complete process for obtaining a categorical model of the quantum lambda cal-culus consists of two stages. In the first stage, we will construct abstract models of
HAPTER 7. PRESHEAF MODELS
B → C → D . This construction is very general, and the basecategories B , C , and D can be viewed as parameters. We will identify the precise con-ditions required of the base categories (and the functors connecting them) in orderto obtain a valid abstract model. This is the content of Chapter 7.In the second stage, we will construct a concrete model of the quantum lambdacalculus by identifying particular base categories so that the remaining conditions ofDefinition 7.1.7 are satisfied. This is the content of Chapter 8.We briefly outline the main steps of the construction; full details will be given inlater sections. • The basic idea of the construction is to lift a sequence of functors B Φ → C Ψ → D into a pair of adjunctions between presheaf categories[ B op , Set ] L / / [ C op , Set ] F / / Φ ∗ ⊥ o o [ D op , Set ] Ψ ∗ ⊥ o o Here, Φ ∗ and Ψ ∗ are the precomposition functors, and L and F are their leftKan extensions. By Remark 7.1.6, such a pair of adjunctions potentially yieldsa linear category for duplication, and therefore, with additional conditions, anabstract model of quantum computation. Our goal is to identify the particularconditions on B , C , D , Φ, and Ψ, that make this construction work correctly. • By Day’s construction, the requirement that [ C op , Set ] and [ D op , Set ] aremonoidal closed can be achieved by requiring C and D to be monoidal. Therequirement that the adjunctions L ⊣ Φ ∗ and F ⊣ Ψ ∗ are monoidal is directlyrelated to the fact that the functors Ψ and Φ are strong monoidal. More pre-cisely, this implies that the left Kan extension is a strong monoidal functorwhich in turn determines the enrichment of the adjunction. We also note thatthe category B must be cartesian. HAPTER 7. PRESHEAF MODELS • One important complication with the model, as discussed so far, is the following.The Yoneda embedding Y : D → [ D op , Set ] is full and faithful, and by Day’sresult, also preserves the monoidal structure ⊗ . Therefore, if one takes D = Q , all but one of the conditions of a concrete model (from Definition 7.1.7)are automatically satisfied. Unfortunately, the Yoneda embedding does notpreserve coproducts, and therefore the remaining condition of Definition 7.1.7fails. For this reason, we modify the construction and use the modified presheafcategory and coproduct-preserving Yoneda embedding from Section 6.9. Ouradjunctions, and the associated Yoneda embeddings, now look like this:[ B op , Set ] L ⊣ Φ ∗ / / [ C op , Set ] F ⊣ G / / [ D op , Set ] Γ B Y O O Φ / / C Y O O Ψ / / D Y Γ O O The second pair of adjoint functors F ⊣ G is generated by the composition oftwo adjunctions:[ C op , Set ] F / / [ D op , Set ] F / / Ψ ∗ ⊥ o o [ D op , Set ] Γ G ⊥ o o Here, the pair of functors F ⊣ G arises as a reflection of [ Q op , Set ] Γ in[ Q op , Set ], and depends on a choice of a certain class Γ of cones. The structuralaspects of the modified Yoneda embedding Q → [ Q op , Set ] Γ depend cruciallyon general properties of the functor categories, which go back to the studyof continuous functors by Lambek (see Section 6.6) and Freyd and Kelly (seeSection 6.9).But, as we mentioned before, at the same time we still require that the reflec-tion functor remain strongly monoidal. Here will will use Day’s results (seeSection 6.10) on the conditions that are needed for the reflection to be strongmonoidal, by inducing a monoidal structure from the category [ Q op , Set ] intoits subcategory [ Q op , Set ] Γ (see Section 6.10). In particular, this induces aconstraint on the choice of Γ considered above: all the cones considered in Γ HAPTER 7. PRESHEAF MODELS D (seeRemark 6.11.1). • Notice that the above adjunctions are examples of what in topos theory is namedan essential geometric morphism, in which both functors are left adjoint to someother two functors: L ⊣ Φ ∗ ⊣ Φ ∗ . Therefore, this shows that the comonad “!”obtained will preserve finite coproducts. • The condition for the comonad “!” to be idempotent turns out to depend onthe fact that the functor Φ is full and faithful. • In addition to the requirement that “!” preserves coproducts, we also need “!” topreserve the tensor, i.e., to be strongly monoidal, as required in Definition 7.1.7.This property is unusual for models of intuitionistic linear logic and puts somerestriction on the range of possible choices we have for the category C . In brief,since the left Kan extension along Φ is a strong monoidal functor we find thata concrete condition in the category C is necessary to ensure that this propertyholds when we lift the functor Φ to the category of presheaves; see Section 7.6. • Once we have constructed this categorical environment our next task is to trans-late these properties to the Kleisli category. To achieve this we use the compari-son Kleisli functor for passing from the framework we have already established tothe Kleisli monoidal adjoint pair of functors. Also, at the same time in this con-text, we shall find it convenient to characterize the functor H : D → [ C op , Set ] T as a strong monoidal functor.All of the above steps yield an abstract model of quantum computation, para-metric on the sequence of functors B → C → D . • Finally, as we shall see in Section 8.2, we will identify specific categories B , C ,and D that yield a concrete model of quantum computation. We let D = Q ,the category of superoperators. The categories B and C must be chosen in sucha way as to satisfy all of the properties outlined above. For B , we take thecategory of finite sets. HAPTER 7. PRESHEAF MODELS C is more tricky. For example,here are two of the requirements directly concerning the semantics: C must beaffine monoidal and must satisfy the condition of equation (19) in Section 7.6.In a series of intermediate steps, with the help of some universal constructions,we introduce a category C = Q ′′ related to the category Q of superoperators.As we have noted, the category Q ′′ plays a central role in our construction. It isin some sense the “barycenter” of our model. While the basic structural propertiesoccur at the level of the functor categories, providing a general mathematical setting,the development of the concrete quantum meaning of the model occurs mostly at thisbase level. The first definition of a categorical model of linear logic was given by Seely [62]. Otherpioneering studies in this area were Lafont’s thesis [50] and Abramsky’s paper [1].Also, Melli`es’ survey [55] is an excellent introduction to the topic.Now we formulate Bierman’s definition of linear category [14] which is based upon theabove-mentioned previous work on the Topic. We also state an equivalent alternativesimplified version that we take from Benton [9] (this is the notion we outlined inRemark 7.1.6). For the purpose of this thesis, since it is clear that the linear fragmentof Definition 7.1.7 does not impose any constraints on the rest of the definition,it follows that it will be more helpful to work with Benton’s version representingthe underlying linear fragment. In any case, to appreciate the details behind thesecategorical models, Bierman’s definition will occupy the rest of the present section.
Definition 7.3.1 (Bierman) . A linear category C consists of a symmetricmonoidal closed category ( C, I, ⊗ , ⊸ , α, λ, ρ, γ ) with a symmetric monoidal comonad(! , ε, δ, m I , m A,B ) defined on C and monoidal natural transformations e :!( − ) → I , d : !( − ) → !( − ) ⊗ !( − ) such that:- e A : !( A ) → I , d A : !( A ) → !( A ) ⊗ !( A ) are coalgebra morphisms for each A ; HAPTER 7. PRESHEAF MODELS
A, δ A ) , e A , d A ) is a commutative comonoid for every free coalgebra (! A, δ A )and- morphisms between free coalgebras f : (! A, δ A ) → (! B, δ B ) are also comonoidcommutative morphisms.We will now consider the meaning of each of these conditions:- for every A ∈ C there exists a commutative comonoid, with d A :! A → ! A ⊗ ! A and e A :! A → I as associated maps. This means the following:The assumption that ((! A, δ A ) , e A , d A ) is a commutative comonoid for every freecoalgebra (! A, δ A ) means that:! A d A / / d A (cid:15) (cid:15) ! A ⊗ ! A d A ⊗ ! A (cid:15) (cid:15) ! A ⊗ ! A ! A ⊗ d A / / ! A ⊗ (! A ⊗ ! A ) α / / (! A ⊗ ! A ) ⊗ ! A ! A ! A ! A / / d A (cid:15) (cid:15) ! A o o ! AI ⊗ ! A λ ! A O O ! A ⊗ ! A ! A ⊗ e A / / e A ⊗ ! A o o ! A ⊗ I ρ ! A O O ! A d A / / d A ●●●●●●●●● ! A ⊗ ! A γ ! A, ! A (cid:15) (cid:15) ! A ⊗ ! A - d A and e A are monoidal natural transformation with respect to the naturaltransformation m .The transformations e :!( − ) → I and d :!( − ) → !( − ) ⊗ ( − ) are monoidalnatural transformations between monoidal functors; if f : A → B then e : (! , m A,B , m I ) → ( I, λ I , I ) is the statement that the following diagramscommute: ! A ! f / / e A ! ! ❇❇❇❇❇❇❇❇❇ ! B e B (cid:15) (cid:15) I HAPTER 7. PRESHEAF MODELS A ⊗ ! B m A,B / / e A ⊗ e B (cid:15) (cid:15) !( A ⊗ B ) e A ⊗ B (cid:15) (cid:15) I ⊗ I λ I / / I I m I / / id I (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ ! I e I (cid:15) (cid:15) I and d : (! , m A,B , m I ) → (! ⊗ ! , t A,B , t I ):! A d A / / ! f (cid:15) (cid:15) ! A ⊗ ! A ! f ⊗ ! f (cid:15) (cid:15) ! B d B / / ! B ⊗ ! B ! A ⊗ ! B m A,B / / d A ⊗ d B (cid:15) (cid:15) !( A ⊗ B ) d A ⊗ B (cid:15) (cid:15) ! A ⊗ ! A ⊗ ! B ⊗ ! B Id ! A ⊗ γ ! A, ! B ⊗ Id ! B / / ! A ⊗ ! B ⊗ ! A ⊗ ! B m A,B ⊗ m A,B / / !( A ⊗ B ) ⊗ !( A ⊗ B )with t A,B = ( m A,B ⊗ m A,B ) ◦ Id ! A ⊗ γ ! A, ! B ⊗ Id ! B and I m I / / λ − I (cid:15) (cid:15) ! I d I (cid:15) (cid:15) I ⊗ I m I ⊗ m I / / ! I ⊗ ! I with t I = ( m I ⊗ m I ) ◦ λ − I .- d A and e A are coalgebra morphisms when we consider (! A, δ A ), (! A ⊗ ! A, m ! A, ! A ◦ ( δ A ⊗ δ A )), and ( I, m I ) as coalgebras:The definition of linear category characterizes e A : (! A, δ A ) → ( I, m I ) and d A : (! A, δ A ) → (! A ⊗ ! A, m ! A, ! A ◦ ( δ A ⊗ δ A )) as coalgebra morphisms which meansthat the following diagrams commute: HAPTER 7. PRESHEAF MODELS A δ A / / e A (cid:15) (cid:15) !! A ! e A (cid:15) (cid:15) I m I / / ! I ! A δ A / / d A (cid:15) (cid:15) !! A ! d A (cid:15) (cid:15) ! A ⊗ ! A m ! A, ! A ◦ ( δ A ⊗ δ A ) / / !(! A ⊗ ! A )- Morphisms between free coalgebras f : (! A, δ A ) → (! B, δ B ) are also comonoidcommutative morphisms. This means that if f :! A → ! B is an arrow with! f ◦ δ A = δ B ◦ f then is also true that f : (! A, d A , e A ) → (! B, d B , e B ) is a mapbetween commutative comonoids that is f :! A → ! B is an arrow that satisfies:! A d A / / f (cid:15) (cid:15) ! A ⊗ ! A f ⊗ f (cid:15) (cid:15) ! B d B / / ! B ⊗ ! B ! A f / / e A ! ! ❇❇❇❇❇❇❇❇❇ ! B e B (cid:15) (cid:15) I To complete the list of conditions let us show the structural conditions. Thenatural transformations ε :!( − ) → I and δ :!( − ) → !!( − ) are monoidal. If (! , m A,B , m I )and ( Id, A ⊗ B , I ) are monoidal functors then ε : (! , m A,B , m I ) → ( Id, A ⊗ B , I ) is amonoidal natural transformation which is compatible in the sense that the followingdiagrams commute:! A ⊗ ! B m A,B / / ε A ⊗ ε B (cid:15) (cid:15) !( A ⊗ B ) ε A ⊗ B (cid:15) (cid:15) A ⊗ B / / A ⊗ B I m I / / (cid:31) (cid:31) ❄❄❄❄❄❄❄❄ ! I ε I (cid:15) (cid:15) I Also δ : (! , m A,B , m I ) → (!! , t A,B , t I ) is a monoidal natural transformation betweenmonoidal functors; with t A,B =!( m A,B ) ◦ m ! A, ! B and t I =!( m I ) ◦ m I :! A ⊗ ! B m A,B / / δ A ⊗ δ B (cid:15) (cid:15) !( A ⊗ B ) δ A ⊗ B (cid:15) (cid:15) !! A ⊗ !! B m ! A, ! B / / !(! A ⊗ ! B ) !( m A,B ) / / !!( A ⊗ B ) HAPTER 7. PRESHEAF MODELS I m I / / t I (cid:31) (cid:31) ❅❅❅❅❅❅❅❅❅❅ ! I δ I (cid:15) (cid:15) !! I Recalling that a symmetric monoidal comonad (! , ε, δ, m
A,B , m I ) is a comonad(! , ε, δ ) equipped with a symmetrical monoidal functor (! , m A,B , m I ), where: ! : C → C is a functor, for every object A and B there is a morphism m A,B :! A ⊗ ! B → !( A ⊗ B )natural in A and B , for the unit I there is a morphism m I : I → ! I .These morphisms with the structural maps α, λ, ρ, γ must make the followingdiagrams commute:! A ⊗ (! B ⊗ ! C ) id ! A ⊗ m B,C / / α (cid:15) (cid:15) ! A ⊗ !( B ⊗ C ) m A,B ⊗ C / / !( A ⊗ ( B ⊗ C ) ! α (cid:15) (cid:15) (! A ⊗ ! B ) ⊗ ! C m A,B ⊗ id ! C / / !( A ⊗ B ) ⊗ ! C m A ⊗ B,C / / !(( A ⊗ B ) ⊗ C )! B ⊗ I ρ ! B / / id ! B ⊗ m I (cid:15) (cid:15) ! B ! B ⊗ ! I m B,I / / !( B ⊗ I ) !( ρ B ) O O I ⊗ ! B λ ! B / / m I ⊗ id ! B (cid:15) (cid:15) ! B ! I ⊗ ! B m I,B / / !( I ⊗ B ) !( λ B ) O O ! A ⊗ ! B γ ! A, ! B / / m A,B (cid:15) (cid:15) ! B ⊗ ! A m B,A (cid:15) (cid:15) !( A ⊗ B ) !( γ A,B ) / / !( B ⊗ A ) Definition 7.3.2 (Benton [9]) . A linear-non-linear category consists of:(1) a symmetric monoidal closed category ( C , ⊗ , I, ⊸ )(2) a category ( B , × ,
1) with finite product(3) a symmetric monoidal adjunction:( B , × , ( F,m ) / / ( C , ⊗ , I ) ( G,n ) ⊥ o o . HAPTER 7. PRESHEAF MODELS
Proposition 7.3.3.
Every linear-non-linear category gives rise to a linear category.Every linear category defines a linear-non-linear category, where ( B , × , is the cat-egory of coalgebras of the comonad (! , ε, δ ) .Proof. See [9] or [55].
Remark 7.3.4.
Kelly’s characterization of monoidal adjunctions (see Proposi-tion 2.3.6) allows us to replace condition (3) in the definition of linear-non-linearcategories by the following new statement in Definition 7.3.2:(3’) an adjunction: ( B , × , F / / ( C , ⊗ , I ) G ⊥ o o and there exist isomorphisms m A,B : F A ⊗ F B → F ( A × B ) , m I : I → F (1)making ( F, m
A,B , m I ) : ( B , × , → ( C , ⊗ , I ) a strong symmetric monoidal func-tor. Our purpose here is to characterize Benton’s linear-non-linear models of intuitionis-tic linear logic, in the sense of Definition 7.3.2, on presheaf categories using Day’smonoidal structure from Section 6.8. This is an application of monoidal enrichmentof the Kan extension see [22]. We use Kelly’s equivalent formulation of monoidaladjunctions from Proposition 2.3.6.
Proposition 7.4.1.
Suppose we have a strong monoidal functor
Φ : ( A , × , → ( B , ⊗ , I ) from a cartesian category to a monoidal category, i.e., we have a naturalisomorphism Φ( a ) ⊗ Φ( b ) ∼ = Φ( a × b ) and I ∼ = Φ(1) . HAPTER 7. PRESHEAF MODELS
Let us consider the left Kan extension along Φ in the functor category [ B op , Set ] where the copower is product on sets: Lan Φ ( F ) = Z a B ( − , Φ( a )) × F ( a ) Then
Lan Φ is strong monoidal.Proof. By the Yoneda Lemma, the strong functor Φ, Fubini and coend properties:
Lan Φ ( F × G ) = Lan Φ ( R a A ( − , a ) × F ( a ) × R b A ( − , b ) × G ( b )) by the Yoneda Lemmaand pointwise product= Lan Φ ( R a b A ( − , a ) × A ( − , b ) × F ( a ) × G ( b )) = Lan Φ ( R a b A ( − , a × b ) × F ( a ) × G ( b )) cartesian product= R c B ( − , Φ( c )) × R a b A ( c, a × b ) × F ( a ) × G ( b ) = R a b ( R c B ( − , Φ( c )) × A ( c, a × b )) × F ( a ) × G ( b ) definition of Kan extension= R a b B ( − , Φ( a × b )) × F ( a ) × G ( b ) = R a b B ( − , Φ( a ) ⊗ Φ( b )) × F ( a ) × G ( b ) Φstrong functor= R a b ( R y B ( y, Φ( a )) × B ( − , y ⊗ Φ( b ))) × F ( a ) × G ( b ) = R a b R y B ( y, Φ( a )) × ( R z B ( − , y ⊗ z ) × B ( z, Φ( b ))) × F ( a ) × G ( b ) by the Yoneda Lemma= R y z ( R a B ( y, Φ( a )) × F ( a )) × ( R b B ( z, Φ( b )) × G ( b )) × B ( − , y ⊗ z ) = R y z (( Lan Φ ( F ))( y )) × (( Lan Φ ( G ))( z )) × B ( − , y ⊗ z ) by Fubini and copower preservescolimits= Lan Φ ( F ) ⊗ D Lan Φ ( G ) by definition of Kan extension and convolutionand also the units: Lan Φ ( I A D ) = Lan Φ ( A ( − , R a B ( − , Φ( a )) × A ( a,
1) = B ( − , Φ(1)) = B ( − , I ) = I B D . Remark 7.4.2.
Note that, in view of the line of arguments used above, the casewhere A is monoidal has the same proof, i.e., if we have Φ( a ) ⊗ Φ( b ) ∼ = Φ( a ⊗ b ) and I ∼ = Φ( I ′ ) we start directly from the convolution product: Lan Φ ( F ⊗ D G ) = Lan Φ ( Z a b A ( − , a ⊗ b ) × F ( a ) × G ( b ))and we repeat the same proof. Also notice that when we have a product in A theconvolution is a pointwise product of functors: F × G = Z a b A ( − , a × b ) × F ( a ) × G ( b )) . HAPTER 7. PRESHEAF MODELS
Remark 7.4.3.
If the unit of a monoidal category C is a terminal object then theunit of the convolution is also terminal. Let us consider a morphism α : F → C ( − , I )in the functor category [ C op , Set ]. Then for every V there is only one way to definethe map α V : F ( V ) → C ( V, I ) which is α V ( x ) = ! for every x ∈ F ( V ) in the categoryof sets. Hence there is a unique α . Therefore it is a terminal object in the functorcategory. A comonad (! , ǫ, δ ) is said to be idempotent if δ : ! ⇒ !! is an isomorphism. Let (! , ǫ, δ )be the comonad generated by the adjunction:( D , × , F / / ( V , ⊗ , I, ⊸ ) G ⊥ o o then δ = F η G with η : A → GF A . Thus if η is an isomorphism then δ is also anisomorphism. Now consider the unit of the Kan extension: G ⇒ F ∗ ( Lan F ( G )) . It is given by: G ( a ) i F ( a ) / / ( η G )( a ) & & ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ B ( F ( a ) , F ( a )) × G ( a ) ( w a ) F ( a ) (cid:15) (cid:15) R a ′ B ( F ( a ′ ) , F ( a )) × G ( a ′ ) , where i is the injection of the copower and w is the wedge of the coend. Proposition 7.5.1. If F is a full and faithful functor then η G : G ⇒ F ∗ ( Lan F ( G )) is an isomorphism.Proof. [15] HAPTER 7. PRESHEAF MODELS
In this section we study conditions that allow us to force the idempotent comonad tobe a strong monoidal functor. This property, part of the model we are building, is amain difference with other previously intuitionistic linear models. In order to achievethis, consider a full and faithful functor Φ :
A → B as in Proposition 7.5.1. Let Φ ∗ be the functor we had seen earlier in Section 6.7:[ B op , Set ] Φ ∗ −→ [ A op , Set ] , i.e., the right adjoint of the left Kan extension. Lemma 7.6.1.
If there exists a natural isomorphism: B (Φ( a ) , b ) × B (Φ( a ) , b ′ ) ∼ = B (Φ( a ) , b ⊗ b ′ ) (19) where a ∈ A and b, b ′ ∈ B and Φ is a fully faithful, strong monoidal functor then Φ ∗ is a strong monoidal functor.Proof. To see this: Φ ∗ ( F ) × Φ ∗ ( G ) = F (Φ( − )) × G (Φ( − )) ∼ = R b F ( b ) × B (Φ( − ) , b ) × R b ′ G ( b ′ ) × B (Φ( − ) , b ′ ) ∼ = by the Yoneda Lemma, definition of Φ ∗ and the fact thatconvolution in [ A op , Set ] is pointwise cartesian product ∼ = R b b ′ F ( b ) × G ( b ′ ) ×B (Φ( − ) , b ) ×B (Φ( − ) , b ′ ) ∼ = by properties of coends (preser-vation) ∼ = R b b ′ F ( b ) × G ( b ′ ) × B (Φ( − ) , b ⊗ b ′ ) = by hypothesis (19) and Lemma 6.3.6= ( F ⊗ G )(Φ( − )) = Φ ∗ ( F ⊗ G ) by definition of convolution in [ B op , Set ] anddefinition of Φ ∗ .Moreover the units are isomorphic,Φ ∗ ( B ( − , I )) = B (Φ( − ) , I ) ∼ = by definition of Φ ∗ ∼ = B (Φ( − ) , Φ(1)) since Φ is strong A ( − ,
1) since Φ is fully faithful.
Remark 7.6.2.
At this point it is useful to mention that the conditions ofLemma 7.6.1 are an example of a multiplicative kernel K : B × A op → Set from
HAPTER 7. PRESHEAF MODELS B to A in the sense of [21]. In fact, K is explicitly defined as K ( b, a ) = B (Φ( a ) , b ) satisfying the two following equations as part of the definition: Z yz K ( a, y ) × K ( b, z ) × A ( x, y × z ) ∼ = Z c K ( c, x ) × B ( c, a ⊗ b ) Z b B (Φ( x ) , b ) × B ( b, I ) ∼ = A ( x, Q ′′ which is a modification of the category Q ofsuperoperators. Also we consider the functor Φ of Section 2.5 where C + = Q ′′ . C has finite coproducts then C T has finitecoproducts An important property of the Kleisli construction is that if we assume that the originalcategory has finite coproducts then we can define finite coproducts in the Kleislicategory.
Proposition 7.7.1.
Kleisli categories inherit coproducts, i.e., if C has finite coprod-ucts then C T also has finite coproducts.Proof. Suppose we have that A f K → C and B g K → C two arrows in the category C T . Wetake A ⊕ K B = A ⊕ B on objects, and A i A " " ❊❊❊❊❊❊❊❊❊ i TA / / T ( A ⊕ B ) B i B " " ❋❋❋❋❋❋❋❋❋ i TB / / T ( A ⊕ B ) A ⊕ B η A ⊕ B qqqqqqqqqq A ⊕ B η A ⊕ B qqqqqqqqqq as injections in the category C T . HAPTER 7. PRESHEAF MODELS A ⊕ K B [ f K ,g K ] K −→ C such that f K = [ f K , g K ] K ◦ K i TA and g K = [ f K , g K ] K ◦ K i TB commute. This is verified by the following diagram: A f ' ' ❖❖❖❖❖❖❖❖❖❖❖❖❖❖❖ i A / / A ⊕ B [ f,g ] (cid:15) (cid:15) η A ⊕ B / / T ( A ⊕ B ) T [ f,g ] (cid:15) (cid:15) A ⊕ B [ f,g ] (cid:15) (cid:15) η A ⊕ B o o B g w w ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ i B o o T C TC ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ η TC / / T C µ C (cid:15) (cid:15) T C η TC o o TC u u ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ T C where [ f, g ] is the unique morphism that defines coproduct in C . This last diagramcommutes by naturality of η with respect to [ f, g ] and by definition of monad.Uniqueness follows from the following reasoning: suppose there is an arrow A ⊕ K B h K −→ C , i.e., A ⊕ B h −→ T C , such that µ C ◦ T ( h ) ◦ η A ⊕ B ◦ i A = f and µ C ◦ T ( h ) ◦ η A ⊕ B ◦ i B = g then by naturality and monad definition we have that h ◦ i A = f and h ◦ i A = g , thus by uniqueness in C we have that h = [ f, g ].We notice that C : C T → D preserves finite coproducts. To see this, by definitionwe have that i TA = η A ⊕ B ◦ i A and i TB = η A ⊕ B ◦ i B . Then C ( i TA ) = c ( η A ⊕ B ◦ i A ) = ε F ( A ⊕ B ) ◦ F ( η A ⊕ B ◦ i A ) == ε F ( A ⊕ B ) ◦ F ( η A ⊕ B ) ◦ F ( i A ) = 1 A ⊕ B ◦ F ( i A ) = F ( i A ) . In the same way C ( i TB ) = F ( i B ).Given that right adjoint preserves coproducts then C ( A ⊕ K B ) = C ( A ⊕ C B ) = F ( A ⊕ K B ) = F A ⊕ D F B and C ( A i TA −→ A ⊕ K B ) = C ( A ) C ( i TA ) −→ C ( A ⊕ K B ) = F ( A ) F ( i A ) −→ F ( A ) ⊕ D F ( B )which is a coproduct in D .In the same way we can apply a similar reasoning with B . H : D → ˆ C T Let C and D be categories, and let ˆ C and T = G ◦ F be defined as in Section 7.2. HAPTER 7. PRESHEAF MODELS C op , Set ] F / / [ D op , Set ] Γ G ⊥ o o In this section we consider the construction of a coproduct and tensor preservingfunctor H : D → ˆ C T with properties similar to the Yoneda embedding. We investigatethe role of a general category D fully embedded into a Kleisli category ˆ C T . Certainproperties of this functor are introduced in order to apply this to the category ofsuperoperators Q as well as to develop a methodology for obtaining higher-ordermodels in the sense of Section 7.1.Let F ⊣ G and F ⊣ G be two monoidal adjoint pairs with associated naturaltransformations ( F , m ), ( G , n ) and ( F , m ), ( G , n ). We shall use the followingnotation F = F ◦ F , G = G ◦ G , T = G ◦ F . We now describe a typical situationof this kind generated by a functor Φ : C → D .Let us consider F = Lan Φ and G = Φ ∗ . With some co-completeness conditionassumed, we can express F ( A ) = R c D ( − , Φ( c )) ⊗ A ( c ) and G = Φ ∗ .On the other hand we consider D Y Γ " " ❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋❋ Y / / [ D op , Set ] F (cid:15) (cid:15) [ D op , Set ] Γ G ⊢ O O where we take F = Lan Y ( Y Γ ) : [ D op , Set ] → [ D op , Set ] Γ , and Y Γ : D → [ D op , Set ] Γ is given by Y Γ ( d ) = D ( − , d ). Thus we have that F ( D ) = D ⋆ Y Γ = R d D ( d ) ⊗ Y Γ ( d ).Assuming that [ D op , Set ] Γ is co-complete and contains the representablepresheaves then the right adjoint is given by G ( F ) = [ D op , Set ] Γ ( Y Γ − , F ) = [ D op , Set ]( Y − , F ) ∼ = F since it is a full subcategory and by the Yoneda Lemma. Therefore we consider G as the inclusion functor up to isomorphism. HAPTER 7. PRESHEAF MODELS H . We want to study the following situation:ˆ C F T (cid:15) (cid:15) F / / ˆ D G ⊥ o o F / / ˆ D Γ G ⊥ o o C Φ & & Y A A ˆ C TG T ⊢ O O C ♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ D H O O ✤✤✤ Y Γ ? ? The goal is to determine a fully faithful functor, H in this diagram, that preservestensor and coproduct.First, notice that the perimeter of this diagram commutes on objects: F ( C ( − , c )) = Z c ′ D ( − , Φ( c ′ )) ⊗ C ( c ′ , c ) = D ( − , Φ( c ))When we evaluate again we obtain: F ( D ( − , Φ( c ))) = Z d ′ D ( d ′ , Φ( c )) ⊗ Y Γ ( d ′ ) = Y Γ (Φ( c )) = D ( − , Φ( c ))Summing up we have that F ( C ( − , c )) = D ( − , Φ( c )) up to isomorphism.Suppose now that Φ is onto on objects. We have that: D ( − , d ) = D ( − , Φ( c ))for some c ∈ C , i.e., we can make a choice, for every d ∈ |D| , of some c ∈ |C| such thatΦ( c ) ∼ = d . Let us call this choice a “choice of preimages”. We can therefore define amap H : |D| → | ˆ C T | by H ( d ) = C ( − , c ) on objects.Hence, we can define a functor H : D → ˆ C T in the following way:let d f → d ′ be an arrow in the category D , then we apply Y Γ obtaining D ( − , d ) Y Γ ( f ) →D ( − , d ′ ). This arrow is equal to D ( − , Φ( c )) Y Γ ( f ) → D ( − , Φ( c ′ )) for some c, c ′ ∈ C and HAPTER 7. PRESHEAF MODELS F ( C ( − , c )) Y Γ ( f ) → F ( C ( − , c ′ )). Now we usethe fact that the comparison functor C : ˆ C T → ˆ D Γ , C : ˆ C T ( C ( − , c ) , C ( − , c ′ )) → ˆ D Γ ( F ( C ( − , c )) , F ( C ( − , c ′ )))is fully faithful, i.e, there is a unique γ : C ( − , c ) → C ( − , c ′ ) such that C ( γ ) = Y Γ ( f ).Then we define: H ( f ) = γ on morphisms and H ( d ) = C ( − , c ) on objects, where c isgiven by our choice of preimages.Explicitly on arrows we have that H : D → ˆ C T is given by H ( f ) = G ( Y Γ ( f )) ◦ η C ( − ,c ) i.e., C ( − , c ) η C ( − ,c ) −→ GF ( C ( − , c )) G ( Y Γ ( f )) −→ GF ( C ( − , c ′ )) Remark 7.8.1.
We notice that: C ◦ H ( d ) = C ( C ( − , c )) = F ( C ( − , c )) = D ( − , Φ( c )) = D ( − , d ) = Y Γ ( d )Also since Φ( c ) = d f −→ d ′ = Φ( c ′ ) then F ( C ( − , c )) = D ( − , Φ( c )) Y Γ ( f )) −→ D ( − , Φ( c ′ )) = F ( C ( − , c ′ ))Moreover, C ◦ H ( f ) = C ( C ( − , c ) η C ( − ,c ) −→ GF ( C ( − , c )) G ( Y Γ ( f )) −→ GF ( C ( − , c ′ )) ) = Y Γ ( f )since F ( C ( − , c )) F ( η C ( − ,c ) ) / / * * ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ F GF ( C ( − , c )) ε F ( C ( − ,c )) (cid:15) (cid:15) F G ( Y Γ ( f )) / / F GF ( C ( − , c ′ )) ε F ( C ( − ,c ′ )) (cid:15) (cid:15) F ( C ( − , c )) Y Γ ( f ) / / F ( C ( − , c ′ ))Thus C ◦ H = Y Γ . Remark 7.8.2.
Suppose that we are in the above situation where ( F , m ) ⊣ ( G , n )is a monoidal adjunction. The Yoneda embedding is a strong monoidal functor re-specting the Day’s convolution monoidal structure. Then we have: HAPTER 7. PRESHEAF MODELS D ( F ,m ) / / ˆ D Γ G ⊥ o o D ( Y,y ) O O Y Γ = = Since the adjunction is monoidal F is a strong monoidal functor. This implies that Y Γ is a strong monoidal functor by composition. C : ˆ C T → ˆ D Γ is a strong monoidal functor We define C ( A ) ⊗ ˆ D Γ C ( B ) u AB −→ C ( A ⊗ C T B ) by the following arrow: F ( A ) ⊗ ˆ D Γ F ( B ) m AB −→ F ( A ⊗ B ). We want to check naturality: for every A f → A ′ , B g −→ B ′ , where f , g ∈ ˆ C T C ( A ) ⊗ ˆ D Γ C ( B ) u AB / / C ( f ) ⊗ ˆ D Γ C ( g ) (cid:15) (cid:15) C ( A ⊗ C T B ) C ( f ⊗ C T g ) (cid:15) (cid:15) C ( A ′ ) ⊗ ˆ D Γ C ( B ′ ) u A ′ B ′ / / C ( A ′ ⊗ C T B ′ ) . This turns out to be F ( A ) ⊗ ˆ D Γ F ( B ) m AB / / ε F A ′ F ( f ) ⊗ ε F B ′ F ( g ) (cid:15) (cid:15) F ( A ⊗ C B ) ε F ( A ′⊗ B ′ ) F ( G ( m A ′ B ′ ) n ( f ⊗ g )) (cid:15) (cid:15) F ( A ′ ) ⊗ ˆ D Γ F ( B ′ ) m A ′ B ′ / / F ( A ′ ⊗ C B ′ )where f K ⊗ C T g K is equal to A ⊗ B f ⊗ g −→ GF A ′ ⊗ C GF B ′ n F A ′ F B ′ −→ G ( F A ′ ⊗ ˆ D Γ F B ′ ) G ( m A ′ B ′ ) −→ GF ( A ′ ⊗ ˆ C B ′ ) . We define I u I = m I −→ C ( I ) = F ( I ). HAPTER 7. PRESHEAF MODELS F ( A ) ⊗ F ( B ) m AB / / F ( f ) ⊗ F ( g ) (cid:15) (cid:15) F ( A ⊗ B ) F ( f ⊗ g ) / / F ( T A ′ ⊗ T B ′ ) ( a ) F ( n ) / / F ( G ( F A ′ ⊗ F B ′ )) F ( G ( m A ′ B ′ )) (cid:15) (cid:15) ε F A ′⊗ F B ′ s s ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ F GF ( A ′ ) ⊗ F GF ( B ′ ) m GF A ′ GF B ′ ❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢ ε F A ′ ⊗ ε F B ′ (cid:15) (cid:15) F G ( F ( A ′ ⊗ B ′ )) ε F ( A ′⊗ B ′ ) (cid:15) (cid:15) ( b ) F A ′ ⊗ F B ′ m A ′ B ′ / / F ( A ′ ⊗ B ′ )(a) commutes since ε is a monoidal natural transformation of the monoidal adjunction( F, m ) ⊣ ( G, n ).(b) ε is natural with m .Since m AB and m I are invertible in ˆ D Γ then u AB and u I are invertible. Thisimplies that ( C, m ) is a strong functor.Now we want to check that C ( A ) ⊗ ˆ D Γ I ρ / / ⊗ u I (cid:15) (cid:15) C ( A ) C ( A ) ⊗ ˆ D Γ C ( I ) u AI / / C ( A ⊗ C T I ) C ( ρ T ) O O since A ⊗ C T I ρ T → A is by definition A ⊗ C I ρ → A η → GF A this implies that C ( ρ T )is F ( A ⊗ I ) F ( ρ ) → F A F ( η ) → F GF A ε F A → F A i.e., C ( ρ T ) = F ( ρ ). Thus we obtain that F ( A ) ⊗ I ρ / / ⊗ m I (cid:15) (cid:15) F ( A ) F ( A ) ⊗ F ( I ) m AI / / F ( A ⊗ I ) F ( ρ ) O O and this is satisfied since ( F, m ) is a monoidal functor. The same is true for the λ axiom. HAPTER 7. PRESHEAF MODELS C ( A ) ⊗ ( C ( A ′ ) ⊗ C ( A ′′ )) α / / ⊗ u A ′ A ′′ (cid:15) (cid:15) ( C ( A ) ⊗ C ( A ′ )) ⊗ C ( A ′′ ) u AA ′ ⊗ (cid:15) (cid:15) C ( A ) ⊗ C ( A ′ ⊗ A ′′ ) u A,A ′⊗ A ′′ (cid:15) (cid:15) C ( A ⊗ A ′ ) ⊗ C ( A ′′ ) u A ⊗ A ′ ,A ′′ (cid:15) (cid:15) C ( A ⊗ ( A ′ ⊗ A ′′ )) C ( α ) / / C (( A ⊗ A ′ ) ⊗ A ′′ )For the same reasons as above we have that C ( α T ) = F ( α ), since α T = η A ⊗ A ′ ,A ′′ ◦ α by definition. H is a strong monoidal functor We want to define a natural transformation H ( A ) ⊗ C T H ( B ) ψ A,B −→ H ( A ⊗ D B ) thatmakes H into a strong monoidal functor.Definition of ψ .We begin by recalling that ( C, u ) and ( Y Γ , y ) are strong monoidal functors, i.e., u and y are isomorphisms, and since C is a fully faithful functor this allows us to define ψ A,B as the unique map making the following diagram commute: Y Γ ( A ) ⊗ Y Γ ( B ) y A,B / / u HA,HB ) ) ❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ Y Γ ( A ⊗ B ) = C ◦ H ( A ⊗ B ) C ( H ( A ) ⊗ H ( B )) C ( ψ A,B ) ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ In the same way we define ψ I as the unique map ψ I : I → H ( I ) making thefollowing diagram commute: I y I / / u I ❆❆❆❆❆❆❆❆❆ Y Γ ( I ) = C ◦ H ( I ) C ( I ) C ( ψ I ) ♦♦♦♦♦♦♦♦♦♦♦♦♦ i.e., since C is fully faithful the unique ψ I such that C ( ψ I ) = y I ◦ u − I . HAPTER 7. PRESHEAF MODELS C T C → D Γ is fully faithful and u and y are invertible maps thisimplies that φ is an invertible map.We shall prove naturality of φ . CH ( A ) ⊗ CH ( B ) = Y Γ ( A ) ⊗ Y Γ ( B ) ( a ) Y Γ ( f ) ⊗ Y Γ ( g ) CH ( f ) ⊗ CH ( g )= (cid:15) (cid:15) y A,B - - ❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩❩ u HA,HB ) ) ❚❚❚❚❚❚❚❚❚❚❚❚❚❚❚ C ( H ( A ) ⊗ H ( B )) ( c ) C ( H ( f ) ⊗ H ( g )) (cid:15) (cid:15) C ( ψ A,B ) / / CH ( A ⊗ B ) = Y Γ ( A ⊗ B ) ( d ) (cid:15) (cid:15) CH ( f ⊗ g )= Y Γ ( f ⊗ g ) (cid:15) (cid:15) C ( H ( A ′ ) ⊗ H ( B ′ )) C ( ψ A ′ ,B ′ ) / / CH ( A ′ ⊗ B ′ ) = Y Γ ( A ′ ⊗ B ′ ) CH ( A ′ ) ⊗ CH ( B ′ ) = Y Γ ( A ′ ) ⊗ Y Γ ( B ′ ) ( b ) y A ′ ,B ′ ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ u HA ′ ,HB ′ ❦❦❦❦❦❦❦❦❦❦❦❦❦❦ (a) and (b) by definition of ψ .(c) naturality of u where ( C, u ) is a monoidal functor.The perimeter of the diagram commutes by naturality of y where ( Y Γ , y ) is a monoidalfunctor. Using the fact that u HA,HB is an iso, all this implies that the interior square( d ) commutes. Thus we obtain that C ( ψ A ′ ,B ′ ) ◦ C ( H ( f ) ⊗ H ( g )) = CH ( f ⊗ g ) ◦ C ( ψ A,B )therefore since C is faithful ψ A ′ ,B ′ ◦ ( H ( f ) ⊗ H ( g )) = H ( f ⊗ g ) ◦ ψ A,B .Now we want to prove that this natural transformation satisfies all the axioms ofa monoidal structure. We start with the following axiom: H ( A ) ⊗ H ( I ) ρ / / ⊗ Ψ I (cid:15) (cid:15) H ( A ) H ( A ) ⊗ H ( I ) ψ AI / / H ( A ⊗ I ) H ( ρ ) O O (20)This turns to be the following diagram: HAPTER 7. PRESHEAF MODELS CH ( A ) ⊗ CH ( I ) ( d )1 (cid:25) (cid:25) ✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷✷ CH ( A ) ⊗ C ( I ) ( c ) u HA,I ( ( ❘❘❘❘❘❘❘❘❘❘❘❘❘ C ⊗ Cψ I h h ❘❘❘❘❘❘❘❘❘❘❘❘❘ C ⊗ Cψ I (cid:15) (cid:15) CH ( A ) ⊗ I ⊗ u I o o ρ (cid:15) (cid:15) ⊗ y I m m ❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬❬ ( a ) C ( H ( A ) ⊗ I )) ( e ) C (1 ⊗ ψ I ) (cid:15) (cid:15) C ( ρ ) / / CH ( A ) C ( H ( A ) ⊗ H ( I )) Cψ A,I / / CH ( A ⊗ I ) CH ( ρ )= O O Y Γ( ρ ) O O CH ( A ) ⊗ CH ( I ) ( b ) u HA,HI ❧❧❧❧❧❧❧❧❧❧❧❧❧ y A,I ❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡❡ We use the same argument again and we show that it satisfies the required equation.(a) C is a monoidal functor.(b) by definition of ψ A,I .(c) naturality of u where ( C, u ) is a monoidal functor.(d) definition of ψ I .The exterior diagram commutes because ( Y Γ , y ) is a monoidal functor. Using the factthat u HA,I is an iso, all this implies that the interior square ( e ) commutes. Again,since C is faithful we get H ( ρ ) ◦ ψ A,I ◦ (1 ⊗ ψ I ) = ρ which is diagram (20).In the same way we can verify that H ( λ ) ◦ ψ I,A ◦ ( ψ I ⊗
1) = λ . HAPTER 7. PRESHEAF MODELS x CHA ⊗ ( CHA ′ ⊗ CHA ′′ ) ( a )1 ⊗ y A ′ ,A ′′ o o ⊗ u t t ❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ ⊗ u (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ α (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄❄❄❄ ( CHA ⊗ CHA ′ ) ⊗ CHA ′′ u ⊗ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ u ⊗ (cid:3) (cid:3) y A,A ′ ⊗ (cid:15) (cid:15) CHA ⊗ C ( HA ′ ⊗ HA ′′ ) u (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ C ( HA ⊗ HA ′ ) ⊗ CHA ′′ ) u (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ CHA ⊗ CH ( A ′ ⊗ A ′′ ) (cid:15) (cid:15) CHA ⊗ C ( HA ′′ ⊗ HA ′′ ) ( i )1 ⊗ C ( ψ A ′ ,A ′′ ) ⑧⑧⑧⑧⑧⑧ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧ u (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄❄❄❄ C ( HA ⊗ ( HA ′ ⊗ HA ′′ )) ( k )( b ) C (1 ⊗ ψ A ′ ,A ′′ ) ⑧⑧⑧⑧⑧⑧ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧ C ( α ) (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄❄❄❄ ( f ) C (( HA ⊗ HA ′ ) ⊗ HA ′′ ) ( j ) C ( ψ A,A ′ ⊗ ⑧⑧⑧⑧⑧⑧ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧ C ( HA ⊗ HA ′ ) ⊗ CHA ′′ ( g )( h ) u _ _ ❄❄❄❄❄❄❄❄❄❄❄❄❄❄ C ( ψ A,A ′ ) ⊗ ⑧⑧⑧⑧⑧⑧ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧ CH ( A ⊗ A ′ ) ⊗ CHA ′′ o o CHA ⊗ CH ( A ′ ⊗ A ′′ ) y A,A ′⊗ A ′′ (cid:15) (cid:15) u (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄❄❄❄ ( c ) C ( HA ⊗ H ( A ′ ⊗ A ′′ )) C ( ψ A,A ′⊗ A ′′ ) ⑧⑧⑧⑧⑧⑧ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧ C ( H ( A ⊗ A ′ ) ⊗ HA ′′ ) C ( ψ A ⊗ A ′ ,A ′′ ) ⑧⑧⑧⑧⑧⑧ (cid:127) (cid:127) ⑧⑧⑧⑧⑧⑧ CH ( A ⊗ A ′ ) ⊗ CHA ′′ y A ⊗ A ′ ,A ′′ o o ( d ) u _ _ ❄❄❄❄❄❄❄❄❄❄❄❄❄❄ CH ( A ⊗ ( A ′ ⊗ A ′′ )) CH ( α ) (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄❄❄❄ CH (( A ⊗ A ′ ) ⊗ A ′′ ) The goal is to prove that the diagram ( k ) commutes. We have that:- (a): ( C, u ) is a monoidal functor.- (b) and (h): u is natural with Ψ and 1.- (c) and (d): definition of ψ A,A ′ ⊗ A ′′ and ψ A ⊗ A ′ ,A ′′ .- (i) and (g): definition of ψ A ′ ,A ′′ and ψ A,A ′ and functoriality of the tensor.- (f) and (j): are equal.- The exterior diagram commutes because ( Y Γ , y ) is a monoidal functor. HAPTER 7. PRESHEAF MODELS ⊗ u HA ′ ,HA ′′ and u HA,HA ′ ⊗ HA ′′ are isos it is enough to check that(top leg of (h)) ◦ u ◦ (1 ⊗ u ) =(bottom leg of (h)) ◦ u ◦ (1 ⊗ u ). Then we use the factthat C is a faithful functor. Remark 7.8.3.
Notice that since C and Y Γ are fully faithful functor, H is fully-faithful as well. H preserves coproducts In this section we focus on the specific problem of the preservation of finite coproductsof the functor H defined in Section 7.8.2. First, we notice that the category [ C op , Set ]has finite coproducts. These coproducts are computed pointwise: if F and G are in[ C op , Set ] then ( F ⊕ G )( C ) = F ( C ) ⊕ G ( C ) for every C ∈ C and with injections as inthe category Set .Also these coproducts are preserved going to the category [ D op , Set ] Γ via the leftadjoint F = F ◦ F , where F = R is the left adjoint of the reflection determinedby the class Γ. The coproducts in [ D op , Set ] Γ are induced by this reflection i ⊢ R .More precisely: A ⊕ Γ B = R ( i ( A ) ⊕ i ( B )) and in Γ = R ( in ) where A and B are in[ D op , Set ] Γ . Then, it makes sense to think about finite coproducts in [ D op , Set ] Γ .Finally, the Kleisli category ˆ C T inherits the coproduct structure from ˆ C as weproved in Section 7.7. Therefore, [ C op , Set ] T has finite coproducts. Recall that thecomparison functor C : [ C op , Set ] T → [ D op , Set ] Γ is fully faithful. Also, by Corollary 6.6.6, H : D → [ C op , Set ] T preserves coproductsiff [ C op , Set ] T ( H − , A ) : D op → Set preserves products for every A ∈ [ C op , Set ] T .But, we have [ C op , Set ] T ( H − , A ) ∼ = [ D op , Set ] Γ ( CH − , CA ) . More precisely, since C is fully faithful then the following functors: D op H op → ˆ C opT ˆ C T ( − ,A ) → Set
HAPTER 7. PRESHEAF MODELS D op ( CH ) op → ˆ D opT ˆ D Γ ( − ,CA ) → Set are naturally isomorphic.Therefore, we have:[ D op , Set ] Γ ( CH − , CA ) = [ D op , Set ] Γ ( Y Γ − , CA )because CH = Y Γ . Also,[ D op , Set ] Γ ( Y Γ − , CA ) ∼ = [ D op , Set ]( Y − , CA )because [ D op , Set ] Γ is a full subcategory of [ D op , Set ] and Y Γ − = Y − evaluated on − . Finally, [ D op , Set ]( Y − , CA ) ∼ = CA holds because by the Yoneda Lemma 6.1.5 there is a bijection which is natural in − , i.e., these functors are naturally isomorphic. But CA ∈ [ D op , Set ] Γ for every A ∈ [ C op , Set ] T , which by definition means that CA satisfies the property of continuityi.e., preserves all the cylinders and limit cones that are in Γ. In particular, sincenatural isomorphisms preserve limits, it will be enough to impose that condition onthe class Γ. From this, we conclude that Γ contains all the finite products. This isanother requirement to obtain a model. F T ⊣ G T is a monoidal adjunction In this section we show how a monoidal adjoint pair (
F, m ) ⊣ ( G, n ) induces amonoidal structure for the adjunction F T ⊣ G T associated with the Kleisli construc-tion, where T = GF . Lemma 7.9.1.
Let F ⊣ G be a monoidal adjunction, let T = GF , and consider theKleisli adjunction C F T / / C TG T ⊥ o o as in Definition 2.1.4. Then C T is a monoidal categoryand F T ⊣ G T is a monoidal adjunction.Proof. Since F ⊣ G is a monoidal adjunction, it follows that T = GF is a monoidalmonad. The result then follows from Lemma 2.3.3. HAPTER 7. PRESHEAF MODELS
To sum up the sections of this chapter we have the following theorem.
Theorem 7.10.1.
Given categories B , C and D , and functors Φ :
B → C , and
Ψ :
C → D , satisfying- B has finite products, C and D are symmetric monoidal,- B , C , and D have coproducts, and they are distributive w.r.t. tensor,- C is affine,- Φ and Ψ are strong monoidal,- Φ and Ψ preserve coproducts,- Φ is full and faithful,- Ψ is essentially surjective on objects,- for every b ∈ B , c, c ′ ∈ C we have C (Φ( b ) , c ) × C (Φ( b ) , c ′ ) ∼ = C (Φ( b ) , c ⊗ c ′ ) . Let Γ be any class of cones preserved by the opposite tensor functor, including all thefinite product cones and Lan Φ , Φ ∗ , F and G be defined as in Section 7.2. Then [ B op , Set ] Lan Φ / / [ C op , Set ] F / / Φ ∗ ⊥ o o [ D op , Set ] Γ G ⊥ o o forms an abstract model of the quantum lambda calculus.Proof. Relevant propositions from sections 6.7, 6.8, 6.9, 6.10, 6.11, 7.3, 7.4, 7.5, 7.6. hapter 8A concrete model of the quantumlambda calculus f n
Before we give the main model for higher-order quantum computation, it is instructiveto consider a simpler model for higher-order probabilistic computation. In the senseof Section 7.2, we let D be the category Srel fn of sets and stochastic relations, seeDefinition 4.2.3, and we let C be the category of finite sets and functions. In thissetting, we let “!” be the identity comonad, i.e., B = C . The latter is justified becausein the context of classical probabilistic computation, there are no quantum types andno no-cloning property; all types are classical and hence ! A = A . Lemma 8.1.1. Srel fn has finite coproducts, satisfying distributivity ( A ⊕ B ) ⊗ C ∼ = A ⊗ C ⊕ B ⊗ C .Proof. The coproduct of two objects is given by their disjoint union, A ⊕ B = ( A × ∪ ( B × i : A → A ⊕ B and i : B → A ⊕ B , where i (( x, j ) , y ) = ( j = 1 and x = y HAPTER 8. A CONCRETE MODEL i (( x, j ) , y ) = ( j = 2 and x = y d A,B,C : A ⊗ C ⊕ B ⊗ C → ( A ⊕ B ) ⊗ C is defined as [ i ⊗ C, i ⊗ C ]. The map d is easily seen to be a natural isomorphism byprecomposing with injections i , i and using the universal property for coproducts. Definition 8.1.2.
Let Ψ :
FinSet → Srel fn be the functor that is the identity onobjects, and defined on morphisms byΨ( f )( x, y ) = ( x = y Remark 8.1.3.
The functor Ψ is strong monoidal and preserves coproducts.
Theorem 8.1.4.
The choice B = FinSet , C = FinSet , D = Srel fn with Φ = id and Ψ as in Definition 8.1.2. Let Γ be the class of all finite product cones in D op .This choice satisfies all the properties required by the Theorem 7.10.1. Therefore, thisgives an abstract model of the quantum lambda calculus.Proof. By Lemma 8.1.1 and Remark 8.1.3.
Remark 8.1.5.
Such a model could be considered to be a concrete model of “prob-abilistic lambda calculus”, i.e., of higher-order probabilistic computation.
Remark 8.1.6.
By Lemma 8.1.1, the functor − ⊗ X preserves finite coproducts forall X ∈ Srel fn . It is possible to show that this functor in fact preserves all existingcolimits (due to the natural isomorphism A ⊗ X ∼ = A ⊕ A ⊕ . . . ⊕ A , | X | times forany fixed X ). Therefore, in Theorem 8.1.4, we could have alternatively defined Γ tobe the class of all limit cones. In fact, any class of limit cones that contains at leastall finite product ones would do. Each such choice yields an a priori different model. HAPTER 8. A CONCRETE MODEL ′′ and the functors Φ and Ψ Recall the definition of the category Q of superoperators from Section 3.2. In thissection, we discuss a category Q ′′ related to superoperators Q , together with functors FinSet Φ −→ Q ′′ Ψ −→ Q . Here, the goal is to choose Q ′′ and the functors Φ and Ψcarefully so as to satisfy the requirement of Theorem 7.10.1.Recall the definition of the free affine monoidal category F wm ( K ) from Sec-tion 2.6. We apply this universal construction to situation where K is a discretecategory. For later convenience, we let K be the discrete category with finite dimen-sional Hilbert spaces as objects. Then F wm ( K ) has sequences of Hilbert spaces asobjects and dualized, compatible, injective functions as arrows:- objects: finite sequences of finite dimensional Hilbert spaces- a morphism from { V , . . . , V n } to { W , . . . , W m } is given by an injective function f : { , . . . , m } → { , . . . , n } , such that for all i, V f ( i ) = W i . Remark 8.2.1.
Since the objects of Q and F wm ( K ) are finite sequences of finite-dimensional Hilbert spaces, and there are only countably many finite-dimensionalHilbert spaces up to isomorphism, we may w.l.o.g. assume that Q and F wm ( K ) aresmall categories.Now consider the identity-on-objects inclusion functor F : K → Q ′ s where Q ′ s isthe category of simple trace-preserving superoperator defined in Section 3.2. Since Q ′ s is affine, by Proposition 2.6.3 there exists a unique (up to natural isomorphism)strong monoidal functor ˆ F such that: K I (cid:15) (cid:15) F / / Q ′ s F wm ( K ) ˆ F : : ✉✉✉✉✉✉✉✉✉✉ Remark 8.2.2.
This reveals the purpose of using the equality instead of ≤ in thedefinition of a trace-preserving superoperator (Definition 3.2.4). When the codomainis the unit, there is only one map f ( ρ ) = tr( ρ ), and therefore Q ′ s is affine. HAPTER 8. A CONCRETE MODEL
Remark 8.2.3.
By definition, Q s is a full subcategory of Q , and the inclusion functor In : Q s → Q is strong monoidal. Also, since every trace preserving superoperator istrace non-increasing, Q ′ s is a subcategory of Q s , and the inclusion functor E : Q ′ s → Q s is strong monoidal as well.Then we apply the machinery of Proposition 2.4.9 to the functor: F wm ( K ) ˆ F → Q ′ s E → Q s In → Q . where In and E are as defined in Remark 8.2.3. Definition 8.2.4.
Let Q ′′ = ( F wm ( K )) + and let Ψ be the unique finite coproductpreserving functor making the following diagram commute: F wm ( K ) I (cid:15) (cid:15) ˆ F / / Q ′ s E / / Q s In / / Q ( F wm ( K )) + Ψ ❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤❤ (21)Note that such a functor exists by Proposition 2.4.4, and it is strong monoidal byProposition 2.4.9. Remark 8.2.5.
SinceΨ {{ V ai } i ∈ [ n a ] } a ∈ A = a a ∈ A { ( V a ⊗ . . . ⊗ V an a ) ∗ } ∗∈ the functor Ψ is essentially onto objects. Specifically, given any object { V a } a ∈ A ∈ | Q | ,we can choose a preimage (up to isomorphism) as follows:Ψ {{ V ai } i ∈ [1] } a ∈ A = a a ∈ A { ( V a ) ∗ } ∗∈ ∼ = { V a } a ∈ A . (22)Here is the full picture of categories and functors: K F / / I (cid:15) (cid:15) Q ′ s E / / Q sIn (cid:15) (cid:15) F wm ( K ) I (cid:15) (cid:15) ˆ F sssssssssss Q ( F wm ( K )) + Ψ ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ HAPTER 8. A CONCRETE MODEL
Remark 8.2.6.
Since
F wm ( K ) is an affine category and Q ′′ = F wm ( K ) + , let usconsider the functor Φ : Finset → Q ′′ defined by Lemma 2.5.2. Theorem 8.2.7.
The choice B = FinSet , C = Q ′′ , D = Q with the functors Φ asin Remark 8.2.6 and Ψ as in Definition 8.2.4 satisfies all the properties required byTheorem 7.10.1.Proof. By relevant propositions from Section 8.2.
Theorem 8.3.1.
Let Q and Q ′′ be defined as in Sections 3.2 and 8.2. Let Γ be theclass of all finite product cones in D op where D = Q . Then [ FinSet op , Set ] Lan Φ / / [ Q ′′ op , Set ] F / / Φ ∗ ⊥ o o [ Q op , Set ] Γ G ⊥ o o forms a concrete model of the quantum lambda calculus.Proof. The proof is by Theorem 8.2.7, by and by Theorem 7.10.1. hapter 9Conclusions and future work
In the first part of this thesis, we established that the partially traced categories, inthe sense of Haghverdi and Scott, are precisely the monoidal subcategories of totallytraced categories. This was proved by a partial version of Joyal, Street, and Verity’s“Int”-construction, and by considering a strict symmetric compact closed version ofFreyd’s paracategories.We also introduced some new examples of partially traced categories, in connectionwith some standard models of quantum computation such as completely positive mapsand superoperators.One question that we did not answer is whether specific partially traced categoriescan always be embedded in totally traced categories in a “natural” way. For example,the category of finite dimensional vector spaces, with the biproduct ⊕ as the tensor,carries a partial trace. By our proof, it follows that it can be faithfully embedded in atotally traced category. However, we do not know any concrete “natural” example ofsuch a totally traced category (i.e., other than the free one constructed in our proof)in which it can be faithfully embedded.In the second part, we constructed mathematical (semantical) models of higher-order quantum computation, and more specifically, for the quantum lambda calculusof Selinger and Valiron. The central idea of our model construction was to applythe presheaf construction to a sequence of three categories and two functors, andto find a set of sufficient conditions for the resulting structure to be a valid model.196 HAPTER 9. CONCLUSIONS AND FUTURE WORK ibliography [1] Samson Abramsky. Computational interpretation of linear logic.
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