A category of quantum posets
aa r X i v : . [ m a t h . OA ] J a n A CATEGORY OF QUANTUM POSETS
ANDRE KORNELL, BERT LINDENHOVIUS, AND MICHAEL MISLOVE
Department of Computer ScienceTulane University
Abstract.
We define a quantum poset to be a hereditarily atomic von Neumann algebraequipped with a quantum partial order in Weaver’s sense. These quantum posets forma category that is complete, cocomplete and symmetric monoidal closed. This yields aquantum analogue of the inclusion order on a powerset. We show that every quantum posetcan be embedded into its powerset via a quantum analogue of the mapping that takes eachelement of a poset to its down set.
Let M be a von Neumann algebra of bounded operators on some Hilbert space H . Aquantum partial order on M may be defined to be an ultraweakly closed algebra V ofbounded operators on the same Hilbert space H such that V ∩ V ∗ = M ′ [8, Definition2.6(d)]. Together M and V might be called a quantum poset: the von Neumann algebra M plays the role of the space that is being order, and V plays the role of the order relation.Of course, von Neumann algebras are typically understood as a quantum generalization ofmeasure spaces, and not of sets. However, hereditarily atomic von Neumann algebras, i.e.,those of the form M = L i ∈ I M d i ( C ) , may be understood as a quantum generalization of sets.Thus, in this paper, we investigate hereditarily atomic von Neumann algebras M equippedwith a quantum partial order V .More precisely, we investigate an equivalent class of objects. Inspired by the very simpleform of a hereditarily atomic von Neumann algebra, we may define a quantum set X tobe simply a set of finite-dimensional Hilbert spaces, called its atoms [3, Definition 2.1]. Toeach quantum set X , we associate the hereditarily atomic von Neumann algebra ℓ ∞ ( X ) := L { L ( X ) : X is an atom of X } ; this is the equivalence between the two notions. Intuitively,the atoms of X are not its elements but rather its indecomposable subsets. One imaginesthat a quantum set consisting of a single d -dimensional atom in fact consists of d elementsthat are inextricably clumped together.In the special case of hereditarily atomic von Neumann algebras, Weaver’s quantumrelations also have a convenient, explicit characterization. A binary relation R from aquantum set X to a quantum set Y may be defined to be simply a choice of subspaces R ( X, Y ) ⊆ L ( X, Y ) , for atoms X of X and Y of Y [3, Definition 3.1]. Together, quantumsets and the binary relations between them form a dagger-compact category [1, Definition12], i.e., a rigid symmetric monoidal category equipped with an involution that is compati-ble with its monoidal structure. This category is also enriched over the category of modularorthomodular lattices and supremum-preserving monotone functions. This dagger-compactcategory qRel forms the basic setting of this paper. e now define quantum posets and monotone functions in terms of this structure, andstate the main result of the paper: Definition.
A quantum poset is a quantum set X equipped with a binary relation R on X that satisfies R ◦ R ≤ R and R ∧ R † = I X , where I X is the identity binary relation on X . Definition.
A monotone function from a quantum poset ( X , R ) to a quantum poset ( Y , S ) is a binary relation F from X to Y such that F † ◦ F ≥ I X , F ◦ F † ≤ I Y and F ◦ R ≤ S ◦ F . Theorem.
The category qPOS of quantum posets and monotone functions is complete,cocomplete and symmetric monoidal closed. Furthermore, the full subcategory of those quan-tum posets that have only one-dimensional atoms is equivalent to the category of posets andmonotone functions.
One consequence of this theorem is that the powerset of a quantum set is canonically aquantum poset. To define the powerset of a quantum set, we first recall that a function froma quantum set X to a quantum set Y is just a binary relation F such that F † ◦ F ≥ I X and F ◦ F † ≤ I Y . This defines the category qSet of quantum sets and functions [3]. Foreach quantum set Y , we have the following sequence of bijections, natural in X as functors qSet → Set : qRel ( X , Y ) ∼ = qRel ( X × Y ∗ , ) ∼ = qSet ( X × Y ∗ , ‘ { , } ) ∼ = qSet ( X , ‘ { , } Y ∗ ) , where ‘ { , } is a quantum set that consists of two one-dimensional atoms [3, Section 2].We have the first bijection because qRel is a dagger-compact category, we have the secondbijection by [3, Theorem B.8], and we have the third bijection because qSet is symmetricmonoidal closed [3, Theorem 9.1]. This justifies the definition Pow( Y ) = ‘ { , } Y ∗ for eachquantum set Y .To define the quantum poset (Pow( Y ) , Incl( Y )) , we form the internal hom of Y ∗ and ‘ { , } in qPOS , rather than in qSet . Intutively, we order Y ∗ trivially, and we order ‘ { , } by ⊏ . Classically, every poset may be embedded into its powerset, by mapping each elementto its down set. We establish a quantum analogue of this fact: Definition.
An order embedding of a quantum poset ( Y , S ) into a quantum poset ( Z , T ) isa function F from Y to Z such that S = F † ◦ T ◦ F . Theorem.
Let ( Y , S ) be a quantum poset. Then, there exists an order embedding of ( Y , S ) into (Pow( Y ) , Incl( Y )) . This research was motivated by the goal of generalizing information orders to the quantumsetting [4]. In computer-scientific semantics, the points of a poset commonly interpret terms.Intuitively, the points are datasets, and strictly higher points are datasets that contain moreinformation. Alternatively, the points are the pure states of an almost static classical physicalsystem, which may undergo unpredictable evolution. If we observe the system after sometime, then states that are lower in the transition order carry less information about thesystem’s initial state.This intuition coheres with Weaver’s discussion of quantum preorders as encoding thepossibility of transition in a quantum system from one state to another in multiple stepstaken from some set of quantum channels [9, Section 1]. For us, these quantum channelsformalize possible evolutions of an almost static quantum physical system. n the sequel, the Dirac delta symbol δ a,b names the complex number if a = b andotherwise names . Similarly, the symbol ∆ a,b names the maximum binary relation on andotherwise names the minimum binary relation on . Thus, δ a,b ∈ ∆ a,b ( C , C ) . The binaryrelation ∆ a,b is a scalar in the category qRel . Hence, for each binary relation R from aquantum set X to a quantum set Y , we write ∆ a,b R for the binary relation from X to Y obtained by composing ∆ a,b × R with unitors in the obvious way.This research was supported by AFOSR under MURI grant FA9550-16-1-0082.1. Definitions and examples
We begin by defining orders and pre-orders on quantum sets, essentially Weaver’s quantumpreorders and quantum partial orders [8, Definition 2.6]. We also define the notion of amonotone function between quantum posets, which reduces to the familiar notion in theclassical case.
Definition 1.1.
Let X be a quantum set. We call a binary relation R on X a pre-order ,and ( X , R ) a quantum pre-ordered set if it satisfies the following two axioms:(1) I X ≤ R ( R is reflexive);(2) R ◦ R ≤ R ( R is transitive).If, in addition, R satisfies(3) R ∧ R † = I X ( R is antisymmetric),then we call R a order , and ( X , R ) a quantum poset . Example . Let X be a quantum set. Then, I X is easily seen to be an order on X , the trivial order. Example . Let ( A, ⊑ ) be any poset. Then, (‘ A, ‘ ⊑ ) is a quantum poset, because ordinaryposets are defined in the category Rel of sets and binary relations by the same three axiomsand the “inclusion” functor ‘( − ) : Rel → qRel preserves all the relevant structure [3, SectionIII]. Furthermore, because this inclusion functor is full and faithful, every order on ‘ A is ofthis form. Example . Let A be a unital algebra of operators on a nonzero finite-dimensional Hilbertspace H that is anti-symmetric in the sense that A ∩ A † = C [6][7][9]. Then, the equation R ( H, H ) = A defines an order on H , the quantum set whose only atom is H . Furthermore,every order on H is of this form, as a simple consequence of the definition of qRel and ofall the relevant structure on this category. Example . Let X be a quantum set with two atoms, X and X . We may define an order R on X as follows: (cid:18) R ( X , X ) R ( X , X ) R ( X , X ) R ( X , X ) (cid:19) = (cid:18) C X L ( X , X ) C X (cid:19) . Intuitively, the structure of the quantum poset ( X , R ) may be described as follows: the atom X represents a subset of X of cardinality dim( X ) that is trivially ordered, the atom X represents a subset of X of cardinality dim( X ) that is trivially ordered, and every elementof the first subset is below every element of the second.We record the following basic facts about orders on quantum sets in a single lemma, whoseproof we omit because it is straightforward. emma 1.6. Let R be a pre-order on a quantum set X . Then, (1) R † is a pre-order, and it is an order if R is an order; (2) R ◦ R = R ; The order R † is called the opposite order, since for any ordinary poset ( S, ⊑ ) , we have ‘( ⊑ ) † = ‘( ⊒ ) on ‘ S . Lemma 1.7.
Let R be a pre-order on X and let X ∝ X . Then R ( X, X ) is a unital subal-gebra of L ( X ) := L ( X, X ) . Moreover, if R is an order, then R ( X, X ) is an antisymmetricsubalgebra, i.e., R ( X, X ) ∧ R ( X, X ) ⊥ = C X .Proof. Since I X ≤ R , we have C X = I X ( X, X ) ≤ R ( X, X ) , implying that R ( X, X ) containsthe identity element of L ( X ) . We also compute that R ( X, X ) · R ( X, X ) ≤ _ X ′ ∝ X R ( X, X ′ ) · R ( X ′ , X ) = ( R ◦ R )( X, X ) ≤ R ( X, X ) . Therefore, R ( X, X ) is an algebra. Finally, if R is an order, then C X = I X ( X, X ) = ( R ∧ R † )( X, X ) = R ( X, X ) ∧ R † ( X, X ) = R ( X, X ) ∧ R ( X, X ) † , so R ( X, X ) is antisymmetric. (cid:3) Lemma 1.8.
Let X be a quantum set, and let { R α } α ∈ A be a collection of pre-orders on X .Then, R = V α ∈ A R α is also a pre-order.Proof. We have I X ≤ R α for each α ∈ A , hence also I X ≤ V α ∈ A R α = R . Furthermore, usingLemma A.1, we find that R ◦ R = ^ α ∈ A R α ! ◦ ^ β ∈ A R β ! ≤ ^ α,β ∈ A R α ◦ R β ≤ ^ α ∈ A R α ◦ R α ≤ ^ α ∈ A R α = R. (cid:3) Definition 1.9.
Let ( X , R ) and ( Y , S ) be quantum pre-ordered sets. Then, a monotone function F : ( X , R ) → ( Y , S ) is a function F : X → Y that satisfies any of the followingequivalent conditions:(1) F ◦ R ≤ S ◦ F ;(2) F ◦ R ◦ F † ≤ S ;(3) R ≤ F † ◦ S ◦ F .The equivalences between these conditions follow directly from the definition of a functionbetween quantum sets. Lemma 1.10.
Let ( X , R ) , ( Y , S ) and ( Z , T ) be quantum posets, and let F : X → Y and G : Y → Z be monotone. Then G ◦ F : X → Z is monotone.Proof.
Since F is monotone, we have F ◦ R ≤ S ◦ F . Monotonicity of G means that G ◦ S ≤ T ◦ G . Hence G ◦ F ◦ R ≤ G ◦ S ◦ F ≤ T ◦ G ◦ F, showing that indeed G ◦ F ismonotone. (cid:3) Since the composition of two monotone functions is monotone, we obtain a category ofquantum posets and monotone functions, which we notate qPOS . xample . Let X be a trivially ordered quantum set (cf. Example 1.2) and let ( Y , S ) bea quantum poset. Then any function F : X → Y is monotone. Indeed, we have F ◦ I X = I Y ◦ F ≤ S ◦ F. Lemma 1.12.
Let ( A, ⊑ A ) and ( B, ⊑ B ) be posets, and let f : A → B be a function. Then f is monotone if and only if ‘ f is monotone.Proof. The function ‘ f : ‘ A → ‘ B is monotone if and only if ‘ f ◦ ‘( ⊑ S ) ≤ ‘( ⊑ B ) ◦ ‘ f , orequivalently ‘( f ◦ ( ⊑ A )) ≤ ‘(( ⊑ B ) ◦ f ) . The functor ‘( − ) : Rel → qRel preserves daggermonoidal structure [3, Section 3]; hence ‘ f is monotone if and only if f ◦ ( ⊑ A ) ≤ ( ⊑ B ) ◦ f .It remains only to show that this inequality is equivalent to the monotonicity of f .Assume that f is monotone, and let ( a, b ) be a pair in the binary relation f ◦ ( ⊑ A ) . Itfollows that b = f ( a ′ ) for some a ′ ⊒ A a . Since f is monotone, we find that b ⊒ B f ( a ) . Inother words, ( a, b ) is in the binary relation ( ⊑ B ) ◦ f . Therefore, f ◦ ( ⊑ A ) ≤ ( ⊑ B ) ◦ f .Now, assume that f satisfies the inequality f ◦ ( ⊑ A ) ≤ ( ⊑ B ) ◦ f , and let a ⊑ A a . Thepair ( a , f ( a )) is in f ◦ ( ⊑ A ) simply by definition of composition. By assumption, it is alsoin ( ⊑ B ) ◦ f . Thus, f ( a ) ⊑ A f ( a ) . Therefore, f is monotone. (cid:3) Proposition 1.13.
The functor ‘( − ) : POS → qPOS , given by ( A, ⊑ ) (‘ A, ‘ ⊑ ) onobjects and by f ‘ f on morphisms is fully faithful.Proof. The functor ‘( − ) : Rel → qRel is fully faithful, and it preserves dagger monoidalstructure [3, Section 3]. Because functions and, moreover, monotone functions are defined interms of this dagger monoidal structure (Lemma 1.12), it follows that this functor restrictsto a functor Set → qSet and, moreover, to a functor POS → qPOS . (cid:3) Subposets
The subsets of a quantum set X correspond to injections into X [3, Proposition 10.1]. Weshow that the subsets of a quantum poset X similarly correspond to order embeddings. Lemma 2.1.
Let X and Y be quantum sets, and let S be a pre-order on Y . Let F : X → Y bea function. Then R = F † ◦ S ◦ F is a pre-order on X , and F : ( X , R ) → ( Y , S ) is monotone.Moreover, if ( Y , S ) is a quantum poset and F is injective, then ( X , R ) is a quantum posetas well.Proof. We have I X ≤ F † ◦ F = F † ◦ I Y ◦ F ≤ F † ◦ S ◦ F = R, so R satisfies the first axiom of a pre-order on a quantum set. Furthermore, we have R ◦ R = F † ◦ S ◦ F ◦ F † ◦ S ◦ F ≤ F † ◦ S ◦ S ◦ F ≤ F † ◦ S ◦ F = R, so R also satisfies the second axiom and hence is a pre-order.Next we show that F is monotone: F ◦ R = F ◦ F † ◦ S ◦ F ≤ I Y ◦ S ◦ F = S ◦ F. Now, assume that F is injective. Furthermore, assume that S satisfies the antisymmetryaxiom, i.e., S ∧ S † = I Y . We check that R satisfies the antisymmetry axiom too: R ∧ R ⊥ = ( F † ◦ S ◦ F ) ∧ ( F † ◦ S ◦ F ) † = ( F † ◦ S ◦ F ) ∧ ( F † ◦ S † ◦ F ) = F † ◦ ( S ∧ S † ) ◦ F = F † ◦ F = I X , where the second equality follows from Proposition A.6, the penultimate equality follows bythe antisymmetry axiom for S , and the last equality follows from the injectivity of F . (cid:3) he previous lemma assures that the following definition is sound: Definition 2.2.
Let ( Y , S ) be a quantum poset. Then a subposet of Y consists of a subset X ⊆ Y equipped with order R = J †X ◦ S ◦ J X , to which we refer as the induced order on X .The quantum generalization of the concept of a subposet leads to the quantum general-ization of the notion of an order embedding: Definition 2.3.
Let ( X , R ) and ( Y , S ) be quantum pre-ordered sets. Then we call a function J : X → Y an order embedding if R = J † ◦ S ◦ J .Just as order embeddings between posets in the classical sense are monotone and injective,this happens to be true for the quantum case as well: Lemma 2.4.
Let ( X , R ) and ( Y , S ) be quantum posets, and let F : X → Y be an orderembedding. Then F is both injective and monotone.Proof. Monotonicity directly follows from Definition 1.9. Since R and S are both orders,and R = F † ◦ S ◦ F , we obtain I X = R ∧ R † = ( F † ◦ S ◦ F ) ∧ ( F † ◦ S † ◦ F ) = F † ◦ ( S ∧ S † ) ◦ F = F † ◦ F, where we used Proposition A.6 in the penultimate equality. (cid:3) Lemma 2.5.
Let ( X , R ) , ( Y , S ) , and ( Z , T ) be quantum posets, and let F : X → Y and F : Y → Z be order embeddings. Then the composition F ◦ F is an order embedding, too.Proof. Since F and F are order embeddings, we have R = F † ◦ S ◦ F and S = F † ◦ T ◦ F .Hence ( F ◦ F ) † ◦ T ◦ ( F ◦ F ) = F † ◦ F † ◦ T ◦ F ◦ F = F † ◦ S ◦ F = R . (cid:3) Definition 2.6.
Let ( X , R ) and ( Y , S ) be quantum posets. A monotone map F : X → Y iscalled an order isomorphism if it is bijective, and its inverse F † is monotone, too. Proposition 2.7.
Let ( X , R ) and ( Y , S ) be quantum posets, and let F : X → Y be afunction. Then the following statements are equivalent: (a) F is an order isomorphism; (b) F is a surjective order embedding; (c) F is a bijection such that F ◦ R = S ◦ F .Proof. First assume that F is an order isomorphism. We have that R ≤ F † ◦ S ◦ F by themonotonicity of F . Since F † is also monotone, we have S ≤ F ◦ R ◦ F † . Since F is injective,it follows that F † ◦ S ◦ F ≤ F † ◦ F ◦ R ◦ F † ◦ F = R. Hence R = F † ◦ S ◦ F , so F is an orderembedding, which is surjective since F is a bijection. Therefore, (a) implies (b).Assume that (b) holds. Then R = F † ◦ S ◦ F . We have that F ◦ R = F ◦ F † ◦ S ◦ F = S ◦ F by the surjectivity of F . Moreover, since F is an order embedding, it is injective by Lemma2.4, and it is hence bijective. So (c) holds.Finally, we show that (c) implies (a). Hence, let F be a bijection such that F ◦ R = S ◦ F .This equality immediately gives that F is monotone. Moreover, the bijectivity of F yields F † ◦ S = F † ◦ S ◦ F ◦ F † = F † ◦ F ◦ R ◦ F † = R ◦ F † , which implies that F † is monotone,too. We conclude that F is an order isomorphism. (cid:3) . Monomorphisms and epimorphisms
We show that the monomorphisms of qPOS are exactly the injective monotone functionsand that all extremal epimorphisms of qPOS are surjective. We will later use both resultsto show that qPOS is cocomplete. We do not characterize arbitrary epimorphisms.
Lemma 3.1.
Let ( X , R ) and ( Y , S ) be quantum posets, and let M : X → Y be monotone.Then M is a monomorphism in qPOS if and only if M is injective.Proof. Let M : X → Y be injective, let ( W , T ) be a quantum poset, and let F, G : W →X be two monotone functions such that M ◦ F = M ◦ G . Since M is injective, it is amonomorphism in qSet (cf. [3, Proposition 8.4]); hence it follows that F = G . Thus M isalso a monomorphism in qPOS .We prove the converse by contraposition, so assume that M is not injective. By [3,Proposition 8.4], M is not an monomorphism in qSet . Hence, there is a quantum set W andthere are functions F, G : W → X such that F = G , but M ◦ F = M ◦ G . Equip W withthe trivial order I W . By Example 1.11, it follows that F and G are monotone. We concludethat M is not a monomorphism in qPOS . (cid:3) Definition 3.2.
Let F : X → Y be a function from a quantum set X to a quantum set Y .We define the range of F to be the subset ran F := Q{ Y ∝ Y : F ( X, Y ) = 0 for some X ∝ X } ⊆ Y . We also define the binary relation F from X to ran F by F ( X, Y ) = F ( X, Y ) , for X ∝ X and Y ∝ ran F . It is routine to verify that F is a surjective function that satisfies F = J ran F ◦ F ,where J ran F : ran F ֒ → Y is the canonical inclusion [3, Definition 8.2].Thus, each function F has a canonical factorization into an inclusion following a surjection.If F is monotone, then both factors are also monotone, provided that we equip the range of F with the induced order. Lemma 3.3.
Let ( X , R ) and ( Y , S ) be quantum posets, and let F be a monotone functionfrom ( X , R ) to ( Y , S ) . Then, F is a monotone function from ( X , R ) to (ran F, J † ◦ S ◦ J ) ,where J = J ran F : ran F ֒ → Y is the canonical inclusion.Proof.
We reason that F ◦ R = J † ◦ J ◦ F ◦ R ≤ J † ◦ F ◦ R ≤ J † ◦ S ◦ F = J † ◦ S ◦ J ◦ F . (cid:3) It is easy to see that any surjective monotone function is an epimorphism in the category qPOS . We do not show the converse; for our purposes, it is sufficient to show that any extremal epimorphism is surjective. Recall that an epimorphism E is said to be extremal ifthe only mononomorphisms M satisfying E = M ◦ F for some morphism F are isomorphisms[2, Definition 7.74]. Lemma 3.4.
Let ( X , R ) and ( Y , S ) be quantum posets, and let E : X → Y be an extremalepimorphism in qPOS . Then, E is surjective.Proof. The monotone function E factors through ran E as E = J ran E ◦ E , with both factorsbeing monotone for the induced order on ran E . Because E is an extremal epimorphism, themonomorphism J ran E must be an isomorphsim in qPOS , and therefore also an isomorphismin qSet , i.e., a bijection. It clearly follows that ran E = Y . We conclude that F = F and,in particular, that F is a surjection. (cid:3) . Order enrichment
We show that for all quantum posets X and Y , the order on Y imposes an order on thehom set qPOS ( X , Y ) , just as it does in the classical case. Let ( Y, ⊑ ) be a poset. Then, forany set X , we can order Set ( X, Y ) by f ⊑ g if and only if f ( x ) ⊑ g ( x ) for all x ∈ X . If weorder the binary relations between two sets by inclusion, and regard f , g and ⊑ as binaryrelations, this condition is equivalent to g ≤ ( ⊑ ) ◦ f and also to ( ⊑ ) ◦ g ≤ ( ⊑ ) ◦ f . The nextlemma shows that these last two inequalities between binary relations can be generalized tothe quantum setting. Lemma 4.1.
Let X be a quantum set, and let ( Y , S ) be a quantum poset. Then, we write F ⊑ G if F, G ∈ qSet ( X , Y ) satisfy any of the following equivalent conditions: (1) G ≤ S ◦ F ; (2) S ◦ G ≤ S ◦ F ; (3) F ≤ S † ◦ G ; (4) G ◦ F † ≤ S .Moreover, the relation ⊑ defines an order on qSet ( X , Y ) .Proof. The equivalence of conditions (1)-(4) follows easily from the definitions of a functionand of an order. In order to show that ⊑ is an order on qSet ( X , Y ) , it is straightforwardto show that ⊑ is reflexive and transitive. For antisymmetry, let F, G ∈ qSet ( X , Y ) , andassume that G ⊑ F and F ⊑ G . Thus, F ≤ S ◦ G and G ≤ S ◦ F , or equivalently, F ≤ S † ◦ G .We now compute that F ≤ ( S ◦ G ) ∧ ( S † ◦ G ) = ( S ∧ S † ) ◦ G = I ◦ G = G, where we useProposition A.6 for the first equality, and axiom (3) of an order for the second equality. ByLemma A.7, we obtain F = G . (cid:3) Lemma 4.2.
Let ( X , R ) be a quantum poset, let Y and Z be quantum sets, and let F : Y → Z be a function. Let K and K be functions Z → X such that K ⊑ K . Then, K ◦ F ⊑ K ◦ F .Proof. Let K , K : Z → X be such that K ⊑ K . Then, K ≤ R ◦ K , and hence K ◦ F ≤ R ◦ K ◦ F, which expresses that K ◦ F ⊑ K ◦ F . (cid:3) Definition 4.3.
Let ( X , R ) and ( Y , S ) be quantum posets. Then, we order qPOS ( X , Y ) by the induced order ⊑ from qSet ( X , Y ) (cf. Lemma 4.1).We note that the order on qPOS ( X , Y ) only depends on S and not on R . It is the samein the classical case, where the order on POS (( X, ⊑ X ) , ( Y, ⊑ Y )) is defined by f ≤ g if andonly if f ( x ) ⊑ Y g ( x ) for all x ∈ X . Lemma 4.4.
Let ( X , R ) , ( Y , S ) and ( Z , T ) be quantum posets, and let F : Y → Z bemonotone. Let K , K : X → Y be functions (not necessarily monotone). If K ⊑ K , then F ◦ K ⊑ F ◦ K Proof.
Recall that, by definition, F is monotone if and only if F ◦ S ≤ T ◦ F . Let K , K ∈ qPOS ( X , Y ) such that K ⊑ K . This means that K ≤ S ◦ K , so we compute that F ◦ K ≤ F ◦ S ◦ K ≤ T ◦ F ◦ K , and we conclude that F ◦ K ⊑ F ◦ K . (cid:3) Lemma 4.5.
Let ( X , R ) , ( Y , S ) and ( Z , T ) be quantum posets, and let F : ( Y , S ) → ( Z , T ) be monotone. Then, qPOS ( X , F ) : qPOS ( X , Y ) → qPOS ( X , Z ) , K F ◦ K, nd qPOS ( F, X ) : qPOS ( Z , X ) → qPOS ( Y , X ) , K K ◦ F are monotone.Proof. This follows from Lemmas 4.2 and 4.4. (cid:3)
Theorem 4.6.
The category qPOS is enriched over
POS .Proof.
For this statement to hold, we must have that each homset in qPOS is a poset,which is the case by Definition 4.3, and that composition is monotone in both arguments.The latter follows directly from Lemma 4.5. (cid:3) Completeness
We show that the category qPOS is complete by defining pre-orders on the limits thatwe have in qSet . The main technical challenge is showing that these pre-orders are in factpartial orders.
Definition 5.1 (cf. [8, Definition 2.6(a)]) . Let X be a quantum set. Then E ∈ qRel ( X , X ) is called an equivalence relation if(1) I X ≤ E (2) E ◦ E ≤ E ;(3) E † = E . Lemma 5.2.
Let E be an equivalence relation on a quantum set X , and let X ∝ X . Then E ( X, X ) is a unital C*-subalgebra of L ( X ) := L ( X, X ) .Proof. The subspace E ( X, X ) is a unital subalgebra of L ( X ) by Lemma 1.7, because E isa pre-order. Condition (3) specializes to the equation E ( X, X ) † = E ( X, X ) , so E ( X, X ) isfurthermore a ∗ -subalgebra. It is automatically closed in the norm topology because L ( X ) is finite-dimensional. (cid:3) Lemma 5.3.
Let X be a quantum set and let R ∈ qRel ( X , X ) be a pre-order. Then E = R ∧ R † is an equivalence relation.Proof. By Lemma 1.6, R † is a pre-order, too, so we have both I X ≤ R and I X ≤ R † , hence I X ≤ R ∧ R † = E . Furthermore, we have E ◦ E = ( R ∧ R † ) ◦ ( R ∧ R † ) ≤ ( R ◦ ( R ∧ R † )) ∧ ( R † ◦ ( R ∧ R † )) ≤ ( R ◦ R ) ∧ ( R ◦ R † ) ∧ ( R † ◦ R ) ∧ ( R † ◦ R † ) ≤ ( R ◦ R ) ∧ ( R † ∧ R † ) ≤ R ∧ R † = E, where we used Lemma A.1 for the first two inequalities, and the fact that both R and R † are pre-orders in the last inequality. Finally, using Lemma A.2, we obtain E † = ( R ∧ R † ) † = R † ∧ R = E. (cid:3) The following lemma is essentially the dual of [3, Lemma 8.5] in qSet . Lemma 5.4.
Let X be a quantum set, and let E ∈ qRel ( X , X ) be an equivalence relationsuch that E = I X . Then there exists a function G : X → X such that G = I X , G ≤ E , G ◦ G = I X and G † = G − = G . roof. We consider two cases. The first is E ( X, X ) = I X ( X, X ) for each X ∝ X ; the secondis E ( X , X ) = I X ( X , X ) for some X ∈ X . We consider the latter case first. By 5.2, E ( X , X ) us a unital C*-subalgebra of L ( X , X ) , and by assumption, it is larger than I X ( X , X ) = C X . Thus, it contains a non-trivial projection p : X → X . Hence, it alsocontains u := 1 − p , which is a unitary X → X that satisfies u = u † = u − . Now define G ∈ qRel ( X , X ) by G ( X, Y ) = C u, X = Y = X ; C X , X = Y = X , , X = Y. Clearly G ( X, Y ) = G ( Y, X ) † = G † ( X, Y ) for all X, Y ∝ X , so G = G † . Moreover, wehave G ( X , X ) ≤ E ( X , X ) by construction, hence G ≤ E . A direct calculation yields ( G ◦ G )( X, X ) = C X for all X ∝ X , with G ( X, Y ) vanishing otherwise; hence G = I X .In the other case, since E = I X but E ( X, X ) = I X ( X, X ) for all X ∝ X , there exist distinct X , X ∝ X such that E ( X , X ) = I X ( X , X ) = 0 . Hence, let a ∈ E ( X , X ) be non-zero.Its adjoint a † is in E ( X , X ) † = E † ( X , X ) = E ( X , X ) . We compute that a † a ∈ E ( X , X ) · E ( X , X ) ≤ _ Y ∝ X E ( Y, X ) · E ( X , Y )= ( E ◦ E )( X , X ) ≤ E ( X , X ) = I ( X , X ) = C X , and similarly, aa † ∈ C X . Let u = a/ k a k ; then it follows that u † u = 1 X and uu † = 1 X .Now define G ∈ qRel ( X , X ) by G ( X, Y ) = C u, X = X , Y = X ; C u † , X = X , Y = X ; C X , X = X = Y = X , , otherwise . Here too, we find that G † ( X, Y ) = G ( Y, X ) † = G ( X, Y ) , so G = G † , and G ( X, Y ) ≤ E ( X, Y ) by construction. A direct calculation yields ( G ◦ G )( X, X ) = C X for all X ∝ X , with G ( X, Y ) vanishing otherwise; so here too, G = I X .We conclude that in both cases we have G ≤ E , and G = G † , and G ◦ G = I X . Hence G † ◦ G = G ◦ G = I X = G ◦ G = G ◦ G † , so G : X → X is a function. (cid:3)
Theorem 5.5.
The category qPOS is complete. More specifically, given a diagram of shape A consisting of objects X α for α ∈ A and monotone maps F f : X α → X β for each morphism f : α → β , the limit ( X , R ) in qPOS consists of the limit X in qSet , and R = ^ α ∈ A F † α ◦ R α ◦ F α , where the functions F α : X → X α are the limiting functions of this diagram in qSet .Proof. Firstly, we note that [3, Proposition 8.7] ensures that the diagram of shape A in thestatement indeed has a limit X in qSet . Let F α : X → X α be the limiting functions, i.e.,such that F f ◦ F α = F β for each function f : α → β in A . et E = V α ∈ A F † α ◦ F α ; we claim that E = I X . Assume otherwise. The binary relation E isa pre-order on X by Lemma 1.8. Furthermore, E clearly satisfies E † = E , and it is thereforean equivalence relation. We apply Lemma 5.4 to obtain a function G : X → X such that G = I X and G ≤ E , and we calculate that F α ◦ G ≤ F ◦ E ≤ F α ◦ F † α ◦ F α ≤ I X ◦ F α = F α for each α ∈ A . We conclude by Lemma A.7 that F α ◦ G = F α . This equality holds for each α ∈ A , and hence G = I X , by the universal property of the limit in qSet , contradicting ourchoice of G . Therefore, E = I .Similarly, let R = V α ∈ A F † α ◦ R α ◦ F α ; we claim that R is an order on X . Indeed, it is apre-order by Lemma 1.8, and it is antisymmetric by the following calculation: R ∧ R † = ^ α,β ∈ A ( F † α ◦ R α ◦ F α ) ∧ ( F † β ◦ R β ◦ F β ) ≤ ^ α ∈ A ( F † α ◦ R α ◦ F α ) ∧ ( F † α ◦ R α ◦ F α )= ^ α ∈ A F † α ◦ ( R α ∧ R † α ) ◦ F α = ^ α ∈ A F † α ◦ F α = E = I X We have used Proposition A.6 in the second equality. We conclude that R is indeed an orderon X .The definition of R trivially implies that R ≤ F † α ◦ R α ◦ F α , which expresses that F α ismonotone. Thus, we have a cone on the given diagram in qPOS , and it remains only toshow that it is a limiting cone.Let ( Y , T ) be a quantum poset, and let C α : Y → X α , for α ∈ A , be monotone mapsthat together form a cone. Since X is the limit of the X α in qSet , it follows that thereis a unique function H : Y → X such that C α = F α ◦ H. By the monotonicity of C α ,we have T ≤ C † α ◦ R α ◦ C α = ( F α ◦ H ) † ◦ R α ◦ F α ◦ H = H † ◦ F † α ◦ R α ◦ F α ◦ H. Hence, T ≤ V α ∈ A H † ◦ F † α ◦ R α ◦ F α ◦ H = H † ◦ (cid:0)V α ∈ A F † α ◦ R α ◦ F α (cid:1) ◦ H = H † ◦ R ◦ H, where wehave used Proposition A.6 in the penultimate equality. Hence, H is monotone. We havethus established that the functions F α , for α ∈ A , together form a limiting cone and that,more generally, the category qPOS is complete. (cid:3) Cocompleteness
We show that the category qPOS is cocomplete. Unlike limits in qPOS , colimits in qPOS cannot be formed simply by ordering the corresponding colimits in qSet . However,coproducts in qPOS are simply coproducts in qPOS , ordered appropriately, and we beginwith this special case.The categories qRel and qSet are cocomplete [3]. The coproduct of an indexed family {X α } α ∈ A of quantum sets is the same in both categories, and it is easiest to characterizewhen the quantum sets X α has no atoms in common. In this special case, the coproduct U α ∈ A = U α ∈ A X α . In the general case, some of the quantum sets X α may have atoms incommon, but we may replaces these quantum sets by isomorphic quantum sets to avoid thisnuisance. This defines the coproduct of an arbitrary family up to isomorphism. Lemma 6.1.
Let {X α } α ∈ A and {Y β } β ∈ B be collections of quantum sets, and let X = U α ∈ A X α and Y = U β ∈ B Y β be their coproduct. Let J α : X α → X and K β : Y β → Y be the canonicalinjections. We have all of the following: (a) For all α ∈ A , J † α ◦ J α = I X α , and for all distinct α , α ∈ A , J † α ◦ J α = ⊥ . (b) Let
R, S ∈ qRel ( X , Y ) . Then, the following are equivalent: R ≤ S , (2) R ◦ J α ≤ S ◦ J α for all α ∈ A , (3) K † β ◦ R ≤ K † β ◦ S for all β ∈ B , and (4) K † β ◦ R ◦ J α ≤ K † β ◦ S ◦ J α for all α ∈ A and β ∈ B . (c) For each α ∈ A and n ∈ N , let T α,n ∈ qRel ( X α , Y ) . Then, " ^ n ∈ N T α,n : α ∈ A = ^ n ∈ N [ T α,n : α ∈ A ] . (d) Assume A = B . For each α ∈ A and n ∈ N , let T α,n ∈ qRel ( X α , Y α ) . Then, ] α ∈ A ^ n ∈ N T α,n ! = ^ n ∈ N ] α ∈ A T α,n ! . Proof.
Without loss of generality, we can assume that the quantum sets X α are pairwisedisjoint; hence, J α = J XX α [3, Definition 8.2]. Similarly, we can assume that the quantum sets Y β are pairwise disjoint; hence, K β = K YY β . Then, (a) follows from a direct calculation.For (b), it is clear that (1) implies (2) and (3). It is also clear both that (2) implies (4)and that (3) implies (4). So we only have to show that (4) implies (1). Assume (4), and fix X ∝ X and Y ∝ Y . The Hilbert space X is an atom of X α for some α ∈ A , and similarly, theHilbert space Y is an atom of Y β for some β ∈ B . Applying Lemma A.5, we compute that R ( X, Y ) = ( K † β ◦ R ◦ J α )( X, Y ) ≤ ( K † β ◦ S ◦ J α )( X, Y ) = S ( X, Y ) . We now vary X ∝ X and Y ∝ Y to conclude that R ≤ S , i.e., to conclude (1).For (c), we compute that for each α ∈ A , " ^ n ∈ N T α,n : α ∈ A ◦ J α = ^ n ∈ N T α ,n = ^ n ∈ N ([ T α,n : α ∈ A ] ◦ J α ) = ^ n ∈ N [ T α,n : α ∈ A ] ! ◦ J α , where we apply Proposition A.6 for the last equality. Claim (c) now follows from (b).For (d), we compute that for each α ∈ A : ] α ∈ A ^ n ∈ N T α,n !! ◦ J α = ^ n ∈ N T α ,n = ^ n ∈ N ] α ∈ A T α,n ! ◦ J α ! = ^ n ∈ N ] α ∈ A T α,n !! ◦ J α , where we used Proposition A.6 in the last equality. Claim (d) now follows from (b). (cid:3) Proposition 6.2.
Let { ( X α , R α ) } α ∈ A be a collection of quantum posets. Let X = U α ∈ A X α be the coproduct of the X α in qSet , and let R = U α ∈ A R α be the coproduct of the R α asmorphisms in qRel , i.e., the unique R ∈ qRel ( X , X ) such that, for each α ∈ A , (1) R ◦ J α = J α ◦ R α . Then ( X , R ) is the coproduct of the ( X α , R α ) in qPOS , and the canonical injections J α : X α → X are order embeddings.Proof. Fix α, β ∈ A . By Lemma 6.1, we have J † β ◦ J α = ∆ α,β I X α ; hence, using that R α is anorder, we compute that J † β ◦ I X ◦ J α = J † β ◦ J α ◦ I X α ≤ J † β ◦ J α ◦ R α = J † β ◦ R ◦ J α , † β ◦ R ◦ R ◦ J α = J † β ◦ R ◦ J α ◦ R α = J † β ◦ J α ◦ R α ◦ R α ≤ J † β ◦ J α ◦ R α = J † β ◦ R ◦ J α ,J † β ◦ ( R ∧ R † ) ◦ J α = ( J † β ◦ R ◦ J α ) ∧ ( J † β ◦ R † ◦ J α ) = ( J † β ◦ R ◦ J α ) ∧ (( R ◦ J β ) † ◦ J α )= ( J † β ◦ J α ◦ R α ) ∧ (( J β ◦ R β ) † ◦ J α ) = ( J † β ◦ J α ◦ R α ) ∧ ( R † β ◦ J † β ◦ J α )= (∆ α,β I X α ◦ R α ) ∧ ( R † β ◦ ∆ α,β I X α ) = ∆ α,β ( R α ∧ R † α )= ∆ α,β I X α = J † β ◦ I X ◦ J α , We have used Proposition A.6 for the calculation of J † β ◦ ( R ∧ R † ) ◦ J α . Since these (in)equalitieshold for each α, β ∈ A , it follows from Lemma 6.1 that I X ≤ R , R ◦ R ≤ R , and R ∧ R † = I X , i.e., that R is an order on X . Now, we find from Equation (1) and Lemma 6.1 that J † α ◦ R ◦ J α = J † α ◦ J α = I X α ; hence, J α is an order embedding.Let ( Y , S ) be a quantum poset, and let F α : X α → Y be a collection of monotone maps.We need to check that [ F α : α ∈ A ] : X → Y is monotone, too. For each β ∈ A , we have [ F α : α ∈ A ] ◦ R ◦ J β = [ F α : α ∈ A ] ◦ J β ◦ R β = F β ◦ R β ≤ S ◦ F β = S ◦ [ F α : α ∈ A ] ◦ J β , where we use Equation (1) in the first and the last equality, whereas the inequality followsby the monotonicity of F β . Since the resulting inequality holds for each β ∈ A , it followsfrom Lemma 6.1 that [ F α : α ∈ A ] ◦ R ≤ S ◦ [ F α : α ∈ A ] . Therefore, [ F α : α ∈ A ] is indeedmonotone, which concludes the proof that ( X , R ) = U α ∈ A ( X α , R α ) in qPOS . (cid:3) Proposition 6.3.
Let {X α } α ∈ A be a collection of quantum sets, let {Y α , S α } be a collectionof quantum posets, let X = U α ∈ A X α in qSet , and let ( Y , S ) = U α ( Y α , S α ) in qPOS . Foreach α ∈ A , let F α and G α be functions from X α to Y α such that F α ⊑ G α . Let F = U α ∈ A F α and G = U α ∈ A G α be functions from X to Y . Then, F ⊑ G .Proof. Since F α ⊑ G α , we have S α ◦ G α ≤ S α ◦ F α . For each β ∈ A , let J β : X β → X be thecanonical injection. We find that S ◦ G ◦ J β = ] α ∈ A S α ◦ G α ! ◦ J β = S β ◦ G β ≤ S β ◦ F β = ] α ∈ A S α ◦ F α ! ◦ J β = S ◦ F ◦ J β . Hence, it follows by Lemma 6.1 that S ◦ G ≤ S ◦ F , i.e., F ⊑ G . (cid:3) Next, we aim to show that qPOS is cocomplete. Since we have already shown that it hasall coproducts, it is sufficient to show that is has coequalizers. Since it is difficult to give anexplicit description of coequalizers in qSet , we choose to follow a more abstract route forwhich we recall that a category is wellpowered if the subobjects of any object form a set, co-wellpowered if the quotient objects of any object form a set, and extremally co-wellpowered if those quotient objects that are represented by extremal epimorphisms form a set. Here, werecall that an epimorphism E is said to be extremal if E = M ◦ F for some monomorphism M implies that M is an isomorphism. For us, the importance of these concepts is that anycategory that is complete, well-powered and extremally co-well powered has all coequalizers.This theorem is originally proven in [5, Theorem 5.11], but also stated in [2] as Exercise 12J. Theorem 6.4.
The category qPOS is wellpowered and extremally co-wellpowered. It istherefore cocomplete.Proof.
The category W ∗ of von Neumann algebras and unital normal ∗ -homomorphismsis wellpowered and co-wellpowered: the subjects of a von Neumann algebra correspond to ts unital ultraweakly closed ∗ -subalgebras, and the quotient objects of a von Neumannalgebra correspond to its ultraweakly closed two-sided ideals. Hence, the full subcategory M ∗ of hereditarily atomic von Neumann algebras is also wellpowered and co-wellpowered.Therefore, qSet is also wellpowered and co-wellpowered [3, Theorem 7.4].Let ( X , R ) be a quantum poset, and choose representatives for the subobjects of X . Asubobject of X is an equivalence class of pairs ( W , M ) , with W a quantum set and M a monomorphism W → X . Two such pairs, ( W , M ) and ( W , M ) , are defined to beequivalent if they are isomorphic as objects in the slice category qSet / X , i.e., if there isan isomorphism F : W → W such that M = M ◦ F . We choose a family of pairs { ( W α , M α ) } α ∈ Sub( X ) to represent these equivalence classes.Let (( W , T ) , M ) represent a subobject of ( X , R ) in qSet . Thus, ( W , T ) is a poset, and M is a monomorphism from ( W , T ) to ( X , R ) . By Lemma 3.1, M is an injection, i.e., amonomorphism in qSet . By our choice of the pairs ( W α , M α ) , for α ∈ Sub( X ) , there exista subobject α ∈ Sub( X ) and an isomorphism F : W α → W such that M α = M ◦ F . Thus, F is an isomorphism in qPOS from the quantum poset ( W α , F † ◦ T ◦ F ) to the quantumposet ( W , T ) such that M α = M ◦ F . The function M is monotone by the definition ofa subobject, and the function F is monotone because it is an isomorphism, so M α is alsomonotone. Therefore, we have show that (( W , T ) , M ) is isomorphic to a pair of the form (( W α , T ′ ) , M α ) for some α ∈ Sub( X ) and some order T ′ on ω α . Since qSet is wellpowered, Sub( X ) is a set, and furthermore, for each α ∈ Sub( X ) , the orders on W α form a set. Weconclude that the subobjects of (( W , T ) , M ) in qPOS form a set, and more generally, that qPOS is well-founded.The proof that qPOS is extremally co-wellpowered is entirely similar. It replaces monomor-phisms into an arbitrary quantum poset (( W , T ) , M ) with extremal epimorphisms out of anarbitrary quantum poset (( W , T ) , M ) , and it appeals to Lemma 3.4 instead of Lemma 3.1.Hence, the category qPOS is both wellpowered and extremally co-wellpowered. By Theorem5.5, it is complete, so by [5, Theorem 5.11], it has all coequalizers. It also has all coproductsby Proposition 6.2. Therefore, qPOS is cocomplete. (cid:3) Monoidal product
We show that qPOS is a symmetric monoidal category, with all monoidal structure inher-ited from qSet . This monoidal structure on qSet , and more generally on qRel , is termedthe Cartesian product, and it is notated × , but it is not the category-theoretic product. Thisterminology and this notation is justified by the fact that this monoidal structure generalizesthe family Cartesian product of ordinary sets in a sense appropriate to the noncommutativedictionary.Let X and Y be quantum sets. Their Cartesian product X × Y is the quantum set definedby
At(
X × Y ) = { X ⊗ Y : X ∈ X , Y ∈ Y } . Furthermore, if R is some binary relation on X and S is some binary relation on Y , then their product R × S is the binary relation on X × Y defined by ( R × S )( X ⊗ Y , X ⊗ Y ) = R ( X , X ) ⊗ S ( Y , Y ) , for X , X ∝ X and Y , Y ∝ Y . Proposition 7.1.
Let ( X , R ) and ( Y , S ) be quantum posets. The, ( X × Y , R × S ) is also aquantum poset.Proof. Verifying that R × S is a quantum pre-order is routine. Furthermore, we computethat ( R × S ) ∧ ( R × S ) † = ( R × S ) ∧ ( R † × S † ) = ( R ∧ R † ) × ( S ∧ S † ) = I X × I Y = I X , Y . The ey, second equality may be checked atom by atom. Thus, the pre-order R × S is in fact anorder on X × Y . (cid:3) Lemma 7.2.
Let ( X , R ) , ( X , R ) , ( Y , S ) , ( Y , S ) be quantum posets, and let F : ( X , R ) → ( X , R ) and G : ( Y , S ) → ( Y , S ) be monotone. Then F × G : ( X × Y , R × S ) → ( X × Y , R × S ) is monotone.Proof. ( F × G ) ◦ ( R × S ) = ( F ◦ R ) × ( G ◦ S ) ≤ ( R ◦ F ) × ( S ◦ G ) = ( R × S ) ◦ ( F × G ) (cid:3) Lemma 7.3.
Let ( X , R ) and ( Y , S ) be quantum pre-orders. Let P : X × Y → X and Q : X × Y → Y be the projection functions, in the sense of Appendix B. Then P ◦ ( R × S ) = R ◦ P, Q ◦ ( R × S ) = Q ◦ S, hence P and Q are monotone.Proof. The proof is by straight-forward calculation, atom by atom. (cid:3)
Proposition 7.4.
Let ( X , R ) , ( Y , S ) and ( Z , T ) be quantum posets, and let M : Z → X × Y be a function. Then M is monotone if and only if P ◦ M and Q ◦ M are both monotone.Proof. Assume M is monotone. By Lemma 7.3, both P and Q are monotone. It now followsfrom Lemma 1.10 that P ◦ M and Q ◦ M are both monotone. For the converse, assume that P ◦ M : Z → X and Q ◦ M : Z → Y are both monotone. By Definition 1.9 this means that P ◦ M ◦ T ◦ M † ◦ P † ≤ R, Q ◦ M ◦ T ◦ M † ◦ Q † ≤ S. Applying Lemma B.6 to the relation M ◦ T ◦ M † on X × Y yields M ◦ T ◦ M † ≤ ( P ◦ M ◦ T ◦ M † ◦ P † ) × ( Q ◦ M ◦ T ◦ M † ◦ Q † ) ≤ R × S, which expresses that M is indeed monotone. (cid:3) Theorem 7.5. ( qPOS , × , ( , I )) is a symmetric monoidal category.Proof. By Proposition 7.1 and Lemma 7.2, the monoidal product × on qSet induces abifunctor × on qPOS . The monoidal structure of qSet is inherited from the monoidalstructure on qRel [3, Theorem 3.6]. The statement follows now from verifying that theassociator, the unitors and the braiding are order isomorphisms, which is routine. (cid:3) The monoidal product × on qSet is not the category-theoretic product. It generalizes thefamiliar Cartesian product of ordinary sets in another sense that is appropriate to quantumphysics and noncommutative geometry. Thus, for example, the quantum set ‘ R × ‘ R modelspairs of real numbers, which must not be product of ‘ R with itself in the category of quantumsets and functions because functions into ‘ R model observables. Most pairs of observablesare not compatible, so most pairs of functions into ‘ R should not correspond to a functioninto ‘ R × ‘ R . For functions F and G from a quantum set Z to ‘ R , we write ( F, G ) for thecorresponding function from Z to ‘ R × ‘ R , if such a function exists . More generally, for eachfunction F : Z → X and each function G : Z → Y , we write ( F, G ) for the unique function Z → X × Y such that P ◦ ( F, G ) = F and Q ◦ ( F, G ) = G if such a function exists , where : X × Y → X and Q : X × Y → Y are the canonical projection functions [3, Sections I.Band X]. In this case, we say that F and G are compatible . Proposition 7.6.
Let ( X , R ) and ( Y , S ) be quantum posets, and let Z be a quantum set.Let F , F : Z → X and G , G : Z → Y be functions. If F is compatible with G and F iscompatible with G , then ( F , G ) ⊑ ( F , G ) if and only if both F ⊑ F and G ⊑ G . Proof.
Assume ( F , G ) ⊑ ( F , G ) . Since P and Q are monotone by Lemma 7.3, it followsthat F = P ◦ ( F , G ) ⊑ P ◦ ( F , G ) = F and G = Q ◦ ( F , G ) ⊑ Q ◦ ( F , G ) = G . Nowassume that F ⊑ F and G ⊑ G . This means that F ◦ F † ≤ R and G ◦ G † ≤ S ; hence, byProposition B.7 it follows that ( F , G ) ≤ (( F ◦ F † ) × ( G ◦ G † )) ◦ ( F , G ) ≤ ( R × S ) ◦ ( F , G ) , which expresses that ( F , G ) ⊑ ( F , G ) . (cid:3) Corollary 7.7.
Let ( X , R ) and ( Y , S ) be quantum posets, and let V , W be quantum sets.Let F , F : V → X and G , G : W → Y be functions. Then F × G ⊑ F × G if and onlyif F ⊑ F and G ⊑ G .Proof. Consider the following diagram for i = 1 , : V V × W WX X × Y Y . F i P PF i × G i G i P Q
We first show that F ⊑ F is equivalent to F ◦ P ⊑ F ◦ P . If F ⊑ F , then F ◦ P ⊑ F ◦ P by Lemma 4.2. Conversely, if F ◦ P ⊑ F ◦ P , then F ◦ P ◦ ( F ◦ P ) † ≤ R ; hence F ◦ F † = F ◦ P ◦ P † ◦ F † = F ◦ P ◦ ( F ◦ P ) † ≤ R, where we appeal to the surjectivityof P (cf. Lemma B.1) in the third equality. Thus F ⊑ F . In a similar way we find that G ⊑ G if and only if G ◦ Q ⊑ G ◦ Q .Since for each i = 1 , , we have F i × G i = ( F i ◦ P, G i ◦ Q ) , it follows now from Proposition7.6 that F × G ⊑ F × G if and only if F ◦ P ⊑ F ◦ P and G ◦ Q ⊑ G ◦ Q . (cid:3) Monoidal closure
We show that the category qPOS is monoidal closed. Let ( Y , S ) and ( Z , T ) be quantumposets. Intuitively, we construct the hom quantum poset [( Y , S ) , ( Z , T )] ⊑ by taking thelargest subset of the quantum function set Z Y that consists of monotone functions andordering them pointwise. Definition 8.1.
Let ( X , R ) and ( Y , S ) be quantum sets equipped with binary relations.Then we say that a function F : X → Y is a homomorphism ( X , R ) → ( Y , S ) if F ◦ R ≤ S ◦ F .Note that if R and S are orders on X and Y , respectively, then F is a homomorphism if andonly if F is monotone.For the next lemma, recall that qSet is monoidal closed [3, Theorem 9.1], so for each pairof quantum sets Y and Z , there is an exponential object Z Y . Since qSet is monoidal closed,the monoidal product × has a right adjoint. Thus, × preserves colimits, in particular themonoidal product distributes over coproducts; this fact will be used in the proof of the nextlemma. For each subset W ⊆ Z Y , we write J W : W ֒ → Z Y for the canonical inclusion. emma 8.2. Let ( Y , S ) and ( Z , T ) be quantum sets equipped with reflexive binary relations.Then, there exists a quantum set W ⊆ Z Y that is the largest subset of Z Y such that Eval ◦ ( J W × I Y ) : ( W × Y , I W × S ) → ( Z , T ) is a homomorphism, and there exists a reflexive binary relation Q on W that is the largestbinary relation on W such that Eval ◦ ( J W × I Y ) : ( W × Y , Q × S ) → ( Z , T ) is a homomorphism. Moreover, W and Q satisfy and are uniquely determined by the followingproperties:(1) The function Eval ◦ ( J W × I Y ) is a homomorphism ( W × Y , Q × S ) → ( Z , T ) .(2) For every quantum set X equipped with a reflexive binary relation R , and everyhomomorphism F : ( X × Y , R × S ) → ( Z , T ) , there exists a unique homomorphism G : ( X , R ) → ( W , Q ) such that F = Eval ◦ ( J W × I Y ) ◦ ( G × I Y ) : ( X × Y , R × S )( W × Y , Q × S ) ( Z , T ) FG × I Y Eval ◦ ( J W × I Y ) Proof.
To show that W is well defined, we show that the set F of subsets V ⊆ Z Y such that Eval ◦ ( J V × I Y ) is a homomorphism from ( V × Y , I V × S ) to ( Z , T ) is closed under subsetsand arbitrary disjoint unions. For all V ∈ F , and all V ′ ⊆ V , we write J VV ′ for the canonicalinclusion of V ′ into V , and we calculate that Eval ◦ ( J V ′ × I Y ) ◦ ( I V ′ × S ) = Eval ◦ ( J V × I Y ) ◦ ( J VV ′ × I Y ) ◦ ( I V ′ × S )= Eval ◦ ( J V × I Y ) ◦ ( I V × S ) ◦ ( J VV ′ × I Y ) ≤ T ◦ Eval ◦ ( J V × I Y ) ◦ ( J VV ′ × I Y )= T ◦ Eval ◦ ( J V ′ × I Y ) . Thus, V ′ ∈ F . We conclude that F is closed under subsets. For all V , V ∈ F , if V and V are disjoint in the sense that they have no atoms in common, then we make the identification V ⊎ V = V ∪ V , and we calculate that Eval ◦ ( J V ∪V × I Y ) ◦ ( I V ∪V × S ) = Eval ◦ ( J V ∪V × S ) = Eval ◦ ([ J V , J V ] × S )= Eval ◦ [ J V × S, J V × S ] = [Eval ◦ ( J V × S ) , Eval ◦ ( J V × S )]= [Eval ◦ ( J V × I Y ) ◦ ( I V × S ) , Eval ◦ ( J V × I Y ) ◦ ( I V × S )] ≤ [ T ◦ Eval ◦ ( J V × I Y ) , T ◦ Eval ◦ ( J V × I Y )]= T ◦ Eval ◦ [ J V × I Y , J V × I Y ]= T ◦ Eval ◦ ([ J V , J V ] × I Y ) = T ◦ Eval ◦ ( J V ∪V × I Y ) . Apart from a single step in which we apply our assumption that V , V ∈ F , the entirecalculation is category-theoretic. It is simplified by the fact that ( V ∪ V ) × Y = ( V ×Y ) ∪ ( V × Y ) . The same argument applies to arbitrarily large families of pair-wise disjointelements of F . We conclude that F is closed under arbitrary disjoint unions. Therefore, F has a largest subset, i.e., W is well defined. e construct the binary relation Q on W directly as the join of all the binary relationson W satisfying the desired condition. For each binary relation Q ′ on W , the function Eval ◦ ( J W × I Y ) is a homomorphism from ( W × Y , Q ′ × S ) to ( Z , T ) if and only if Eval ◦ ( J W × I Y ) ◦ ( Q ′ × S ) ≤ T ◦ Eval ◦ ( J W × I Y ) , simply by definition. The composition of binary relations respects arbitrary joins, so thejoin of all binary relations Q ′ satisfying this inequality also satisfies this inequality. This isexactly the binary relation Q . It is reflexive because Eval ◦ ( J W × I Y ) is a homomorphismfrom ( W × Y , I W × S ) to ( Z , T ) , simply by the definition of W .Property (1) holds by construction. To establish property (2), let F be a homomorphismfrom ( X × Y , R × S ) to ( Z , T ) . The universal property of the quantum function set Z Y guarantees the existence of a function G : X → Z Y satisfying Eval ◦ ( G × I Y ) = F . Thefunction G factors through its range V ⊆ Z Y , yielding a surjective function G : X → V that satisfies J V ◦ G = G . The surjectivity of G is equivalent to the equation G ◦ G † = I V ,which we apply in the following calculation: Eval ◦ ( J V × I Y ) ◦ ( I V × S ) = Eval ◦ ( J V × I Y ) ◦ ( G × I Y ) ◦ ( G † × I Y ) ◦ ( I V × S )= Eval ◦ ( G × I Y ) ◦ ( I X × S ) ◦ ( G † × I Y )= F ◦ ( I X × S ) ◦ ( G † × I Y )= F ◦ ( R × S ) ◦ ( G † × I Y ) ≤ T ◦ F ◦ ( G † × I Y )= T ◦ Eval ◦ ( G × I Y ) ◦ ( G † × I Y )= T ◦ Eval ◦ ( J V × I Y ) ◦ ( G × I Y ) ◦ ( G † × I Y )= T ◦ Eval ◦ ( J V × I Y ) Thus,
Eval ◦ ( J V × I Y ) is a homomorphism from ( V × Y , I V × S ) to ( Z , T ) , i.e., V ∈ F .The maximality of W now implies that V is a subset of W , so we can define G = J WV ◦ G .The function G makes the diagram commute on the level of quantum sets and functions: Eval ◦ ( J W × I Y ) ◦ ( G × I Y ) = Eval ◦ (( J W ◦ G ) × I Y ) = Eval ◦ (( J W ◦ J WV ◦ G ) × I Y )= Eval ◦ (( J V ◦ G ) × I Y ) = Eval ◦ ( G × I Y ) = F For any other function G ′ : X → W making the diagram commute, we have
Eval ◦ (( J W ◦ G ′ ) × I Y ) = Eval ◦ (( J W ◦ G ) × I Y ) . The universal property of the evaluation function
Eval : Z Y × Y → Z then implies that J W ◦ G ′ = J W ◦ G . The inclusion J W is injective, so we conclude that G ′ = G . Therefore, G is the unique function making the diagram commute.To show that G is a homomorphism, we first show that Eval ◦ ( J W × I Y ) is a homomorphismfrom ( W × Y , ( G ◦ R ◦ G † ) × S ) to ( Z , T ) : Eval ◦ ( J W × I Y ) ◦ (( G ◦ R ◦ G † ) × S ) = Eval ◦ ( J W × I Y ) ◦ ( G × I Y ) ◦ ( R × S ) ◦ ( G † × I Y )= F ◦ ( R × S ) ◦ ( G † × I Y ) ≤ T ◦ F ◦ ( G † × I Y ) T ◦ Eval ◦ ( J W × I Y ) ◦ ( G × I Y ) ◦ ( G † × I Y )= T ◦ Eval ◦ ( J W × I Y ) ◦ (( G ◦ G † ) × I Y ) ≤ T ◦ Eval ◦ ( J W × I Y ) By definition of Q , we find that G ◦ R ◦ G † ≤ Q . Composing on the right by G , we concludethat G ◦ R ≤ G ◦ R ◦ G † ◦ G ≤ Q ◦ G . In other words, we conclude that G is a homomorphismfrom ( X , R ) to ( W , Q ) .We have shown that for every homomorphism F : ( X × Y , R × S ) → ( Z , T ) , there is indeeda unique homomorphism G : ( X , R ) → ( W , Q ) making the diagram commute. This universalproperty determines the structure ( W , Q ) up to canonical isomorphism, in the usual way. Toestablish equality, let ( W ′ , Q ′ ) be another structure with W ⊆ Z Y that satisfies properties(1) and (2). The canonical isomorphism G : ( W ′ , Q ′ ) → ( W , Q ) makes the appropriatediagram commute, and therefore, it satisfies Eval ◦ ( J W ′ × I Y ) = Eval ◦ ( J W × I Y ) ◦ ( G × I Y ) .As before, we appeal to the universal property of Eval to infer that J W ′ = J W ◦ G . Byproposition 10.1 of [3], we conclude that W ′ is a subset of W . Similarly, W is a subsetof W ′ , so the two quantum sets are equal. The equation J W ′ = J W ◦ G now gives us J W ◦ G = J W ′ = J W = J W ◦ I W . Appealing to the injectivity of J W , we conclude that G = I W . This bijection is a homomorphism in both directions, so we have the followingchain of inequalities: Q = Q ◦ I W ≤ I W ◦ Q ′ = Q ′ = Q ′ ◦ I W ≤ I W ◦ Q = Q Therefore, W ′ = W and Q ′ = Q . (cid:3) Theorem 8.3.
The category qPOS is monoidal closed with respect to the monoidal product × , i.e., for each pair of quantum posets ( Y , S ) and ( Z , T ) , there exist a quantum poset ([ Y , Z ] ⊑ , Q ) and a monotone function Eval ⊑ : [ Y , Z ] ⊑ × Y → Z such that for each quantumposet ( X , R ) and each monotone function F : X × Y → Z , there is a unique monotonefunction G : X → [ Y , Z ] ⊑ satisfying Eval ⊑ ◦ ( G × I Y ) = F : ( X × Y , R × S )([ Y , Z ] ⊑ × Y , Q × S ) ( Z , T ) . FG × I Y Eval ⊑ Furthermore, [ Y , Z ] ⊑ is the largest subset of Z Y such that evaluation restricted to [ Y , Z ] ⊑ ×Y is a monotone function ([ Y , Z ] ⊑ × Y , I × S ) → ( Z , T ) , and Q is the largest binary relationon [ X , Y ] ⊑ such that (2) Eval ⊑ ◦ ( Q × S ) ≤ T ◦ Eval ⊑ . Proof.
We define [ Y , Z ] ⊑ to be the quantum set W in Lemma 8.2, and Q is obtained via thesame lemma. Furthermore, we define Eval ⊑ := Eval ◦ ( J W × I Y ) . By definition of Eval ⊑ and Q it follows that Q is the largest binary relation on [ Y , Z ] ⊑ such that Equation (2) holds.We need to show that Q is an order on W . It satisfies I W ≤ Q by Lemma 8.2. To establishtransitivity, we compute that Eval ⊑ ◦ (( Q ◦ Q ) × S ) = Eval ⊑ ◦ (( Q ◦ Q ) × ( S ◦ S )) = Eval ⊑ ◦ ( Q × S ) ◦ ( Q × S ) ≤ T ◦ Eval ⊑ ◦ ( Q × S ) ≤ T ◦ Eval ⊑ ≤ T ◦ Eval ⊑ , here we use Equation (2) of Lemma 1.6 in the first equality. Since Q is the largest binaryrelation on W satisfying Equation (2), we obtain Q ◦ Q ≤ Q . Thus, we have proven that Q is a pre-order.Assume that Q is not an order, so Q ∧ Q † = I W . Let E = Q ∧ Q † . Then E is an equivalencerelation by Lemma 5.3. Since Q is not an order, we have E = I W ; hence Lemma 5.4 yieldsa function K : W → W such that K = I W and K ≤ E .We have that Q × S ≤ Eval †⊑ ◦ Eval ⊑ ◦ ( Q × S ) ≤ Eval †⊑ ◦ T ◦ Eval ⊑ by definition ofa function and Equation (2), whence Q × I Y ≤ Eval †⊑ ◦ T ◦ Eval ⊑ . Applying the adjointoperation to both sides of this inequality, we obtain Q † × I Y ≤ Eval †⊑ ◦ T † ◦ Eval ⊑ . Thus, K × I Y ≤ E × I Y = ( Q ∧ Q † ) × I Y = ( Q × I Y ) ∧ ( Q † × I Y ) ≤ (Eval †⊑ ◦ T ◦ Eval ⊑ ) ∧ (Eval †⊑ ◦ T † ◦ Eval ⊑ )= Eval †⊑ ◦ ( T ∧ T † ) ◦ Eval ⊑ = Eval †⊑ ◦ Eval ⊑ , where we appeal to Lemma A.3 in the second equality and to Proposition A.6 in the penul-timate equality. Hence, Eval ⊑ ◦ ( K × I Y ) ≤ Eval ⊑ ◦ Eval †⊑ ◦ Eval ⊑ ≤ Eval ⊑ , and by LemmaA.7 it follows that Eval ⊑ ◦ ( K × I Y ) = Eval ⊑ , or equivalently, Eval ◦ ( J W × I Y ) ◦ ( E × I Y ) =Eval ◦ ( J W × I Y ) ◦ ( I W × I Y ) . By the universal property in Lemma 8.2, it follows that K × I Y = I W × I Y . By Lemma B.1 and Proposition B.2, we obtain K = K ◦ P ◦ P † = P ◦ ( K × I Y ) ◦ P † = P ◦ ( I W × I Y ) ◦ P † = I W ◦ P ◦ P † = I W , contradicting our choice of K . We conclude that Q must be an order. It then followsimmediately from Equation (2) that Eval ⊑ is monotone. The claimed universal property of ([ Y , Z ] ⊑ , Eval ⊑ ) is just the universal property of Lemma 8.2 because a monotone function isjust a homomorphism between two quantum sets equipped with preorders. (cid:3) Embedding into the powerset
Each ordinary poset ( A, ⊑ ) may be embedded into the poset (Pow( A ) , ⊆ ) by mappingeach element a ∈ A to its principal down set ↓ a := { a ′ ∈ A : a ′ ⊑ a } . We generalize thisproposition to the quantum setting. Definition 9.1.
Let X be a quantum set. Its powerset is defined to be the quantum set Pow( X ) := ‘ B X ∗ , with B = { , } . The membership relation is defined to be the uniquebinary relation Elem from X to Pow( X ) such that the following diagram in qRel commutes: X × X ∗ Pow( X ) × X ∗ ‘ B . E EvalElem † × I ‘1 † Here, {∗} → B is the function corresponding to ∈ B , and E : X × X ∗ → is the counitof the duality between X and X ∗ . The relation Elem † exists and is unique by the universalproperty of the counit.Ordering X ∗ flatly and B by ⊏ , we obtain a quantum poset [ X ∗ , ‘ B ] ⊑ , whose underlyingquantum set is Pow( X ) (Theorem 8.3). Hence, Pow( X ) is canonically ordered. ix a quantum set X and an order R on X . Then, P := E ◦ ( R † × I ) is a binary relationfrom X × X ∗ to , and there is a unique function ˜ P : X × X ∗ → ‘ B such that ‘1 † ◦ ˜ P = P [3, Theorem B.8]. Intuitively, ˜ P yields if and only if its first argument is above its secondargument. Appealing to the universal property of the evaluation function, we obtain afunction G : X →
Pow( X ) such that the following diagram in qSet commutes: X × X ∗ Pow( X ) × X ∗ ‘ B . G × I ˜ P Eval
We will show that this function G is an order embedding ( X , R ) → [( X ∗ , I ) , (‘ B , ‘ ⊑ )] ⊑ . Proposition 9.2.
Let ( Y , S ) be a quantum poset, and let Q be a binary relation from Y to . Then, Q ◦ S † = Q if and only if the unique function ˜ Q : Y → ‘ B such that ‘1 † ◦ ˜ Q = Q ismonotone.Proof. We reason in terms of the trace on qRel (Appendix C), as follows: ˜ Q is monotone ⇐⇒ ˜ Q ◦ S ◦ ˜ Q † ≤ ‘( ⊑ ) ⇐⇒ ˜ Q ◦ S ◦ ˜ Q † ⊥ ‘( ) = ‘0 ◦ ‘1 † ⇐⇒ Tr(( ˜ Q ◦ S ◦ ˜ Q † ) † ◦ ‘0 ◦ ‘1 † ) = ⊥ ⇐⇒ Tr(‘1 † ◦ ˜ Q ◦ S † ◦ ˜ Q † ◦ ‘0) = ⊥⇐⇒ Tr( Q ◦ S † ◦ ¬ Q † ) = ⊥ ⇐⇒ Q ◦ S † ⊥ ¬ Q ⇐⇒ Q ◦ S † ≤ Q ⇐⇒ Q ◦ S † = Q In this computation, we use that fact that ˜ Q † ◦ ‘0 = (‘0 † ◦ ˜ Q ) = ¬ ( ¬ ‘0 † ◦ ˜ Q ) = ¬ (‘1 † ◦ ˜ Q ) = ¬ Q ,because binary relations from ‘ B to correspond to projections in ℓ ∞ (‘ B ) and precompositionby functions corresponds to application of unital normal ∗ -homomorphisms [3, TheoremB.8]. (cid:3) Lemma 9.3.
The function ˜ P is monotone ( X × X ∗ , R × I ) → (‘ B , ‘ ⊑ ) . Furthermore, forall orders T on X , if ˜ P is monotone ( X × X ∗ , T × I ) → (‘ B , ‘ ⊑ ) , then T ≤ R .Proof. We calculate that P ◦ ( R × I ) † = E ◦ ( R † × I ) ◦ ( R † × I ) = E ◦ ( R † × I ) = P . Therefore,by Proposition 9.2, ˜ P is a monotone function ( X × X ∗ , R × I ) → (‘ B , ‘ ⊑ ) . Now, let T bean order on X , and assume that ˜ P is a monotone function ( X × X ∗ , T × I ) → (‘ B , ‘ ⊑ ) . ByProposition 9.2, we find that P ◦ ( T × I ) † = P . We now calculate that E ◦ ( R † × I ) = P = P ◦ ( T × I ) † = E ◦ ( R † × I ) ◦ ( T × I ) † = E ◦ (( T ◦ R ) † × I ) . We conclude that T ◦ R = R . Therefore T = T ◦ I ≤ T ◦ R = R , as claimed. (cid:3) Theorem 9.4.
Let X be a quantum set equipped with an order R . Let ˜ P : X × X ∗ → ‘ B bethe function defined by ‘1 † ◦ ˜ P = E ◦ ( R † × I ) . Equip X ∗ with the trivial order I X ∗ . Theunique monotone function G that makes the following diagram in qPOS commute is anorder embedding: X × X ∗ [ X ∗ , ‘ B ] ⊑ × X ∗ ‘ B . G × I ˜ P Eval ⊑ roof. By Lemma 9.3, ˜ P is monotone, so Theorem 8.3 guarantees the existence of such amonotone function G . We claim that G is injective. Let W be a quantum set, and let F and F be functions W → X . Assume that G ◦ F = G ◦ F . We now reason as follows: G ◦ F = G ◦ F = ⇒ ‘1 † ◦ Eval ⊑ ◦ ( G × I ) ◦ ( F × I ) = ‘1 † ◦ Eval ⊑ ◦ ( G × I ) ◦ ( F × I ) ⇐⇒ ‘1 † ◦ ˜ P ◦ ( F × I ) = ‘1 † ◦ ˜ P ◦ ( F × I ) ⇐⇒ E ◦ ( R † × I ) ◦ ( F × I ) = E ◦ ( R † × I ) ◦ ( F × I ) ⇐⇒ R † ◦ F = R † ◦ F ⇐⇒ F ⊑ F and F ⊑ F ⇐⇒ F = F . We conclude that G is monic in qSet , and it is therefore injective [3, Proposition 8.4].Let Q be the binary relation that orders [ X ∗ , ‘ B ] ⊑ . Since G is injective, the binary relation T = G † ◦ Q ◦ G is an order on X (Lemma 2.1). The function G is then an order embedding ( X , T ) → ([ X ∗ , ‘ B ] ⊑ , Q ) , and in particular, it is monotone. Hence, ˜ P is a monotone function ( X × X ∗ , T × I ) → (‘ B , ‘ ⊑ ) . By Lemma 9.3, T ≤ R . In other words, G † ◦ Q ◦ G ≤ R .However, since G is monotone, we also have G † ◦ Q ◦ G ≥ R . Therefore, G † ◦ Q ◦ G = R ; inother words, G is an order embedding ( X , R ) → ([ X ∗ , ‘ B ] ⊑ , Q ) . (cid:3) The quantum poset [ X ∗ , ‘ B ] ⊑ is nothing but Pow( X ) equipped with its canonical order.Thus, G is an order embedding X →
Pow( X ) . Appendix A. Quantum sets
We record a number of basic facts about quantum sets and the binary relations betweenthem in the sense of [3], which serves as our basic reference.
Lemma A.1.
Let W , X , Y and Z be quantum sets, let R ∈ qRel ( W , X ) , let { S i } i ∈ I ⊆ qRel ( X , Y ) and let T ∈ qRel ( Y , Z ) . Then, _ i ∈ I ( S i ◦ R ) = _ i ∈ I S i ! ◦ R, _ i ∈ I ( T ◦ S i ) = T ◦ _ i ∈ I S i ! , ^ i ∈ I ( S i ◦ R ) ≥ ^ i ∈ I S i ! ◦ R, ^ i ∈ I ( T ◦ S i ) ≥ T ◦ ^ i ∈ I S i ! . Proof.
We prove the last formula. For all atoms X ∝ X and Z ∝ Z , we calculate that ^ i ∈ I ( T ◦ S i ) ! ( X, Z ) = ^ i ∈ I ( T ◦ S i )( X, Z ) = ^ i ∈ I _ Y ∝ Y ( T ( Y, Z ) · S i ( X, Y )) ≥ _ Y ∝ Y ^ i ∈ I ( T ( Y, Z ) · S i ( X, Y )) ≥ _ Y ∝ Y T ( Y, Z ) · ^ i ∈ I S i ( X, Y ) ! = T ◦ ^ i ∈ I S i ! ( X, Z ) . The other three formulas are proved similarly. (cid:3)
Lemma A.2.
Let X and X be quantum sets, and let R , S , { T i } i ∈ I be binary relations from X to X . Then, (1) R ≤ S if and only if R † ≤ S † ; (cid:0)V i ∈ I T i (cid:1) † = V T † i ; (3) (cid:0)W i ∈ I T i (cid:1) † = W T † i ; (4) ( ¬ S ) † = ¬ ( S † ) .Proof. In each case, the proof proceeds by fixing arbitrary atoms X ∝ X and X ∝ X andthen verifying the equivalence or equality in question in the ( X , X ) -component, appealingto the same equivalence or equality for arbitrary subspaces of L ( X , X ) . (cid:3) Lemma A.3.
Let X , X , Y and Y be quantum sets. Let R ∈ qRel ( X , Y ) , and S ∈ qRel ( X , Y ) . Let { R α } α ∈ A be an indexed family in qRel ( X , Y ) , and let { S β } β ∈ B be anindexed family in qRel ( X , Y ) . Then, (a) ( R × S ) † = R † × S † ; (b) (cid:0)V α ∈ A R α (cid:1) × (cid:16)V β ∈ B S β (cid:17) = V α ∈ A V β ∈ B ( R α × S β ) ; (c) (cid:0)W α ∈ A R α (cid:1) × S = W α ∈ A ( R α × S ) .Proof. In each case, the proof proceeds by fixing arbitrary atoms X ∝ X , X ∝ X , Y ∝ Y and Y ∝ Y and then verifying the equality in question in the ( X ⊗ X , Y ⊗ Y ) -component,appealing to the same equality for arbitrary subspaces of L ( X , Y ) and L ( X , Y ) . (cid:3) The next lemma concerns inclusion functions [3, Definition 8.2]. For each atom X of aquantum set X we abbreviate J X := J XQ{ X } [3, Definition 2.3]. The function J X : Q{ X } → X is defined by J X ( X, X ) = C · X , with all other components vanishing. Lemma A.4.
Let X be a quantum set. Then I X = W X ∝ X J X ◦ J † X .Proof. It is sufficient to observe that for all X ∝ X , we have that ( J X ◦ J † X )( X, X ) = C · X ,with all the other components of J X ◦ J † X vanishing. (cid:3) The next lemma refers subsets of quantum sets [3, Definition 2.2(3)]. A quantum set X is said to be subset of a quantum set X if each atom of X is also an atom of X . Lemma A.5.
Let X and Y be quantum sets, let X ⊆ X , let Y ⊆ Y , and let R ∈ qRel ( X , Y ) . Then, ( R ◦ J X )( X, Y ) = R ( X, Y ) for all atoms X ∝ X and Y ∝ Y , and ( J Y ◦ R )( X, Y ) = R ( X, Y ) for all atoms X ∝ X and Y ∝ Y .Proof. Both equalities follow easily by direct computation. (cid:3)
Proposition A.6.
Let W , X , Y and Z be quantum sets, and let { S i } i ∈ I ⊆ qRel ( X , Y ) .Then, for all functions F : W → X and G : Z → Y , we have V i ∈ I ( S i ◦ F ) = (cid:0)V i ∈ I S i (cid:1) ◦ F and V i ∈ I ( G † ◦ S i ) = G † ◦ (cid:0)V i ∈ I S i (cid:1) . Proof.
The inequality V i ∈ I ( S i ◦ F ) ≥ (cid:0)V i ∈ I S i (cid:1) ◦ F follows from Lemma A.1, as does theinequality V i ∈ I ( S i ◦ F ◦ F † ) ≥ (cid:0)V i ∈ I ( S i ◦ F (cid:1) ) ◦ F † . We apply the latter inequality in thefollowing calculation: ^ i ∈ I ( S i ◦ F ) ≤ ^ i ∈ I ( S i ◦ F ) ! ◦ F † ◦ F ≤ ^ i ∈ I ( S i ◦ F ◦ F † ) ! ◦ F ≤ ^ i ∈ I S i ! ◦ F. Thus, we establish the first equality of the proposition. The second equality can be obtainedfrom the first by taking its adjoint and replacing S i and F by S † i and G , respectively. (cid:3) emma A.7. Let X and Y be quantum sets, and let F and G be functions X → Y . If F ≤ G , then F = G .Proof. Since we also have that F † ≤ G † , we find that G = G ◦ I X ≤ G ◦ F † ◦ F ≤ G ◦ G † ◦ F ≤ I Y ◦ F = F, whence F = G . (cid:3) Appendix B. Projection functions
The monoidal product of quantum sets generalizes the ordinary Cartesian product inthe sense that we have a natural isomorphism ‘ S × ‘ T ∼ = ‘( S × T ) for all sets S and T .Furthermore, for all quantum sets X and Y , we have projection functions P XX ×Y : X × Y → X and P YX ×Y : X × Y → Y [3, Section 10]. Explicitly, P XX ×Y and P YX ×Y are defined by theequations P XX ×Y ( X ⊗ Y, X ) = ρ X ( C X ⊗ L ( Y, C )) and P YX ×Y ( X ⊗ Y, Y ) = λ Y ( L ( X, C ) ⊗ C Y ) ,for X ∝ X and Y ∝ Y , with the other components vanishing, where where λ and ρ denotethe left and right unitors in FdHilb . For brevity, we will sometimes write P X = P XX ×Y and P Y = P YX ×Y . Lemma B.1.
Let X and Y be nonempty quantum sets, and let P X : X × Y → X and P Y : X × Y → Y be the two projection functions. Then, P X and P Y are surjective.Proof. For all atoms X ∝ X and Y ∝ Y , we have that ( P X ◦ P X † )( X, X ) ≥ P X ( X ⊗ Y, X ) · P X ( X ⊗ Y, X ) † = ρ X ( C X ⊗ L ( Y, C ))( C X ⊗ L ( Y, C ) † ) ρ † X = ρ X ( C X ⊗ C ) ρ †X = C X = I X ( X, X ) . Hence, P ◦ P † ≥ I X , that is, P is surjective. Similarly, Q is surjective. (cid:3) Let W be a quantum set, and let F : W → X and G : W → Y be functions. If thereexists a function ( F, G ) :
W → X × Y such that P X ◦ ( F, G ) = F and P Y ◦ ( F, G ) = G ,then it is clearly unique (Lemma B.1) [3, Theorem 7.4, Proposition 8.1]. This justifies thenotation ( F, G ) . However, such a function ( F, G ) need not exist; if it does, we say that F and G are compatible [3, Definition 10.3]. For this and other reasons, this generalizationof the Cartesian product to quantum sets is at once conceptually natural and technicallychallenging. In this subsection, we resolve a few basic questions about it. Proposition B.2.
Let V , W , X and Y be quantum sets, and let F : V → X and G : W → Y be functions. Then F × G : V × W → X × Y is the unique function such that the followingdiagram commutes:
V V × W WX X × Y Y . F P V P W F × G GP X P Y In other words, F ◦ P V and F ◦ P V are compatible, and F × G = ( F ◦ P V , F ◦ P W ) .Proof. This follows by direct calculation, using the naturality of the unitors ρ and λ . (cid:3) Proposition B.3.
Let X and Y be quantum sets, and let V ⊆ X and
W ⊆ Y . Then J V×W = J V × J W .Proof. For all V ∝ V and W ∝ W , we calculate that J V×W ( V ⊗ W, V ⊗ W ) = C V ⊗ W = C V ⊗ C W = J V ( V, V ) ⊗ J W ( W, W ) = ( J V × J W )( V ⊗ W, V ⊗ W ) . Reasoning similarly, wemay show that the other components of both J V×W and J V × J W are zero. (cid:3) orollary B.4. Let W , X , X , Y , and Y be quantum sets. Let F : W → X , F : W →X , G : X → Y and G : X → Y be functions. Then (a) If ( F , F ) exists, so does ( G ◦ F , G ◦ F ) ; (b) If ( G ◦ F , G ◦ F ) exists, and both G and G are injective, then ( F , F ) exists.In both cases, we have ( G ◦ F , G ◦ F ) = ( G × G ) ◦ ( F , F ) . Proof.
For (a), we have P Y ◦ ( G × G ) ◦ ( F , F ) = G ◦ P X ◦ ( F , F ) = G ◦ F byProposition B.2, and similarly, P Y ◦ ( G × G ) ◦ ( F , F ) = G ◦ P X ◦ ( F , F ) = G ◦ F .Again appealing to Proposition B.2, we conclude that ( G ◦ F , G ◦ F ) exists and is equalto ( G × G ) ◦ ( F , F ) .For (b), assume that ( G ◦ F , G ◦ F ) exists. By [3, Lemma 10.4], this is equivalent to thestatement that each element in the image of ( G ◦ F ) ⋆ = F ⋆ ◦ G ⋆ commutes with each elementin the image of ( G ◦ F ) ⋆ = F ⋆ ◦ G ⋆ . Since G and G are injective, G ⋆ and G ⋆ are surjective[3, Propositions 8.1 and 8.4]; hence each element in the image of F ⋆ and each element in theimage of F ⋆ commute with each other. Again appealing to [3, Lemma 10.4], we concludethat ( F , F ) exists. By (a), it follows that ( G ◦ F , G ◦ F ) = ( G × G ) ◦ ( F , F ) . (cid:3) We now verify an elementary inequality for operators subspaces, whose ultimate purposeis to facilitate computations involving binary relations between products of quantum sets.For this verification, we introduce the notations ˇ x and ˆ x for vectors x ∈ X . Specifically,for each finite-dimensional Hilbert space X and each vector x ∈ X , let ˇ x ∈ L ( C , X ) bedefined by λ λx , and let ˆ x ∈ L ( X, C ) be defined by y
7→ h x, y i . Furthermore, we write ˇ X = { ˇ x : x ∈ X } = L ( C , X ) and ˆ X = { ˆ x : x ∈ X } = X ∗ = L ( X, C ) . Lemma B.5.
Let X , X , Y and Y be finite-dimensional Hilbert spaces, and let V bea subspace of L ( X ⊗ X , Y ⊗ Y ) . Let V = ρ Y ( C Y ⊗ ˆ Y ) V ( C X ⊗ ˇ X ) ρ − X , and let V = λ Y ( ˆ Y ⊗ C Y ) V ( ˇ X ⊗ C X ) λ − X , where ρ and λ are the right and left unitors in thecategory FdHilb . Then, V ≤ V ⊗ V .Proof. Fix v ∈ V . Choose orthonormal bases { x i } n i =1 , { x i } n i =1 , { y i } m i =1 , { y i } m i =1 for X , X , Y , Y , respectively. Since v ∈ L ( X , Y ) ⊗ L ( X , Y ) , we have v = P kℓ =1 b ℓ ⊗ c ℓ for some b ℓ ∈ L ( X , Y ) and c ℓ ∈ L ( X , Y ) . Since for any basis { e i } ni =1 of any n -dimensional Hilbertspace H , we have that H = P ni =1 ˇ e i ˆ e i , we find that v = (1 Y ⊗ Y ) v (1 X ⊗ X ) = Y ⊗ m X j =1 ˇ y j ˆ y j ! v X ⊗ n X i =1 ˇ x i ˆ x i ! = m X j =1 n X i =1 (1 Y ⊗ ˇ y j ˆ y j ) v (1 X ⊗ ˇ x i ˆ x i ) = m X j =1 n X i =1 (1 Y ⊗ ˇ y j ˆ y j ) k X ℓ =1 b ℓ ⊗ c ℓ ! (1 X ⊗ ˇ x i ˆ x i )= m X j =1 n X i =1 k X ℓ =1 b ℓ ⊗ ˇ y j ˆ y j c ℓ ˇ x i ˆ x i = m X j =1 n X i =1 k X ℓ =1 [ ρ Y ( b ℓ ⊗ C ) ρ − X ] ⊗ ˇ y j ˆ y j c ℓ ˇ x i ˆ x i = m X j =1 n X i =1 k X ℓ =1 [ ρ Y ( b ℓ ⊗ ˆ y j c ℓ ˇ x i ) ρ − X ] ⊗ ˇ y j ˆ x i m X j =1 n X i =1 k X ℓ =1 [ ρ Y (1 Y ⊗ ˆ y j )( b ℓ ⊗ c ℓ )(1 X ⊗ ˇ x i ) ρ − X ] ⊗ ˇ y j ˆ x i = m X j =1 n X i =1 [ ρ Y (1 Y ⊗ ˆ y j ) v (1 X ⊗ ˇ x i ) ρ − X ] ⊗ ˇ y j ˆ x i where the third-to-last equality follows because each operator ˆ y j c ℓ ˇ x i is a scalar. We con-clude that v ∈ V ⊗ L ( X , Y ) , and similarly, v ∈ L ( X , Y ) ⊗ V . The intersection of V ⊗ L ( X , Y ) and L ( X , Y ) ⊗ V is of course V ⊗ V , and thus v ∈ V ⊗ V . We vary v ∈ V to conclude that V ≤ V ⊗ V . (cid:3) Lemma B.6.
Let X , X , Y and Y be quantum sets, and let R ∈ qRel ( X × X , Y × Y ) .Then, R ≤ ( P Y ◦ R ◦ P X † ) × ( P Y ◦ R ◦ P X † ) . Proof.
Fix X ∝ X , Y ∝ Y , X ∝ X and Y ∝ Y . Let V = R ( X ⊗ Y , X ⊗ Y ) , and define V and V as in Lemma B.5. We calculate that ( P Y ◦ R ◦ P X † )( X , Y ) ≥ P Y ( Y ⊗ Y , Y ) · R ( X ⊗ Y , X ⊗ Y ) · P ( X ⊗ X , X ) † = ρ Y ( C Y ⊗ ˆ Y ) · V · ( C X ⊗ ˇ X ) ρ − X = V Similarly, ( P Y ◦ R ◦ P X † )( X , Y ) ≥ V . We now apply Lemma B.5 to calculate that R ( X ⊗ X , Y ⊗ Y ) ≤ V ⊗ V ≤ ( P Y ◦ R ◦ P X † )( X , Y ) ⊗ ( P Y ◦ R ◦ P X )( X , Y )= (( P Y ◦ R ◦ P X † ) × ( P Y ◦ R ◦ P X † ))( X ⊗ X , Y ⊗ Y ) We vary X ∝ X , X ∝ X , Y ∝ Y and Y ∝ Y to conclude that R ≤ ( P Y ◦ R ◦ P X † ) × ( P Y ◦ R ◦ P X † ) , as claimed. (cid:3) Proposition B.7.
Let W , X , X , Y and Y be quantum sets. Let F : W → X , F : W →X , G : W → Y and G : W → Y be functions. If ( F , F ) and ( G , G ) both exist, then ( G , G ) ≤ (( G ◦ F † ) × ( G ◦ F † )) ◦ ( F , F ) . Proof.
By Lemma B.6, we have ( G , G ) ◦ ( F , F ) † ≤ ( P Y ◦ ( G , G ) ◦ ( F , F ) † ◦ P X † ) × ( P Y ◦ ( G , G ) ◦ ( F , F ) † ◦ P X † )= ( G ◦ F † ) × ( G ◦ F † ); hence ( G , G ) ≤ ( G , G ) ◦ ( F , F ) † ◦ ( F , F ) ≤ (cid:0) ( G ◦ F † ) × ( G ◦ F † ) (cid:1) ◦ ( F , G ) . (cid:3) Appendix C. The trace on binary relations
Every compact closed category has a canonically defined trace on each endomorphismset. For each quantum set X , the trace Tr X : qRel ( X , X ) → qRel ( ; ) is defined by Tr X ( R ) = E X ◦ ( R × I X ∗ ) ◦ E †X . The orthomodular lattice qRel ( , ) consists of two elements: ⊥ ≤ ⊤ . Note that Tr is just the identity on qRel ( , ) . Lemma C.1.
Let X be a quantum set. For each binary relation R from X to X , theequation Tr X ( R ) = ⊥ is equivalent to the equation Tr X ( R ( X, X )) = 0 for each Hilbert space X ∈ At( X ) . roof. The equation Tr X ( R ) = ⊥ is equivalent to X X ,X ∈ At( X ) X X ,X ∈ At( X ) E X ( X ⊗ X ∗ , C ) · ( R ( X , X ) ⊗ I X ∗ ( X ∗ , X ∗ )) · E †X ( C , X ⊗ X ∗ ) = 0 , by definition of composition and product for binary relations [3, section 3]. Terms for which X = X do not contribute because E X ( X ⊗ X ∗ , C ) = 0 , terms for which X = X do notcontribute because E †X ( C , X ⊗ X ∗ ) = 0 , and terms for which X = X do not contributebecause I X ∗ ( X ∗ , X ∗ ) = 0 . Thus, Tr X ( R ) = ⊥ if and only if X X ∈ At( X ) E X ( X ⊗ X ∗ , C ) · ( R ( X, X ) ⊗ I X ∗ ( X ∗ , X ∗ )) · E †X ( C , X ⊗ X ) = 0 . A sum of subspaces is equal to zero if and only if each subspace is equal to zero. Furthermore,the operator spaces E X ( X ⊗ X ∗ , C ) , E †X ( C , X ⊗ X ) , and I X ∗ ( X ∗ , X ∗ ) are each spanned by asingle operator. Thus, Tr X ( R ) = ⊥ if and only if ǫ X · ( R ( X, X ) ⊗ X ∗ ) · ǫ † X = 0 for each atom X ∈ At( X ) , where ǫ X denotes the unit of the dagger compact category of finite-dimensionalHilbert spaces and linear operators. This completes the proof, because it is well known that ǫ X · ( r ⊗ X ∗ ) · ǫ † X = Tr X ( r ) for each operator r on a finite-dimensional Hilbert space X . (cid:3) Proposition C.2.
Let X and Y be quantum sets, and let R and S be binary relations from X to Y . Then, R ⊥ S if and only if Tr X ( S † ◦ R ) = ⊥ .Proof. We follow a chain of equivalences, with Lemma C.1 used for the first equivalence. Tr X ( S † ◦ R ) = ⊥ ⇔ ∀ X ∈ At( X ) . Tr X (( S † ◦ R )( X, X )) = 0 ⇔ ∀ X ∈ At( X ) . Tr X X Y ∈ At( Y ) S ( X, Y ) † · R ( X, Y ) = 0 ⇔ ∀ X ∈ At( X ) . ∀ Y ∈ At( Y ) . Tr X ( S ( X, Y ) † · R ( X, Y )) = 0 ⇔ ∀ X ∈ At( X ) . ∀ Y ∈ At( Y ) . R ( X, Y ) ⊥ S ( X, Y ) = 0 ⇔ R ⊥ S (cid:3) For
P, Q ∈ qRel ( Y , ) , we set X = , R = P † , and S = Q † , to find that P ⊥ Q if andonly if Q ◦ P † = ⊥ . References [1] S. Abramsky and B. Coecke,
Categorical quantum mechanics , arXiv:0808.1024 .[2] J. Adámek, H. Herrlich and G. E. Strecker, Abstract and Concrete Categories: The Joy of Cats , Wiley,1990.[3] A. Kornell,
Quantum sets , J. Math. Phys. (2020).[4] A. Kornell, B. Lindenhovius and M. Mislove, Quantum CPOs , to appear in Proceedings 17th InternationalConference on Quantum Physics and Logic.[5] R. Nakagawa,
Categorical topology , Topics in General Topology, North-Holland, 1989.[6] S. Wacław,
Antisymmetric operator algebras I , Ann. Polon. Math. (1980).[7] S. Wacław, Antisymmetric operator algebras II , Ann. Polon. Math. (1980).[8] N. Weaver, Quantum relations , Mem. Amer. Math. Soc. (2012).[9] N. Weaver,
Hereditarily antisymmetric operator algebras , J. Inst. Math. Jussieu (2019)., J. Inst. Math. Jussieu (2019).