aa r X i v : . [ qu a n t - ph ] O c t Coalgebras, Chu Spaces, and Representations ofPhysical Systems
Samson AbramskyOxford University Computing LaboratoryNovember 11, 2018
Abstract
We revisit our earlier work on the representation of quantum systems asChu spaces, and investigate the use of coalgebra as an alternative framework.On the one hand, coalgebras allow the dynamics of repeated measurement tobe captured, and provide mathematical tools such as final coalgebras, bisim-ulation and coalgebraic logic. However, the standard coalgebraic frameworkdoes not accommodate contravariance, and is too rigid to allow physical sym-metries to be represented. We introduce a fibrational structure on coalgebrasin which contravariance is represented by indexing. We use this structureto give a universal semantics for quantum systems based on a final coalge-bra construction. We characterize equality in this semantics as projectiveequivalence. We also define an analogous indexed structure for Chu spaces,and use this to obtain a novel categorical description of the category of Chuspaces. We use the indexed structures of Chu spaces and coalgebras over acommon base to define a truncation functor from coalgebras to Chu spaces.This truncation functor is used to lift the full and faithful representation of thegroupoid of physical symmetries on Hilbert spaces into Chu spaces, obtainedin our previous work, to the coalgebraic semantics.
Chu spaces and universal coalgebra are two general formalisms for systems mod-elling in a broad sense. Both have been studied quite extensively in ComputerScience over the past couple of decades. Recently, we showed how quantum sys-tems with their symmetries have a full and faithful representation as Chu spaces[1]. We had in fact originally intended to use coalgebras as the vehicle for thiswork. This did not prove satisfactory, for reasons which will be explained later.1ut coalgebras have many features which make them promising for studies of thiskind. Moreover, as we shall show, the problems which arise can in fact be over-come to a considerable degree, in a fashion which brings to light some interestingand novel aspects of these two well-studied models, and in particular of the re-lationships between them — which have, to the best of our knowledge, not beenstudied at all previously.The purpose of the present paper is thus to develop some systematic connec-tions and contrasts between Chu spaces and coalgebras, the modelling issues whicharise, what can be done to resolve them, and which problems remain outstanding.The main results of our investigations can be summarized as follows: • Firstly, at the general level, we look at the comparative strengths and weak-nesses of the two formalisms. On our analysis, the key feature that Chuspaces have and coalgebras lack is contravariance ; the key feature whichcoalgebras have and Chu spaces lack is extension in time . There are someinteresting secondary issues as well, notably symmetry vs. rigidity . • Formally, we introduce an indexed structure for coalgebras to compensatefor the lack of contravariance, and show how this can be used to represent awide class of physical systems in coalgebraic terms. In particular, we showhow a universal model for quantum systems can be constructed as a finalcoalgebra . This opens the way to the use of methods such as coalgebraiclogic in the study of physical systems. It also suggests how coalgebra canmediate between ontic and epistemic views of the states of physical systems. • We also define an analogous indexed structure for Chu spaces, and use thisto obtain a novel categorical description of the category of Chu spaces. Weuse the indexed structures of Chu spaces and coalgebras over a common baseto define a truncation functor from coalgebras to Chu spaces. • We use this truncation functor to lift the full and faithful representation of thegroupoid of physical symmetries on Hilbert spaces into Chu spaces, obtainedin [1], to the coalgebraic semantics.The further contents of the paper are organized as follows. In Section 2 wereview some background on Chu spaces and coalgebras. In section 3 we make afirst comparison of Chu spaces and coalgebras. Then in Section 4 we discuss themodelling issues, the problems which arise, and the strengths and weaknesses ofthe two approaches. In Section 5 we develop the technical material on indexedstructure for coalgebras. A similar development for Chu spaces is carried out inSection 6, and the truncation functor is defined. In Section 7 we show how a uni-versal model for quantum systems can be constructed as a final coalgebra; equality2n the coalgebraic semantics is characterized as projective equivalence, and therepresentation theorem for the symmetry groupoid on Hilbert spaces is lifted fromChu spaces to the coalgebraic category. Section 8 outlines the general scheme of‘bivariant coalgebra’ underlying our approach.
Coalgebra has proved to be a powerful and flexible tool for modelling a wide rangeof systems. We shall give a very brief introduction. Further details may be founde.g. in the excellent presentation in [25].Category theory allows us to dualize algebras to obtain a notion of coalgebrasof an endofunctor . However, while algebras abstract a familiar set of notions, coal-gebras open up a new and rather unexpected territory, and provides an effectiveabstraction and mathematical theory for a central class of computational phenom-ena: • Programming over infinite data structures : streams, infinite trees, etc. • A novel notion of coinduction . • Modelling state-based computations of all kinds. • The key notion of bisimulation equivalence between processes. • A general coalgebraic logic can be read off from the functor, and used tospecify and reason about properties of systems.Let F : C → C be a functor. An F -coalgebra is a pair ( A, α ) where A is anobject of C , and α is an arrow α : A → F A . We say that A is the carrier of thecoalgebra, while α is the behaviour map .An F -coalgebra homomorphism from ( A, α ) to ( B, β ) is an arrow h : A → B such that A α ✲ F ABh ❄ β ✲ F BF h ❄ -coalgebras and their homomorphisms form a category F − Coalg .An F -coalgebra ( C, γ ) is final if for every F -coalgebra ( A, α ) there is a uniquehomomorphism from ( A, α ) to ( C, γ ) , i.e. if it is the terminal object in F − Coalg . Proposition 2.1
If a final F -coalgebra exists, it is unique up to isomorphism. Proposition 2.2 (Lambek Lemma) If γ : C → F C is final, it is an isomorphism
Chu spaces are a special case of a construction which originally appeared in [7],written by Po-Hsiang Chu as an appendix to Michael Barr’s monograph on ∗ -autonomous categories [4].Chu spaces have several interesting aspects: • They have a rich type structure, and in particular form models of LinearLogic [9, 26]. • They have a rich representation theory; many concrete categories of interestcan be fully embedded into Chu spaces [17, 23]. • There is a natural notion of ‘local logic’ on Chu spaces [6], and an interestingcharacterization of information transfer across Chu morphisms [28].Applications of Chu spaces have been proposed in a number of areas, includingconcurrency [24], hardware verification [16], game theory [29] and fuzzy systems[21, 19]. Mathematical studies concerning the general Chu construction include[22, 5, 10].We briefly review the basic definitions.Fix a set K . A Chu space over K is a structure ( X, A, e ) , where X is a set of‘points’ or ‘objects’, A is a set of ‘attributes’, and e : X × A → K is an evaluationfunction.A morphism of Chu spaces f : ( X, A, e ) → ( X ′ , A ′ , e ′ ) is a pair of functions f = ( f ∗ : X → X ′ , f ∗ : A ′ → A ) such that, for all x ∈ X and a ′ ∈ A ′ : e ( x, f ∗ ( a ′ )) = e ′ ( f ∗ ( x ) , a ′ ) . f : ( X , A , e ) → ( X , A , e ) and g : ( X , A , e ) → ( X , A , e ) , then ( g ◦ f ) ∗ = g ∗ ◦ f ∗ , ( g ◦ f ) ∗ = f ∗ ◦ g ∗ . Chu spaces over K and their morphisms form a category Chu K . Our basic paradigm for representing physical systems, as laid out in [1], is as fol-lows. We take a system to be specified by its set of states S , and the set of questions Q which can be ‘asked’ of the system. We shall consider only ‘yes/no’ questions;however, the result of asking a question in a given state will in general be proba-bilistic . This will be represented by an evaluation function e : S × Q → [0 , where e ( s, q ) is the probability that the question q will receive the answer ‘yes’when the system is in state s . Thus a system is represented directly as a Chu space.In particular, a quantum system with a Hilbert space H as its state space willbe represented as ( H ◦ , L ( H ) , e H ) where H ◦ is the set of non-zero vectors of H , L ( H ) is the set of closed subspacesof H , and the evaluation function e H is the basic ‘statistical algorithm’ of QuantumMechanics: e H ( ψ, S ) = h ψ | P S ψ ih ψ | ψ i = h P S ψ | P S ψ ih ψ | ψ i = k P S ψ k k ψ k . For a more detailed discussion see [1]. That paper goes on to show that: • The biextensional collapse of this Chu space yields the usual projective rep-resentation of states as rays. • The Chu morphisms between these spaces are exactly the unitaries and uni-taries, yielding a full and faithful functor from the groupoid of physical sym-metries on Hilbert spaces to Chu spaces. • This representation is preserved by collapsing the unit interval to three val-ues, but not by the further collapse by either of the standard ‘possibilistic’reductions to two values.This yields quite a pleasant picture. We would now like to investigate to whatextent we can use coalgebras as an alternative setting for such representations; whatproblems arise, and on the other hand, what new possibilities become available.5
Comparison: A First Attempt
We shall begin by showing that a subcategory of Chu spaces can be captured incompletely equivalent form as a category of coalgebras.Fix a set K . We can define a functor on Set : F K : X K P X . If we use the contravariant powerset functor, F K will be covariant. Explicitly, for f : X → Y : F K f ( g )( S ) = g ( f − ( S )) , where g ∈ K P X and S ∈ P Y . A coalgebra for this functor will be a map of theform α : X → K P X . Consider a Chu space C = ( X, A, e ) over K . We suppose furthermore that thisChu space is normal (cf. [20] for a related but not identical use of this term), mean-ing that A = P X . Given this normal Chu space, we can define an F K -coalgebraon X by α ( x )( S ) = e ( x, S ) . We write GC = ( X, α ) .A coalgebra homomorphism from ( X, α ) to ( Y, β ) is a function h : X → Y such that X α ✲ K P X Yh ❄ β ✲ K P Y F h ❄ Proposition 3.1
Suppose we are given a Chu morphism f : C → C ′ , where C and C ′ are normal Chu spaces, such that f ∗ = f − ∗ . Then f ∗ : GC → GC ′ is an F K -algebra homomorphism. Conversely, given any F K -algebra homomorphism f : GC → GC ′ , then ( f, f − ) : C → C ′ is a Chu morphism. roof Let ( f ∗ , f − ∗ ) : C → C ′ be a Chu space morphism. Then ( F f ∗ ◦ α )( x )( S ) = F f ∗ ( α ( x ))( S )= α ( x )( f − ∗ S )= e ( x, f − ∗ S )= e ( x, f ∗ S )= e ′ ( f ∗ ( x ) , S )= β ◦ f ∗ ( x )( S ) so f ∗ is a F K -coalgebra homomorphism. The converse is verified similarly (in factby a cyclic permutation of the steps of the above proof). (cid:3) Let
NChu K be the category of normal Chu spaces and Chu morphisms of theform ( f, f − ) . Then by the Proposition, G extends to a functor G : NChu K → F K − Coalg , with G ( f, f − ) = f . Conversely, given an F -coalgebra ( X, α ) , wecan define a normal Chu space H ( X, α ) = ( X, P X, e ) , where e ( x, S ) = α ( x )( S ) ,and given a coalgebra homomorphism f : ( X, α ) → ( Y, β ) , Hf = ( f, f − ) : H ( X, α ) → ( Y, β ) will be a Chu morphism; this is verified in entirely similar fashion to Proposi-tion 3.1.Altogether, we have shown: Theorem 3.2
NChu K and F K − Coalg are isomorphic categories, with the iso-morphism witnessed by G and H = G − . Of course, the assumption of normality for Chu spaces is very strong;although it is worth mentioning that we have assumed nothing about either thevalue set or the evaluation function, in contrast to the notion of normality usedin [20] (for quite different purposes), which allows the attributes to be any subsetof the powerset, but stipulates that K = and that the evaluation function is thecharacteristic function for set membership. One would like to extend the abovecorrespondence to allow for wider classes of Chu spaces, in which the attributesneed not be the full powerset. This is probably best done in an enriched setting ofsome kind. 7t should also be said that the use of powersets, full or not, to represent ‘ques-tions’ is fairly crude and ad hoc. The degree of freedom afforded by Chu spacesto choose both the states and the questions appropriately is a major benefit to con-ceptually natural and formally adequate modelling of a wide range of situations. The Type Functor
The experienced coalgebraist will be aware that the functors F K are problematic from the point of view of coalgebra. In particular, they failto preserve weak pullbacks, and hence F K − Coalg will lack some of the nicestructural properties one would like a category of coalgebras to possess. In fact, F K is a close cousin of the ‘double contravariant powerset’, which is a standardcounter-example for these properties [25]. However, much coalgebra can be donewithout this property [12], and recent work has achieved interesting results forcoalgebras over the double contravariant powerset [14].A secondary problem is that as it stands, F K − Coalg cannot have a final coal-gebra, for mere cardinality reasons. In fact, this issue can be addressed in a stan-dard way. We can replace the contravariant powerset by a bounded version P κ .We can also replace the function space by the partial function space Pfn ( X, Y ) .Thinking of partial functions in terms of their graphs, there is a set inclusion Pfn ( X, Y ) ⊆ P ( X × Y ) . Hence we can use a bounded version of the partialfunction functor, say Pfn λ ( X, Y ) , yielding those partial functions whose graphshave cardinality < λ . The resulting modified version of F K : X Pfn λ ( P κ ( X ) , K ) is bounded, and admits a final coalgebra. Moreover, by choosing κ and λ suf-ficiently large, we can still represent a large class of systems whose behaviourinvolves total functions. Behaviours vs. Symmetries
However, there is a deeper conceptual problemwhich militates against the use of coalgebras in our context. An important prop-erty of physical theories is that they have rich symmetry groups (and groupoids),in which the key invariants are found, and from which the dynamics can be ex-tracted. The main result of [1] was to recover these symmetries in the case ofquantum systems as Chu morphisms. The picture in coalgebra is rather different.One is concerned with behavioural or observational equivalence, as encapsulatedby bisimulation, and the final coalgebra gives a ‘fully abstract’ model of behaviour,in which bisimulation turns into equality. Moreover, every coalgebra morphism is afunctional bisimulation. If we consider the class of strongly extensional coalgebras[25], those which have been quotiented out by bisimulation, they form a preorder ,8nd essentially correspond to the subcolagebras of the final coalgebra. Thus in asense coalgebras are oriented towards maximum rigidity, and minimum symmetry.From this point of view, it would seem more desirable to have a universalhomogeneous model , with a maximum degree of symmetry, as a universal modelfor a large class of physical systems, rather than a final coalgebra. Such a modelhas been constructed for bifinite Chu spaces in [8]. That context is too limited forour purposes here. It remains to be seen if universal homogeneous models can beconstructed for larger subcategories of Chu spaces, encompassing those involvedin our representation results.In the present paper, we shall develop an alternative resolution of this problemby using a fibred category of coalgebras, in which there is sufficient scope forvariation to allow for the representation of symmetries. We shall use this to lift therepresentation theorem of [1] from Chu spaces to coalgebras. • The coalgebraic point of view can be described as state-based , but in a waythat emphasizes that the meaning of states lies in their observable behaviour .Indeed, in the “universal model” we shall construct, the states are determinedexactly as the possible observable behaviours — we actually find a canon-ical solution for what the state space should be in these terms. States areidentified exactly if they have the same observable behaviour.We can see this as a kind of reconciliation between the ontic and epistemic standpoints, in which moreover operational ideas are to the fore. • Coalgebras allow us to capture the ‘dynamics of measurement’ — what hap-pens after a measurement — in a way that Chu spaces don’t. They have extension in time [3]. We explain what we mean by this in more detail be-low.
Extension in Time
Consider a coalgebraic representation of stochastic transduc-ers : F : X Prob ( O × X ) I where I is a fixed set of inputs , O a fixed set of outputs , and Prob ( S ) is the set ofprobability distributions of finite support on S . This expresses the behaviour of astate x ∈ X in terms of how it responds to an input i ∈ I by producing an output o ∈ O and evolving into a new state x ′ ∈ X . Since the automaton is stochastic,what is specified for each input i is a probability distribution over the pairs ( o, x ′ ) comprising the possible responses. 9e can think of I as a set of questions , and O as a set of answers (which wecould standardize by only considering yes/no questions). Thus we can see sucha stochastic automaton as a variant of the representation of physical systems wediscussed previously, with the added feature of extension in time — the capacity torepresent behaviour under repeated interactions.What we can learn from this observation, incidentally, is thatQM is less nondeterministic/probabilistic than stochastic transducerssince in Quantum Mechanics, if we know the preparation and the outcome of themeasurement, we know (by the projection postulate) exactly what the resultingquantum state will be. In automata theory, by contrast, even if we know the currentstate, the input, and which observable output was produced in response, we still donot know in general what the next state will be. Could there be physical theoriesof this type? As a first step to developing a viable coalgebraic approach to representing physicalsystems, we shall hold a single system fixed, and see how we can represent thiscoalgebraically. This simple step eliminates most of the problems with coalgebraswhich we encountered in the previous Section. We will then have to see howvariation of the system being represented can be reintroduced.
We fix attention on a single Hilbert space H . This determines a set of questions Q = L ( H ) . We now define an endofunctor on Set : F Q : X ( { } + (0 , × X ) Q . A coalgebra for this functor is then a map α : X → ( { } + (0 , × X ) Q The interpretation is that X is a set of states; the coalgebra map sends a state toits behaviour, which is a function from questions in Q to the probability that theanswer is ‘yes’; and, if the probability is not 0 , to the successor state following a‘yes’ answer.Unlike the functors F K , the functors F Q are very well-behaved from the pointof view of coalgebra (they are in fact polynomial functors [25]). They preserve10eak pull-backs, which guarantees a number of nice properties, and they are boundedand admit final coalgebras γ Q : U Q → ( { } + (0 , × U Q ) Q . The elements of U Q can be visualized as ‘ Q -branching trees’, with the arcs labelledby probabilities.The F Q -coalgebra which is of primary interest to us is a H : H ◦ → ( { } + (0 , × H ◦ ) Q defined by: a H ( ψ )( S ) = , e H ( ψ, S ) = 0( r, P S ψ ) , e H ( ψ, S ) = r > . The new ingredient compared with the Chu space representation of H is the statewhich results in the case of a ‘yes’ answer to the question, which is computedaccording to the (unnormalized) L¨uders rule .This system will of course have a representation in the final coalgebra ( U Q , γ Q ) ,specified by the unique coalgebra homomorphism h : ( H ◦ , a H ) → ( U Q , γ Q ) . Our strategy will now be to externalize contravariance as indexing . This will al-low us to alleviate many of the problems we encountered with using coalgebras torepresent physical systems, and to access the power of the coalgebraic framework.In particular, we will be able to construct a single universal model for quantumsystems .We shall define a functor F : Set op → CAT where
CAT is the ‘superlarge’ category of categories and functors. F is definedon objects by Q F Q − Coalg . For those concerned with set-theoretic foundations, we shall on a couple of occasions refer to‘superlarge’ categories such as
CAT , the category of ‘large categories’ such as
Set . If we thinkof large categories as based on classes, superlarge categories are based on entities ‘one size up’ —‘conglomerates’ in the terminology of [15]. This can be formalized in set theory with a couple ofGrothendieck universes. f : Q ′ → Q , we define t fX : F Q ( X ) → F Q ′ ( X ) :: Θ Θ ◦ f and F ( f ) = f ∗ : F Q − Coalg → F Q ′ − Coalg f ∗ : ( X, α ) ( X, t fX ◦ α ) , f ∗ : ( h : ( X, α ) → ( Y, β )) h. Proposition 5.1
For each f : Q ′ → Q , t f is a natural transformation, and f ∗ isa functor. Proof
The naturality of t f is the diagram X F Q X t fX ✲ F Q ′ XYg ❄ F Q YF Q g ❄ t fY ✲ F Q ′ YF Q ′ g ❄ This diagram commutes because t f acts by pre-composition and F Q , F Q ′ by post-composition. For any Θ ∈ F Q X , we obtain the common value (1 + (1 × g )) ◦ Θ ◦ f. It is a general fact [25] that a natural transformation t : F → G induces afunctor between the coalgebra categories in the manner specified above. The factthat the coalgebra homomorphism condition is preserved follows from the commu-tativity of X α ✲ F X t X ✲ GXYh ❄ β ✲ F YF h ❄ t Y ✲ GYGh ❄ The left hand square commutes because h is an F -coalgebra homomorphism; theright hand square is naturality of t . (cid:3) strict indexed category of coalgebra categories, with contravari-ant indexing. We now recall an important general construction. Where we have an indexed cat-egory, we can apply the
Grothendieck construction [11], to glue all the fibres to-gether (and get a fibration).Given a functor I : C op → CAT we define R I with objects ( A, a ) , where A is an object of C and a is an object of I ( A ) . Arrows are ( G, g ) : (
A, a ) → ( B, b ) , where G : B → A and g : I ( G )( a ) → b . Composition of ( G, g ) : (
A, a ) → ( B, b ) and ( H, h ) : (
B, b ) → ( C, c ) is givenby ( G ◦ H, h ◦ I ( H )( g )) : ( A, a ) → ( C, c ) . Applying the Grothendieck construction to F , we can now put all our categoriesof coalgebras, indexed by the sets of questions, together in one category. We willuse this to get our universal model for quantum systems.Before turning to this, we will consider an analogous indexed structure forChu spaces, which will allow us to define a comparison functor between the twomodels. For each Q , we define Chu QK to be the subcategory of Chu K of Chu spaces ( X, Q, e ) and morphisms of the form ( f ∗ , id Q ) .This doesn’t look too exciting. In fact, it is just the comma category ( − × Q, ˆ K ) where ˆ K : → Set picks out the object K .Given f : Q ′ → Q , we define a functor f ∗ : Chu QK → Chu Q ′ K :: ( X, Q, e ) ( X, Q ′ , e ◦ (1 × f )) and which is the identity on morphisms. To verify functoriality, we only need tocheck that the Chu morphism condition is preserved. That is, we must show, for13ny morphism ( f ∗ , id Q ) : ( X, Q, e ) → ( X ′ , Q, e ′ ) , x ∈ X , and q ′ ∈ Q , that e ( x, f ( q ′ )) = e ′ ( f ∗ ( x ) , f ( q ′ )) which follows from the Chu morphism condition on ( f ∗ , id Q ) .This gives an indexed category Chu K : Set op → CAT . The fibre categories
Chu QK are pale reflections of the full category of Chu spaces,trivialising the contravariant component of morphisms. However, the Grothendieckconstruction gives us back the full category. Proposition 6.1 Z Chu K ∼ = Chu K . Proof
Expanding the definitions, we see that objects in R Chu K have the form ( Q, ( X, Q, e : X × Q → K )) while morphisms have the form ( f, ( f ∗ , id Q ′ )) : ( Q, ( X, Q, e )) → ( Q ′ , ( X, Q ′ , e ′ )) where f : Q ′ → Q , and ( f ∗ , id Q ′ ) : ( X, Q ′ , e ◦ (1 × f )) → ( X ′ , Q ′ , e ′ ) is a morphism in Chu Q ′ K . The morphism condition is: e ( x, f ( q ′ )) = e ′ ( f ∗ ( x ) , q ′ ) . This is exactly the Chu morphism condition for ( f ∗ , f ) : ( X, Q, e ) → ( X ′ , Q ′ , e ′ ) . Composition of ( f, ( f ∗ , id Q ′ )) with ( g, ( g ∗ , id Q ′′ )) is given by ( f ◦ g, ( g ∗ ◦ f ∗ , id Q ′′ )) .The isomorphism with Chu K is immediate from this description. (cid:3) .3 The Truncation Functor The relationship between coalgebras and Chu spaces is further clarified by an in-dexed truncation functor T : F → Chu .For each set Q there is a functor T Q : F Q − Coalg → Chu QK This is defined on objects by T Q ( X, α ) = (
X, Q, e ) where e ( x, q ) = , α ( x )( q ) = 0 r, α ( x )( q ) = ( r, x ′ ) The action on morphisms is trivial: T Q : ( h : ( X, α ) → ( Y, β )) ( h, id Q ) . The verification that coalgebra homomorphisms are taken to Chu morphisms isstraightforward. The fact that each T Q is a faithful functor is then immediate.For each f : Q ′ → Q , we have the naturality square F Q − Coalg T Q ✲ Chu QK F Q ′ − Coalg F ( f ) ❄ T Q ′ ✲ Chu Q ′ K Chu K ( f ) ❄ On objects, both paths around the diagram carry a coalgebra ( X, α ) to the Chuspace ( X, Q ′ , e ) , where e ( x, q ′ ) = , α ( x )( f ( q ′ )) = 0 r, α ( x )( f ( q ′ )) = ( r, x ′ ) The action on morphisms in both cases is trivial: a coalgebra homomorphism h issent to the Chu morphism ( h, id Q ′ ) .We can summarize this as follows: 15 roposition 6.2 T : F −→ Chu is a strict indexed functor, which is faithful oneach fibre.
As an immediate corollary, we obtain:
Proposition 6.3
There is a faithful functor R T : R F −→ R Chu ∼ = Chu K . We can also refine the isomorphism of Theorem 3.2. We say that an F Q -coalgebra ( X, α ) is static if for all x ∈ X : α ( x )( q ) = ( r, x ′ ) ⇒ x ′ = x. Thus in a static coalgebra, observing an answer to a question has no effect on thestate. We write S Q − Coalg for the full subcategory of F Q − Coalg determinedby the static coalgebras. This extends to an indexed subcategory S of F , since thefunctors f ∗ , for f : Q ′ → Q , carry S Q − Coalg into S Q ′ − Coalg . Proposition 6.4
For each set Q , Chu QK is isomorphic to S Q − Coalg . Moreoverthis is an isomorphism of strict indexed categories.
Proof
We can define an indexed functor E Q : Chu QK → S Q − Coalg E Q : ( X, Q, e ) ( X, α ) where α ( x )( q ) = , e ( x, q ) = 0( r, x ) , e ( x, q ) = r > . E Q takes a Chu morphism ( f, id Q ) to f .It is straightforward to verify that this is an indexed functor, and inverse to therestriction of T to S . (cid:3) We can combine this with Proposition 6.1 to obtain:
Theorem 6.5
The category of Chu spaces
Chu K is isomorphic to a full subcate-gory of R F , the Grothendieck category of an indexed category of coalgebras. This gives a clear picture of how coalgebras extend Chu spaces with some‘observational dynamics’. 16
A Universal Model
We can now define a single coalgebra which is universal for quantum systems .We proceed in a number of steps:1. Fix a countably-infinite-dimensional Hilbert space, e.g. H U = ℓ ( N ) , withits standard orthonormal basis { e n } n ∈ N . Take Q = L ( H U ) . Let ( U Q , γ Q ) bethe final coalgebra for F Q .2. Any quantum system is described by a separable Hilbert space K . In prac-tice, the Hilbert space chosen to represent a given system will come with apreferred orthonormal basis { ψ n } . This basis will induce an isometric em-bedding i : K ✲ ✲ H U :: ψ n e n . Taking Q ′ = L ( K ) , this induces a map f = i − : Q → Q ′ . This in turninduces a functor f ∗ : F Q ′ − Coalg → F Q − Coalg .3. This functor can be applied to the coalgebra ( K ◦ , a K ) corresponding to theHilbert space K to yield a coalgebra in F Q − Coalg .4. Since ( U Q , γ Q ) is the final coalgebra in F Q − Coalg , there is a unique coal-gebra homomorphism J · K K ◦ : f ∗ ( K ◦ , a K ) → ( U Q , γ Q ) .5. This homomorphism maps the quantum system ( K ◦ , a K ) into ( U Q , γ Q ) in a fully abstract fashion , i.e. identifying states precisely according to observa-tional equivalence.6. This homomorphism is an arrow in the Grothendieck category R F .7. This works for all quantum systems, with respect to a single final coalgebra.This is a ‘Big Toy Model’ in the sense of [1].We shall now investigate the nature of this coalgebraic semantics for physicalsystems in more detail. Our first aim is to characterize when two states of a physical system are sent to thesame element of the final coalgebra by the semantic map J · K . We can call on somegeneral coalgebraic notions for this purpose.We shall begin with one of the key ideas in the theory of coalgebra, bisimilarity .This can be defined in generality for coalgebras over any endofunctor [25], but we17hall just give the concrete definition as it pertains to F Q − Coalg . Given F Q -coalgebras ( X, α ) and ( Y, β ) , a bisimulation is a relation R ⊆ X × Y such that: xRy ⇒ ∀ q ∈ Q. α ( x )( q ) = 0 ⇒ β ( y )( q ) = 0 ∧ α ( x )( q ) = ( r, x ′ ) ⇒ β ( y )( q ) = ( r, y ′ ) ∧ x ′ Ry ′ . We say that x and y are bisimilar , and write x ∼ b y , if there is some bisimulation R with xRy . Note that bisimilarity can hold between elements of different coal-gebras. This means that states of different systems can be compared in terms of acommon notion of observable behaviour.The above definition is given in an apparently asymmetric form, but ∼ b is easilyseen to be a symmetric relation, since the cases α ( x )( q ) = 0 and α ( x )( q ) = ( r, x ′ ) are mutually exclusive and exhaustive. Proposition 7.1
Bisimilarity is an equivalence relation.
Proof
The main point is transitivity, which follows automatically since the poly-nomial functor F Q preserves pullbacks [25]. (cid:3) The key feature of bisimilarity is given by the following proposition, which isalso standard for functors preserving weak pullbacks [25]. We consider coalgebrasfor such a functor F for which a final coalgebra exists. Given an F -coalgebra ( X, α ) and x ∈ X , we write J x K for the denotation of x in the final coalgebra. Proposition 7.2
For any F -coalgebras ( X, α ) and ( Y, β ) , and x ∈ X , y ∈ Y : J x K = J y K ⇐⇒ x ∼ b y. Thus bisimilarity characterizes equality of denotation in the final coalgebra seman-tics.We begin by characterizing bisimilarity in the coalgebra ( K ◦ , a K ) arising fromthe Hilbert space K , for the functor F Q , where Q = L ( K ) .We define the usual projective equivalence on the non-zero vectors of a Hilbertspace K ◦ by: ψ ∼ p φ ⇐⇒ ∃ λ ∈ C . ψ = λφ. Thus two vectors are projectively equivalent if they belong to the same ray or one-dimensional subspace.
Proposition 7.3
For any vectors ψ, φ ∈ K ◦ : ψ ∼ p φ ⇐⇒ ψ ∼ b φ. roof Firstly, recall the definition of e K from Section 2.3. We can describethe bisimilarity condition on a relation R ⊆ K ◦ for the coalgebra ( K ◦ , a K ) moredirectly as follows: ψ R φ ⇒ ∀ S ∈ L ( H ) . e K ( ψ, S ) = e K ( φ, S ) ∧ ( P S ψ ) R ( P S φ ) . Thus if ψ ∼ b φ , then for all S ∈ L ( K ) , e K ( ψ, S ) = e K ( φ, S ) , and hence ψ ∼ p φ by Proposition 3.2 of [1]. For the converse, it suffices to show that the relation ∼ p ⊆ K ◦ is a bisimulation. If ψ = λφ , then for all S , e K ( ψ, S ) = e K ( φ, S ) byProposition 3.2 of [1], and P S ψ = λP S φ , so ∼ p is a bisimulation as required. (cid:3) We now show that bisimilarity in Hilbert spaces is stable under transport acrossfibres by isometries.Firstly, we have a general property of fibred coalgebras.
Proposition 7.4 If f : Q ′ → Q is surjective, then bisimulation on the F Q ′ -coalgebra f ∗ ( X, α ) coincides with bisimulation on the F Q -coalgebra ( X, α ) . Proof
Unwinding the definitions of the two bisimulation conditions on relations,the only difference is that one quantifies over questions q ∈ Q , and the other overquestions f ( q ′ ) , for q ′ ∈ Q ′ . If f is surjective, these are equivalent. (cid:3) Given a Hilbert space K and an isometric embedding i : K ✲ ✲ H U , let Q = L ( H U ) , Q ′ = L ( K ) , f = i − : Q → Q ′ . Then the F Q -coalgebra f ∗ ( K ◦ , a K ) is ( K ◦ , β ) , where: β ( ψ )( S ) = a K ( ψ )( i − ( S )) . Proposition 7.5
Bisimulation on the elements of the F Q -coalgebra ( K ◦ , β ) co-incides with bisimulation on the F Q ′ -coalgebra ( K ◦ , a K ) . If we identify K withthe subspace H ′ ⊂ ✲ H U determined by the image of i , it also coincides withbisimulation on H ′ . It is also the restriction of bisimulation on H U . Proof
Since i is an isometry, the direct image i ( S ) of a closed subspace of K is a closed subspace of H U , and since i is injective, i − ( i ( S )) = S . Thus i − is surjective, yielding the first statement by Proposition 7.4. The fact that i is anisometric embedding also guarantees that e K ( ψ, S ) = e H U ( ψ, S ) for ψ ∈ H ′ , S ∈ L ( H ′ ) . Finally, by Proposition 7.3, bisimulation on Hilbert spaces coincideswith projective equivalence, and projective equivalence on H ′ is the restriction ofprojective equivalence on H U . (cid:3) Theorem 7.6
Let J · K K ◦ : f ∗ ( K ◦ , a K ) → ( U Q , γ Q ) be the final coalgebra seman-tics for K ◦ with respect to the isometric embedding i : K ✲ ✲ H U . Then for any ψ, φ ∈ K ◦ : J ψ K K ◦ = J φ K K ◦ ⇐⇒ ψ ∼ b φ ⇐⇒ ψ ∼ p φ. Thus the strongly extensional quotient [25] of the coalgebra ( K ◦ , a K ) is the pro-jective coalgebra ( P ( K ) , ¯ a K ) , where P ( K ) is the set of rays or one-dimensionalsubspaces of K , and ¯ a K is defined by: ¯ a K ( ¯ ψ ) = , α ( ψ ) = 0( r, ¯ φ ) α ( ψ ) = ( r, φ ) . Here ¯ ψ = { λψ | λ ∈ C } is the ray generated by ψ . Remark
There is a subtlety lurking here, which is worthy of comment. When weconsider an extension of a Hilbert space to a larger one, H ′ ⊂ ✲ H , the character-istic quantum phenomenon of incompatibility can arise; a subspace S of H may beincompatible with the subspace H ′ (so that e.g. the corresponding projectors do notcommute). The characterization of bisimulation as projective equivalence showsthat this notion is nevertheless stable under such extensions. However, we can ex-pect incompatibility to be reflected in some fashion in the coalgebraic approach, inparticular in the development of a suitable coalgebraic logic . We shall now show that the passage to the Grothendieck category of coalgebrasdoes succeed in alleviating the problem of excessive rigidity of coalgebras as dis-cussed in Section 3.1.1. Our strategy will be to lift the Representation Theo-rem 3.15 from [1] from Chu spaces to coalgebras, using the results of Section 6.3.We consider a morphism in R F between representations of Hilbert spaces.Such a morphism has the form h : f ∗ ( H ◦ , a H ) → ( K ◦ , a K ) where H and K are any Hilbert spaces, and writing Q = L ( H ) , Q ′ = L ( K ) , thefunctor f ∗ is induced by a map f : Q ′ → Q , and h is a homomorphism of F Q ′ -coalgebras.By Proposition 6.3, ( h, f ) : ( H ◦ , L ( H ) , e H ) → ( K ◦ , L ( K ) , e K )
20s a Chu morphism. By Proposition 3.2 and the remark following Theorem 3.10 of[1], the Chu morphism induced by the biextensional collapse of these Chu spacesis ( P h, f ) : ( P ( H ◦ ) , L ( H ) , ¯ e H ) → ( P ( K ◦ ) , L ( K ) , ¯ e K ) where P ( h )( ¯ ψ ) = h ( ψ ) . By Theorem 7.6, the induced coalgebra homomorphismon the strongly extensional quotients of the corresponding coalgebras is P h : f ∗ ( P ( H ) , ¯ a H ) → ( P ( K ) , ¯ a K ) . We can now use Theorem 3.12 of [1]:
Theorem 7.7
Let H , K be Hilbert spaces of dimension greater than 2. Considera Chu morphism ( f ∗ , f ∗ ) : ( P ( H ) , L ( H ) , ¯ e H ) → ( P ( K ) , L ( K ) , ¯ e K ) . where f ∗ is injective. Then there is a semiunitary (i.e. a unitary or antiunitary) U : H → K such that f ∗ = P ( U ) . U is unique up to a phase. Moreover, f ∗ is thenuniquely determined as U − . Since any coalgebra homomorphism gives rise to a Chu morphism, this will al-low us to lift fullness of the representation in Chu spaces to the coalgebraic setting.
Proposition 7.8 If U : H → K is a semiunitary, then U ◦ : f ∗ ( H ◦ , a H ) → ( K ◦ , a K ) is a coalgebra homomorphism, where f ∗ = U − . Proof
This follows by the same argument as Proposition 3.13 of [1]. In particu-lar, the fact that U ◦ is a coalgebra homomorphism follows from the relation P S ( U ψ ) = U ( P U − ( S ) ψ ) which is shown there. (cid:3) We must now account for the injectivity hypothesis in Theorem 7.7. The fol-lowing properties of coalgebras and Chu spaces respectively are standard.
Proposition 7.9 If F preserves weak pullbacks, the kernel of an F -coalgebrahomomorphism is a bisimulation. Hence if ( A, α ) is a strongly extensional F -coalgebra, on which bisimilarity is equality, then any homomorphism with ( A, α ) as domain must be injective. Proposition 7.10 If f : C → C is a morphism of separated Chu spaces, and f ∗ is surjective, then f ∗ is injective.
21e shall write sF for the restriction of F to sSet , the category of sets and surjectivemaps. Similarly, we write sChu for the restriction of Chu to sSet . Clearly T cutsdown to these restrictions. Moreover, the isomorphism of Chu K with R Chu ofProposition 6.1 cuts down to an isomorphism of R sChu with sChu K , the subcat-egory of Chu spaces and morphisms f with f ∗ surjective.Thus if we define the category P SymmH as in [1], with objects Hilbert spacesof dimension > , and morphisms semiunitaries quotiented by phases, we obtainthe following result: Theorem 7.11
There is a full and faithful functor P C : P SymmH → R sF .Moreover, the following diagram commutes: P SymmH > P C >> Z sF sChu [0 , P R ∨∨∨ < ∼ = Z sChu R T ∨∨ Here P R is the full and faithful functor of Theorem 3.15 of [1]. This result confirms that our approach of expressing contravariance throughindexing over a base does succeed in allowing sufficient scope for the representa-tion of physical symmetries, while also allowing for the construction of a universalmodel as a final coalgebra, and for the expression of the dynamics of repeatedmeasurements.
Our development of ‘coalgebra with contravariance’ can be carried out quite gen-erally. We shall briefly sketch this general development.Suppose we have a functor G : C op × C −→ C . Since
CAT is cartesian closed, we can curry G to obtain ˆ G : C op −→ [ C , C ] where [ C , C ] is the (superlarge) functor category on C . There is also a functor [ C , C ] −→ CAT F to its category of coalgebras, and a natural transformation t : F → G to the corresponding functor between the categories of coalgebras, as inProposition 5.1. Composing these two functors, we obtain a strict indexed category G : C op −→ CAT . We can then form the Grothendieck category R G .The indexed category F arises in exactly this way, from the functor G : Set op × Set −→ Set :: (
Q, X ) ( { } + (0 , × X ) Q . We have found this combination of fibrational and coalgebraic structure a con-venient one for our objective in the present paper of representing physical sys-tems. In particular, the fibrational approach to contravariance allows enough ‘el-bow room’ for the representation of symmetries. We also used the fibrationalstructure in formulating the connection to Chu spaces, which proved to be bothtechnically useful and conceptually enlightening. A natural follow-up would be todevelop a fibred version of coalgebraic logic, which we plan to do in a sequel.We note that a quite different, and in some sense more direct approach to coal-gebra for bivariant functors has been developed by Tews [27]. A viable approachis developed in [27] only for a limited class of functors, the ‘extended polyno-mial functors’. Moreover, the issues of rigidity vs. symmetry which we have beenconcerned with are not addressed in this approach, which is also technically fairlycomplex. Of course, there is a beautiful theory of the solution of reflexive equa-tions for mixed-variance functors provided by
Domain theory [13, 2]. The valueof coalgebras, in our view, is that they provide a simpler setting in which a greatdeal can be very effectively accomplished, without the need for the introduction ofpartial elements and the like.The need for contravariance in our context, motivated by the representation ofphysical systems, appears to be of a different nature, and hence better met by thefibrational methods we have introduced in the present paper.A deeper understanding of the issues here will, we hope, shed interesting lighton each of the topics we have touched on in this paper: foundations of physics,computational models, and the mathematics of coalgebras.
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