Dagger linear logic for categorical quantum mechanics

Categorical quantum mechanics exploits the dagger compact closed structure of finite dimensional Hilbert spaces, and uses the graphical calculus of string diagrams to facilitate reasoning about finite dimensional processes. A significant portion of quantum physics, however, involves reasoning about infinite dimensional processes, and it is well-known that the category of all Hilbert spaces is not compact closed. Thus, a limitation of using dagger compact closed categories is that one cannot directly accommodate reasoning about infinite dimensional processes. A natural categorical generalization of compact closed categories, in which infinite dimensional spaces can be modelled, is *-autonomous categories and, more generally, linearly distributive categories. This article starts the development of this direction of generalizing categorical quantum mechanics. An important first step is to establish the behaviour of the dagger in these more general settings. Thus, these notes simultaneously develop the categorical semantics of multiplicative dagger linear logic. The notes end with the definition of a mixed unitary category. It is this structure which is subsequently used to extend the key features of categorical quantum mechanics.


Introduction
Categorical quantum mechanics (CQM), as described in [CK17,HJ19], employs a graphical calculus for †-compact closed categories ( †-KCCs) to study quantum processes within the †-KCC of finite dimensional Hilbert spaces (FHilb). From a logical perspective, the graphical calculus is the proof theory of a compact fragment of multiplicative †-linear logic [Dun06]. This programme of CQM was initiated by Abramsky and Coecke's seminal paper [AC04] and it has allowed much of the structure of FHilb to be abstracted away and absorbed into the graphical calculus.
A well-known limitation of compact closed categories, is that, while they model finite dimensional Hilbert spaces, they do not model infinite dimensional spaces [Heu08]. A categorical generalization of compact closed categories, in which infinite dimensions spaces can be modelled, however, is *-autonomous categories. Thus, from a categorical perspective, One may think of the unitary category as acting on the larger category, much as a field K acts on a K-algebra as scalars. Expressing these categories in this manner allows an obvious notion of functor as a square of †-Frobenius functors, whose component on the unitary categories preserves unitary structure, and which commutes up to a linear natural isomorphism. For any †-isomix category one can build a unitary category by collecting the "pre-unitary" objects which are in the core. This give a way of generating a mixed unitary category from a †-isomix category, which we call the unitary construction.
We have provided examples throughout the text. An important example, closely related to traditional CQM, is the * -autonomous category of "finiteness matrices", FMat(C), over the complex numbers [Ehr05] (see Example 3.5.2). Here the maps are infinite dimensional matrices whose support is carefully controlled by the finiteness structure. The dagger on the category is given by simultaneously transposing and conjugating the matrices: on objects it is given by taking the dual finiteness space. FMat(C) forms a †-isomix category which, furthermore, is †- * -autonomous. An object is in the core if and only if it's underlying finiteness space is finite and these objects are also exactly the unitary objects. The unitary

Linearly distributive categories
This section recalls some background concepts from the theory of linearly distributive categories. The definition of linearly distributive categories is available in [CS97b,BCST96].
Here we briefly recall the definition of linear functors and their transformations [CS99], the notion of a linear adjoint [CKS00] -which we shall refer to as a linear dual -and the notion of a mix category and its core [CS97a,BCS00].
2.1. Linearly distributive categories, functors, and transformations. A linearly distributive category (LDC) is a category, X, with two monoidal structures (⊗, , a ⊗ , u L ⊗ , u R ⊗ ) and (⊕, ⊥, a ⊕ , u L ⊕ , u R ⊕ ) linked by natural transformations called the linear distributors: such that the monoidal natural isomorphisms -associators and unitors -interact coherently with the linear distributors, see [BCST96,CS97b] for more details. A symmetric LDC is an LDC in which both monoidal structures are symmetric, with symmetry maps c ⊗ and c ⊕ , such that ∂ R = c ⊗ (1 ⊗ c ⊕ )∂ L (c ⊗ ⊕ 1)c ⊕ . LDCs provide categorical semantics for the multiplicative linear logic (MLL). LDCs come equipped with a graphical calculus [BCST96] that contains the calculus for monoidal categories.
In this section, we review the fundamentals of the graphical calculus for LDCs. For detailed exposition, see [BCST96,CS97b]. The following are the generators of LDC circuits: wires represent objects and circles represent maps. The input wires of a map are tensored (with ⊗), and the output wires are "par"ed (with ⊕). The following diagram represents a The ⊗-associator, the ⊕-associator, the left linear distributor, and the right linear distributors are, respectively, drawn as follows: is the ⊕-introduction rule, is ⊗-introduction rule, is the ⊗-elimination rule, is the ⊕-elimination rule.
The unitors are drawn as follows: Diagram (a) is called the left -introduction, (b) is called the left -elimination, (c) is the left ⊥-introduction, and (d) is the left ⊥-elimination. The unit is introduced, and the counit ⊥ is eliminated using the thinning links which are shown using dotted wires in the diagrams.
The following are a set of circuit equalities (which when oriented become reduction rewrite rules): The following are also circuit equalities (and when oriented become expansion rules): As in linear logic, not all circuit diagrams constructed from these basic components represent a valid LDC circuit. In his seminal paper on linear logic, [Gir87], Girard introduced a criterion for the correctness of his representation of proofs using proof nets based on switching links. A valid proof structure must be connected and acyclic for all the switching link choices. Using this correctness criterion has the disadvantage of requiring exponential time in the number of switching links. Danos and Regnier [DR89] improved this situation significantly by providing an algorithm for correctness which takes linear time (see [Gue99]) on the size of the circuit. To verify the validity of the circuit diagrams of LDCs, Blute et.al. [BCST96], provided a boxing algorithm which was based on Danos and Regnier's more efficient algorithm which we now describe.
In order to verify that an LDC circuit is valid, circuit components are "boxed" using the below rules. The primitive generating maps are automatically boxed.
Double lines refer to multiple number of wires. ⊗-introduction and ⊕-elimination are boxed in (a 1 ) and (a 2 ) respectively. In (b 1 ), it is shown how a box 'eats' the ⊗-elimination: in (b 2 ) the dual rule shows a ⊕-introduction being eaten. (c) shows how boxes can be amalgamated when they are connected by a single wire. In (e 1 )-(e 4 ), it is shown how the thinning links can be boxed. By progressively enclosing the components of the circuit in boxes using these rules, if we end up with a single box (or a wire), precisely when the circuit is valid.
As an example, we verify the validity of the left linear distributor: In the firts step the ⊗-introduction and ⊕-elimination are boxed. In the second step the boxes are amalgamated along the single wire joining them. In the third step, the box absorbs the ⊗-elimination and ⊕-introduction.
In contrast, we now show that the reverse of the linear distributor is invalid as the boxing process gets stuck: The first diagram for the monoidal functor is the associative law, and the other two diagrams are the right and the left unit laws respectively.
Definition 2.1. [CS99, Definition 1] Given linearly distributive categories X and Y, a linear functor F : X − → Y consists of (i) a pair of functors F = (F ⊗ , F ⊕ ): (F ⊗ , m ⊗ , m ) which is monoidal with respect to ⊗ and (F ⊕ , n ⊕ , n ⊥ ) which is comonoidal with respect to ⊕. We refer to m ⊗ and n ⊕ as tensor laxors, and m and n ⊥ as unit laxors. (ii) natural transformations: such that the following coherence conditions hold: In the graphical calculus, functors are represented by linear functor boxes [CS99]. A linear functor box can be monoidal or comonoidal. When the functor box is monoidal (F ⊗ ), it has one principal output wire (represented by a port where the wire exits the box) and the other wires are auxiliary. When the box is comonoidal (F ⊕ ), it has one principal input wire with a port and the other wires are auxiliary. The functor boxes are subject to a very natural "box eats box" calculus described in [CS99]. A box can eat another box only when a ported wire meets an auxiliary wire.
The linear strengths are drawn in the graphical calculus as follows: When working in the categorical doctrine of symmetric LDCs we will expect the linear functors to preserve the symmetry. Thus, a symmetric linear functor is a linear functor F = (F ⊗ , F ⊕ ) which satisfies in addition: Natural transformations between linear functors also break into two components linking respectively the tensor functors and, in the opposite direction, the par functors: a comonoidal transformation satisfying the following coherence conditions: An adjunction of linear functors, (η, ) : F G is an adjunction in the usual sense (i.e. satisfying the triangle equalities) in the 2-category of LDCs with linear functors and linear natural transformations. In particular, such an adjunction yields a pair of adjunctions: (η ⊗ , ⊗ ) : F ⊗ G ⊗ which is a monoidal adjunction, and ( ⊕ , η ⊕ ) : G ⊕ F ⊕ which is a comonoidal adjunction. By Kelly's results [Kel74], a functor with a right adjoint is comonoidal if and only if its right adjoint is monoidal. This leads to the observation that: Lemma 2.3. If (η, ) : F G is an adjunction of linear functors, then F ⊗ is iso-monoidal (or strong) with respect to ⊗ and F ⊕ is iso-comonoidal making the linear functor F strong.
A linear equivalence is a linear adjunction in which the unit and counit are linear natural isomorphisms.
2.2. Mix categories. In this paper we shall be predominately concerned with LDCs which have a mix map: The map mx A,B is natural and is called the mixor. The coherence condition for the mix map has the following form in string diagrams (where the mix map is represented by an empty box): In a mix category, the associator, the distributor and the mix maps interact as follows. See Lemma 2, and proposition 3 in [BCS00] for a proof.
There are many examples of mix categories including Coherence spaces [Gir87], and Finiteness spaces [Ehr05].
When the mix map m is an isomorphism, then X is said to be an isomix category. Recall that, when m is an isomorphism, the coherence requirement for the mixor is automatically satisfied (see [CS97a, Lemma 6.6]). Finiteness spaces [Ehr05] and Chu Spaces [Bar06] provide examples of isomix categories.
Proof. We show that Mx ↓ : (X, ⊗, ⊗) − → (X, ⊗, ⊕) is a linear functor: the monoidal and comonoidal components of the functor are given by (1, 1, 1) and (1, mx, m −1 ) respectively. The linear strenghts are ν L • The associative law for monoidal functors, (mx ⊗ 1) mx a ⊕ = a ⊗ (1 ⊗ mx) mx, is satisfied: • The unit laws for monoidal functors hold. Here is the pictorial proof of (1 ⊗ m −1 )mx = u L ⊗ (u L ⊕ ) −1 , where the filled rectangles represent m −1 : The other unit law holds similarly. Definition 2.6. [BCS00] The core of a mix category, Core(X) ⊆ X, is the full subcategory with objects U such that the mixors Proposition 2.7. [BCS00, Proposition 3] If X is a mix-LDC and A, B ∈ Core(X) then A⊕B and A ⊗ B ∈ Core(X) (and A ⊕ B A ⊗ B). If X is an isomix-LDC, then , ⊥ ∈ Core(X).
Corollary 2.8. When X is a compact LDC, the mix functors, Mx ↓ and Mx ↑ , are linear isomorphisms. Consequently, compact LDCs are linearly equivalent to monoidal categories.
We shall denote the inverse of Mx ↓ by Mx * ↓ : (X, ⊗, ⊕) − → (X, ⊕, ⊕): this is the identity functor as a mere functor, strict on the par structure, and on the tensor structure having as the unit laxor m and as the tensor laxor mx −1 . Similarly, we shall denote the inverse of Mx ↑ by Mx * ↑ .

Linear duals.
A key notion in the theory of LDCs is the notion of a linear adjoint [CKS00]. Here we shall refer to linear adjoints as "linear duals" in order to avoid any confusion with an adjunction of linear functors.
The commuting diagrams are called often referred to as "snake diagrams" because of their shape when drawn in string calculus: (iii) In a mix-LDC if B ∈ Core(X) and B A, then A ∈ Core(X). Proof. The unit and counit of the adjunction (η , ) : F ⊗ (A) F ⊕ (B) is given as follows: The unit and counit of the other adjunction is given similarly, however using the right linear strengths (ν R ⊗ and ν R ⊕ ). An LDC in which every object has a chosen left and right linear dual, respectively (η * , * ) : A * A and ( * η, * ) : A * A, is a * -autonomous category. In the symmetric case a left linear dual gives a right linear dual using the symmetry: thus, it is standard to assume the existence of just the left dual with the right being the same object with the unit and counit given by symmetry (as above).
Just as compact LDCs are linearly equivalent to monoidal categories so compact *autonomous categories are linearly equivalent to compact closed categories. The equivalence is given by Mx ↑ which spreads the par onto two tensor structures (or, indeed, by Mx ↓ which shows how to spread out a compact closed structure on the tensor).
In a symmetric * -autonomous category the left dual of an object is always canonically isomorphic to the right dual. Moreover, even in non-symmetric * -autonomous categories, it is often the case that the two duals are coherently isomorphic: Definition 2.12. [EM12] A cyclor in a * -autonomous category (X, ⊗, , ⊕, ⊥, * ( ), ( ) * ) is a natural isomorphism A * ψ − − → * A satisfying the following coherence conditions: A * -autonomous category with a cyclor is said to be cyclic. The requirement [C.2] implies: Symmetric * -autonomous categories always have a canonical cyclor: * η * We shall use the cyclor in Section 4 to show how conjugation and dagger are related in the presence of dualization.

Frobenius functors and daggers
We shall be interested here in linear functors between LDCs called Frobenius functors which come in various flavours, including mix functors and isomix functors, as illustrated in Figure  1. These functors are directly related to the Frobenius monoidal functors of [DP08] and they are referred to as degenerate linear functors in [BPS12]. Furthermore, we have already seen two rather basic examples, namely, Mx ↑ and Mx ↓ .  Frobenius functors preserve linear duals and with an additional coherence condition they preserve the mix map. The coherence requirements for a dagger on an LDC are implied by requiring that the dagger functor be a Frobenius involutive equivalence. Once the dagger is understood we can consider †-mix categories and their functors which we shall take to be mix Frobenius functors with a further requirement concerning the preservation of the dagger.
Definition 3.1. A Frobenius functor is a linear functor F such that: The left and right linear strengths of ⊗ and ⊕ coinciding with the m ⊗ and n ⊕ respectively means that in the diagrammatic calculus, ports can be moved around freely:

This implies that the ports can be omitted in the circuits.
A Frobenius functor is symmetric if as a linear functor it preserves the symmetries of the tensor and par.
It is immediate from Lemma 2.11 that Frobenius functors preserve linear duals. In fact if F : X − → Y is a Frobenius functor and A B is a linear dual, as the duals F ⊗ (A) F ⊕ (B) and F ⊕ (A) F ⊗ (B) now coincide, we just obtain the one dual F (A) F (B). In the case when the Frobenius functor is between cyclic * -autonomous categories we expect the functor to be cyclor-preserving in the following sense: where the left and right vertical arrows are respectively the maps: The cyclor preserving condition may be pictorially represented as follows: Lemma 3.3. Suppose F is a cyclor preserving Frobenius linear functor, then Proof.
Definition 3.4. Suppose X and Y are mix categories. F : X − → Y is a mix functor if it is a Frobenius functor such that This is diagrammatically represented as follows: Lemma 3.5. Mix functors preserve the mix map: Proof.
Linear natural isomorphisms between Frobenius functors (α ⊗ , α ⊕ ) : F − → G often take a special form with α ⊗ = α −1 ⊕ : this allows the coherence requirements to be simplified. The next results describe some basic circumstances in which this happens: Lemma 3.6. Suppose F : X − → Y, and G : X − → Y are Frobenius linear functors and α := (α ⊗ , α ⊕ ) : F ⇒ G is a linear natural transformation. Then, the following are equivalent: (i) One of [nat.1](a) or [nat.1](b) holds, and one of α ⊗ or α ⊕ is an isomorphism. holds.
is a linear transformation.
Frobenius functors between isomix categories are especially important in the development of dagger linearly distributive categories and they often satisfy an additional property: Definition 3.7. A Frobenius functor between isomix categories is an isomix functor in case it is a mix functor which satisfies, in addition, the following diagram: ecall that a linear functor is normal in case both m and n ⊥ are isomorphisms. We observe: Lemma 3.8. For a mix Frobenius functor, F : X − → Y, between isomix categories the following are equivalent: As the mix map m is an isomorphism so is F (m) which implies that if n ⊥ is an isomorphism then m must be an isomorphism and vice versa. Thus, F will be a normal functor. (ii) ⇒ (iii): If F is normal then n ⊥ and m are isomorphisms and so The mix-preservation for F makes n ⊥ a section (and m a retraction) while the isomix-preservation makes m ⊥ a retraction (and m a section). This means n ⊥ is an isomorphism (m is an isomorphism).
In Lemma 3.6, when A B ∈ X, then α ⊕ is defined as follows: For these special linear isomorphisms with α ⊗ = α −1 ⊕ we can simplify the coherence requirements: Lemma 3.10. Suppose F and G are Frobenius functors and α : F − → G is a natural isomorphism then: is a linear transformation; (ii) If F and G are strong Frobenius functors and α is ⊗-monoidal and ⊕-monoidal then (α, α −1 ) is a linear transformation. Proof.
(i) If α is ⊗-monoidal and ⊕-comonoidal then so is α −1 supporting the possibility that it is a component of a linear transformation. Considering [LT.1] we show that (α, α −1 ) satisfies this requirement as: The remaining requirements follow in a similar manner. (ii) When the laxors for the functors are isomorphisms then being monoidal implies being comonoidal. 3.2. Dagger mix categories. Conventionally, in categorical quantum mechanics a dagger is defined as a contravariant functor which is an involution that is stationary on objects (A † = A). Before proceeding to define the dagger functor for LDCs, the notion of the opposite LDC and whence the notion of a contravariant linear functors have to be developed. For LDCs we cannot expect the dagger to be stationary on objects, however, it is still possible that it can act like an involution. If (X, ⊗, , ⊕, ⊥) is a linear distributive category, the opposite linear distributive category is (X, ⊗, , ⊕, ⊥) op := (X op , ⊕, ⊥, ⊗, ) where X op is the usual opposite category with the monoidal structures are flipped as follows: ( ) op is an endo functor for the category of LDCs and linear functors. It is an involution: First note that saying this is an involutive equivalence asserts that the unit and counit of the equivalence are the same (although one is in the opposite category). Thus, the adjunction expands to take the form (ı, ı) : ( ) † ( ) † op : X op − → X. However, the unit and counit are linear natural transformations so ı expands to ı = (ı ⊗ , ı ⊕ ). As the dagger functor is a left adjoint, it is strong and, thus, is normal. Furthermore, as the unit of an equivalence, ı is a linear natural isomorphism. This means ı = (ı ⊗ , ı ⊕ ) satisfies the requirements of Lemma 3.6, implying that ı −1 ⊗ = ı ⊕ . Simplifying notation we shall set ι := ı ⊕ so the unit linear transformation is ı := (ι −1 , ι). We then can simplify the requirements of ı to the map ι : A − → (A † ) † which we refer to as the involutor.
A symmetric †-LDC is a †-LDC which is a symmetric LDC for which the dagger is a symmetric linear functor. A cyclic †- * -autonomous category is a †-LDC with chosen left are right duals and a cyclor which is preserved by the dagger. A †-mix category is a †-LDC for which ( ) † : X op − → X is a mix functor. As the dagger functor is strong (and so normal) if the category is an isomix category then being †-mix already implies that the dagger is an isomix functor. Thus, a †-isomix category is a †-mix category which happens to be an isomix category.
In the remainder of the section, we unfold the definition of a †-isomix category and give the coherence requirements explicitly.
Proposition 3.12. A dagger linearly distributive category is an LDC with a functor ( ) † : X op − → X and natural isomorphisms such that the following coherences hold: [ †-ldc.1] Interaction of λ ⊗ , λ ⊕ with associators: and two symmetric diagrams for u L ⊗ and u L ⊕ must also be satisfied.
The structure is presented using strong monoidal laxors: to form a linear functor the laxor λ ⊕ needs to be reversed by taking its inverse. Then, we have ν l ⊗ = ν r ⊗ := λ −1 ⊕ and Once this adjustment is made all the required coherences for † to be a linear functor are present.
Note that [ †-ldc.6] equivalently expresses the triangle identities of the adjunction (ι, ι) :: The coherences for the involutor asserts that it is a monoidal transformation for both the tensor and par: by Lemma 3.10 (ii) this suffices to show that it is a linear transformation.
A symmetric †-LDC is a †-LDC which is a symmetric LDC and for which the following additional diagrams commute: [ †-ldc.7] Interaction of λ ⊗ , λ ⊕ with symmetry maps: A †-mix category is a †-LDC which has a mix map and satisfies the following additional coherence: If m is an isomorphism, then X is a †-isomix category and, since ( ) † is normal, ( ) † is an isomix Frobenius functor.
Lemma 3.13. Suppose X is a †-mix category then the following diagram commutes: Proof. The proof follows directly from Lemma 3.5.
With respect to its applications to quantum theory, this article primarily focuses on †-isomix categories. As we will see in Section 5, the notion of unitary objects and unitary isomorphisms is supported only within a †-isomix category.
It is useful to observe that objects in the core are closed under taking the dagger and duals.
Lemma 3.14. Suppose X is a †-mix category and A ∈ Core(X) then A † ∈ Core(X).
Proof. The natural transformation A † ⊗ X mx − − → A † ⊕ X is an isomorphism as follows: A †- * -autonomous category is a cyclic †- * -autonomous category when the dagger preserves the cyclor in the following sense.
Lemma 3.16. In a cyclic, †- * -autonomous category, The rectangle is a functor box for the †-functor. Notice how we use vertical mirroring to express the contravariance of the †-functor. By the functoriality of ( ) † , we have: = . These contravariant functor boxes compose... contravariantly. Given maps f : A − → B and g : B − → C: The following are the representations of the basic natural isomorphisms of a †-LDC: It is important to note that one may not have a legal proof net inside a †-box. This complicates the correctness criterion. However, the required correctness criterion is discussed in [MP05].
3.4. Functors for †-linearly distributive categories. Clearly the functors and transformations between †-LDCs must "preserve" the dagger in some sense. Precisely we have: ) called the preservator, such that the following diagrams commute: In case that F is a normal mix functor between †-isomix categories, then by Lemma 3.8, F is an isomix functor and, therefore by Corollary 3.9, the preservators become inverses, This means the squares [ †-LF.1] and [ †-LF.2] coincide to give a single condition for the tensor preservator: In case when F is an isomix functor, by Lemma 3.6, ρ := ρ ⊗ is monoidal on ⊗ and comonoidal on ⊕: For linear natural transformations (β ⊗ , β ⊕ ) : F − → G between †-linear functors, we demand that β ⊗ and β ⊕ are related by: Notice that this means that β ⊗ is completely determined by β ⊕ in the following sense: Because the vertical maps are isomorphisms, this diagram can be used to express β ⊗ in terms of β ⊕ . Similarly, β ⊕ can be expressed in terms of β ⊗ . Thus, it is possible to express the coherences in terms of just one of these transformations. 3.5.1. Finite dimensional framed vector spaces. In this section we describe the category of "framed" finite dimensional vector spaces, where a frame in this context is just a choice of basis. Thus, the objects in this category are vector spaces with a chosen basis while the maps, ignoring the basis, are simply homomorphisms of the vector spaces. The category of finite dimensional framed vectors spaces, FFVec K , is a monoidal category defined as follows: where e is the unit of the field K.
To define the "dagger" we must first choose a conjugation ( ) : K − → K (see more details in Section 4.2), that is a field homomorphism with k = (k). The canonical example being conjugation of the complex numbers, however, the conjugation can be arbitrarily chosen -so could also, for example, be the identity. This conjugation then can be extended to a (covariant) functor: where (V, V) is the vector space with the same basis but with the conjugate action The conjugate homomorphism, f , is then the same underlying map which is homomorphism between the conjugate spaces.
This makes ( ) * : FFVec op K − → FFVec K a contravariant functor whose action is determined by precomposition. Finally, we define the "dagger" to be the composite (V, This is a compact LDC with tensor and par being identified (so the linear distribution is the associator) and is isomix. We must show that it is a †-LDC. Towards this aim we define the required natural transformations on the basis: Note that the last transformation is given in a basis independent manner. Importantly, it may also be given in a basis dependent manner as ι(v i ) = v i as the behaviour of these two maps is the same when applied to the basis of (V, V) † namely the elements v j : We must show (identifying tensor and par) that ). Now the result is a higher-order term so it suffices to show the evaluations on basis elements are Thus, FFVec K is a compact †-isomix category where the † functor shifts objects i.e., A = A † .
3.5.2. Finiteness matrices. Finiteness spaces were introduced by Ehrhard, [Ehr05], as a model of linear logic. The type system can be used to produce a typed system for infinite dimensional matrix multiplication in which no sums become infinite. This system of infinite dimensional matrices forms an isomix * -autonomous category. If these matrices have entries in the complex numbers then there is a natural notion of conjugation and this gives a †-isomix category. We shall see that by taking the core of this category, which is the category of finite dimensional matrices, one obtains a basic example of a mixed unitary category as described in Section 5 of this paper. This example is also explored further in the sequels to this paper [CS19a,CS19b].
Definition 3.18. A finiteness space is a pair X := (|X|, F) with |X| a set, called the web, and F be a subset of P(|X|) such that F = F ⊥⊥ where The elements of F are called the finitary sets of the finiteness space of X, and the elements of F ⊥ are called cofinitary sets.
Observe that if the web of X is finite, then F is forced to be the whole powerset of X. This is well-defined as F × G = (F × G) ⊥⊥ is proven in [Ehr05] Lemma 2. Given maps R : X 1 − → Y 1 and S : X 2 − → Y 2 : 3.5.3. Category of abstract state spaces. This model is inspired by the category of convex operational models [BW11]. The following is a way to construct new †-isomix categories from an exisiting one.

Daggers, duals, and conjugation
The goal of this section is to review the interaction of the dualizing, conjugation and dagger functors. In dagger compact closed categories, the dagger functor ( ) † , and the dualizing functor ( ) * commute with each other and their composite gives the conjugate functor ( ) * . Similary, ( ) * and ( ) * when composed gives the dagger functor. Our aim is to generalize these interactions to †-LDCs and to achieve this at a reasonable level of abstraction. To achieve this we shall need the notion which we here refer to as "conjugation" but was investigated by Egger in [Egg11] under the moniker of "involution" (which clashes with our usage). In a * -autonomous category, taking the left (or right) linear dual of an object extends to a Frobenius linear functor as follows: The ( ) * functor is both contravariant and, op-monoidal and op-comonoidal: These maps are op-monoidal and op-comonoidal laxors, hence are isomorphisms, which satisfy the obvious coherences. Thus, ( ) * is a strong Frobenius linear functor.
In the rest of the section, we will write (X op ) rev as X oprev .
Lemma 4.1. If X is an isomix category, then ( ) * : X oprev − → X is an isomix functor. Proof. Because, ( ) * is a strong Frobenius functor, by Lemma 3.8, it suffices to prove that ( ) * preserves mix, i.e., ( ) * is a mix functor i.e., we need to show that n ⊥ m m : * − → ⊥ * = m * . The proof is as follows: is a linear equivalence of Frobenius linear functors.
Proof. The proof is straightforward in the graphical calculus.
For a cyclic * -autonomous category, we can straighten out this equivalence to be a dualizing involutive equivalence (i.e. so that the unit and counit are equal): Proof. The unit and counit are drawn as follows: The cyclor is a linear transformation which is an isomorphism as it is monoidal with respect to both tensor and par and adjoints are determined only upto isomorphism. It remains to check that the triangle identities hold: The other triangle identity holds similarly.
The equality of η and is immediate from [C.2] for cyclors with the map η = being the dualizor. In the symmetric case, the dualizor of this equivalence may be drawn as: Conjugation. Recall the following structure from Egger [Egg11]: Definition 4.4. A conjugation for a monoidal category (X, ⊗, I) consists of a functor ( ) : X rev − → X with natural isomorphisms: called respectively the (tensor reversing) conjugating laxor and the conjugator such that A monoidal category is conjugative when it has a conjugation functor. A symmetric monoidal category, which is conjugative, is symmetric conjugative in case it satisfies the additional coherence: A conjugative LDC is a linearly distributive category (X, ⊗, , ⊕, ⊥) together with a conjugating functor ( ) : X − → X and natural isomorphisms: such that (X, ⊗, , χ ⊗ , ε) and (X, ⊕, ⊥, χ −1 ⊕ , ε) are conjugative (symmetric) monoidal categories with respect to the conjugating functor and the following diagrams commute: Note, by Lemma 4.5, there exists canonical isomorphisms conjugation is a normal functor. However, the conjugation is not necessarily a mix functor when X is a mix category. For conjugation to be a mix functor, the following extra condition must be satisfied: Proposition 4.7. A conjugative LDC is precisely a LDC, X, with a Frobenius adjoint ( −1 , ) : ( ) ( ) rev : X rev − → X where := (ε, ε −1 ). Furthermore, if X is an isomix category and conjugation is a mix functor then conjugation is an isomix equivalence.
Proof. It is clear that ( ) is a strong Frobenius functor so being mix implies isomix. Also, ε is clearly monoidal for tensor and par. The triangle equalities give ε −1 ε = 1 : A − → A thus ε = ε.
Clearly conjugation should flip left duals into right duals: Lemma 4.8. If B A is a linear dual then A B is a linear dual. When a * -autonomous category is cyclic one expects that conjugation should interact with the cyclor in a coherent fashion: Definition 4.9. [EM12] A conjugative cyclic * -autonomous category is a conjugative * -autonomous category together with a cyclor A * ψ − − → * A such that which gives a map σ : (A) * − → (A * ).
The above condition is drawn as follows: When the * -autonomous category is symmetric, conjugation automatically preserves the canonical cyclor.
= η * (ε −1 ⊕ 1) 4.3. Dagger and conjugation. The interaction of the dagger and conjugation for cyclic * -autonomous categories in the presence of the dualizing functor is illustrated by the following diagram: Specifically we have: Theorem 4.11. Every cyclic, †- * -autonomous category is a conjugative * -autonomous category.
Proof. Let X be a cyclic, †- * -autonomous category. Then composing adjoints gives the equivalence ( ) † * ( ) * † . To build a conjugation, however, we need an equivalence between the same functors: to obtain such an equivalence we use the natural equivalence ω : ( ) † * − → ( ) * † from the cyclor preserving condition for Frobenius linear functors. A conjugative equivalence, in addition, requires that the unit and counit of the equivalence be inverses of each other. The unit and counit of the equivalence are given by (a) and (b) respectively; (a) X rev where the isomorphism ω : ( ) † * − → ( ) * † is from the cyclor preserving condition, [CFF], for Frobenius linear functors: It remains to show that the unit and the counit maps are inverses of each other in X: ( * ) holds because † preserves the cyclor. Thus, (a) and (b) are inverses of each other.
Next, we show that a conjugation functor together with a dualizing functors gives a †: Theorem 4.12. Every cyclic, conjugative * -autonomous category is also a †- * -autonomous category.
To build a dagger we need an equivalence on the same functor: we obtain this by using the natural equivalence σ : ( ) * − → ( ) * from Definition 4.9. An involutive equivalence, in addition, requires the unit and counit of the (contravariant) equivalence to be the same map (which we called the involutor, ι). We show that this is the case: The unit and counit of the equivalence is given by (a) and (b) respectively; (a) X op  Observe that for composition of the dualizing functor and the conjugation functor to yield a dagger, and vice versa, a * -autonomous category is required to be cyclic with the cyclor being preserved by the conjugation (see Definition 4.9) and the dagger (see just before Lemma 3.16).

Examples.
In this section, we cover examples of †-isomix categories where the † is given by conjugation and the dualizing functor.

Category of a group with conjugation.
Definition 4.13. A group with conjugation is a group (G, ., e) together with a function ( ) : G − → G such that, for all g ∈ G, g = g, and for all g, h ∈ G, g.h = hg, and e = e.
Let (G, ., e) be a group with conjugation. The discrete category D(G, ., e) whose objects are the elements of the group is a monoidal category with the tensor product given by g ⊗ h := g.h, and the monoidal unit e. Moreover, D(G, ., e) is a compact closed category where g * := g −1 and it has a trivial conjugative cyclor (See Definition 4.9). Thus, D(G, ., e) is a compact †-isomix- * -autonomous category with g † := g * gives an example of how the conjugation gives rise to a dagger.
Here are some examples of groups with conjugation and the discrete categories given by them: • Suppose we fix the group to be (C, +, 0) where the objects are complex numbers and the tensor product is addition. The dual and conjugation of complex numbers are given as follows: (a + ib) * = −a − ib and a + ib := a − ib. Hence, • Consider the multiplicative group (C * , ., 1) where the objects are non-zero complex numbers and the tensor product is given by multiplication. The dualizing and the conjugation functors are given as follows: (a + ib) * = c + id, where ac − bd = 1 and ad + bc = 0 a + ib := a − ib (a + ib) † is given by (a + ib) * .
• Suppose the group is fixed to be D(P (x), +, 0) where P (x) is a polynomial ring.  • Consider the general linear group of degree 2, (M 2 , ., I 2 ) over complex numbers. Then, the discrete category D(M 2 , ., I 2 ) has a dualizing functor given by matrix inverse and conjugation is given by conjugate transpose: Then, D(M 2 , ., I 2 ) is a †-isomix * -autonomous category with: Chu Spaces. Applications of Chu Spaces to represent quantum systems have been studied in [Abr12], [Abr13]. In this section we show that the Chu construction over a closed conjugative monoidal category, which has pullbacks, produces a †-isomix LDC, Chu X (I). To get the * -autonomous category and †-structure on Chu X (I) we shall start by explaining how one can produce conjugative structure on the Chu category. To achieve this we iteratively develop the structure of this category, starting with a conjugative closed monoidal category, X, which is not necessarily symmetric. Note that the fact that it is conjugative means that it is both left and right closed which allows us to consider the non-commutative Chu construction: in this regard we shall follow Jürgen Koslowski's construction [Kos06] using simplified "Chu-cells" on the same dualizing object to obtain not a * -linear bicategory but a cyclic * -autonomous category. Furthermore, we shall choose a dualizing object which is conjugative in order to obtain a conjugative cyclic * -autonomous category.
In addition, Chu X (D) is conjugative with (A, B, ψ 0 , ψ 1 ) := (A, B, χψ 1 d, χψ 0 d) and (f, g) = (f , g). Finally, being conjugative cyclic * -autonomous implies that one has a dagger! In the case that X is a symmetric monoidal closed category we may recapture the usual Chu construction [Bar06], which we denote Chus X (D). Consider the full subcategory of Chu-objects with special Chu-cells of the form (A, B, ψ, c ⊗ ψ) in which the symmetry map is used to obtain the second cell, this gives an inclusion Chus X (D) − → Chu X (D).
We observe that X is symmetric conjugative when this subcategory is closed to the conjugation: Lemma 4.14. If X is a conjugative symmetric monoidal closed category and d : D − → D is an involutive object, then Chus X (D) is a conjugative symmetric * -autonomous category.
Proof. It suffices to observe that the Chu-cells of (A, B, ψ, c ⊗ ψ) have the right form. Using the coherence of the involution with symmetry, the first Chu-cell of this object has χc ⊗ ψd = c ⊗ χψd which is exactly the symmetry map applied to the second Chu-cell of the object as desired. To obtain an isomix category one can choose D = I. Chus X (I) is an isomix category because the unit for tensor and par are the same (namely = ⊥ = (I, I, u l ⊗ = u r ⊗ )). The tensor unit is always a conjugative object since (χ • ) −1 : I − → I; therefore, this is immediately a conjugative symmetric * -autonomous category. Composing the conjugation with the dualizing functor gives us a dagger.

Category of Hopf
Modules of a * -autonomous category. In the previous example, starting from a conjugative closed monoidal category with pullbacks, we built a †- * -autonomous category using the Chu construction. In this example 3 , we start with any symmetric *autonomous category, X, and build the category of modules over a Hopf Algebra which is in turn a †- * -autonomous category.
First of all, it has been already proven in [PS09], that the category of Hopf modules over a ⊗-Hopf algebra in any symmetric * -autonomous category is also a * -autonomous category. Then we note that, whenever the Hopf Algebra is cocommutative, the resulting * -autonomous category has a conjugation functor. One can construct the dagger functor by composing the conjugation functor and dualizing functor as in Theorem 4.12. We establish some basic definitions before describing the category of modules over a Hopf Algerba, H-Mod X . Note that instead of requiring that ∇ and e are coalgebra homomorphisms, one could equivalently require ∆ and u are algebra homomorphims with respect to the multiplication and the unit.
The components of a bialgebra are graphically depicted as follows: This gives a succinct graphical depiction of the coalgebra homomorphism laws; namely: A Hopf algebra is a bialgebra with an antipode. An involutive Hopf algebra is a hopf algebra where the antipode is self-inverse.
A standard example of a Hopf algebra is a group algebra over a field: for all group elements g, ∇ : g → g ⊗ g, : g → 1, ∆ : g ⊗ h → gh and s : g → g −1 .
Lemma 4.17. Suppose X is a symmetric monoidal category, then: (i) [Blu96, Theroem 3.5] If H is a commutative or a cocommutative Hopf Algebra in X, then s 2 = 1 where s is the antipode: so it is an involutive Hopf algebra.
This is graphically depicted as follows: Observe that any left action is indeed a module homomorphism.
Theorem 4.20 . [PS09] Let X be symmetric * -autonomous category and H be a ⊗-Hopf Algebra in X with bijective antipode (s 2 = 1). Then, H-Mod X is a * -autonomous category. If the Hopf Algebra, H, is cocommutative, then H-Mod X is a symmetric * -autonomous category.
and the unit of ⊕ is All the basic natural isomorphisms are inherited directly from X and they are module homomorphisms. Thus, HMod X is a LDC.
The dualizing functor ( ) * is given as follows:(A, : The cups and caps are inherited directly from X, hence the snake diagrams hold. The antipode in the definition of * : H ⊗ A * − → A * makes the cup and cap module morphisms.  In that case, H-Mod X is a symmetric * -autonomous category.

Suppose (A, )
Futhermore, we can show that the category of Hopf modules is conjugative.
Lemma 4.21. Let X be a symmetric * -autonomous category. H-Mod X , the category of modules over a cocommutative Hopf Algebra H is a conjugative symmetric * -autonomous category.
Proof. We already know that H-Mod X is a symmetric * -autonomous category. We define the conjugation functor ( ) : H-Mod X − → H-Mod X as follows: • The basic natural isomorphisms are given by: The natural isormorphisms satisfy all the coherences of conjugative symmetric *autonomous category.
Lemma 4.22. Suppose X is a symmetric (iso)mix * -autonomous category, then H-Mod X , the category of Hopf modules over a cocommutative Hopf Algebra H is a (iso)mix conjugative symmetric * -autonomous category. Proof. The mix map m : ⊥ − → is inherited directly from X.

Unitary structure and mixed unitary categories
The objective of this section is to introduce mixed unitary categories (MUCs) and their morphisms. A mixed unitary category consists of a unitary category, U, with a †-isomix Frobenius functor M : U − → C into a "large" † isomix category C. We refer to U as the unitary core of the MUC. The unitary core is to be regarded as providing the analogue of scalars for the larger category much as a field provides scalars for an algebra over that field. The section starts by describing the general notion of unitary structure in a †-isomix category. This allows the definition of a unitary category as a compact †-isomix category in which all objects have unitary structure satisfying certain coherence conditions. We then show how to extract a unitary category from any compact †-isomix category using pre-unitary objects. This is a useful construction in practice. However, it can just deliver a trivial unitary category -trivial in the sense that all objects are isomorphic to the units. This means that, in applying the construction, it is important to identify non-trivial pre-unitary objects to ensure that one is getting something worthwhile out.
Next we show, using the isomix functors Mx ↑ (or Mx ↓ ) that unitary categories are †linearly equivalent to †-monoidal categories and, furthermore, that closed unitary categories are equivalent to †-compact closed categories. This provides an explicit connection from MUCs to the standard notions from categorical quantum mechanics. One contribution of this more general perspective is that through the constructions in this section one can obtain examples not only of mixed unitary categories but also of †-monoidal and †-compact closed categories which might otherwise have been difficult to realize.
The final subsection introduces mixed unitary categories (MUCs). These form the basis for our approach to infinite dimensional categorical quantum mechanics. A MUC has a unitary core which is a model of classical categorical quantum mechanics extended by a larger setting in which infinite dimensional objects can be modelled. 5.1. Unitary structure. The notion of unitary maps is central to both quantum information theory as well as quantum mechanics since the evolution of a closed quantum system is described by such maps. Categorically, within a †-category, a unitary map is an isomorphism f : A − → B such that f −1 = f † . This definition of unitary isomorphism cannot be used directly within the framework of †-LDCs since the types of f −1 : B − → A and f † : B † − → A † are different. It is therefore apparent that one can only ask to have unitary isomorphisms between certain objects, which we call "unitary objects": Definition 5.1. A †-isomix category, X has unitary structure in case there is an essentially small class of objects U, called the unitary objects of X such that [U.1] for all A ∈ U, A ∈ Core(X), and A is equipped with an isomorphism, ϕ A : 3] for all A ∈ U, the following diagram commutes: R. Cockett, C. Comfort, and P. V. Srinivasan Vol. 17:4 [U.4] ⊥, ∈ U such that: Lemma 5.2. When A and B is a unitary object in a †-isomix category then, Often we shall want the unitary objects to have linear adjoints (or duals) but we shall need the analogue of †-duals (η † = c ⊗ and † = ηc ⊗ ) from categorical quantum mechanics: Definition 5.3. A unitary linear duality (η, ) : A u B between unitary objects A and B is a linear duality satisfying in addition: . In a compact †-LDC, u ⊥.
[Udual] (a) is shown diagrammatically as follows: . This is easily checked to be a unitary linear adjoint. We can now define what it means for an isomorphism to be unitary: Definition 5.5. Suppose A and B are unitary objects. An isomorphism A f − → B is said to be a unitary isomorphism if the following diagram commutes: Observe that ϕ is "twisted" natural for all unitary isomorphisms, thus, unitary isomorphisms compose and contain the identity maps. In a category in which the unitary structure maps are identity morphisms, one recovers the usual notion of unitary isomorphisms.
Our next objective is to show that all the coherence isomorphisms between unitary objects are unitary maps. First a warm up: Lemma 5.6. In a †-isomix category with unitary structure: (i) If f is a unitary isomorphism, then so is f † ; (ii) If f and g are unitary, then so are f ⊗ g and f ⊕ g; (iii) Unitary isomorphisms are closed under composition. Proof. ( is just the dagger functor applied to the unitary diagram of f . (ii) Suppose f and g are unitary morphisms, then: The inner square commutes because f and g are unitary maps. Similarly, using [U.5(b)], one can show that if f and g are unitary, then f ⊕ g is unitary.
(iii) The proof is trivial.
The following lemma will be used to prove that the associator natural isomorphisms are unitary. Lemma 5.7. The following diagram commutes: Proof.
Lemma 5.8. Suppose X is a †-isomix category with unitary structure and A, B, and C are unitary objects then the following are unitary maps: (ii) λ ⊕ is unitary because: Lems. 5.8 (vi), 5.6 (i) the left triangle commutes by [U.3] and the right triangle commutes by: Lem. 5.7 where the left square commutes because where the left square commutes for the same reason and the right square is the dagger of the left square. (xiii) ∂ L is unitary (see Figure 2). (xiv) ∂ R is unitary because: Definition 5.9. A unitary category is a †-isomix category with unitary structure such that every object in the category is a unitary object.
Clearly, a unitary category must be a compact †-LDC, because every object is in the core.
A †-monoidal category is a strict unitary category in which the unitary structure map and the mix map are identity morphisms. Similarily, a †-compact closed category is a strict unitary category in which all objects have unitary duals.
In the rest of this subsection, we show that any unitary category is †-linearly equivalent to a conventional dagger monoidal category. A unitary category being a compact LDC is linearly equivalent, using Mx * ↑ : (X, ⊗, ⊕) − → (X, ⊕, ⊕) (see Corollary 2.8) to the underlying monoidal category based on the par (and the tensor). We now show that for a unitary category one can induce a stationary on objects dagger on (X, ⊕, ⊕). We denote this dagger by ( ) ‡ and define it by f ‡ := ϕ B f † ϕ −1 A as illustrated by the left diagram below: (id)⊗(Lem. 5.8 (vii)) This new dagger clearly preserves composition and is also a stationary on objects involution as proven by the second diagram: the lower square of this diagram is the dagger of the inverted definition and the resulting outer square is the naturality of ι forcing f ‡ ‡ = f .
Next, we observe that u : X − → Y is a unitary isomorphism in X if and only if u −1 = u ‡ . This makes unitary isomorphisms in the traditional sense of categorical quantum mechanics coincide with the notion introduced here. Thus, u is unitary in the sense here if and only if the diagram below commutes Notice that if F preserves unitary structure, it must be an isomix functor by Lemma 3.8. Also, when A ∈ X is a unitary object, then F (A) must be a unitary object, and so F (A) is in the core.
We now show that Mx ↑ : (X, ⊕, ⊕) − → (X, ⊗, ⊕) provides a unitary structure preserving equivalence of a dagger monoidal category into a unitary category: Proposition 5.11. Unitary categories are †-linearly equivalent via the mix functor Mx ↑ : (X, ⊕, ⊕) − → (X, ⊗, ⊕) to the underlying dagger monoidal category on the par. Furthermore, closed unitary categories under this equivalence become dagger compact closed categories.
Proof. We must exhibit a preservator, that is a natural transformation showing that the involution is preserved: Note that ϕ is a natural transformation by the definition of ( ) ‡ and its coherence requirements make it a linear natural equivalence. Making this the preservator immediately means that unitary structure is preserved. Finally, we must show that unitary linear duals under Mx * ↑ become ‡-duals. Given (η, ) : A u B we must show that under Mx * ↑ this produces a dagger dual. Mx * ↑ (η) = m η : ⊥ − → A ⊕ B and Mx * ↑ ( ) = mx −1 : B ⊕ A − → ⊥ We then require that c ⊕ Mx * ↑ ( ) = Mx * ↑ (η) ‡ . This is provided by:

The unitary construction.
A †-isomix category can have many different unitary structures, as we shall describe in this section, thus it is structure, and not a property. The requirements, however, do mean that for a †-isomix category, X, there is always the smallest unitary structure, referred to as the "trivial" unitary structure, that produces a full unitary subcategory in X. In this subsection, we provide a construction that produces this unitary category from any †-isomix category. This construction, which we call the unitary construction provides an important technique for building unitary categories. The construction is based on identifying objects with pre-unitary structure: the tensor units always have a canonical pre-unitary structure so the construction always produces a non-empty category. However, to ensure that an application of the construction yields a unitary category in which there are objects which are not isomorphic to the units, one must exhibit concretely such objects. Fortunately this is often not difficult to do, making the construction quite applicable. Definition 5.12.
(i) In a †-isomix category, a pre-unitary object is an object U ∈ Core(X), together with an isomorphism α : U − → U † such that α(α −1 ) † = ι. (ii) Suppose X is a †-isomix category, then define Unitary(X), the unitary core of X, as follows: Objects: Pre-unitary objects (U, α), We note that any object which is isomorphic to a preunitary object is also pre-unitary: Our objective is to show that Unitary(X) is endowed with all the structure of a unitary category.
Proof. The proof uses the techniques of Lemma 5.2.
Note that, as the map and tensor structure is inherited from X, it suffices to show that these objects are all pre-unitary objects. Starting with (U α) † we have: For the tensor and par we have: This makes Unitary(X) into a compact †-LDC with all the structure inherited directly from X. However, more is true: each object now has an obvious unitary structure. This gives: Proposition 5.15. For any †-isomix category, X, Unitary(X) is a unitary category with a full and faithful underlying †-isomix functor U : Unitary(X) − → X. Proof. The laxors are all identity maps so that the underlying functors is immediately a †-mix functor. It remains to show that every object is unitary: we set the unitary structure of an object to be α : (X, α) − → (X, α) † . However, [U.1] -[U.5] are immediately satisfied by construction implying this provides unitary structure for every object.
Next, we prove the couniversal property of the unitary construction. Define UCat to be the category of unitary categories and †-isomix functors that preserve unitary structure in the sense of Definition 5.10, thus, whenever ϕ A is the unitary structure then F (ϕ A )ρ F is unitary structure. Define Kompact to be the category of compact †-LDCs and †-isomix functors.
We now show that the unitary construction produces a right adjoint to the underlying functor U : UCat − → Kompact which is the identity functor. Preliminary to this result we prove that Frobenius functors preserve preunitary objects: Proof. To prove that (F (A), F (ϕ)ρ) is a preunitary object, one has the following computation: Proof. The couniversal diagram is as follows: Since F is a †-isomix functor it preserves preunitary structure (see Lemma 5.16). This means that each (U, ϕ U ) in U is carried by F onto a preunitary object in C, (F (U ), F (ϕ)ρ F ). But a preunitary object in C is an object of Unitary(C) and this determines F . The functor F is uniquely determined as it must preserve the unitary structure. 5.4.1. Category of abstract state spaces. In Section 3.5.3, we discussed a construction on a †-isomix category, X, that produces a category of abstract state spaces, Asp(X), which is a †-isomix category. In this section, we examine the preunitary objects of Asp(X). Since all the basic natural isomorphisms are inherited from X, Core(X) determine Core(Asp(X)).
If (A, α) is a preunitary object for X, and (A, e A , u A ) ∈ Asp(X) then, ((A, e A , u A ), α) is a preunitary object for Asp(X) if u A α = λ e † A .

5.4.2.
Category of a group with involution. We discussed a source of examples of compact †-LDCs which are given by groups with conjugation. Applying unitary construction to each of the example categories results in the following unitary categories. It could be noticed that the preunitary objects in each of these categories includes those group elements such that g −1 = g. More explicitly, the preunitary objects are (g, 1) such that g −1 = g.
• In the discrete category of complex numbers, D(C, +, 0), The preunitary objects in this category are given by all complex numbers, i.e., (ib, 1). • In the discrete category of non-zero complex numbers, D(C, ., 1), the preunitary objects are given by complex numbers on a unit circle. • In the discrete category, D(P (x), +, 0), where P (x) is a polynomial ring, P (x) † = −P (−x) and the preunitary objects are polynomials P (x) = n a n x n such that n is odd. • In D(M 2 , ·, I 2 ) where M 2 is the group of 2 × 2 invertible matrices over C. The † structure is as follows: The preunitary objects in this category are given by unitary matrices.

Category of Hopf
Modules in a * -autonomous category. In Section 4.4.3, we described a construction of †-isomix categories from any symmetric isomix * -autonomous category, X, by choosing the Hopf Modules over a cocommutative ⊗-Hopf Algebra. We referred to the resulting category as H-Mod X . Now we shall look at the preunitary objects in H-Mod X in order to apply the unitary construction to this category. We begin by identifying the objects in the core of H-Mod X : Lemma 5.18. Suppose X is a mix * -autonomous category and H is a cocommutative Hopf Algebra in X. If (A, ) is a H-Module and A ∈ Core(X), then (A, ) ∈ Core(H-Mod X ).
Now that we identified the objects in the core, we prove a lemma that will be used later to identify the preunitary objects from the core: Proof.
Corollary 5.21. (((A, , , , ), ), 1) is a preunitary object. Thus, we have a source of non-trivial preunitary objects so that we can form a non-trivial unitary category. 5.5. Mixed unitary categories. We are now ready for the definition of mixed unitary categories, which is the key structure developed in this paper.
The functor F : U − → V is between unitary categories and we demand of it that it preserves unitary structure in the sense of Definition 5.10, thus, whenever ϕ A is the unitary structure then F (ϕ A )ρ F is unitary structure. 2-cells: These are "pillows" of natural transformations. (β, β ) : (F, F , γ F ) ⇒ (G, G , γ G ) is a 2-cell if and only if it satisfies the following equality: We remark that we have observed that any MUC can be "simplified" to a dagger monoidal category with a strong †-mix Frobenius functor into a †-isomix category: this is achieved by precomposing with Mx ↓ . This may seem a worthwhile simplification, but it should be recognized that it simply transfers complexity from the unitary category itself onto the preservator which must now "create" unitary structure: Here U ↓ = (U, ⊕, ⊕) is viewed as a dagger monoidal category and Mx * ↓ is the inverse of Mx ↓ . The point is that the preservator of the lower arrow Mx ↓ ; M is non-trivial as it must encode the unitary structure of U.
Our objective is now to show that the unitary construction of the previous section gives rise to a right adjoint to the underlying 2-functor U : MUC − → MCC where the 2-category MCC is defined as: 0-cells: Its objects are mixed †-compact categories (MCC), that is strong †-Frobenius functors V : C − → Y where C is a compact †-LDC, Y is a †-isomix category, and V factors through the core of Y i.e, for all ∀ objects C ∈ C, Y ∈ Y, ∃ mx : V (C) ⊕ Y − → V (C) ⊗ Y such that mx mx = 1 and mx mx = 1. 1-cells: The 1-cells are squares of mix Frobenius functors which commute up to a linear natural isomorphism; 2-cells: Are pillows of natural transformations (which we shall ignore).
An example of a mix †-compact category is, of course, the inclusion of the core into a †-isomix category C : Core(X) → X; Proof. The couniversal diagram is as follows: where is the square on the left and (F , G, γ ) is the square on the right: It follows from Proposition 5.17 that the couniversal diagram commute.
This proposition means that in building a non-trivial MUC from a mixed †-compact category it suffices to show that the compact †-LDC contains non-trivial pre-unitary objects. We have already noted that dagger monoidal categories are automatically unitary categories in which the unitary structure is given by identity maps. The identity functors then give a rather trivial MUC. More excitingly one can take the bicompletion of the †-monoidal category: this is a non-trivial †-isomix * -autonomous category extension of the original †-monoidal category and provides, thus, an interesting example of how MUCs arise.
Our purpose in this section is to exhibit some non-trivial manifestations of the various structural components of a MUC. To this end we discuss in some detail three basic examples. 5.6.1. Finite dimensional framed vector spaces. In this section we show that the example FFVec K , the category of finite dimensional framed vector spaces defined in Section 3.5.1 is a unitary category (hence is immediately a mixed unitary category). The unitary structure map on each object (V, V) is defined as follows: 3] we require that ϕ A † ( a i ) = (ϕ −1 A ) † ( a i ) the result is a higher-order term, so we may check that the evaluations are the same on basis elements: (ϕ A † ( a i ))( a j ) = a i ( a j ) = ∂ i,j ((ϕ −1 A ) † ( a i ))( a j ) = a i (ϕ −1 A ( a j )) = a i (a j ) = ∂ i,j Note that [U.5](a) and [U.5](b), in this example, require λ = ϕ which can easily be verified as each reduces to conjugation. [U.6](a) and [U.6](b), in this example, are the same requirement which is verified by: This gives: Proposition 5.24. FFVec K with the unitary structure above is a MUC.
This raises the question of what precisely the unitary maps of this example are. To elucidate this we note that a functor can easily be constructed U : FFVec K − → Mat(K) where, for each object in FFVec K we choose a total order on the elements of the basis and note that any map is then given by a matrix acting on the bases: thus a matrix in Mat(K) with the appropriate dimensions. We now observe: Proof. While U does not preserve ( ) † on the nose it does so up to the natural equivalence determined by U (ϕ A ) which being a basis permutation is a unitary equivalence. Thus, it is not hard to see that the following diagram commutes: Recall that in the category of matrices, the dagger is stationary on objects so U (B, B) = U (B, B) † . Now suppose u is unitary in FFVec K then u −1 = ϕ B u † ϕ −1 A so that U (u) −1 = U (u −1 ) = U (ϕ B u † ϕ −1 A ) = U (ϕ B )U (u † )U (ϕ −1 A ) = U (u) † so that its underlying map is unitary. Conversely, if U (u) is unitary then A ) which immediately implies, as U is faithful, that u is unitary in FFVec K .
One might reasonably regard this as a rather roundabout way to describe the standard notion of a unitary map. However, two things of importance have been achieved. First an example of a unitary category with a non-stationary dagger and, thus, a non-identity unitary structure, has been exhibited. Second we have shown how the standard unitary structure may be re-expressed in this formalism using non-stationary constructs. 5.6.2. Finiteness matrices. In Section 3.5.2, we described the category of finiteness matrices, FMat(C). The core of FMat(C) is the subcategory determined by objects whose webs are finite sets, that is the objects are X = (|X|, P (X)) where |X| is a finite set. Clearly, Core(FMat(C)) is then equivalent to the category of finite dimensional matrices, Mat(C). This is a well-known †-compact closed category, which is a unitary category with unitary structure given by identity maps (as ( ) † is stationary on objects).
The inclusion I : Mat(C) − → FMat(C) provides an important example of a MUC. 5.6.3. The embedding of finite-dimensional Hilbert Spaces into Chu spaces. In Section 4.4.2, we showed that the Chu construction applied to a symmetric conjugative closed monoidal category, X, with pullbacks gives a †-isomix category. Recall that the dagger in the resulting category of Chu spaces is given by composing the conjugation with the dualizing functor. In this section, we start by discussing, in general, the construction of a mixed unitary category from a Chu category Chus X (I). A crucial step in this is to identify objects which are in the core of this category.
Recall that a symmetric monoidal closed category, X, is (degenerately) a compact linearly distributive category and, thus, there may be objects which have linear adjoints: these are called nuclear objects [HR89]. Explicitly a nuclear object A in a symmetric monoidal closed category is an object with A B ∼ = A * ⊗ B, where A * := A I. The nuclear objects form a compact closed subcategory of X which is conjugative when X is conjugative. In Vec C the nucleus consist precisely of the finite dimensional vector spaces. If (η, ) : A B is witness that A (and B) are nuclear in X then the object (A, B, , c ⊗ ) is in the core of Chus X (I) because in the second component of the tensor product with any other object (X, Y, ν, c ⊗ ν) one has the degenerate pullback: