Theory and Applications of Categories, Vol. 25, No. 13, 2011, pp. 342–367.

COVARIANT PRESHEAVES AND SUBALGEBRAS

ULRICH HÖHLE

Abstract. For small involutive and integral quantaloids Q it is shown that covariant presheaves on symmetric Q-categories are equivalent to certain subalgebras of a specific monad on the category of symmetric Q-categories. This construction is related to a weakening of the subobject classifier axiom which does not require the classification of all subalgebras, but only guarantees that classifiable subalgebras are uniquely classifiable. As an application the identification of closed left ideals of non-commutative C^{∗}-algebras with certain «open» subalgebras of freely generated algebras is given.

Introduction

Let Q be a small involutive quantaloid. Then Q induces an involution on the quantaloid of symmetric Q-categories and distributors. In particular the involute of every contravariant presheaf is covariant and vice versa. In this framework we ask the question whether there exists a concept of a weak subobject classifier in the sense that a subobject classified by a covariant presheaf is always uniquely classified. For this purpose we introduce a special kind of presheaves on symmetric Q-categories which we call weak singletons. As a first property we note that weak singletons form a non-idempotent monad.

Weak singletons appear already in the theory of metric spaces. If symmetric Qcategories are metric spaces, then maximal weak singletons coincide with extremal functions (cf. [11]). Further there is a close relationship between weak singletons and the type of singletons considered by H. Heymans (cf. [8]). In fact, if the Cauchy completion preserves the symmetry axiom (cf. [9], see also Proposition 3.1), then the monad associated with the Cauchy completion is a submonad of the weak singleton monad.

After this brief historical digression, we return to the problem of unique classification of subobjects. Let W denote the weak singleton monad. For every object a of the given involutive quantaloid Q there exists a W-algebra structure on the free cocompletion of the trivial Q-category \underline{a}. If Q is integral, this W-algebra serves as a weak subobject classifier in the category of W-algebras. Under the assumption of the integrality of Q we show that classifiable subalgebras are uniquely classifiable (cf. Theorem 4.3). Moreover, the classifiable hull of every subalgebra exists (cf. Section 5). Since in general there exist more subalgebras than «characteristic morphisms» , this property might be of some importance for a non-commutative model theory based on integral and involutive, small quantaloids.

Received by the editors 2010-04-23 and, in revised form, 2011-07-06.

Transmitted by Ieke Moerdijk. Published on 2011-07-14.

2000 Mathematics Subject Classification: 06F07, 18C15, 18D20, 18F20.

Key words and phrases: Involutive quantale, involutive quantaloid, symmetric Q-category, covariant presheaf, monad of weak singletons, classifiable subalgebra, closed left ideal.

\begin{matrix}c \\ ◯ \end{matrix} Ulrich Höhle, 2011. Permission to copy for private use granted.

The previous theory has an immediate application to Q-enriched presheaves on symmetric Q-categories. Because of the universal property of free W-algebras every covariant presheaf on a symmetric Q-category 𝕏 has a unique extension to a W-homomorphism on the free W-algebra generated by 𝕏. Then we conclude from Theorem 4.3 that covariant presheaves on symmetric Q-categories are equivalent to classifiable subalgebras of free W-algebras. Moreover there exists a functor from the category of W-algebras to Set^{Q} preserving monomorphisms. Hence covariant presheaves are not only subalgebras, but also traditional set-valued functors with domain Q.

As a simple application of the previous results we make a contribution to the problem of constructing a non-commutative topological space for spectra of non-commutative C^{∗}-algebras. Since Hilbert spaces occurring in the Gelfand-Naimark-Segal-construction depend usually on the respective pure states, it seems to be reasonable to choose an integral and involutive quantaloid as basis for the necessary considerations. Then on the set of all locally irreducible representations of a given unital, non-commutative C^{∗}-algebra (cf. Section 6) we introduce an appropriate structure of a symmetric Q-category. This approach leads to the Q-category 𝕀_{loc}. Then closed left ideals are identified with certain covariant presheaves on 𝕀_{loc}. Subsequently, based on the equivalence between covariant presheaves and classifiable subalgebras of free algebras we obtain that closed left ideals can be interpreted as «open» subalgebras of the free algebra generated by 𝕀_{loc}. In this sense free algebras of the weak singleton monad can be understood as non-commutative spaces of non-commutative C^{∗}-algebras.

1. Quantaloids

In order to fix notation we begin with some basic definitions and properties from the theory of quantaloids (cf. [23]). A small quantaloid is a small category Q provided with the following additional properties:

• each hom-set is a complete lattice,

• the composition · of morphisms preserves arbitrary joins in both variables.

Objects (resp. morphisms) of Q are denoted by small Roman (resp. Greek) letters. Further, we write Q(a, b) for hom-sets of the form hom(a, b).

By the special adjoint functor theorem, for every morphism a ^{\overline{}^{λ}} b both maps

λ · : Q(c, b) → Q(c, a) and · λ : Q(a, c) → Q(b, c)

have right adjoints λ ↘ and ↙ λ which are determined by

λ ↘ α = ⋁ {β ∈ Q(c, b) | λ · β ≤ α}, β ↙ λ = ⋁ {γ ∈ Q(a, c) | γ · λ ≤ β}.

It is easily seen that the following relations hold:

(λ ↘ α) · (α ↘ ν) ≤ λ ↘ ν and (β ↙ λ) · (λ ↙ ν) ≤ β ↙ ν.

Further, Q(a, a) is always a unital quantale. A quantaloid Q is integral if for all objects a of Q the unit 1_{a} is the universal upper bound in Q(a, a).

A quantaloid Q is called involutive iff there exists a contravariant functor j : Q → Q satisfying the following conditions (cf. [6, 23]):

(q1) j(a) = a (i.e. j(1_{a}) = 1_{a}) for all object a in Q .

(q2) j^{2}(α) = α for every morphism α in Q .

(q3) If α, β ∈ Q(a, b) with α ≤ β, then j(α) ≤ j(β).

In order to shorten the notation we write α^{o} for j(α). In any involutive quantaloid the operations ↘ and ↙ are mutually determined by each other (cf. Proposition 2.9 in [8])

— i.e.

(α ↘ β)^{o} = β^{o} ↙ α^{o}.  (1.1)

With regard to the standing assumption in Section 4 we give here three examples of integral and involutive quantaloids which will be of some importance throughout this paper. We begin with a remark devoted to involutive quantales.

1.1. Remark. Let Q = (Q, ≤, & ) be an arbitrary quantale according to the terminology in [22]. In this context quantales are not necessarily unital (i.e. quantales are not the same as one-object quantaloids (cf. Example 1.2)). The left- and right implication in Q are determined by:

α ↙ β = ⋁ {γ ∈ Q | γ & β ≤ α}, α ↘ β = ⋁ {γ ∈ Q | α & γ ≤ β}.

An element α is a left-divisor of an element β iff β = α & (α ↘ β); and β is a right-divisor of α iff α = (α ↙ β) & β. Further, a unital quantale Q is integral if the unit coincides with the universal upper bound in Q. A quantale Q is involutive if Q is provided with an order preserving involution ^{o} s.t. (cf. [19, 21]):

(α & β)^{o} = β^{o} & α^{o}.

An element α of an involutive quantale is symmetric if α^{o} = α. Obviously, the universal lower (resp. upper) bound ⊥ (resp. ⊤) is always symmetric. In the event that Q is unital, then also the unit is symmetric. An important source of involutive and unital quantales are complete lattices L with a duality (resp. order reversing involution), that is, with a unary operation ′ satisfying the following conditions:

ℓ′′ = ℓ,  ( ⋁_{i∈I} ℓ_{i})′ = ⋀_{i∈I} ℓ′_{i},  ℓ ∈ L, {ℓ_{i} | i ∈ I} ⊆ L.

The details of this situation are as follows (cf. Example 4 and 5 in [21]). On the set Q(L) of all join preserving maps L ^{\overline{}^{σ}} L we define a partial ordering ≤ by:

σ_{1} ≤ σ_{2} ⟺ ∀ ℓ ∈ L : σ_{1}(ℓ) ≤ σ_{2}(ℓ).

Then Q(L) = (Q(L), ≤, & ) is a unital quantale where & is determined by the usual composition of maps — i.e.

σ_{1} & σ_{2}(ℓ) = σ_{1}(σ_{2}(ℓ)), ℓ ∈ L,

and the unit is given by the identity map id_{L}. Further, the order reversing involution on L induces a self-mapping σ ⟼ σ^{o} on Q(L) by σ^{o}(ℓ) = σ_{∗}(ℓ′)′, ℓ ∈ L where σ_{∗} is the right adjoint of σ — i.e.

σ_{∗}(b) = ⋁{ℓ ∈ L | s(ℓ) ≤ b}, b ∈ L.

Obviously, σ ⟼ σ^{o} is an order preserving involution on Q(L). Since adjoint situations can be composed, we obtain (σ_{1} & σ_{2})^{o} = σ_{2}^{o} & σ_{1}^{o}. Hence Q(L) = (Q(L), ≤, & ,^{o}) is involutive and unital quantale. If L has at least three elements, then Q(L) is never integral. Therefore our interest goes to the involutive and integral subquantale Q(L)_{id} of Q(L) whose support set is given by

Q(L)_{id} = {σ ∈ Q(L) | σ ≤ id_{L}}.

The answer to the question, whether Q(L)_{id} is commutative or not, depends on the structure of the underlying lattice. If (L,′) is the ortho-lattice of all closed subspaces of a Hilbert space (see also Section 6), then Q(L)_{id} is commutative and is isomorphic to the power set of the set of all atoms of L. On the other hand, if (L,′) is given by the real unit interval [0, 1] provided with the order reversing involution x ↦ 1 − x, then it is easily seen that Q([0, 1])_{id} is non-commutative.

The following list of examples shows that there exist at least three different procedures of constructing integral and involutive quantaloids from quantales.

1.2. Example. Let Q be an integral and involutive quantale. Then Q induces an integral and involutive quantaloid as follows: The set of objects is a singleton — i.e. obj(Q) = {∗}, and the set of morphism is given by mor(Q) = Q(∗, ∗) = Q where the composition, the involution and the lattice structure on Q(∗, ∗) is determined by Q.

1.3. Example. Let Q be an arbitrary involutive quantale. Since Q is not necessarily unital, we need some more terminology. An element α ∈ Q is stable, if α is left- and right-divisor of α. Obviously, α is stable iff the relation

α & (α ↘ α) = α = (α ↙ α) & α  (1.2)

holds. Typical examples of stable elements are the following ones: every element in a unital quantale is stable. In non-unital quantales every idempotent element is stable.

After these preparations we can construct a small, involutive and integral quantaloid Q from Q as follows^{1}:

• The set of objects of Q is the set SH(Q) of all stable and symmetric elements of Q.

• For a, b ∈ SH(Q) the hom-set Q(a, b) consists of all λ ∈ Q satisfying the conditions:

λ ≤ a ∧ b, a is left divisor and b is right divisor of λ.

• For every triple (a, b, c) the composition law Q(a, b) × Q(b, c) \overline{}Q(a, c) is determined by:

λ_{2} · λ_{1} = λ_{1} & (b ↘ λ_{2}), λ_{1} ∈ Q(a, b), λ_{2} ∈ Q(b, c),

where ↘ denotes the right implication in Q (cf. Remark 1.1).

• The partial ordering on Q(a, b) is inherited from Q. In particular, the universal upper bound in Q(a, a) coincides with a.

Because of

a & (⋁_{j∈J}(a ↘ λ_{j}) = ⋁_{j∈J} λ_{j} = (⋁_{j∈J}(λ_{j} ↙ b)) & b

Q(a, b) is a complete lattice and joins are computed in Q. For (λ_{1}, λ_{2}) ∈ Q(a, b) × Q(b, c) the relation

(λ_{1} ↙ b) & λ_{2} = λ_{1} & (b ↘ λ_{2})  (1.3)

holds. Thus λ_{2} · λ_{1} ∈ Q(a, c) follows from (1.2), (1.3) and the stability of b. The associativity axiom is evident, and the universal upper bound in Q(a, a) is the unit in Q(a, a). Referring again to (1.3) it is easily seen that the composition preserves arbitrary joins in both variables. Hence Q is an integral quantaloid.

A contravariant endofunctor j on Q is determined by the involution ^{o} on Q — i.e. j(λ) = λ^{o}, λ ∈ mor(Q). Thus Q is also an involutive quantaloid.

Finally, if Q is integral, then the universal upper bound ⊤ of Q plays also a special role in Q. For every object b of Q the hom-set Q(b, ⊤) is dominating — this means that the following relation holds:

Q(b, ⊤) = ⋃_{a∈SH(Q)} Q(b, a).  (1.4)

1Here we make use of a more set-theoretical language in the presentation of categorical axioms (see e.g. p. 4 in [3]).

1.4. Example. We maintain the notation from Example 1.3. Further, let Q be an involutive quantale which is not integral. Typical examples are involutive and unital quantales having left-sided elements which are not right-sided. In this framework the formula (1.4) is in general not valid — e.g. ⋃_{a∈SH(Q)} Q(⊤, a) is not always a hom-set. This observation is

a motivation to enlarge the quantaloid Q constructed in Example 1.3. We add a further object ω to Q and introduce the following additional hom-sets:

• Q(ω, a) = {λ ∈ Q | λ ≤ a, a is a right divisor of λ}, a ∈ SH(Q),

• Q(a, ω) = {λ ∈ Q | λ ≤ a, a is a left divisor of λ}, a ∈ SH(Q),

• Q(ω, ω) = {0, 1}.

The extension of the composition law is determined by:

• Q(ω, b) × Q(b, c) \overline{}Q(ω, c) is defined by: λ_{2} · λ_{1} = λ_{1} & (b ↘ λ_{2}),

• Q(a, b) × Q(b, ω) \overline{}Q(a, ω) is defined by: λ_{2} · λ_{1} = λ_{1} & (b ↘ λ_{2}),

• Q(a, ω) × Q(ω, c) \overline{}Q(a, c) is defined by: λ_{2} · λ_{1} = ⊥,

• Q(ω, b) × Q(b, ω) \overline{}Q(ω, ω) is defined by: λ_{2} · λ_{1} = 0,

• 1 acts as unit in Q(ω, ω), and consequently 0 is the zero element in Q(ω, ω).

It is a matter of routine to check that the associativity and the identity axioms hold. Finally, the involution has the following extension:

λ ∈ Q(ω, b) ∪ Q(b, ω), j(λ) = λ^{o}, j(1) = 1, j(0) = 0.

Hence the enlargement of Q by the addition of ω leads again to an integral and involutive quantaloid satisfying the following additional property:

⋃_{a∈SH(Q)} Q(b, a) ⊆ Q(b, ω), b ∈ SH(Q).

2. Presheaves on symmetric Q-categories and the weak envelope

Let Q be a small quantaloid. A Q-category is a triple 𝕏 = (X, e_{X}, d_{X}) where X is a set, X ^{\overline{}^{e_{X}}} obj(Q) and X × X ^{\overline{}^{d_{X}}} mor(Q) are maps subjected to the following axioms for all x, y, z ∈ X (cf. [2, 26]):

(Q1) d_{X}(x, y) ∈ Q(e_{X}(y), e_{X}(x)),

(Q2) 1_{e_{X}(x)} ≤ d_{X}(x, x),

(Q3) d_{X}(x, y) · d_{X}(y, z) ≤ d_{X}(x, z).

A Q-category 𝕏 is skeletal iff for all x, y ∈ X the following implication holds:

e_{X}(x) = e_{X}(y) and 1_{e_{X}(x)} ≤ d_{X}(x, y) ∧ d_{X}(y, x) imply x = y.

A Q-functor 𝕏 ^{\overline{}^{φ}} 𝕐 between Q-categories is a map X ^{\overline{}^{φ}} Y satisfying the following axioms for all x, y ∈ X:

(M1) e_{X}(x) = e_{Y} (φ(x)),

(M2) d_{X}(x, y) ≤ d_{Y} (φ(x), φ(y)).

Q-categories and Q-functors form a category Cat(Q) in an obvious way. It is not difficult to show that Cat(Q) is wellpowered, complete and cocomplete. Hence Cat(Q) is an (epi, extremal mono)-category. In particular, a Q-functor 𝕏 ^{\overline{}^{φ}} 𝕐 is an extremal monomorphism iff φ is an injective map provided with the following property for all x_{1}, x_{2} ∈ X:

(EM) d_{X}(x_{1}, x_{2}) = d_{Y} (φ(x_{1}), φ(x_{2})). (Fully Faithfulness)

The terminal object 1 in Cat(Q) has the form (obj.Q, e_{1}, d_{1}) where e_{1} = id_{objQ} and d_{1}(a, b) = ⋁ Q(b, a).

From now on, let Q be an involutive small quantaloid. With the involution on Q, it now makes sense to define a Q-category 𝕏 to be symmetric when the following axiom holds (cf. [8]):

(Q4) d_{X}(x, y) = d_{X}(y, x)^{o} for all x, y ∈ X. (Symmetry)

The terminal object 1 in Cat(Q) is symmetric. Further, symmetric Q-categories form a full subcategory sCat(Q) of Cat(Q). It is easily seen that sCat(Q) is also wellpowered, complete and cocomplete. Moreover, sCat(Q) is a coreflective subcategory of Cat(Q), and the corresponding coreflector is given by the symmetrization of Q-categories (X, e_{X}, d_{X}) — i.e. d_{X} is replaced by d_{X}^{s} defined by (cf. [9]):

d_{X}^{s}(x, y) = d_{X}(x, y) ∧ d_{X}(y, x)^{o}, x, y ∈ X.

Let a be an object of Q. In order to recall the concept of covariant presheaves of type a we first have to specify the Q-category 𝕊_{a} = (S_{a}, e_{a}, d_{a}) of «morphisms of type a» :

S_{a} = {λ ∈ mor(Q) | codom(λ) = a}, d_{a}(λ_{1}, λ_{2}) = λ_{1} ↘ λ_{2}, e_{a}(λ) = dom(λ).

Obviously, 𝕊_{a} is the free cocompletion of the trivial Q-category \underline{a}. Further, let 𝕏 be an arbitrary Q-category. A Q-functor 𝕏 ^{\overline{}^{φ}} 𝕊_{a} is called a covariant presheaf on 𝕏 of type a ∈ Q (cf. [25]). A simple characterization of covariant presheaves can be given as follows: a map X ^{\overline{}^{φ}} mor(Q) is a covariant presheaf on 𝕏 of type a iff φ satisfies the following conditions:

(P1) φ(x) ∈ Q(e_{X}(x), a) for all x ∈ X.

(P2) φ(x) · d_{X}(x, y) ≤ φ(y) for all x, y ∈ X. (Right Extensionality)

Hence covariant presheaves on 𝕏 of type a and distributors from 𝕏 to the trivial Q-category \underline{a} are equivalent concepts. It is not surprising that covariant presheaves on Q-categories constitute again a Q-category. Therefore let 𝕏 be a Q-category and P_{ℓ}(𝕏) be the set of all pairs (a, φ) where a is an object of Q and φ is a covariant presheaf on 𝕏 of type a. We specify the following maps P_{ℓ}(𝕏) ^{\overline{}^{e_{ℓ}}} obj(Q) and P_{ℓ}(𝕏) × P_{ℓ}(𝕏) ^{\overline{}^{d_{ℓ}}} mor(Q) :

e_{ℓ}(a, φ) = a, d_{ℓ}((a, φ), (b, ψ)) = ⋀_{x∈X} φ(x) ↙ ψ(x).

Then it is easily seen that ℙ_{ℓ}(𝕏) = (P_{ℓ}(𝕏), e_{ℓ}, d_{ℓ}) is a Q-category and the (covariant) Yoneda embedding 𝕏 ^{\overline{}^{η_{𝕏}}} ℙ_{ℓ}(𝕏) (enriched in Q) has the form

η_{𝕏}(x) = (e_{X}(x), x̃), where x̃(z) = d_{X}(x, z), x, z ∈ X.

It is well known that (η_{𝕏}, ℙ_{ℓ}(𝕏)) is the free completion of 𝕏 (cf. [25]) and the following important formula

d_{ℓ}(a, φ), η_{𝕏}(x)) = φ(x),  x ∈ X,  (2.1)

holds for all covariant presheaves on 𝕏. In particular, 𝕊_{a} and ℙ_{ℓ}(𝕏) are skeletal.

In the following considerations we will introduce weak singletons of symmetric Qcategories as a special kind of covariant presheaves. For this purpose we choose a symmetric Q-category and consider the symmetrization ℙ_{ℓ}^{s}(𝕏) of ℙ_{ℓ}(𝕏). In particular, d_{ℓ}^{s} is given by:

d_{ℓ}^{s}((a, φ), (b, ψ)) = ⋀_{x∈X}(φ(x) ↙ ψ(x)) ∧ (φ^{0}(x) ↘ ψ^{0}(x)), φ, ψ ∈ P_{ℓ}(𝕏).

Since the relation (2.1) does not remain valid in ℙ_{ℓ}^{s}(𝕏), we are interested in the largest full sub-category of ℙ_{ℓ}^{s}(𝕏) satisfying (2.1). This approach leads to the full subcategory W(𝕏) of ℙ_{ℓ}^{s}(𝕏) whose objects are all those covariant presheaves σ of type a satisfying the following condition for all x ∈ X:

d_{ℓ}^{s}((a, σ), (e_{X}(x), x̃)) = σ(x)  (2.2)

Since 𝕏 is symmetric, (2.2) is equivalent to σ(x) ≤ ⋀_{z∈X} σ^{o}(z) ↘ d_{X}(z, x). Thus we

introduce the following terminology.

2.1. Definition. Let 𝕏 be a symmetric Q-category. A covariant presheaf σ on 𝕏 of type a is called a weak singleton of 𝕏 of type a if σ satisfies the following condition for all x, y ∈ X:

(S1) σ^{o}(x) · σ(y) ≤ d_{X}(x, y). (Singleton Condition)

A weak singleton σ of 𝕏 of type a is a singleton of 𝕏, if (σ, σ^{o}) is an adjoint pair of bimodules between \underline{a} and 𝕏 (cf. [8]) — this means in this context that σ fulfills the additional property :

(S2) 1_{a} ≤ ⋁_{x∈X} σ(x) · σ(x)^{o}.

2.2. Remark. (a) In any symmetric Q-category 𝕏 for all x ∈ X the covariant presheaf η_{𝕏}(x) is always a singleton of 𝕏.

(b) Let Q be an integral and involutive, small quantaloid. If σ is a weak singleton of 𝕏 of type a, then for every κ ∈ Q(a, b) the pair κ · σ is again a weak singleton of 𝕏 of type b.

(c) In general there exist weak singletons which are not singletons.

On the set W_{a}(𝕏) of all weak singletons on 𝕏 of type a we introduce a partial ordering ⪯ by:

σ_{1} ⪯ σ_{2} ⟺ ∀ x ∈ X : σ_{1}(x) ≤ σ_{2}(x).

A maximal element in (W_{a}(𝕏), ⪯) is called a maximal weak singleton of type a. The axiom of choice guarantees the existence of maximal weak singletons. In the case of integral quantaloids we can give a characterization.

2.3. Proposition. Let Q be an integral quantaloid, and σ be a weak singleton of 𝕏 of type a. Then the following assertions are equivalent:

(i) σ is maximal in (W_{a}(𝕏), ⪯).

(ii) For all y ∈ X the following relation holds:

σ(y) = ⋀_{x∈X} σ^{o}(x) ↘ d_{X}(x, y).

Proof. The implication (ii) ⟹ (i) is obvious. Further, the singleton condition (S1) implies:

σ(y) ≤ ⋀_{x∈X} σ^{o}(x) ↘ d_{X}(x, y).

In order to establish the inverse inequality, we fix y_{0} ∈ Y and consider the following weak singleton \overline{σ} of type a defined by:

\overline{σ}(y) = σ(y) ∨ ( ⋀_{X∈X} σ^{o}(x) ↘ d_{X}(x, y_{0})) · d_{X}(y_{0}, y), y ∈ X.

In fact, because of the integrality of Q it is not difficult to show that (S1) holds. Then the relation ⋀_{X∈X} σ^{o}(x) ↘ d_{X}(x, y_{0}) ≤ σ(y_{0}) follows from the maximality of σ.

2.4. Example. (a) Let I be the real unit interval provided with the usual multiplication. Then I is an integral and commutative quantale with the identity map as involution. Further, let Q be the quantaloid induced by I in the sense of Example 1.2. Referring to F.W. Lawvere’s fundamental paper [14] it is easily seen that symmetric Q-categories 𝕏 = (X, d_{X}, e_{X}) are equivalent to pseudo-metric spaces (X, ϱ_{X}). The correspondence between d_{X} and ϱ_{X} is given by − ln (d_{X}(x, y)) = ϱ_{X}(x, y), x, y ∈ X. Moreover, maximal weak singletons and extremal functions (cf. [11]) are equivalent concepts.

(b) Let Ω be a frame. Then the quantaloid Q_{Ω} given by Ω in the sense of Example 1.3 is exactly the construction invented by R.F.C. Walters 1981 (cf. [26]). In particular the involution on Q_{Ω} coincides with the identity functor. In this context, symmetric Q_{Ω}-categories and Ω-sets (cf. [7]) are equivalent concepts. A weak singleton σ of type a is a singleton iff a coincides with the height of σ — i.e a = ⋁_{x∈X} σ(x).

Let 𝕏 be a symmetric Q-category. The full subcategory of ℙ_{ℓ}^{s}(𝕏) whose objects are all weak singletons of 𝕏 coincides with W(𝕏). In particular, the underlying set W (𝕏) of all weak singletons of 𝕏 is given by:

W (𝕏) = ⋃_{a∈obj(Q)}{a} × W_{a}(𝕏).  (2.3)

In this context the symmetric Q-category W(𝕏) is called the weak envelope of 𝕏.

Since 𝕏 is symmetric, the Yoneda embedding η_{𝕏} factors through W(𝕏). Hence the formation of weak envelopes gives rise to a monad W on sCat(Q) whose clone-composition function ∘ is determined by:

[figure: 𝕏; W(𝕐),; Φ; 𝕐; W(ℤ),; Ψ; 𝕏; W(ℤ),; Ψ∘Φ]

Φ(x) = (e_{X}(x), σ_{x}), Ψ(x) = (e_{Y} (y), σ_{y}), Ψ ∘ Φ(x) = (e_{X}(x), ξ_{x}), x ∈ X, y ∈ Y,

ξ_{x}(z) = ⋁_{y∈Y} σ_{x}(y) · σ_{y}(z), z ∈ Z.

As usually the object function 𝕏 ⟼ W(𝕏) of W can be completed to an endofunctor W of sCat(Q). In particular, the action of W on Q-functors is given as follows:

𝕏 ^{\overline{}^{Φ}} 𝕐,  W(Φ) = (η_{𝕐} · Φ) ∘ id_{W(𝕏)},  W(Φ)(a, σ) = (a, τ) with

τ(y) = ⋁_{x∈X} σ(x) · d_{Y} (φ(x), y),  (a, σ) ∈ W (𝕏), y ∈ Y.  (2.4)

Also for later purposes we quote the explicit form of the multiplication μ corresponding to W

μ_{𝕏} = id_{W(𝕏)} ∘ id_{W(W(𝕏))}, μ_{𝕏}(b, Σ) = (b, 𝔰), 𝔰(x) = ⋁_{(a,σ)∈W (𝕏)} Σ(a, σ) · σ(x). (2.5)

In the following considerations W is called the weak singleton monad.

3. Cauchy completion and the weak envelope of symmetric Q-categories

It is well known that the Cauchy completion of a symmetric Q-category is again symmetric iff for all objects a of Q every adjoint pair (τ, σ) between the trivial Q-category \underline{a} and 𝕏 is a singleton of 𝕏 — i.e. τ = σ^{o} (cf. [2]). Moreover, we can find in [9] the following result. If Q is an involutive and integral quantaloid provided with the property:

for any object a_{0} of Q and for any family (a_{i} ^{\overline{}^{α_{i}}} a_{0}, a_{0} ^{\overline{}^{β_{i}}} a_{i})_{i∈I} of morphisms of Q the following implication holds:

1_{a_{0}} ≤ ⋁_{i∈I} α_{i} · β_{i} ⟹ 1_{a_{0}} ≤ ⋁_{i∈I}(α_{i} ∧ β_{i}^{0}) · (α_{i}^{0} ∧ β_{i}),  (3.1)

then the symmetric completion of any symmetric Q-category coincides with its Cauchy completion (cf. Proof of Corollary 3.8 in [9]) — i.e. the Cauchy completion preserves the symmetry axiom. Even though (3.1) covers a large class of examples (cf. Section 4 in [9]), I conjecture that in the case of integral and involutive quantaloids (3.1) is not necessary for the preservation of symmetry under the Cauchy completion. In the next proposition we give now a necessary and sufficient condition for this property.

3.1. Proposition. Let Q be an integral and involutive quantaloid. Then the following assertions are equivalent:

(i) The Cauchy completion preserves the symmetry axiom of Q-categories.

(ii) For any family (a_{i} ^{\overline{}^{α_{i}}} a_{0}, a_{0} ^{\overline{}^{β_{i}}} a_{i})_{i∈I} of morphisms in Q provided with the properties

∀ i, j ∈ I : β_{i} · α_{j} · α_{j}^{o} ≤ α_{i}^{o}, β_{i}^{o} · β_{i} · α_{j} ≤ β_{j}^{o},  1_{a_{0}} ≤ ⋁_{i∈I} α_{i} · β_{i} (3.2)

the following relations hold:

1_{a_{0}} ≤ ⋁_{i∈I} α_{i} · α_{i}^{o} and 1_{a_{0}} ≤ ⋁_{i∈I} β_{i}^{o} · β_{i}.  (3.3)

Proof. (a) ((i) ⟹ (ii)) Let (a_{i} ^{\overline{}^{α_{i}}} a_{0}, a_{0} ^{\overline{}^{β_{i}}} a_{i})_{i∈I} be a family of morphisms in

Q satisfying (3.2). First we construct a symmetric Q-category as follows: 𝕀 = (I, e_{I}, d_{I}) where

e_{I}(i) = a_{i}, d_{I}(i, j) = (α_{i} ↘ α_{j}) ∧ (β_{i}^{o} ↘ β_{j}^{o}) ∧ (α_{i}^{o} ↙ α_{j}^{o}) ∧ (β_{i} ↙ β_{j}).

Secondly, we define a covariant (resp. contravariant) presheaf σ (resp. τ) on 𝕀 of type a_{0} by:

σ(i) = α_{i}, τ(i) = β_{i}, i ∈ I.

Since Q is integral, we conclude from (3.2) that (σ, τ) forms an adjoint pair of bimodules between \underline{a_{0}} and 𝕀. Since the Cauchy completion preserves the symmetry axiom, we obtain τ = σ^{o}. Hence (3.3) holds.

(b) ((ii) ⟹ (i)) We choose an adjoint pair (σ, τ) of bimodules between \underline{a_{0}} and a symmetric Q-category 𝕏. Then we conclude from the symmetry axiom and the extensionality axioms that

τ(x) · σ(y) ≤ (τ^{o}(x) ↘ τ^{o}(y)) ∧ (σ^{o}(x) ↙ σ^{o}(y))

holds for all x, y ∈ X. Hence the family (e_{X}(x) ^{\overline{}^{σ(x)}} a_{0}, a_{0} ^{\overline{}^{τ(x)}} e_{X}(x))_{x∈X} of mor-

phisms in Q satisfies (3.2). Now we invoke (ii) and obtain:

1_{a_{0}} ≤ ⋁_{y∈X} σ(y) · σ^{o}(y) ≤ ⋀_{x∈X} τ(x) ↘ σ^{o}(x), 1_{a_{0}} ≤ ⋁_{x∈X} τ^{o}(x) · τ(x) ≤ ⋀_{y∈Y} τ^{o}(y) ↙ σ(y).

Hence τ = σ^{0} follows.

Now we turn to the question under which condition the monad associated with the Cauchy completion is a submonad of the weak singleton monad. We begin with a technical lemma.

3.2. Lemma. Let 𝕏 be a symmetric Q-category. If (a_{1}, σ_{1}) and (a_{2}, σ_{2}) are singletons of 𝕏, then the following relation holds:

d_{ℓ}^{s}((a_{1}, σ_{1}), (a_{2}, σ_{2})) = ⋁_{x∈X} σ_{1}(x) · σ_{2}^{o}(x).

Proof. Because of the singleton condition (S1) and the right extensionality of σ_{1} the relation

( ⋁_{x∈X} σ_{1}(x) · σ_{2}^{o}(x)) · σ_{2}(z) ≤ σ_{1}(z)

holds. Interchanging the role of σ_{1} and σ_{2} we obtain: ( ⋁_{x∈X} σ_{2}(x) · σ_{1}^{o}(x)) · σ_{1}(z) ≤ σ_{2}(z).

Hence

⋁_{x∈X} σ_{1}(x) · σ_{2}^{o}(x) ≤ d_{ℓ}^{s}((a_{1}, σ_{1}), (a_{2}, σ_{2}))

follows. Now we assume that σ_{1} is singleton and not only a weak singleton. Then we conclude from (S2):

d_{ℓ}^{s}((a_{1}, σ_{1}), (a_{2}, σ_{2})) = 1_{a_{1}} · d_{ℓ}^{s}((a_{1}, σ_{1}), (a_{2}, σ_{2}))
≤ ⋁_{x∈X} σ_{1}(x) · σ_{1}^{o}(x) · (σ_{1}^{o}(x) ↘ σ_{2}^{o}(x))

≤ ⋁_{x∈X} σ_{1}(x) · σ_{2}^{o}(x).

Hence the assertion is verified.

If the Cauchy completion preserves symmetry, then Lemma 3.2 shows that the Cauchy completion of 𝕏 is a full subcategory of the weak envelope W(𝕏) of 𝕏. In particular, the «clone-composition» function specified above factors through the Cauchy completion.

To sum up we have the following result.

3.3. Proposition. Let Q be an integral and involutive quantaloid. If the Cauchy completion preserves symmetry (i.e. the assertion (ii) in Proposition 3.1 holds), then the Cauchy completion can be restricted to sCat(Q) and the monad associated with the Cauchy completion is a submonad of the weak singleton monad.

4. Presheaves on symmetric Q-categories and W-subalgebras

We begin with the classification problem of presheaves on symmetric Q-categories. Classification Problem. For every object a ∈ Q let 1 ^{\overline{}^{t_{a}}} 𝕊_{a} be the Q-functor defined by

t_{a}(b) = ⋁ Q(b, a),  b ∈ obj(Q).  (4.1)

Since 1 is symmetric t_{a} can also be viewed as a Q-functor from 1 to the symmetrization 𝕊_{a}^{s} of 𝕊_{a}. Further, let 𝕏 be a symmetric Q-category and φ be a covariant presheaf on 𝕏 of type a . Then φ can also be considered as Q-functor from 𝕏 to 𝕊_{a}^{s}. Since sCat(Q) is complete, we can form the pullback of t_{a} along φ in sCat(Q) — i.e.

[figure: 𝕌; 1; 𝕏; 𝕊_{a}^{s}; !; ι; t_{a}; φ]  (4.2)

Hence the carrier set U of the subobject 𝕌 ^{\overline{}^{ι}} 𝕏 classified by φ is given by:

U = {x ∈ X | φ(x) = t_{a}(e_{X}(x))}

where ι denotes the inclusion map from U to X. On the other hand, the covariant presheaf ψ on 𝕏 of type a defined by

g(x) = ⋁_{u∈U} t_{a}(e_{X}(u)) · d_{X}(u, x), x ∈ X

classifies also 𝕌 ^{\overline{}^{ι}} 𝕏 in the sense of (4.2). But, unfortunately this classification is not unique. In general, the presheaves φ and ψ are different. Simple counterexamples can be given among other things in the context of pseudo-metric spaces (cf. Example 2.4(a)).

The aim of the following considerations is to overcome the non-uniqueness of the classification in sCat(Q) where we might pay the price that not every subobject is classifiable.

We will show that in the case of integral and involutive, small quantaloids there exists a bijection between covariant presheaves on symmetric Q-categories and special subalgebras (cf. [15, 17]) in the sense of the weak singleton monad (cf. Section 2). For this purpose we exhibit a W-algebra structure on the symmetric Q-category 𝕊_{a}^{s}.

We fix an object a of Q and define a map ξ from W (𝕊_{a}^{s}) to S_{a} as follows:

ξ(b, σ) = ⋁ {κ ∈ Q(b, a) | ∀ λ ∈ S_{a} : κ · σ(λ) ≤ λ}.  (4.3)

Because of

ξ(b_{1}, σ_{1}) · d_{ℓ}^{s}((b_{1}, σ_{1}), (b_{2}, σ_{2})) · σ_{2}(λ) ≤ ξ(b_{1}, σ_{1}) · σ_{1}(λ) ≤ λ

the map ξ is a Q-functor W(𝕊_{a}^{s}) ^{\overline{}^{ξ}} 𝕊_{a}^{s}.

In the next remark we explain the relationship between ξ and the free completion of the trivial Q-category \underline{a}.

4.1. Remark. First we notice that 𝕊_{a}^{s} is not necessarily complete. But there exists a Q-functor 𝕊_{a}^{s} ^{\overline{}^{φ}} ℙ_{ℓ}(\underline{a}) induced by the involution of the underlying quantaloid — i.e. φ(λ) = λ^{o}, λ ∈ S_{a}. Moreover, the evaluation of covariant presheaves determines a distributor Θ from 𝕊_{a}^{s} to W(𝕊_{a}^{s}) by:

Θ((b, σ), λ) = σ(λ), ((b, σ), λ) ∈ W (𝕊_{a}^{s}) × S_{a}.

Then the Θ-weighted limit W(𝕊_{a}^{s}) ^{\overline{}^{ψ}} ℙ_{ℓ}(\underline{a}) of φ has the following form:

ψ(b, σ) = ⋁_{λ∈S_{a}} σ(λ) · λ^{0}, (b, σ) ∈ W (𝕊_{a}^{s}).

In particular, ψ factors through φ. Hence we conclude from (S1) that ξ is the unique Q-functor satisfying the condition that φ · ξ coincides with the Θ-weighted limit of φ.

4.2. Proposition. The pair (𝕊_{a}^{s}, ξ) is a W-algebra.

Proof. The relation η_{𝕏} · ξ = id_{𝕏} follows immediately from the definition of η_{𝕏} and ξ. In order to verify the commutativity of the following diagram

[figure: W(W(𝕊_{a}^{s})); W(𝕊_{a}^{s}); W(𝕊_{a}^{s}); 𝕊_{a}^{s}; W(ξ); μ_{𝕊^{s}_{a}}; ξ; ξ]

we choose a weak singleton (b, Σ) of W(𝕊_{a}^{s}) and conclude from (2.2), (2.4) and (2.5):

μ_{𝕊_{a}^{s}}(b, Σ)(z) ≤ [W(ξ)(b, Σ)](z), z ∈ X.  (4.4)

Then the commutativity of the previous diagram follows from (4.4) and the following implication:

(∀ λ ∈ S_{a} : κ · μ_{𝕊_{a}^{s}}(b, Σ)(λ) ≤ λ) ⟹ (∀ λ ∈ S_{a} : κ · [W(ξ)(b, Σ)](λ) ≤ λ)

where κ ∈ Q(b, a). In fact, if κ · μ_{𝕊_{a}^{s}}(b, Σ)(λ) ≤ λ, then the relations (2.5) and (4.3) imply: κ · Σ(c, σ) ≤ ξ(c, σ). Now we obtain:

κ · [W(ξ)(b, Σ)](λ) ≤ ⋁_{(c,σ)∈W (𝕊_{a}^{s})} ξ(c, σ) · d_{a}^{s}(ξ(c, σ), λ) ≤ λ.

Hence the previous implication has been established.

Further, the Q-functor 1 ^{\overline{}^{t_{a}}} 𝕊_{a}^{s} turns out to be W-homomorphism

(1,!_{W(1)}) \overline{}^{t_{a}} (𝕊_{a}^{s}, ξ).

In fact, because of (4.1) the relation κ · σ(c) · d_{a}^{s}(t_{a}(c), λ) ≤ λ holds for all (b, σ) ∈ W (1) and for all κ ∈ Q(b, a).

Now we make the following

Standing Assumption. Q is always an integral and involutive, small quantaloid.

4.3. Theorem. Let (𝕏, ϑ) be a T-algebra and (𝕏, ϑ) ^{\overline{}^{φ}} (𝕊_{a}^{s}, ξ) be a W-homomorphism.

Further, let ((𝕌, ζ), ι) be the subalgebra of (𝕏, ϑ) determined by the following pullback diagram

[figure: (𝕌, ζ); (1,!_{W(1)}); (𝕏, ϑ); (𝕊_{a}^{s}, ξ); ι; !_{𝕌}; t_{a}; φ]

in the category of W-algebras. Then the following relation holds:

φ(x) = ⋁_{u∈U} t_{a}(e_{X}(u)) · d_{X}(ι(u), x),  x ∈ X.  (4.5)

Proof. Since in the category of W-algebras pullbacks are computed at the level of sCat(Q) (cf. [17]), the carrier set U of the subalgebra (𝕌, ζ) together with its embedding W-homomorphism ι is given by:

U = {x ∈ X | φ(x) = t_{a}(e_{X}(x))}, ι = inclusion map.

Then for all x ∈ X the inequality

⋁_{u∈U} t_{a}(e_{X}(u)) · d_{X}(u, x) ≤ φ(x)  (4.6)

follows from the right extensionality of φ. On the other hand, if we fix x_{0} ∈ X, then we conclude from the standing assumption that σ with

σ(x) = φ(x_{0}) · d_{X}(x_{0}, x), x ∈ X,

is a weak singleton of 𝕏 of type a (cf. Remark 2.2(b)). Referring again to the right extensionality axiom we obtain:

[W(φ)(a, σ)](λ) = ⋁_{x∈X} φ(x_{0}) · d_{X}(x_{0}, x) · d_{a}^{s}(φ(x), λ) ≤ λ.

Since 1_{a} is the universal upper bound in Q(a, a), the relation ξ(W(φ)(a, σ)) = t_{a}(a) follows immediately from the definition of ξ. Now we invoke the property that φ is a W-homomorphism and obtain:

φ(ϑ(a, σ)) = t_{a}(a), e_{X}(ϑ(a, σ)) = a.  (4.7)

Hence ϑ(a, σ) ∈ U. Finally, we conclude from (2.2):

φ(x_{0}) = σ(x_{0}) ≤ t_{a}(a) · d_{X}(ϑ(a, σ), x_{0}).  (4.8)

Thus (4.5) follows from (4.6), (4.7) and (4.8).

Theorem 4.3 suggests the following terminology: A subalgebra (𝕌, ζ) ^{\overline{}^{ι}} (𝕏, ϑ)

is called t_{a}-classifiable iff there exists a W-homomorphism (𝕏, ϑ) ^{\overline{}^{φ}} (𝕊_{a}^{s}, ξ) s.t. the diagram

[figure: (𝕌, ζ); (1,!_{W(1)}); (𝕏, ϑ); (𝕊_{a}^{s}, ξ); ι; !_{𝕌}; t_{a}; φ]

is a pullback square. Then Theorem 4.3 implies that every t_{a}-classifiable subalgebra is uniquely t_{a}-classifiable.

As an application of the previous construction we describe those classifiable subalgebras which are classified by covariant presheaves. Because of the universal property of free W-algebras every covariant presheaf φ on a symmetric Q-category 𝕏 of type a can uniquely be extended to a W-homomorphism

(W(𝕏), μ_{𝕏}) \overline{}^{φ^{♯}} (𝕊_{a}^{s}, ξ).

It is well known that φ^{♯} is given by (cf. I.4.12 in [17]): φ^{♯} = ξ · W(φ). Then the carrier set U of the t_{a}-classifiable subalgebra of (W(𝕏), μ_{𝕏}) classified by φ^{♯} contains all weak singletons σ of 𝕏 of type b with

t_{a}(b) · σ(x) · d_{a}^{s}(φ(x), λ) ≤ λ.

In particular, U has the following form:

U = {(b, σ) ∈ W (𝕏) | ∀ x ∈ X : t_{a}(b) · σ(x) ≤ φ(x)}.  (4.9)

4.4. Remark. (a) In the case of pseudo-metric spaces (X, ϱ_{X}) (cf. Example 2.4(a)) it follows immediately from Theorem 4.3 and the subsequent considerations that every non-expansive map X ^{\overline{}^{f}} [0, +∞] can be identified with the metric space of all non-

expansive functions X ^{\overline{}^{s}} [0, +∞] provided with the following additional properties:

f(x) ≤ s(x), ϱ_{X}(x, y) ≤ s(x) + s(y), x, y ∈ X.

(b) Let Ω be a frame, and Q_{Ω} be the quantaloid induced by Ω in the sense of Example 1.3. If ⊤ denotes the universal upper bound in Ω, then (t_{⊤}, 𝕊_{⊤}^{s}) is the subobject classifier in the category CΩ-Set of complete Ω-sets (cf. [7]). Since in the case of Q_{Ω} the Cauchy completion preserves the symmetry axiom (cf. [26] or Proposition 3.1), the monad associated with the Cauchy completion is a submonad of W (cf. Proposition 3.3). Hence the W-algebra (𝕊_{⊤}^{s}, ξ) and the W-homomorphism t_{⊤} form a modification of (t_{⊤}, 𝕊_{⊤}^{s}). Moreover, the formula (4.5) in Theorem 4.3 can be seen as an enlargement of the construction provided by the subobject classifier axiom in CΩ-Set to the more general setting determined by the category of W-algebras.

The next proposition shows that every W-algebra gives rise to a set-valued functor with domain Q.

4.5. Proposition. Every W-algebra (𝕏, ϑ) induces a functor Q ^{\overline{}^{F}} Set by:

\begin{matrix}F(a) = X_{a} = \{x ∈ X | e_{X}(x) = a\}, a ∈ obj(Q), \\ a ^{\overline{}^{λ}} b (in Q), X_{a} ^{\overline{}^{F(λ)}} X_{b}, F(λ)(x) = ϑ(b, λ · x̃). \end{matrix}

Proof. It is easily seen that F preserves the respective identities. In order to show that F preserves also the composition, we fix x ∈ X_{a} and choose arrows [figure: a; b; c; λ_{1}; λ_{2}] in Q. Then we introduce a weak singleton Σ of W(𝕏) of type c as follows:

Σ(k, σ) = λ_{2} · d_{ℓ}^{s}((b, λ_{1} · x̃), (k, σ)), (k, σ) ∈ W (𝕏).

Obviously, W(ϑ)(c, Σ) = (c, σ_{0}) has the form

\begin{aligned}σ_{0}(z) & = & ⋁_{(k,σ)∈S(𝕏)} λ_{2} · d_{ℓ}^{s}((b, λ_{1} · x̃), (k, σ)) · d_{X}(ϑ(k, σ), z), \\ & = & λ_{2} · d_{X}(ϑ(b, λ_{1} · x̃), z). \end{aligned}

Further, we obtain from (2.2):

⋁_{(k,σ)∈S(𝕏)} Σ(k, σ) · σ(z) = λ_{2} · λ_{1} · d_{X}(x, z) = (λ_{2} · λ_{1}) · x̃(z), z ∈ X.

Then F(λ_{2} · λ_{1}) = F(λ_{2}) ∘ F(λ_{1}) follows from ϑ · μ_{𝕏} = ϑ · W(ϑ).

It is not difficult to see that the object function specified in Proposition 4.5 can be completed to a functor from the category of all W-algebras to Set^{Q} which obviously preserves monomorphisms. Hence covariant presheaves are not only subalgebras, but also set-valued functors on Q from which the given covariant presheaf can be reconstructed. In fact, if F is the set-valued functor on Q induced by the free W-algebra (W(𝕏), μ_{𝕏})

— i.e.

F(b) = \{σ | (b, σ) ∈ W (𝕏)\},  b ^{\overline{}^{λ}} c ,  F(λ)(σ) = λ · σ,

and S be the subfunctor of F induced by the subalgebra (𝕌, ζ) ^{\overline{}^{ι}} (W(𝕏), μ_{𝕏}) cor-

responding to a covariant presheaf φ on 𝕏 of type a, then the W-homomorphism φ^{♯} can be reconstructed from S in the following way. Since S is given by (cf. (4.9))

S(b) = {σ ∈ F(b) | t_{a}(b) · σ ≤ φ}, b ∈ obj(Q),

the corresponding characteristic morphism F ^{\overline{}^{χ}} Ω has the form (cf. pp. 38-39 in [16]):

F(b) ^{\overline{}^{χ_{b}}} Ω(b), χ_{b}(σ) = \{λ ∈ mor(Q) | dom(λ) = b, λ · σ ∈ S(codom(λ)\}, σ ∈ F(b)

where Ω is the subobject classifier in Set^{Q}. Then we reconstruct φ^{♯} from χ as follows:

φ^{♯}(b, σ) = ξ(W(φ))(b, σ) = ⋁_{λ∈χ_{b}(s)} t_{a}(codom(λ)) · λ, (b, σ) ∈ W (𝕏).

5. Classifiable subalgebras

Let a be an object of the underlying quantaloid Q. Since not every subalgebra is t_{a}-classifiable (i.e. there exist more subalgebras than «characteristic morphisms»), it is important to show that the t_{a}-classifiable hull of subalgebras exists.

Since the maximal covariant presheaf φ_{a} on 𝕏 of type a is the composition of e_{X} with t_{a}, it is easily seen that φ_{a} is always a W-homomorphism for any structure morphism W(𝕏) ^{\overline{}^{ϑ}} 𝕏. Hence every W-algebra (𝕏, ϑ) viewed as subalgebra of itself is trivially t_{a}-classifiable.

In the next step we show that the non empty intersection of t_{a}-classifiable subalgebras is again t_{a}-classifiable.

We begin with the internalization of the conjunction of type a. The product

(𝕊_{a}^{s}, ξ) × (𝕊_{a}^{s}, ξ) = (𝕋_{a}, ζ)

of the W-algebra (𝕊_{a}^{s}, ξ) with itself can be specified as follows. First we compute the product 𝕊_{a}^{s} × 𝕊_{a}^{s} = 𝕋_{a} in sCat(Q):

\begin{matrix}T_{a} = \{(λ_{1}, λ_{2}) ∈ S_{a} × S_{a} | e_{a}(λ_{1}) = e_{a}(λ_{2}\}, \\ d_{T_{a}}((λ_{1}, λ_{2}), (λ̂_{1}, λ̂_{2})) = d_{a}(λ_{1}, λ̂_{1}) ∧ d_{a}(λ_{2}, λ̂_{2}), e_{T_{a}}(λ_{1}, λ_{2}) = e_{a}(λ_{1}) (= e_{a}(λ_{2})). \end{matrix}

Then the structure morphism ζ is given by ⟨ξ · W(π_{1}), ξ · W(π_{2})⟩ where T_{a} ^{\overline{}^{π_{i}}} S_{a} is the projection onto the i-th coordinate (i = 1, 2). Finally, we define a covariant presheaf f_{∧} on 𝕊_{a}^{s} × 𝕊_{a}^{s} of type a as follows:

f_{∧}(λ_{1}, λ_{2}) = λ_{1} ∧ λ_{2}, (λ_{1}, λ_{2}) ∈ T_{a}.

Since for any weak singleton (b, σ) of 𝕋_{a} the implication

κ_{1} · σ(λ_{1}, λ_{2}) ≤ λ_{1}, κ_{2} · σ(λ_{1}, λ_{2}) ≤ λ_{2} ⟹ (κ_{1} ∧ κ_{2}) · σ(λ_{1}, λ_{2}) ≤ λ_{1} ∧ λ_{2}.

is obvious, it is easily seen that f_{∧} is a W-homomorphism from (𝕊_{a}^{s}, ξ)×(𝕊_{a}^{s}, ξ) to (𝕊_{a}^{s}, ξ). In particular, the diagram

[figure: 1; (𝕊_{a}^{s}, ξ); 1; (𝕊_{a}^{s}ξ) × (𝕊_{a}^{s}, ξ); t_{a}; ⟨t_{a},t_{a}⟩; f_{∧}]

is a pullback square. Hence, by analogy to topos theory we call f_{∧} the conjunction of type a in the category of W-algebras.

5.1. Theorem. The binary intersection of t_{a}-classifiable W-subalgebras is again t_{a}-classifiable.

Proof. Let (𝕌_{i}, ζ_{i}) ^{\overline{}^{ι_{i}}} (𝕏, ϑ) be a t_{a}-classifiable W-subalgebra classified by a W-

homomorphism (𝕏, ϑ) ^{\overline{}^{φ_{i}}} (𝕊_{a}^{s}, ξ) (i = 1, 2). Then we form the pullback square

[figure: (𝕌_{0}, ζ_{0}); (𝕌_{2}, ζ_{2}); (𝕌_{1}, ζ_{1}); (𝕏, ϑ); ι_{0}; ι_{2}; ι_{1}]

and observe that the outer rectangle of the following diagram

[figure: (𝕌_{0}, ζ_{0}); (𝕏, ϑ); 1; (𝕊_{a}^{s}, ξ) × (𝕊_{a}^{s}, ξ); 1; (𝕊_{a}^{s}, ξ); ι_{0}; !; ⟨φ_{1},φ_{2}⟩; ⟨t_{a},t_{a}⟩; f_{∧}; t_{a}]

is also a pullback square.

Since the category of W-algebras is complete, the previous proof holds also for any non empty family of t_{a}-classifiable W-subalgebras. Just for the record we state:

5.2. Corollary. The non empty intersection of t_{a}-classifiable W-subalgebras is again t_{a}-classifiable.

We can summarize the previous results in the following statement:

For any subalgebra (𝕍, ζ) of (𝕏, ϑ) the t_{a}-classifiable hull of (𝕍, ζ) exists and coincides with the intersection of all t_{a}-classifiable subalgebras of (𝕏, ϑ) containing (𝕍, ζ).

We hope that the previous result might play a significant role in the development of a non-commutative model theory.

6. An application to the spectrum of C^{∗}-algebras

We begin with a concretization of Example 1.4. For this purpose let H be a Hilbert space, and M_{H} be the complete lattice of all closed, linear subspaces ℓ of H provided with the ortho-complement^{⊥} as order reversing involution. In particular 1 (0) denotes the universal upper (lower) bound in M_{H}. Now we consider the involutive and unital quantale Q(M_{H}) induced by (M_{H},^{⊥}) in the sense of Remark 1.1. Since the universal upper bound in Q(M_{H}) has the form:

⊤(ℓ) = \{ \begin{matrix}1 & : & ℓ ≠ 0, \\ 0 & : & ℓ = 0, \end{matrix}

we can give a simple description of left- and right-sided elements in Q(M_{H}). An element σ ∈ Q(M_{H}) is left-sided iff there exists b ∈ M_{H} s.t. for all ℓ ∈ M_{H}:

σ(ℓ) = \{ \begin{matrix}1 & : & ℓ ≰ b, \\ 0 & : & ℓ ≤ b. \end{matrix}

An element σ ∈ Q(M_{H}) is right-sided iff there exists a ∈ M_{H} s.t. for all ℓ ∈ M_{H}:

σ(ℓ) = \{ \begin{matrix}a & : & ℓ ≠ ⊥, \\ ⊥ & : & ℓ = 0. \end{matrix}

Obviously, the subquantale of all two-sided elements coincides with {⊥, ⊤} — i.e. Q(M_{H}) is simple.

In order to avoid superfluous information we wish that the semigroup operation is only determined by the multiplication of right-sided elements with left-sided ones. Hence we replace & by the semigroup operation ⊙ defined by

σ_{1} ⊙ σ_{2} = σ_{1} & ⊤ & σ_{2}

Again we obtain an involutive and simple quantale Q(H) = (Q(M_{H}), ⊙, ≤,^{o}). Even though Q(H) is not unital, the universal upper bound is still extensive — i.e.

σ ≤ (⊤ ⊙ σ) ∧ (σ ⊙ ⊤).

Moreover, stable and symmetric elements in Q(H) have a simple form. The details are as follows: since ⊙ coincides with & on all left-sided (resp. right-sided) elements, the subquantale of all left-sided (resp. right-sided) elements is idempotent. Hence ⊙ fulfills the following properties:

(σ_{1} ⊙ σ_{2}) ⊙ (σ_{3} ⊙ σ_{4}) = (σ_{1} ⊙ σ_{3}) ⊙ (σ_{2} ⊙ σ_{4}),  (6.1)
σ_{1} ⊙ σ_{1} ⊙ σ_{2} = σ_{1} ⊙ σ_{2} = σ_{1} ⊙ σ_{2} ⊙ σ_{2}.  (6.2)

Property (6.1) means bi-symmetry, while (6.2) is a kind of generalized idempotency of ⊙. An important corollary from these properties is the fact that σ ∈ Q(H) is stable (cf. Example 1.3) iff σ is idempotent w.r.t. ⊙. Hence σ is stable and symmetric in Q(H) iff there exists a ∈ M_{H} s.t. σ has the following form for all ℓ ∈ M_{H}:

σ(ℓ) = σ_{a}(ℓ) = \{ \begin{matrix}a & : & ℓ ≰ a^{⊥}, \\ 0 & : & ℓ ≤ a^{⊥}. \end{matrix}  (6.3)

Finally, with every pair (a, b) ∈ M_{H} × M_{H} we associate an element σ_{b}^{a} of Q(M_{H}) defined by:

σ_{b}^{a}(ℓ) = \{ \begin{matrix}a & : & ℓ ≰ b, \\ 0 & : & ℓ ≤ b. \end{matrix}  (6.4)

The product w.r.t. ⊙ has always the form described in (6.4). In fact, if σ_{1}(⊤) = a and b = ⋁{ℓ ∈ M_{H} | σ_{2}(ℓ) = 0}, then σ_{1} ⊙ σ_{2} = σ_{b}^{a}. Also the following relations hold:

σ_{a} = σ_{a^{⊥}}^{a},  (σ_{b}^{a})^{o} = σ_{a^{⊥}}^{b^{⊥}},  ⊤ = σ_{0}^{1}.

After these preparations we consider the involutive and integral quantaloid Q_{H} induced by Q(M_{H}) in the sense of Example 1.4. It is not difficult to see that the set of objects and hom-sets have the form:

• obj(Q_{H}) = {σ_{a} | a ∈ M_{H}} ⋃^{·}{ω}.

• Q_{H}(σ_{a}, σ_{b}) = \{ \begin{matrix}\{⊥, σ_{a}\}, & a & = & b, \\ \{⊥\}, & a & ≠ & b. \end{matrix}

• Q_{H}(ω, σ_{a}) = {σ_{a^{⊥}}^{b} | b ≤ a}, Q_{H}(σ_{a}, ω) = {σ_{b}^{a} | a^{⊥} ≤ b}, Q_{H}(ω, ω) = {0, 1}.

• The partial ordering on the hom-sets is inherited from Q(M_{H}).

As special properties of the composition in Q_{H} we first note that all hom-sets Q_{H}(α, α) are isomorphic to the Boolean algebra 2. Hence Q_{H} can be viewed as the quantisation of 2. Furthermore, in the case of b ≠ c it is remarkable to see that Q_{H}(b, c) has the structure of an idempotent quantale. Of course, the multiplication in Q_{H}(b, c) is induced by ⊙ and is compatible with the composition in the following sense:

(σ_{1} ⊙ σ_{2}) · τ = (σ_{1} · τ) ⊙ (σ_{2} · τ), (σ_{1}, σ_{2}) ∈ Q_{H}(β, γ), τ ∈ Q_{H}(α, β), β ≠ γ (6.5)

where we have made use of (6.1) and (6.2).

Now we turn to spectra of unital C^{∗}-algebras which are understood as the quantale of all closed left-ideals (see also [22]). Further, we agree with the conception that irreducible representations play the role of «points» for non-commutative C^{∗}-algebras (cf. pp. 14 in [1]). By means of the Gelfand-Neumark-Segal-construction every pure state ϱ of a C^{∗}-algebra A induces an irreducible representation (H_{ϱ}, ϑ_{ϱ}) of A (cf. [12]). In contrast to the commutative setting it is interesting to note that in the case of non-commutative C^{∗}-algebras the Hilbert space dimension of the underlying Hilbert space H_{ϱ} depends on the pure state ϱ and might possibly vary. In order to overcome this obstacle and to choose an underlying Hilbert space which is independent from the respective pure states, we first recall some terminology from [10].

6.1. Remark. Let P(A) be the set of all pure states of a unital C^{∗}-algebra A. With every pure state ϱ we associated the irreducible representation (H_{ϱ}, ϑ_{ϱ}) given by the Gelfand-Naimark-Segal-construction.

(a) A Hilbert space H is admissible for A iff for every pure state ϱ ∈ P(A) there exists an isometry H_{ϱ} ^{\overline{}^{Φ_{ϱ}}} H. It is easily seen that for every C^{∗}-algebra an admissible Hilbert space exists.

(b) A representation (H, ϑ) of A is called locally irreducible iff H is admissible for A and there exists a pure state ϱ of A and an isometry H_{ϱ} ^{\overline{}^{Φ_{ϱ}}} H s.t. for all a ∈ A the following relation holds:

π(a) = Φ_{ϱ} ∘ ϑ_{ϱ}(a) ∘ Φ_{ϱ}^{∗},  a ∈ A  (6.6)

where H ^{\overline{}^{Φ_{ϱ}^{∗}}} H_{ϱ} denotes the adjoint operator corresponding to Φ_{ϱ}.

In the following considerations we fix a unital C^{∗}-algebra A. Then we choose an admissible Hilbert space H for A and consider the involutive and unital quantale Q(M_{H})

— the Hilbert quantale associated with H (cf. Remark 1.1, [20]). Further, let sp(A) be the spectrum of A, and L(H) be the C^{∗}-algebra of all bounded linear operators on H. From [10] we quote the following result.

6.2. Theorem. Let A ^{\overline{}^{ϑ}} L(H) be a ∗-homomorphism s.t. (H, ϑ) is a locally irredu-

cible representation of A. Then ϑ induces a quantale homomorphism σ(A) ^{\overline{}^{h_{ϑ}}} Q(M_{H}) defined by

[h_{ϑ}(I)](ℓ) = top. closure(lin. hull{ϑ(a)(x) | a ∈ I, x ∈ ℓ}), I ∈ sp(A), ℓ ∈ M_{H}. (6.7)

Moreover h_{ϑ} fulfills the following properties:

(i) h_{ϑ}(A) is an idempotent and symmetric element of Q(M_{H}).

(ii) h_{ϑ}(I_{1} · I_{2}) = h_{ϑ}(I_{1}) & ⊤ & h_{ϑ}(I_{2}) where I_{1}, I_{2} ∈ sp(A).

Further, let X_{A} be the set of all ∗-homomorphisms A ^{\overline{}^{ϑ}} L(H) s.t. (H, ϑ) is a locally irreducible representation of A. We conclude from Theorem 6.2 that every element ϑ ∈ X_{A} induces a quantale homomorphism h_{ϑ} from sp(A) to Q(H) s.t. h_{ϑ}(A) is symmetric and stable in the sense of Q(H). Hence there exists a map X_{A} ^{\overline{}^{e}} obj(Q_{H}) determined by

e(ϑ) = h_{ϑ}(A),  ϑ ∈ X_{A}.  (6.8)

Moreover, every closed left-ideal I of A induces a map X_{A} ^{\overline{}^{f_{I}}} mor(Q_{H}) satisfying the following property

f_{I}(ϑ) = h_{ϑ}(I) ∈ Q_{H}(e(ϑ), ω),  ϑ ∈ X_{A}.  (6.9)

Now we provide (X_{A}, e) with the structure of a symmetric Q_{H}-category and define a map X_{A} × X_{A} ^{\overline{}^{d}} mor(Q_{H}) by:

\begin{matrix}d(ϑ_{1}, ϑ_{2}) = ⋁ \{σ ∈ Q_{H}(e_{0}(ϑ_{2}), e_{0}(ϑ_{1})) | ∀I ∈ sp(A) : \\ f_{I}(ϑ_{1}) · σ ≤ f_{I}(ϑ_{2}), σ · f_{I}(ϑ_{2})^{o} ≤ f_{I}(ϑ_{1})^{o}\} \end{matrix}

Then 𝕀_{loc} = (X_{A}, e, d) is a symmetric Q_{H}-category, and every map f_{I} is a covariant presheaf on 𝕀_{loc} of type ω.

Further, the correspondence I ⟼ f_{I} is injective, because every closed left-ideal is an intersection of maximal left-ideals and every maximal left-ideal can be identified with a pure state (cf. 10.2.10 in [13]). If we make use of the quantale structure on Q_{H}(e(ϑ), ω) induced by ⊙ (see above), then the property (ii) of Theorem 6.2 means: f_{I_{1}·I_{2}} = f_{I_{1}} ⊙ f_{I_{2}}. Hence the spectrum of A is isomorphic to {f_{I} | I ∈ sp(A)} — a result which is closely related to Proposition 4 in [4] (see also Remark (ii) in [24]).

If we now speculate and view closed left-ideals I (resp. f_{I}) as «open subsets» of a non-commutative topological space (cf. [5, 18]), then the problem arises that all f_{I} are covariant presheaves (in particular Q_{H}-functors) and not Q_{H}-subcategories. Because of the methods in Section 4 we can solve this problem. First, we extend the covariant presheaf f_{I} to a W-homomorphism f_{I}^{♯} on the free W-algebra (W(𝕀_{loc}), μ_{𝕀_{loc}}). Secondly, we identify f_{I}^{♯} with a specific subalgebra of (W(𝕀_{loc}), μ_{𝕀_{loc}}) whose carrier set is determined by formula (4.9). Hence (W(𝕀_{loc}), μ_{𝕀_{loc}}) is the non-commutative space of the C^{∗}-algebra A. In particular, «open subsets» corresponding to closed left-ideals of A are certain subalgebras of (W(𝕀_{loc}), μ_{𝕀_{loc}}).

Acknowledgement. I am very grateful to the referee for his helpful suggestions and his reference to [9].

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Fachbereich C Mathematik und Naturwissenschaften, Bergische Universität, Gaußstraße 20, D-42097 Wuppertal, Germany

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