Theory and Applications of Categories, Vol. 16, No. 14, 2006, pp. 283–306.

CATEGORICAL STRUCTURES ENRICHED IN A QUANTALOID: TENSORED AND COTENSORED CATEGORIES

ISAR STUBBE

Abstract. A quantaloid is a sup-lattice-enriched category; our subject is that of categories, functors and distributors enriched in a base quantaloid Q. We show how cocomplete Q-categories are precisely those which are tensored and conically cocomplete, or alternatively, those which are tensored, cotensored and ‘order-cocomplete’. In fact, tensors and cotensors in a Q-category determine, and are determined by, certain adjunctions in the category of Q-categories; some of these adjunctions can be reduced to adjuctions in the category of ordered sets. Bearing this in mind, we explain how tensored Q-categories are equivalent to order-valued closed pseudofunctors on Q^{op}; this result is then finetuned to obtain in particular that cocomplete Q-categories are equivalent to sup-lattice-valued homomorphisms on Q^{op} (a.k.a. Q-modules).

Introduction

The concept of “category enriched in a bicategory W” is as old as the definition of bicategory itself [Bénabou, 1967]; however, J. Bénabou called them “polyads”. Taking a W with only one object gives a monoidal category, and for symmetric monoidal closed V the theory of V-categories is well developed [Kelly, 1982]. But also categories enriched in a W with more than one object are interesting. R. Walters [1981] observed that sheaves on a locale give rise to bicategory-enriched categories: “variation” (sheaves on a locale Ω) is related to “enrichment” (categories enriched in Rel(Ω)). This insight was further developed in [Walters, 1982], [Street, 1983] and [Betti et al., 1983]. Later [Gordon and Power, 1997, 1999] complemented this work, stressing the important rôle of tensors in bicategory-enriched categories.

Here we wish to discuss “variation and enrichment” in the case of a base quantaloid Q (a small sup-lattice-enriched category). This is, of course, a particular case of the above, but we believe that it is also of particular interest; many examples of bicategoryenriched categories (like Walters’) are really quantaloid-enriched. Since in a quantaloid Q every diagram of 2-cells commutes, many coherence issues disappear, so the theory of Qenriched categorical structures is very transparent. Moreover, by definition a quantaloid Q has stable local colimits, hence (by local smallness) it is closed; this is of great help when working with Q-categories. The theory of quantaloids is documented in [Rosenthal, 1996]; examples and applications of quantaloids abound in the literature; and [Stubbe, 2005a] provides a reference on Q-category theory.

Received by the editors 2005-03-14 and, in revised form, 2006-06-14.

Transmitted by Ross Street. Published on 2006-06-19.

2000 Mathematics Subject Classification: 06F07, 18D05, 18D20.

Key words and phrases: quantaloid, enriched category, weighted (co)limit, module.

\begin{matrix}c \\ ◯ \end{matrix} Isar Stubbe, 2006. Permission to copy for private use granted.

Let us illustrate the questions that this paper is concerned with. Consider a right action of a monoid K on some object M in the monoidal category Sup of sup-lattices and sup-morphisms; call it α: M ⊗ K [figure] M. Surely K may be viewed as a (posetal) monoidal category, and M determines a K-enriched category 𝕄: its set of objects is M, and 𝕄(y, x) = ⋁{f ∈ K | α(y ⊗ f) ≤ x} is the hom-object for x, y ∈ M. Now what are the particular properties of this K-category? Can one characterize K-categories arising in this way? Reckoning that a quantaloid Q is the “many-object version” of a monoid in Sup, can we generalize this to Q-modules (i.e. homomorphisms from Q^{op} to Sup) and Q-categories? And instead of looking at Q-modules, are there less stringent forms of variation, e.g. certain order-valued functors on Q^{op}, for which we can do the same trick?

We give affirmative answers to all these questions, and to that end the notion of (co)tensor in a Q-category is crucial: because it is the Q-categorical way of speaking about an “action” of Q.

Overview of contents.

To make this paper self-contained, the first section contains a brief review of some basic facts on quantaloids and quantaloid-enrichment.

The starting point in section 2 is the notion of weighted colimit in a Q-category ℂ [Kelly, 1982; Street, 1983]. Two particular cases of such weighted colimits are tensors and conical colimits; then ℂ is cocomplete (i.e. it admits all weighted colimits) if and only if it is tensored and has all conical colimits [Kelly, 1982; Gordon and Power, 1999] (see also 2.7 below). But we may consider the family of ordered sets of objects of the same type in ℂ; we call ℂ order-cocomplete when these ordered sets admit arbitrary suprema. This is a weaker requirement than for ℂ to have conical colimits, but for cotensored ℂ they coincide. Now ℂ is cocomplete if and only if it is tensored, cotensored and ordercocomplete (as in 2.13). Put differently, for a tensored and cotensored Q-category ℂ, order-theoretical content (suprema) can be “lifted” to Q-categorical content (weighted colimits).

Then section 3 is devoted to adjunctions. We see how, at least for tensored Qcategories, order-adjunctions can be “lifted” to Q-enriched adjunctions, and how (co)tensoredness may be characterized by enriched adjunctions (analogously to V-categories). As a result, for a tensored ℂ, its cotensoredness is equivalent to certain order-adjunctions (cf. 3.7). With this in mind we analyze in section 4 the basic biequivalence between tensored Q-enriched categories and closed pseudofunctors on Q^{op} with values in Cat(2) (as in 4.5, a particular case of results in [Gordon and Power, 1997]). A finetuned version thereof (our main theorem 4.12) says in particular that right Q-modules are the same thing as cocomplete Q-enriched categories.

These results are used in three forthcoming papers: “Causal duality: what it is and what it is good for”, on the relation between (co)tensored Q-categories and the notion of ‘causal duality’ (essentially making use of 3.7); “Towards dynamic domains”, which deals with a Q-categorical version of the ‘totally-below’ relation and is a follow-up to [Stubbe, 2005b] (see [Stubbe, 2006] for an extended abstract); and “Q-modules are Qsup-lattices”, where it is argued that Q-modules are to be thought of as the ‘internal sup-lattices’ amongst the ordered sheaves on Q (see also [Stubbe 2005c]).

Acknowledgement. The better part of this article was written during my time at the Université Catholique de Louvain in Louvain-la-Neuve, in the spring of 2004.

1. Quantaloid-enriched categories

Here is a brief summary of some basic facts concerning quantaloids and quantaloidenriched categories; for details, examples and the appropriate historical references, see [Rosenthal, 1996; Stubbe, 2005a].

Let Sup denote the category of complete lattices and functions that preserve suprema (“sup-lattices and sup-morphisms”): for the usual tensor product, this is a symmetric monoidal closed category. A quantaloid Q is a Sup-enriched category, and a homomorphism H: Q [figure] Q′ of quantaloids is a Sup-enriched functor. In other words, Q is a category whose hom-sets are complete lattices and in which composition distributes on both sides over suprema, and H: Q [figure] Q′ is a functor that preserves suprema of morphisms. It follows in particular that composition with a morphism f: X [figure] Y in a quantaloid Q gives rise to adjunctions

Q(A, X) [figure: ⊥; f ∘ −; [f, −]]Q(A, Y) and Q(Y, A) [figure: ⊥; − ∘ f; {f, −}]Q(X, A);  (1)

these right adjoints are respectively called lifting and extension (through f).

A quantaloid Q is a bicategory and therefore it may serve as base for enrichment; to avoid size-issues (alluded to further on, but see also 2.3), we shall from now on suppose that Q is small. A Q-category 𝔸 is determined by: a set 𝔸_{0} of objects so that to each a ∈ 𝔸_{0} is assigned an object ta of Q (called the type of a); and for any two objects a, a′ ∈ 𝔸_{0}, a morphism 𝔸(a′, a): ta [figure] ta′ in Q, called a hom-arrow of 𝔸. These data are required to satisfy unit and composition inequalities in Q: for all a, a′, a′′ ∈ 𝔸_{0},

1_{ta} ≤ 𝔸(a, a) and 𝔸(a′′, a′) ∘ 𝔸(a′, a) ≤ 𝔸(a′′, a).

A functor F: 𝔸 [figure] 𝔹 between Q-categories is, in the same vein, a map 𝔸_{0} [figure] 𝔹_{0}: a ↦ F a satisfying, for all a, a′ ∈ 𝔸_{0},

ta = t(F a) and 𝔸(a′, a) ≤ 𝔹(F a′, F a).

Q-categories and functors form a category Cat(Q) for the obvious composition law and identities.

For two objects a, a′ in a Q-category 𝔸 we write a ≤ a′ when ta = ta′ and 1_{ta} ≤ 𝔸(a, a′). Due to the composition and unit inequalities, (𝔸_{0}, ≤) is an ordered^{1} set, and if the order relation is moreover anti-symmetric, we say that 𝔸 is skeletal. Further, for two functors F, G: 𝔸 [figure] 𝔹 we put that F ≤ G whenever F a ≤ Ga holds in 𝔹 for all a ∈ 𝔸_{0}; thus Cat(Q) becomes a locally ordered 2-category, which in fact is biequivalent to its full sub-2-category Cat_{skel}(Q) of skeletal Q-categories.

1An order is transitive and reflexive, and a partial order is moreover anti-symmetric.

A distributor (or module or profunctor) Φ: 𝔸 [figure] 𝔹 between Q-categories is a matrix of Q-morphisms Φ(b, a): ta [figure] tb, one for each (a, b) ∈ 𝔸_{0} × 𝔹_{0}, satisfying action inequalities

𝔹(b, b′) ∘ Φ(b′, a) ≤ Φ(b, a) and Φ(b, a′) ∘ 𝔸(a′, a) ≤ Φ(b, a)

for every a, a′ ∈ 𝔸_{0} and b, b′ ∈ 𝔹_{0}. The set of distributors from 𝔸 to 𝔹 is a complete lattice: for (Φ_{i}: 𝔸 [figure] 𝔹)_{i∈I} we naturally define ⋁_{i} Φ_{i}: 𝔸 [figure] 𝔹 by

( ⋁_{i} Φ_{i})(b, a) = ⋁_{i} Φ_{i}(b, a).

Two distributors Φ: 𝔸 [figure] 𝔹, Ψ: 𝔹 [figure] ℂ compose: we write Ψ⊗Φ: 𝔸 [figure] ℂ for the distributor with elements

(Ψ ⊗ Φ)(c, a) = ⋁_{b∈𝔹_{0}} Ψ(c, b) ∘ Φ(b, a).

The identity distributor on a Q-category 𝔸 is 𝔸: 𝔸 [figure] 𝔸 itself, i.e. the distributor with elements 𝔸(a′, a): ta [figure] ta′, and we get a quantaloid Dist(Q) of Q-categories and distributors. Dist(Q) being a quantaloid, we may compute liftings and extensions of distributors between Q-categories; these actually reduce to liftings and extensions in Q as follows: for Θ: 𝔸 [figure] ℂ and Ψ: 𝔹 [figure] ℂ, [Ψ, Θ]: 𝔸 [figure] 𝔹 has elements

[Ψ, Θ](b, a) = ⋀_{c∈ℂ_{0}}[Ψ(c, b), Θ(c, a)],

where the liftings on the right are calculated in Q (and similarly for extensions).

Every functor F: 𝔸 [figure] 𝔹 between Q-categories induces (or represents) an adjoint pair of distributors:

- the left adjoint 𝔹(−, F −): 𝔸 [figure] 𝔹 has elements 𝔹(b, F a): ta [figure] tb,

- the right adjoint 𝔹(F −, −): 𝔹 [figure] 𝔸 has elements 𝔹(F a, b): tb [figure] ta.

The assignment F ↦ 𝔹(−, F −) is a faithful 2-functor from Cat(Q) to Dist(Q). Thus, whenever a distributor Φ: 𝔸 [figure] 𝔹 is represented by a functor F: 𝔸 [figure] 𝔹, this F is essentially unique.

Given a distributor and a functor as in

[figure: 𝔸; Φ; 𝔹; F; ℂ,]

a functor K: 𝔸 [figure] ℂ is the Φ-weighted colimit of F when ℂ(K−, −) = [Φ, ℂ(F −, −)]; if this colimit exists, we write it as colim(Φ, F). Dually, for

[figure: 𝔸; 𝔹; Ψ; G; ℂ,]

L: 𝔸 [figure] ℂ is the Ψ-weighted limit of G if ℂ(−, L−) = {Ψ, ℂ(−, G−)}; lim(Ψ, G) is its usual notation. The Q-category ℂ is (co)complete when it admits all such weighted (co)limits. A simple argument shows that (co)completeness only makes sense for our small Q-categories if Q itself is small (see also 2.1 further on); that is why we made that assumption. Moreover, a Q-category is complete if and only if it is cocomplete.

With notations of the preceding paragraph, a functor H: ℂ [figure] ℂ′ is cocontinuous^{2} when it preserves all colimits that happen to exist in ℂ: H ∘ colim(Φ, F) ≅ colim(Φ, H ∘ F). A left adjoint functor^{3} is always cocontinuous; conversely, if the domain of a cocontinuous functor is cocomplete, then that functor is left adjoint. Cocomplete Q-categories and cocontinuous functors form a sub-2-category Cocont(Q) of Cat(Q), and the biequivalence Cat(Q) ≃ Cat_{skel}(Q) reduces to a biequivalence Cocont(Q) ≃ Cocont_{skel}(Q). One can show that Cocont(Q) has stable local colimits, which makes Cocont_{skel}(Q) a quantaloid.

Every object X of a quantaloid Q determines a one-object Q-category ∗_{X} whose single hom-arrow is 1_{X}. A contravariant presheaf of type X on a Q-category 𝔸 is a distributor ϕ: ∗_{X} [figure] 𝔸; these are the objects of a cocomplete Q-category P𝔸 whose hom-arrows are given by lifting in Dist(Q). Every object a ∈ 𝔸_{0} determines, and is determined by, a functor ∗_{ta} [figure] 𝔸; thus a ∈ 𝔸_{0} also represents a (left adjoint) presheaf 𝔸(−, a): ∗_{ta} [figure] 𝔸. The Yoneda embedding Y_{𝔸}: 𝔸 [figure] P𝔸: a ↦ 𝔸(−, a) is a fully faithful^{4} continuous functor. The presheaf construction 𝔸 ↦ P𝔸 extends to a 2-functor Cat(Q) [figure] Cocont(Q) which is left biadjoint to the inclusion 2-functor, with the Yoneda embeddings as unit; thus presheaf categories are the freely cocomplete ones. Dually to this, a covariant presheaf of type X on 𝔸 is a distributor ψ: 𝔸 [figure] ∗_{X}; using extensions in Dist(Q) for hom-arrows, these form a freely complete Q-category P^{†}𝔸 and there is a cocontinuous embedding Y^{†}_{𝔸}: 𝔸 [figure] P^{†}𝔸.

Finally a word on duality. If 𝔸 is a Q-category, then 𝔸^{op}, defined to have the same object set but with hom-arrows 𝔸^{op}(a′, a) = 𝔸(a, a′), is a Q^{op}-category (but not a Qcategory in general). Doing the natural thing, one easily sees that “applying op twice” gives isomorphisms Cat(Q) ≅ Cat(Q^{op})^{co} and Dist(Q) ≅ Dist(Q^{op})^{op} of 2-categories (the co denotes the reversal of the local order) that allow us to dualize all notions and results concerning Q-categories. For example, colim(Φ, F) = (lim(Φ^{op}, F^{op}))^{op}, or also P^{†}𝔸 = (P(𝔸^{op}))^{op}.

2. More on weighted (co)limits

In this section we recall the special cases of weighted (co)limits called (co)tensor and conical (co)limit in a Q-enriched category ℂ ; in fact, ℂ is cocomplete if and only if it is tensored and conically cocomplete [Kelly, 1982; Street, 1983; Gordon and Power, 1999]. Then we introduce the notion of order-(co)completeness, which is strictly weaker than conical (co)completeness but quite useful in practice, and prove that ℂ is cocomplete if and only if it is tensored, cotensored and order-cocomplete.

2Continuous is synonymous for limit preserving, and one can develop dual results.

3F: 𝔸 [figure] 𝔹 is left adjoint to G: 𝔹 [figure] 𝔸 if 1_{𝔸} ≤ G ∘ F and F ∘ G ≤ 1_{𝔹}.

4A functor F: 𝔸 [figure] 𝔹 is fully faithful when 𝔸(a′, a) = 𝔹(F a′, F a) for every a, a′ ∈ 𝔸_{0}.

(Co)tensors.

An arrow f: X [figure] Y in Q may be viewed as a distributor (f): ∗_{X} [figure] ∗_{Y} between one-object Q-categories. An object y of type Y of a Q-category ℂ may be identified with the functor Δy: ∗_{Y} [figure] ℂ: ∗ ↦ y.

For a Q-arrow f: X [figure] Y and an object y ∈ ℂ_{0} of type ty = cod(f), the tensor of y and f is, by definition, the (f)-weighted colimit of Δy; it will be denoted y ⊗ f. Thus, whenever it exists, y ⊗ f is the (necessarily essentially unique) object of ℂ (necessarily of type t(y ⊗ f) = dom(f)) such that

for all z ∈ ℂ, ℂ(y ⊗ f, z) = [f, ℂ(y, z)] in Q.

Dually, for an arrow f: X [figure] Y in Q and an object x ∈ ℂ of type tx = dom(f), the cotensor of f and x, denoted ⟨f, x⟩, is the (f)-weighted limit of Δx: whenever it exists, it is the object of ℂ of type t⟨f, x⟩ = cod(f) with the universal property that

for all z ∈ ℂ, ℂ(z, ⟨f, x⟩) = {f, ℂ(z, x)} in Q.

A Q-category ℂ is tensored when for all f ∈ Q and y ∈ ℂ_{0} with ty = cod(f), the tensor y ⊗ f exists; cotensored is the dual notion.

Because ‘colimit’ and ‘limit’, and a fortiori ‘tensor’ and ‘cotensor’, are dual notions in the rigorous sense explained at the end of section 1, all we say about one also holds “up to duality” for the other; we do not always bother spelling this out, even though we make use of it.

When making a theory of (small) tensored Q-categories, there are some size issues to address, as the following indicates.

2.1. Lemma. A tensored Q-category has either no objects at all, or at least one object of type X for each Q-object X.

Proof. The empty Q-category is trivially tensored. Suppose that ℂ is non-empty and tensored; say that there is an object y of type ty = Y in ℂ. Then, for any Q-object X the tensor of y with the zero-morphism 0_{X,Y} ∈ Q(X, Y) must exist, and is an object of type X in ℂ.

This motivates why we work over a small base quantaloid Q.

2.2. Example. The two-element Boolean algebra is denoted 2; we may view it as a one-object quantaloid so that 2-categories are ordered sets, functors are order-preserving maps, and distributors are ideal relations. A non-empty 2-category, i.e. a non-empty order, is tensored if and only if it has a bottom element, and cotensored if and only if it has a top element.

2.3. Example. For any object A in a quantaloid Q, PA denotes the Q-category of contravariant presheaves on the one-object Q-category ∗_{A} whose hom-arrow is 1_{A}. In practice, the objects of PA are the Q-arrows with codomain A, the types of which are their domains, and the hom-arrows in PA are given by lifting in Q: PA(f′, f) = [f′, f]. Like any presheaf category it is cocomplete, thus complete, thus both tensored and cotensored. Explicitly, for an object f ∈ PA of type tf = Y and Q-arrows g: X [figure] Y and h: Y [figure] Z, one verifies that f ⊗ g = f ∘ g: X [figure] A seen as object of type X in PA, and ⟨h, f⟩ = {h, f}: Z [figure] A as object of type Z in PA:

[figure: X; g; f ⊗ g = f ∘ g; Y; f; h; Z; ⟨f, h⟩ = {f, h}; A]

Similarly, P^{†}A is the Q-category of covariant presheaves on ∗_{A}: its objects are Q-arrows with domain A, the type of such an object is its codomain, and the hom-arrows are given by extension: P^{†}X(f′, f) = {f, f′} (note the reversal of the variables, which is needed to have a composition inequality). Further, for f: A [figure] Y, k: X [figure] Y and l: Y [figure] Z, f ⊗ l = [l, f] and ⟨k, f⟩ = k ∘ f in P^{†}A.

2.4. Example. More general than the above, consider any presheaf category P𝔸; an object of type Y of P𝔸 is precisely a distributor ψ: ∗_{Y} [figure] 𝔸. It is easily verified by calculations with liftings and extensions in Dist(Q) that for a Q-arrow f: X [figure] Y, which we may view as a one-element distributor (f): ∗_{X} [figure] ∗_{Y}, the tensor ψ ⊗ f is precisely the composition ψ ⊗ (f) in Dist(Q). For an object of type X of P𝔸, i.e. a distributor ϕ: ∗_{X} [figure] 𝔸, the cotensor ⟨f, ϕ⟩ is precisely the extension {(f), ϕ} in Dist(Q). (Similar calculations can be made for P^{†}𝔸.)

Conical (co)limits.

A Q-category ℂ has an underlying order (ℂ_{0}, ≤), as recalled in section 1. Conversely, on an ordered set (A, ≤) we may consider the free Q(X, X)-category 𝔸:

- 𝔸_{0} = A, all objects are of type X;

- 𝔸(a′, a) = \{ \begin{matrix}1_{X} & if a′ ≤ a, \\ 0_{X,X} & otherwise. \end{matrix}

To give a functor F: 𝔸 [figure] ℂ is to give objects F a, F a′, ... of type X in ℂ such that F a′ ≤ F a in the underlying order of ℂ whenever a′ ≤ a in (A, ≤). Consider furthermore the weight ϕ: ∗_{X} [figure] 𝔸 whose elements are ϕ(a) = 1_{X} for all a ∈ 𝔸_{0}. The ϕ-weighted colimit of F: 𝔸 [figure] ℂ (which may or may not exist) is the conical colimit of F. (Notwithstanding the adjective “conical”, this is still a weighted colimit!) A conically cocomplete Q-category is one that admits all conical colimits^{5}.

5Analogously to 2.1, a conically cocomplete Q-category ℂ has, for each Q-object X, at least one object of type X: the conical colimit on the empty functor from the empty free Q(X, X)-category into ℂ.

The dual notions are those of conical limit and conically complete Q-category. We do not bother spelling them out.

The following will help us calculate conical colimits.

2.5. Proposition. Consider a free Q(X, X)-category 𝔸 and a functor F: 𝔸 [figure] ℂ. An object c ∈ ℂ_{0}, necessarily of type tc = X, is the conical colimit of F if and only if ℂ(c, −) = ⋀_{a∈𝔸_{0}} ℂ(F a, −) in Dist(Q)(ℂ, ∗_{X}).

Proof. For the conical colimit weight ϕ: ∗_{X} [figure] 𝔸, ϕ(a) = 1_{X} for all a ∈ 𝔸, thus c = colim(ϕ, F) if and only if

ℂ(c, −) = [ϕ, ℂ(F −, −)]
= ⋀_{a∈𝔸_{0}} [ϕ(a), ℂ(F a, −)]
= ⋀_{a∈𝔸_{0}} [1_{X}, ℂ(F a, −)]
= ⋀_{a∈𝔸_{0}} ℂ(F a, −).

To pass from the first line to the second, we used the explicit formula for liftings in the quantaloid Dist(Q)

2.6. Proposition. A Q-category ℂ is conically cocomplete if and only if for any family (c_{i})_{i∈I} of objects of ℂ, all of the same type, say tc_{i} = X, there exists an object c in ℂ, necessarily also of that type, such that ℂ(c, −) = ⋀_{i∈I} ℂ(c_{i}, −) in Dist(Q)(ℂ, ∗_{X}).

Proof. One direction is a direct consequence of 2.5. For the other, given a family (c_{i})_{i∈I} of objects of ℂ, all of type tc_{i} = X, consider the free Q(X, X)-category 𝕀 on the ordered set (I, ≤) with i ≤ j ⟺ c_{i} ≤ c_{j} in ℂ. The conical colimit of the functor F: 𝕀 [figure] ℂ: i ↦ c_{i} is an object c ∈ ℂ_{0} such that ℂ(c, −) = ⋀_{i∈I} ℂ(c_{i}, −), precisely what we wanted.

In what follows we will often speak of “the conical (co)limit of a family of objects with the same type”, referring to the construction as in the proof above.

2.7. Theorem. A Q-category ℂ is cocomplete if and only if it is tensored and conically cocomplete.

Proof. For the non-trivial implication, the alternative description of conical cocompleteness in 2.6 is useful. If ϕ: ∗_{X} [figure] ℂ is any presheaf on ℂ, then the conical colimit of the family (x ⊗ ϕ(x))_{x∈ℂ_{0}} is the ϕ-weighted colimit of 1_{ℂ}: for this is an object c ∈ ℂ_{0} such that

ℂ(c, −) = ⋀_{x∈ℂ_{0}} ℂ(x ⊗ ϕ(x), −)
= ⋀_{x∈ℂ_{0}} [ϕ(x), ℂ(x, −)]

= [ϕ, ℂ(1_{ℂ}−, −)].

Hence ℂ is cocomplete, for it suffices that ℂ admit presheaf-weighted colimits of 1_{ℂ} [Stubbe, 2005a, 5.4].

Tensors and conical colimits allow for a very explicit description of colimits in a cocomplete category.

2.8. Corollary. If ℂ is a cocomplete Q-category, then the colimit of

[figure: 𝔸; Φ; 𝔹; F; ℂ]

is the functor colim(Φ, F): 𝔸 [figure] ℂ sending an object a ∈ 𝔸_{0} to the conical colimit of the family (F b ⊗ Φ(b, a))_{b∈𝔹_{0}}. A functor F: ℂ [figure] ℂ′ between cocomplete Q-categories is cocontinuous if and only if it preserves tensors and conical colimits.

In 2.15 we will discuss a more user-friendly version of the above: we can indeed avoid the conical colimits, and replace them by suitable suprema.

A third kind of (co)limit.

It makes no sense to ask for the underlying order (ℂ_{0}, ≤) of a Q-category ℂ to admit arbitrary suprema: two objects of different type cannot even have an upper bound! So let us now denote ℂ_{X} for the ordered set of ℂ-objects with type X (which is thus the empty set when ℂ has no such objects); in these orders it does make sense to talk about suprema. We will say that ℂ is order-cocomplete when each ℂ_{X} admits all suprema^{6}.

The dual notion is that of order-complete Q-category; but of course “order-complete” and “order-cocomplete” are always equivalent since each order ℂ_{X} is small. Nevertheless we will pedantically use both terms, to indicate whether we take suprema or infima as primitive structure.

2.9. Example. For the category Pℂ of contravariant presheaves on ℂ, the ordered set (Pℂ)_{X} is precisely the sup-lattice Dist(Q)(∗_{X}, ℂ); so Pℂ is order-cocomplete. When considering covariant presheaves, we get that (P^{†}ℂ)_{X} is Dist(Q)(ℂ, ∗_{X})^{op} (the “op” meaning that the order is reversed). In particular is (PA)_{X} = Q(X, A) and (P^{†}A)_{X} = Q(A, X)^{op}.

2.10. Proposition. Let ℂ be a Q-category. The conical colimit of a family (c_{i})_{i∈I} ∈ ℂ_{X} is also its supremum in ℂ_{X}.

Proof. Use that ℂ(c, −) = ⋀ ℂ(c_{i}, −) in Dist(Q)(ℂ, ∗_{X}) for the conical colimit c ∈ ℂ_{0} of the given family to see that c = ⋁_{i} c_{i} in ℂ_{X}.

6An order-cocomplete Q-category ℂ has, for each Q-object X, at least one object of type X. Namely, each ℂ_{X} contains the empty supremum, i.e. has a bottom element.

So if ℂ is a conically cocomplete Q-category, then it is also order-cocomplete. The converse is not true in general without extra assumptions.

2.11. Example. Consider the Q-category ℂ that has, for each Q-object X, precisely one object of type X; denote this object as 0_{X}. The hom-arrows in ℂ are defined as ℂ(0_{X}, 0_{X}) = 1_{X} (the identity arrow in Q(X, X)) and ℂ(0_{Y} , 0_{X}) = 0_{X,Y} (the bottom element in Q(X, Y)). Then each ℂ_{X} = {0_{X}} is a sup-lattice, so ℂ is order-cocomplete. However the conical colimit of the empty family of objects of type X does not exist as soon as the identity arrows in Q are not top elements, or as soon as Q has more than one object.

2.12. Proposition. Let ℂ be a cotensored Q-category. The supremum of a family (c_{i})_{i∈I} ∈ ℂ_{X} is also its conical colimit in ℂ.

Proof. By hypothesis the supremum ⋁_{i} c_{i} in ℂ_{X} exists, and by 2.10 it is the only candidate to be the wanted conical colimit. Thus we must show that ℂ(⋁_{i} c_{i}, −) = ⋀_{i} ℂ(c_{i}, −). But this follows from the following adjunctions between orders:

for any y ∈ ℂ_{Y}, ℂ_{X} [figure: ⊥; ℂ(−, y); ⟨−, y⟩]Q(Y, X)^{op} in Cat(2).

A direct proof^{7} for this adjunction is easy: one uses cotensors in ℂ to see that, for any x ∈ ℂ_{X},

- 1_{X} ≤ {ℂ(x, y), ℂ(x, y)} = ℂ(x, ⟨ℂ(x, y), y⟩) hence x ≤ ⟨ℂ(x, y), y⟩ in ℂ_{X};
- 1_{X} ≤ ℂ(⟨f, y⟩, ⟨f, y⟩) = {f, ℂ(⟨f, y⟩, y)} hence ℂ(⟨f, y⟩, y) ≤^{op} f in Q(Y, X).

Any left adjoint between orders preserves all suprema that happen to exist, so for any y ∈ ℂ_{Y}, ℂ(⋁_{i} c_{i}, y) = ⋀_{i} ℂ(c_{i}, y) in Q(Y, X), hence – since infima of distributors are calculated elementwise – ℂ(⋁_{i} c_{i}, −) = ⋀_{i} ℂ(c_{i}, −) in Dist(Q)(ℂ, ∗_{X}).

So if ℂ is cotensored and order-cocomplete, then it is also conically cocomplete. Put differently, a cotensored Q-category is conically cocomplete if and only if it is ordercocomplete. Dually, a tensored category is conically complete if and only if it is ordercomplete.

2.13. Theorem. For a tensored and cotensored Q-category, all notions of completeness and cocompleteness coincide.

As usual, for orders the situation is much simpler than for general Q-categories.

7Actually these adjunctions in Cat(2) follow from adjunctions in Cat(Q) which are due to the cotensoredness of ℂ—see 3.2.

2.14. Example. For any 2-category (be it a priori tensored and cotensored or not) all notions of completeness and cocompleteness coincide: an order is order-cocomplete if and only if it is order-complete, but it is then non-empty and has bottom and top element, thus it is tensored and cotensored, thus it is also conically complete and cocomplete, thus also complete and cocomplete tout court.

In 2.8 arbitrary colimits in a cocomplete Q-category are reduced to tensors and conical colimits. But a cocomplete Q-category is always complete too; so in particular cotensored. By cotensoredness the conical colimits may be further reduced to suprema.

2.15. Corollary. If ℂ is a cocomplete Q-category, then the colimit of the diagram

[figure: 𝔸; Φ; 𝔹; F; ℂ]

is the functor colim(Φ, F): 𝔸 [figure] ℂ sending an object a ∈ 𝔸_{0} to the supremum in ℂ_{ta} of the family (F b ⊗ Φ(b, a))_{b∈𝔹_{0}}. And a functor F: ℂ [figure] ℂ′ between cocomplete Q-categories is cocontinuous if and only it preserves tensors and suprema in each of the ℂ_{X}.

3. (Co)tensors and adjunctions

We establish a relation between adjunctions in Cat(Q) and adjunctions in Cat(2), and use this to express (co)tensors in a Q-category ℂ in terms of adjoints in Cat(Q). Further in this section we then prove that a tensored ℂ is cotensored too if and only if for each Q-arrow f: X [figure] Y the map ℂ_{Y} [figure] ℂ_{X}: y ↦ y ⊗ f is a left adjoint in Cat(2), namely with right adjoint ℂ_{X} [figure] ℂ_{Y}: x ↦ ⟨f, x⟩.

Adjunctions and adjunctions are two.

An adjunction of functors between Q-categories, like

𝔸 [figure: ⊥; F; G]𝔹,

means that G∘F ≥ 1_{𝔸} and F ∘G ≤ 1_{𝔹} in Cat(Q); equivalently, 𝔹(F a, b) = 𝔸(a, Gb) for all a ∈ 𝔸_{0} and b ∈ 𝔹_{0}. Since functors are type-preserving, this trivially implies adjunctions

for any Q-object X, 𝔸_{X} [figure: ⊥; F; G]𝔹_{X} in Cat(2).

Now we are interested in the converse: how do adjunctions in Cat(2) determine adjunctions in Cat(Q)? The pertinent result is the following.

3.1. Proposition. Let F: 𝔸 [figure] 𝔹 be a functor between Q-categories, with 𝔸 tensored. Then the following are equivalent:

(i) F is a left adjoint in Cat(Q);

(ii) F preserves tensors and, for all Q-objects X, F: 𝔸_{X} [figure] 𝔹_{X} is a left adjoint in Cat(2).

Proof. One direction is trivial. For the other, write the assumed adjunctions in Cat(2) as

\xymatrix{ & F \ar[dr] \\ 𝔸_{X} & ⊥ & 𝔹_{X} \\  & G_{X} \ar[ul]} , one for each Q-object X;

we shall prove that 𝔸_{0} [figure] 𝔹_{0}: b ↦ G_{tb} is (the object map of) the right adjoint to F in Cat(Q).

First, for any a ∈ 𝔸_{X} and b ∈ 𝔹_{Y},

𝔸(a, G_{Y} b) ≤ 𝔹(F a, F G_{Y} b)
= 𝔹(F a, F G_{Y} b) ∘ 1_{Y}
≤ 𝔹(F a, F G_{Y} b) ∘ 𝔹(F G_{Y} b, b)
≤ 𝔹(F a, b).

(The first inequality holds by functoriality of F; to pass from the second to the third line, use the pertinent adjunction F ⊣ G_{Y}: F G_{Y} b ≤ b in 𝔹_{Y}, so 1_{Y} ≤ 𝔹(F G_{Y} b, b).)

Next, using tensors in 𝔸 and the fact that F preserves them, plus the adjunction F ⊣ G_{Y} where appropriate, to see that for a ∈ 𝔸_{X} and b ∈ 𝔹_{Y},

𝔹(F a, b) ≤ 𝔸(a, G_{Y} b) ⟺ 1_{Y} ≤ [𝔹(F a, b), 𝔸(a, G_{Y} b)]
⟺ 1_{Y} ≤ 𝔸(a ⊗ 𝔹(F a, b), G_{Y} b)
⟺ 1_{Y} ≤ 𝔹(F (a ⊗ 𝔹(F a, b)), b)
⟺ 1_{Y} ≤ 𝔹(F a ⊗ 𝔹(F a, b), b)
⟺ 1_{Y} ≤ [𝔹(F a, b), 𝔹(F a, b)]

which is certainly true.

It remains to prove that G: 𝔹 [figure] 𝔸: b ↦ G_{tb}b is a functor; but for b ∈ 𝔹_{Y} and b′ ∈ 𝔹_{Y′},

𝔹(b′, b) = 1_{Y′} ∘ 𝔹(b′, b)
≤ 𝔹(F G_{Y′}b′, b′) ∘ 𝔹(b′, b)
≤ 𝔹(F G_{Y′}b′, b)
= 𝔸(G_{Y′}b′, G_{Y} b).

Here we use once more the suitable F ⊣ G_{Y′}, and also the composition in 𝔹.

In a way, 3.1 resembles 2.12: in both cases 2-categorical content is “lifted” to Qcategorical content (suprema are “lifted” to conical colimits, adjunctions between orders are “lifted” to adjunctions between categories), and in both cases the price to pay has to do with (existence and preservation of) (co)tensors.

There is a “weaker” version of 3.1: given two functors F: 𝔸 [figure] 𝔹 and G: 𝔹 [figure] 𝔸, F ⊣ G in Cat(Q) if and only if, for each Q-object X, F_{X} ⊣ G_{X} in Cat(2). Here one need not ask 𝔸 to be tensored nor F to preserve tensors (although it does a posteriori for it is a left adjoint). But the point is that for this “weaker” proposition one assumes the existence of some functor G and one proves that it is the right adjoint to F, whereas in 3.1 one proves the existence of the right adjoint to F.

Were we to prove 3.1 under the hypothesis that 𝔸, 𝔹 are cocomplete Q-categories, we simply could have applied 2.15: for such categories, F: 𝔸 [figure] 𝔹 is left adjoint if and only if it is cocontinuous, if and only if preserves tensors and each 𝔸_{X} [figure] 𝔹_{X}: a ↦ F a preserves suprema, if and only if it preserves tensors and each 𝔸_{X} [figure] 𝔹_{X}: a ↦ F a is left adjoint in Cat(2) (for each 𝔸_{X} is a cocomplete order). The merit of 3.1 is thus to have generalized 2.15 to the case of a tensored 𝔸 and an arbitrary 𝔹.

Adjunctions from (co)tensors, and vice versa.

Consider a Q-category ℂ and an object x ∈ ℂ_{X}; for any y, y′ ∈ ℂ the composition inequality says that ℂ(y′, y) ∘ ℂ(y, x) ≤ ℂ(y′, x), or equivalently ℂ(y′, y) ≤ {ℂ(y, x), ℂ(y′, x)}. By definition of P^{†}X, cf. 2.3, there is thus a functor^{8} ℂ(−, x): ℂ [figure] P^{†}X: y ↦ ℂ(y, x).

3.2. Proposition. For a Q-category ℂ and an object x ∈ ℂ_{X}, all cotensors with x exist if and only if the functor ℂ(−, x): ℂ [figure] P^{†}X: y ↦ ℂ(y, x) is a left adjoint in Cat(Q). In this case its right adjoint is ⟨−, x⟩: P^{†}X [figure] ℂ: f ↦ ⟨f, x⟩.

Proof. If for any f: X [figure] Y in Q the cotensor ⟨f, x⟩ exists, then ⟨−, x⟩: P^{†}X [figure] ℂ is a functor: for f: X [figure] Y, f′: X [figure] Y′, i.e. two objects of P^{†}X,

P^{†}X(f′, f) ≤ ℂ(⟨f′, x⟩, ⟨f, x⟩) ⟺ {f, f′} ≤ {f, ℂ(⟨f′, x⟩, x)}
⟸ f′ ≤ ℂ(⟨f′, x⟩, x)
⟺ 1_{Y′} ≤ ℂ(⟨f′, x⟩, ⟨f′, x⟩)

which is true. And ℂ(−, x) ⊣ ⟨−, x⟩ holds by the universal property of the cotensor itself.

Conversely, suppose that ℂ(−, x): ℂ [figure] P^{†}X is a left adjoint; let R_{x}: P^{†}X [figure] ℂ denote its right adjoint. Then in particular for all f: X [figure] Y in Q, R_{x}(f) is an object of type Y in ℂ, satisfying

for all y ∈ ℂ, ℂ(y, R_{x}(f)) = P^{†}X(ℂ(y, x), f) = {f, ℂ(y, x)},

which says precisely that R_{x}(f) is the cotensor of x with f.

8There is a “deeper” reason for this too: in principle, ℂ(−, x): ∗_{X} [figure] ℂ is a contravariant presheaf on ℂ, i.e. a distributor; but these correspond precisely to functors from ℂ to the free completion of ∗_{X}, which is P^{†}X (see section 6 of [Stubbe, 2005a] for details).

In the situation of 3.2 it follows from 3.1 that

for each Q-object Z, ℂ_{Z} [figure: ⊥; ℂ(−, x); ⟨−, x⟩]Q(X, Z)^{op} in Cat(2),  (2)

for each z ∈ ℂ_{Z}, ℂ(z, x) = ⋀{f: X [figure] Z in Q | z ≤ ⟨f, x⟩ in ℂ_{Z}}.  (3)

The dual version of the above will be useful too: it says that tensors with y ∈ ℂ_{Y} exist if and only if ℂ(y, −): ℂ [figure] PY is a right adjoint in Cat(Q), in which case its left adjoint is y ⊗ −: PY [figure] ℂ. And then moreover

for each Q-object Z, ℂ_{Z} [figure: ⊥; ℂ(y, −); y ⊗ −]Q(Z, Y) in Cat(2),  (4)

for each z ∈ ℂ_{Z}, ℂ(y, z) = ⋁{f: Z [figure] Y in Q | y ⊗ f ≤ z in ℂ_{Z}}.  (5)

Here is a useful application of the previous results. For any Q-category ℂ the Yoneda embedding Y^{†}_{ℂ}: ℂ [figure] P^{†}ℂ: c ↦ ℂ(c, −) is a cocontinuous functor; in particular it follows that for any x ∈ ℂ_{X} the functor ℂ(−, x): ℂ [figure] P^{†}X preserves tensors. (A direct proof of this latter fact is easy too: for f: Y [figure] Z in Q and z ∈ ℂ_{Z}, suppose that z ⊗ f exists in ℂ. Then ℂ(z ⊗ f, x) = [f, ℂ(z, x)] = ℂ(z, x) ⊗ f in P^{†}X, because this is how tensors are calculated in P^{†}X.)

3.3. Corollary. If ℂ is a tensored Q-category, then the following are equivalent:

(i) for all Q-objects X and Y and each x ∈ ℂ_{X}, ℂ(−, x): ℂ_{Y} [figure] Q(X, Y)^{op} is a left adjoint in Cat(2);

(ii) for each x ∈ ℂ_{X}, ℂ(−, x): ℂ [figure] P^{†}X is a left adjoint in Cat(Q);

(iii) ℂ is cotensored.

In 3.2 we have results about “(co)tensoring with a fixed object”; now we are interested in studying “tensoring with a fixed arrow”. Recall that a tensor is a colimit of which such an arrow is the weight. So we may apply general lemmas on weighted colimits [Stubbe, 2005a, 5.2 and 5.3] to obtain the following particular results; however we shall give a quick ad hoc proof too.

3.4. Proposition. Let ℂ denote a Q-category.

(i) For all y ∈ ℂ_{Y}, y ⊗ 1_{Y} ≅ y.

(ii) For g: W [figure] X and f: X [figure] Y in Q and y ∈ ℂ_{Y}, if all tensors involved exist then y ⊗ (f ∘ g) ≅ (y ⊗ f) ⊗ g.

(iii) for (f_{i}: X [figure] Y)_{i∈I} in Q and y ∈ ℂ_{Y}, if all tensors involved exist then y ⊗ (⋁_{i} f_{i}) ≅ ⋁_{i}(y ⊗ f_{i}).

(iv) For f: X [figure] Y in Q and y, y′ ∈ ℂ_{Y}, if all tensors involved exist then y ≤ y′ in ℂ_{Y} implies y ⊗ f ≤ y′ ⊗ f in ℂ_{X}.

Proof. We make calculations using liftings in Q and the universal property of tensors:

(i) for all z ∈ ℂ, ℂ(y ⊗ 1_{Y} , z) = [1_{Y} , ℂ(y, z)] = ℂ(y, z), so y ⊗ 1_{Y} ≅ y; (ii) for all z ∈ ℂ, ℂ(y ⊗ (f ∘ g), z) = [f ∘ g, ℂ(y, z)] = [g, [f, ℂ(y, z)]] = ℂ((y ⊗ f) ⊗ g, z), so y ⊗ (f ∘ g) ≅ (y ⊗ f) ⊗ g; (iii) for all z ∈ ℂ, ℂ(y ⊗ (⋁_{i} f_{i}), z) = [⋁_{i} f_{i}, ℂ(y, z)] = ⋀_{i}[f_{i}, ℂ(y, z)] = ⋀_{i} ℂ(y ⊗ f_{i}, z), so y ⊗ ⋁_{i} f_{i} ≅ ⋁_{i}(y ⊗ f_{i}); (iv) from y ≤ y′ we get ℂ(y, −) ≥ ℂ(y′, −), hence for all z ∈ ℂ, ℂ(y⊗f, z) = [f, ℂ(y, z)] ≥ [f, ℂ(y′, z)] = ℂ(y′ ⊗ f, z), so y ⊗ f ≤ y′ ⊗ f.

Of course there is a dual version about cotensors, but we do not bother spelling it out. However, there is an interesting interplay between tensors and cotensors.

3.5. Proposition. Let f: X [figure] Y be a Q-arrow and suppose that all tensors and all cotensors with f exist in some Q-category ℂ. Then

\xymatrix{ & − ⊗ f \ar[dr] \\ ℂ_{Y} & ⊥ & ℂ_{X} \\ & ⟨f, −⟩ \ar[ul]} in Cat(2).

Proof. It follows from 3.4 (and its dual) that − ⊗ f: ℂ_{Y}  [figure] ℂ_{X} and ⟨f, −⟩: ℂ_{X}  [figure] ℂ_{Y} are

order-preserving morphisms. Furthermore, for x ∈ ℂ_{X} and y ∈ ℂ_{Y},

y ⊗ f ≤ x ⟺ 1_{X} ≤ ℂ(y ⊗ f, x) = [f, ℂ(y, x)]
⟺ f ≤ ℂ(y, x)
⟺ 1_{Y} ≤ {f, ℂ(y, x)} = ℂ(y, ⟨f, x⟩)
⟺ y ≤ ⟨f, x⟩.

3.6. Example. Recall from 2.3 and 2.9 that, for any object A of Q, the Q-category PA is tensored and cotensored, and that (PA)_{X} = Q(X, A). Let f: X [figure] Y be a Q-arrow: with the explicit formulas for tensors and cotensors in this case, the general adjunction in 3.5 becomes in this particular example precisely the adjunction on the right hand side of (1) that defines extensions in Q:

(PA)_{Y} [figure: ⊥; − ⊗ f; ⟨f, −⟩] (PA)_{X} is Q(Y, A) [figure: ⊥; − ∘ f; {f, −}] Q(X, A) in Cat(2).

Similarly liftings in Q are recovered by considering P^{†}A (up to reversal of orders):

(P^{†}A)_{Y} [figure: ⊥; − ⊗ f; ⟨f, −⟩] (P^{†}A)_{X} is Q(A, Y)^{op} [figure: ⊥; [f, −]; f ∘ −] Q(A, X)^{op} in Cat(2).

We can push this further.

3.7. Proposition. A tensored Q-category ℂ is cotensored if and only if, for every f: X [figure] Y in Q, − ⊗ f: ℂ_{Y} [figure] ℂ_{X} is a left adjoint in Cat(2). In this case, its right adjoint is ⟨f, −⟩: ℂ_{X} [figure] ℂ_{Y}.

Proof. Necessity follows from 3.5. As for sufficiency, by 3.3 it suffices to show that for all Q-objects X and Y and every x ∈ ℂ_{X},

ℂ(x, −): ℂ_{Y} [figure] Q(X, Y)^{op}: y ↦ ℂ(x, y)

has a right adjoint in Cat(2). Writing, for a Q-arrow f: X [figure] Y, the right adjoint to − ⊗ f: ℂ_{Y} [figure] ℂ_{X} in Cat(2) as R_{f}: ℂ_{X} [figure] ℂ_{Y}, the obvious candidate right adjoint to y ↦ ℂ(x, y) is f ↦ R_{f}(x). First note that, if f ≤^{op} f′ in Q(X, Y) then R_{f}(x) ⊗ f′ ≤ R_{f}(x) ⊗ f ≤ x using − ⊗ f ⊣ R_{f}, which implies by − ⊗ f′ ⊣ R_{f′} that R_{f}(x) ≤ R_{f′}(x): so

R_{(−)}(x): Q(X, Y)^{op} [figure] ℂ_{Y}: f ↦ R_{f}(x)

preserves order. Further, for f ∈ Q(X, Y) and y ∈ ℂ_{Y},

ℂ(y, x) ≤^{op} f ⟺ f ≤ ℂ(y, x)
⟺ y ⊗ f ≤ x
⟺ y ≤ R_{f}(x),

so indeed ℂ(x, −) ⊣ R_{(−)}(x) in Cat(2). Now ℂ is tensored and cotensored, so by 3.5 it follows that R_{f}(x) must be ⟨f, x⟩ (since both are right adjoint to − ⊗ f).

4. Enrichment and variation

The efforts made in the previous sections pay off in this section relating categories enriched in a quantaloid Q (“enrichment”) with (contravariant) order-valued pseudofunctors on Q (“variation”). For completeness’ sake we shall first recall some points regarding the probably best-known form of “variation” in this context, namely Sup-valued homomorphisms on Q^{op} (i.e. Q-modules); at the end of this section these turn out to be equivalent to cocomplete Q-categories. But interestingly enough, also less stringent forms of variation are equivalent to certain (not-so-cocomplete) Q-categories, as our main theorem 4.12 spells out.

Action, representation and variation.

Let K denote a quantale, i.e. a one-object quantaloid. Now thinking of K as a monoid in Sup, let “unit” and “multiplication” in K (the single identity arrow and the composition in the one-object quantaloid) correspond to sup-morphisms ε: I [figure] K and γ: K ⊗K [figure] K. A right action of K on some sup-lattice M is a sup-morphism ϕ: M ⊗ K [figure] M such that the diagrams

[figure: M ⊗ K ⊗ K; 1 ⊗ γ; ϕ ⊗ 1_{K}; M ⊗ K; ϕ; M ⊗ I; 1 ⊗ ε; M ⊗ K; ϕ; M]

commute (we do not bother writing the associativity and unit isomorphisms in the symmetric monoidal closed category Sup); (M, ϕ) is then said to be a right K-module. In elementary terms we have a set-mapping M × K [figure] M: (m, f) ↦ ϕ(m, f), preserving suprema in both variables, and such that (with obvious notations)

ϕ(m, 1) = m and ϕ(m, g ∘ f) = ϕ(ϕ(m, g), f).

By closedness of Sup, to the sup-morphism ϕ: M ⊗ K [figure] M corresponds a unique supmorphism ϕ̄: K [figure] Sup(M, M). In terms of elements, this ϕ̄ sends every f ∈ K to the sup-morphism ϕ(−, f): M [figure] M; it satisfies

ϕ̄(1) = 1_{M} and ϕ̄(g ∘ f) = ϕ̄(f) ∘ ϕ̄(g).

That is to say, ϕ̄: K [figure] Sup(M, M) is a reversed representation of the quantale K by endomorphisms on the sup-lattice M: a homomorphism of quantales that reverses the multiplication (where Sup(M, M) is endowed with composition as binary operation and the identity morphism 1_{M} as unit to form a quantale). Recalling that K is a one-object quantaloid Q, such a multiplication-reversing homomorphism ϕ̄: K [figure] Sup(M, M) is really a Sup-valued quantaloid homomorphism F: Q^{op} [figure] Sup: ∗ ↦ M, f ↦ ϕ̄(f).

In the same way it can be seen that morphisms between modules correspond to Supenriched natural transformations between Sup-presheaves. Explicitly, for two right modules (M, ϕ) and (N, ψ), a module-morphism α: M [figure] N is a sup-morphism that makes

[figure: M ⊗ K; ϕ; α ⊗ 1_{K}; N ⊗ K; ψ; M; α; N]

commute. In elementary terms, such a sup-morphism α: M [figure] N: m ↦ α(m) satisfies

α(ϕ(m, f)) = ψ(α(m), f).

By adjunction – and with notations as above – this gives for any f ∈ K the commutative square

[figure: M; ϕ̄(f); α; N; ψ̄(f); M; α; N]

which expresses precisely the naturality of α viewed as (single) component of a natural transformation α: F [figure] G, where F, G: Q^{op} [figure] Sup denote the homomorphisms corresponding to M and N.

Conclusively, actions, representations and Sup-presheaves are essentially the same thing. The point now is that the latter presentation straightforwardly makes sense for any quantaloid, and not just those with only one object.

4.1. Definition. A right Q-module M is a homomorphism M: Q^{op} [figure] Sup. And a module-morphism α: M [figure] N between two right Q-modules M and N is an enriched natural transformation between these homomorphisms.

That is to say, QUANT(Q^{op}, Sup) is the quantaloid of right Q-modules. Of course, a right module on Q^{op} is called a left module on Q; and “by duality” it is clear that left Q-modules, i.e. covariant Sup-presheaves, correspond to straight representations and left actions.

4.2. Example. An object A of Q represents both the right Q-module Q(−, A): Q^{op} [figure] Sup and the left Q-module Q(A, −): Q [figure] Sup.

M. Kelly’s [1982] contains a wealth of information on this subject in the much greater generality of V-enriched categories. Examples of Q-modules can be found in pure mathematics, for instance in [Mulvey and Pelletier, 2001; Paseka, 2002] to name but a few recent references, but also in computer science [Abramsky and Vickers, 1993; Resende, 2000] and in theoretical physics [Coecke et al., 2000].

Enrichment and variation: terminology and notations.

We must introduce some notations concerning Q-categories. By Cat_{⊗}(Q) we denote the full sub-2-category of Cat(Q) whose objects are tensored categories, and Tens(Q) the sub-2-category whose objects are tensored categories and morphisms are tensor-preserving functors. Similarly we use Cat_{⟨⟩}(Q) for the full sub-2-category of Cat(Q) whose objects are cotensored categories, and moreover the obvious combination Cat_{⊗,⟨⟩}(Q). As usual we write Map(Cat_{⊗}(Q) for the category of left adjoints (“maps”) in Cat_{⊗}(Q), and similarly Map(Cat_{⊗,⟨⟩}(Q)) etc. Recall also that Cocont(Q) denotes the locally completely ordered 2-category whose objects are cocomplete Q-categories and morphisms are cocontinuous (equivalently, left adjoint) functors; and Cocont_{skel}(Q) denotes its biequivalent full subquantaloid whose objects are skeletal.

4.3. Example. Cat(2) is the locally ordered 2-category of orders and order preserving maps. Cat_{⊗}(2) has orders with bottom element as objects and all order-preserving maps as morphisms, whereas Tens(2) has the same objects but the morphisms are required to send bottom onto bottom. Cocont(2) is biequivalent to the quantaloid of sup-lattices and sup-morphisms; taking only skeletal 2-categories (i.e. antisymmetric orders) we have Cocont_{skel}(2) = Sup.

Some more notions and notations, now from the realm of “variation”: Let A and B be locally ordered 2-categories (i.e. Cat(2)-enriched categories). A pseudofunctor F: A [figure] B is an action on objects and morphisms that respects the local order and such that functoriality holds up to local isomorphism (we need not require any coherence because our 2-categories are locally ordered). For two such pseudofunctors F, F′: A [figure] B, a lax natural transformation φ: F [figure] F′ is a family of B-morphisms (φ_{X}: FX [figure] F′X)_{X∈A_{0}} satisfying, for any f: X [figure] Y in A, F′f ∘ φ_{X} ≤ φ_{Y} ∘ Ff in B(FX, F′Y). Such a transformation is pseudonatural when these inequalities are isomorphisms. Lax natural transformations are ordered componentwise. There are locally ordered 2-categories Psd_{lax}(A, B), resp. Psd(A, B), with pseudofunctors as objects and lax natural transformations, resp. pseudonatural transformations, as arrows.

Now consider a pseudofunctor F: A [figure] Cat(2); it is closed when, for every X, Y in A and x ∈ FX,

F(−)(x): A(X, Y) [figure] FY: f ↦ F(f)(x)

is a left adjoint in Cat(2). We write ClPsd_{lax}(A, Cat(2)) and ClPsd(A, Cat(2)) for the full sub-2-categories of Psd_{lax}(A, Cat(2)) and Psd(A, Cat(2)) determined by the closed pseudofunctors.

We will be interested in closed pseudofunctors on the opposite of a quantaloid Q; the closedness of a pseudofunctor F: Q^{op} [figure] Cat(2) reduces to the fact that, for each X, Y in Q and y ∈ Y,

F(−)(y): Q(X, Y) [figure] FX: f ↦ F(f)(y)  (6)

preserves arbitrary suprema (for Q(X, Y) is a sup-lattice). When we replace Cat(2) by any of its sub-2-categories like Cat_{⊗}(2), Tens(2) and so on, the closedness condition for pseudofunctors still makes sense: we will mean precisely that the order-morphisms in (6) preserve suprema (i.e. are left adjoints in Cat(2)).

4.4. Example. A right Q-module is precisely a closed pseudofunctor on Q^{op} with values in Sup = Cocont_{skel}(2), and a module-morphism is the same thing as a pseudonatural transformation (because isomorphisms in the hom-sup-lattices of Sup are necessarily identities); that is to say, QUANT(Q^{op}, Sup) = ClPsd(Q^{op}, Cocont_{skel}(2)).

The basic biequivalence.

4.5. Proposition. A tensored Q-category ℂ determines a closed pseudofunctor

F_{ℂ}: Q^{op}  [figure] Cat(2): (f: X [figure] Y ) ↦ ( − ⊗f: ℂ_{Y}  [figure] ℂ_{X}).  (7)

And a functor F: ℂ [figure] ℂ′ between tensored Q-categories determines a lax natural transformation

φ^{F}: F_{ℂ}  [figure] F_{ℂ′} with components φ_{X}^{F}: ℂ_{X}  [figure] ℂ′_{X}: x ↦ F x.  (8)

Proof. For a tensored Q-category ℂ, F_{ℂ} as in the statement of the proposition is welldefined: each ℂ_{X} is an order and each −⊗f: ℂ_{Y} [figure] ℂ_{X} preserves order (by 3.4). Moreover, this action is pseudofunctorial (again by 3.4). And from (the dual of) 3.2 we know that, for each X, Y in Q and y ∈ ℂ_{Y},

y ⊗ −: Q(X, Y) [figure] ℂ_{X}: f ↦ y ⊗ f

is a left adjoint; so F_{ℂ} is a closed pseudofunctor.

A functor F: ℂ [figure] ℂ′ is a type-preserving mapping F: ℂ_{0} [figure] ℂ′_{0}: x ↦ F x of objects such that ℂ(y, x) ≤ ℂ′(F y, F x) for all x, y ∈ ℂ_{0}. With (5), this functor-inequality may be rewritten as

for all x, y ∈ ℂ_{0}, ℂ(y, x) ≤ ℂ′(F y, F x)

⟺ for all f: X [figure] Y in Q, x ∈ ℂ_{X} and y ∈ ℂ_{Y}, y ⊗ f ≤ x implies F y ⊗ f ≤ F x

⟺ \{ \begin{matrix}for all X in Q and x, x′ ∈ ℂ_{X}, x ≤ x′ implies F x ≤ F x′, \\ for all f: X [figure] Y in Q and y ∈ ℂ_{Y}, F y ⊗ f ≤ F (y ⊗ f). \end{matrix}

(For the last equivalence, necessity follows by looking first at f = 1_{X} and then at x = y⊗f, whereas for sufficiency one has that y ⊗ f ≤ x implies F y ⊗ f ≤ F (y ⊗ f) ≤ F x.) Thus, a functor F: ℂ [figure] ℂ′ is really just a family of mappings ℂ_{X} [figure] ℂ′_{X}: x ↦ F x, one for each Q-object X, which are all order-preserving and satisfy furthermore for any f: X [figure] Y in Q and y ∈ ℂ_{Y} that F y ⊗ f ≤ F (y ⊗ f). Having defined components φ_{X}^{F} as in (8), this says that F_{ℂ′}(f) ∘ φ_{Y}^{F} ≤ φ_{X}^{F} ∘ F_{ℂ}(f), for any f: X [figure] Y in Q, i.e. φ: F_{ℂ} [figure] F_{ℂ′} is a lax natural transformation.

4.6. Theorem. For any quantaloid Q, the action

Cat_{⊗}(Q) [figure] ClPsd_{lax}(Q^{op}, Cat(2)): (F: ℂ [figure] ℂ′) ↦ (φ^{F}: F_{ℂ}  [figure] F_{ℂ′})  (9)

is an equivalence of 2-categories.

Proof. Straightforwardly the action in (9) is functorial: the lax natural transformation corresponding to an identity functor is an identity lax natural transformation; the lax natural transformation corresponding to the composition of functors is the composition of the lax natural transformations corresponding to each of the functors involved.

Now let F: Q^{op} [figure] Cat(2) be any closed pseudofunctor; then define a Q-category ℂ^{F} by:

- for each Q-object X, ℂ_{X}^{F} := FX,
- for x ∈ ℂ_{X}^{F} and y ∈ ℂ_{Y}^{F}, ℂ^{F}(y, x) = ⋁{f: X [figure] Y in Q | F(f)(y) ≤ x in ℂ_{X}^{F}}.

The supremum involved is really an expression of the closedness of the pseudofunctor: x ↦ ℂ^{F}(y, x) is the right adjoint to f ↦ F(f)(y) in Cat(2). Then ℂ^{F} is a tensored Q-category: the tensor of some f: X [figure] Y and y ∈ FY is precisely F(f)(y), by (the dual of) 3.2. It is clear that F ≅ F_{ℂ^{F}}. So far for essential surjectivity of (9).

Finally, given tensored Q-categories ℂ and ℂ′, the ordered sets Cat_{⊗}(Q)(ℂ, ℂ′) and Psd_{lax}(Q^{op}, Cat(2))(F_{ℂ}, F_{ℂ′}) are isomorphic: a functor F: ℂ [figure] ℂ′ between (tensored) Qcategories is completely determined by its action on objects, hence by the family of (orderpreserving) mappings ℂ_{X} [figure] ℂ′_{X}: x ↦ F x, hence by the components of the corresponding transformation φ^{F}: F_{ℂ} [figure] F_{ℂ′}. From the proof of 4.5 it is clear that F is a functor if and only if φ^{F} is lax natural (thanks to tensoredness of ℂ and ℂ′). Furthermore, to say that F ≤ G: ℂ [figure] ℂ′ in Cat(Q) means that, for any Q-object X and any x ∈ ℂ_{X}, F x ≤ Gx in ℂ′_{X}. For the lax natural transformations φ^{F} , φ^{G} corresponding to F, G this is really the same thing as saying that φ_{X}^{F} ≤ φ_{X}^{G} in Cat(2), in other words, φ^{F} ≤ φ^{G} as arrows between (closed) pseudofunctors.

It follows from 2.1 and 4.6 that a closed pseudofunctor F: Q^{op} [figure] Cat(2) either has all of the FX empty, or none of them. A direct proof is easy too (it is of course a transcription of 2.1 modulo the equivalence in 4.6): if y ∈ FY, then F(0_{X,Y})(y) ∈ FX, where 0_{X,Y} ∈ Q(X, Y) is the bottom element. So as soon as one of the FX is non-empty, all of them are. And the empty pseudofunctor is trivially closed.

Finetuning.

Here are some seemingly innocent specifications concerning the 2-functor in 4.6.

4.7. Lemma. Any closed pseudofunctor F: Q^{op} [figure] Cat(2) lands in Cat_{⊗}(2). And any lax natural transformation φ: F [figure] F′: Q [figure] Cat(2) between closed pseudofunctors has components in Cat_{⊗}(2) rather than Cat(2).

Proof. For any closed pseudofunctor F: Q^{op} [figure] Cat(2), for every X in Q and x ∈ FX, F(−)(x): Q(X, X) [figure] FX preserves all suprema, thus in particular the empty supremum, i.e. the bottom element 0_{X,X} ∈ Q(X, X). This implies that every non-empty FX must have a bottom element. Thus F lands in Cat_{⊗}(2) rather than Cat(2). But precisely because of this, the components φ_{X}: FX [figure] F′X of a lax natural transformation φ^{F}: F_{ℂ}X [figure] F_{ℂ′}X live in Cat_{⊗}(2) rather than Cat(2).

From this proof it follows that, for a closed pseudofunctor F: Q^{op} [figure] Cat_{⊗}(2), the bottom element in a non-empty order FX may be calculated as: 0_{X} := F(0_{X,X})(x), where x is an arbitrary element in FX. This allows for the following.

4.8. Lemma. A pseudonatural transformation φ: F [figure] F′: Q^{op} [figure] Cat_{⊗}(2) between closed pseudofunctors has components in Tens(2).

Proof. If FX is non-empty, take any x ∈ FX, then by pseudonaturality of φ,

φ_{X}(0_{X}) = φ_{X}(F(0_{X,X})(x)) ≅ F′(0_{X,X})(φ_{X}(x)) = 0′_{X}.

So each component φ_{X}: FX [figure] F′X, a priori in Cat_{⊗}(2), preserves the bottom element if there is one, thus lives in Tens(2).

4.9. Lemma. Any closed pseudofunctor F: Q^{op} [figure] Map(Cat_{⊗}(2)) lands in Map(Cat_{⊗,⟨⟩}(2)).

Proof. Taking an arbitrary x ∈ FX (presumed non-empty), F(0_{X,X})^{∗}(x) gives the top element of FX. Here F(0_{X,X})^{∗} denotes the right adjoint to F(0_{X,X}) in Cat_{⊗}(2). So each FX is an object of Cat_{⊗,⟨⟩}(2) rather than Cat_{⊗}(2).

Now we can apply all this to finetune 4.6.

4.10. Proposition. Let ℂ be a tensored Q-category.

(i) The associated pseudofunctor F_{ℂ}: Q^{op} [figure] Cat(2) factors through Cat_{⊗}(2).

(ii) ℂ is moreover cotensored if and only if F_{ℂ} factors through Map(Cat_{⊗,⟨⟩}(2)).

(iii) ℂ is cocomplete if and only if F_{ℂ} factors through Cocont(2).

(iv) ℂ is skeletal and cocomplete if and only if F_{ℂ} factors through Cocont_{skel}(2). Proof. (i) Is the content of 4.7. (ii) Is a combination of 3.7, 4.6 and 4.9. (iii) By 2.13 a tensored and cotensored ℂ is cocomplete if and only if it is order-cocomplete, i.e. each ℂ_{X} is a cocomplete order. Now apply (ii), recalling that Cocont(2) is precisely the full sub-2-category of Map(Cat_{⊗,⟨⟩}(2)) determined by the (order-)cocomplete objects. (iv) Is a variation on (iii): a Q-category ℂ is skeletal if and only if each ℂ_{X} is an antisymmetric (i.e. skeletal) order.

4.11. Proposition. Let F: ℂ [figure] ℂ′ be a functor between tensored Q-categories. (i) The lax natural transformation φ^{F}: F_{ℂ} [figure] F_{ℂ′} has components in Cat_{⊗}(2).

(ii) F is tensor-preserving if and only if φ^{F} is pseudonatural.

(iii) F is left adjoint if and only if φ^{F} is pseudonatural and its components are in Map(Cat_{⊗}(2)).

(iv) If ℂ and ℂ′ are moreover cotensored, then F is left adjoint if and only if φ^{F} is pseudonatural and its components are in Map(Cat_{⊗,⟨⟩}(2)).

(v) If ℂ and ℂ′ are cocomplete, then F is left adjoint if and only if φ^{F} is pseudonatural and its components are in Cocont(2).

(vi) If ℂ and ℂ′ are skeletal and cocomplete, then F is left adjoint if and only if φ^{F} is

pseudonatural and its components are in Cocont_{skel}(2). Proof. (i) Is the content of 4.7. (ii) To say that F: ℂ [figure] ℂ′ preserves tensors, means that for any f: X [figure] Y in Q and y ∈ ℂ_{Y}, F (y ⊗ f) ≅ F y ⊗ f in ℂ_{X}. In terms of the transformation φ^{F} this means that φ_{X}^{F} ∘F_{ℂ}(f) ≅ F_{ℂ′}∘φ_{Y}^{F} instead of merely the inequality “≥”; hence it is pseudonatural instead of merely lax natural. (iii) By 3.1 and the previous point. (iv) Is a variation on the previous point, using 4.10 (ii). (v) Follows from (iv), taking into account that all ℂ_{X} and ℂ′_{X} are cocomplete orders. (vi) Is a variation on (v).

We may now state our conclusion.

4.12. Theorem. The equivalence in 4.6 reduces to the following equivalences of locally ordered 2-categories:

(i) Cat_{⊗}(Q) ≃ ClPsd_{lax}(Q^{op}, Cat_{⊗}(2)),

(ii) Tens(Q) ≃ ClPsd(Q^{op}, Tens(2)),

(iii) Map(Cat_{⊗,⟨⟩}(Q)) ≃ ClPsd(Q^{op}, Map(Cat_{⊗,⟨⟩}(2))),

(iv) Cocont(Q) ≃ ClPsd(Q^{op}, Cocont(2)),

(v) Cocont_{skel}(Q) ≃ ClPsd(Q^{op}, Cocont_{skel}(2)).

By 4.4 and the biequivalence of Cocont_{skel}(Q) with Cocont(Q), we may end with the following.

4.13. Corollary. The quantaloid QUANT(Q^{op}, Sup) of right Q-modules and modulemorphisms is biequivalent to the locally ordered 2-category Cocont(Q) of cocomplete Qcategories and cocontinuous functors.

4.14. Example. With 2.3, 2.9 and 3.6 it is easy to check that, given some object A of Q, the closed pseudofunctor F_{ℂ}: Q^{op} [figure] Cocont_{skel}(Q) for ℂ = PA is precisely the representable right Q-module Q(−, A) of 4.2; and ℂ = P^{†}A gives the closed pseudofunctor F_{ℂ}: Q^{op} [figure] Sup sending an f: X [figure] Y in Q to [f, −]: Q(A, Y)^{op} [figure] Q(A, X)^{op} in Sup. It is more familiar to consider the dual of the latter: the closed pseudofunctor F_{ℂ}: Q [figure] Sup corresponding to the Q^{op}-category ℂ = (P^{†}A)^{op} is the representable left Q-module Q(A, −).

References

[1] Samson Abramsky and Steve Vickers (1993), Quantales, observational logic and process semantics, Math. Struct. in Comput. Sci. 3, 161–227.

[2] Jean Bénabou (1967), Introduction to bicategories, Lecture Notes in Math. 47, 1–77.

[3] Renato Betti, Aurelio Carboni, Ross H. Street and Robert F. C. Walters (1983), Variation through enrichment, J. Pure Appl. Algebra 29, 109–127.

[4] Bob Coecke, David Moore and Isar Stubbe (2000), Quantaloids describing causation and propagation of physical properties, Found. Phys. Lett. 14, 133–145.

[5] Robert Gordon and A. John Power (1997), Enrichment through variation, J. Pure Appl. Algebra 120, 167–185.

[6] Robert Gordon and A. John Power (1999), Gabriel-Ulmer duality for categories enriched in bicategories, J. Pure Appl. Algebra 137, 29–48.

[7] G. Max Kelly (1982), Basic concepts of enriched category theory. London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge. Also available as: Reprints Theory Appl. Categ. 10, 1–136 (2005).

[8] Christopher J. Mulvey and Joan W. Pelletier (2001), On the quantisation of points, J. Pure Appl. Algebra 159, 231–295.

[9] Jan Paseka, (2002), Interior tensor product of Hilbert modules, Contributions to general algebra 13, Heyn, Klagenfurt, 253–263.

[10] Pedro Resende (2000), Quantales and observational semantics, Fund. Theories Phys. 111 (“Current research in operational quantum logic”), Kluwer Academic Publishers, 263–288.

[11] Kimmo I. Rosenthal (1996), The theory of quantaloids. Pitman Research Notes in Mathematics Series. Longman, Harlow.

[12] Ross H. Street (1983), Enriched categories and cohomology, Questiones Math. 6, 265–283. Also available as: Reprints Theory Appl. Categ. 14, 1–18 (2005).

[13] Isar Stubbe (2005a), Categorical structures enriched in a quantaloid: categories, distributors and functors, Theory Appl. Categ. 14, 1–45.

[14] Isar Stubbe (2005b), Categorical structures enriched in a quantaloid: regular presheaves, regular semicategories, Cahiers Topol. Géom. Différ. Catég. 46, 99–121.

[15] Isar Stubbe (2005c), Categorical structures enriched in a quantaloid: orders and ideals over a base quantaloid, Appl. Categ. Structures 13, 235–255.

[16] Isar Stubbe (2006), Towards ‘dynamic domains’: totally continuous cocomplete Qcategories (Extended Abstract), Electronic Notes in Theoret. Comput. Sci. 155, 617– 634.

[17] Robert F. C. Walters (1981), Sheaves and Cauchy-complete categories, Cahiers Topologie Géom. Différentielle 22, 283–286.

[18] Robert F. C. Walters (1982), Sheaves on sites as Cauchy-complete categories, J. Pure Appl. Algebra 24, 95–102.

Centro de Matemática da Universidade de Coimbra, Portugal.

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