Applying enriched categories to quasi-uniform spaces.

Vincent Schmitt, Department of mathematics and computer science, University of Leicester, University road, LEICESTER LE1 7RH, Email: vs27@mcs.le.ac.uk

03/07/00

Abstract: The represention of complete metric spaces of [Law73] by enrichments is extented to quasi-uniform spaces. Moreover quasi-uniformly continuous maps are described as enriched functors. The quasi-uniform space completion is also viewed as a Cauchy-completion. Super monoidal functors are introduced to obtain these results. A 2-category of enrichments over different bases is defined. In this general context the Cauchy-completion is still a universal construction.

keywords: Enriched categories, change of base, Cauchy-completion, uniform and quasi-uniform spaces.

1 Introduction

In [Law73] the Cauchy completion for enrichments over monoidal closed categories is defined. This completion process is shown to capture the metric space completion: generalised metric spaces correspond to enrichments and the complete metric spaces correspond to Cauchy-complete enrichments. In this paper, these results are extended to quasi-uniform spaces. Representations in terms of enrichments are given for quasi-uniformly continuous maps ^{1} and quasi-uniform space completion.

The notions of enrichments and Cauchy-completion are general enough and have been applied succesfully in various areas. Let us cite [Wal81] for sheaves, [Be-Ca-St-Wa83] for fibrations, [Rut96] for ultra-metric spaces, [Amb-Ver96] for ring representation... many others exist. Nevertheless it seems that the theory should be further developed. The original example of metric spaces is relevant.

(1) The enriched functors in [Law73] are less general than uniformly continuous maps, hence enriched functors may be too particular to describe morphisms.

(2) Given some quasi-uniform space, one could imagine that it occurs as some V-enrichment for some V — and it does! — but this V should be built from the quasi-uniformity of that space. Therefore it seems unreasonable to represent the category of quasi-uniform spaces and uniformly continuous maps as the category of V-enrichments for a fixed V.

1For short we call “uniformly continuous maps” the quasi-uniformly continuous maps of [FL82]

Indeed quasi-uniform spaces (and also uniform ones) occur as particular enrichments over different bases. In our opinion the study of uniformly continuous maps and quasi-uniform space completion, may help to determine notions of morphisms between enrichments and Cauchy-completion, which are independent from base categories. Let us underline some results of this investigation. Super monoidal functors were introduced. These functors are interesting change of base categories. They serve to define a 2-category of enrichments over different bases. In this more general context the Cauchy-completion is still a universal construction. In this context also, the uniformly continuous maps occur as particular enriched functors and the quasi-uniform space completion is a Cauchy-completion.

The enriched category theory used is very simple: base categories are partial orders. “Nonpure” categorical concepts of smallness and skeletality have nice interpretations. The fact that Cauchy-completion preserves smallness is crucial. Skeletality for enrichments is the counterpart of separation for spaces. We also introduced the notion of reversibility for base categories and enrichments. There are open questions related to the axiomatisation of the base monoidal category. (For example what is a reasonable class F for theorem 4.9 ?)

We also had to look carefully at some topological aspects. According to the usual definition, a quasi-uniformity (X, U) consists in a filter U on X × X with extra properties. As suggested by the definition of enrichments, it seems natural to distinguish quasi-uniformities U from the set X of elements of the space. The idea is to define a lattice of properties U and a relation of satisfaction “(x, y) ⊧ u” between pairs (x, y) in X and properties u ∈ U. This yields to abstract-quasi-uniformities. They occured as intermediate objects to relate quasi-uniformities to enrichments. Nevertheless to make clear the connection with enrichments on monoidal category, it was necessary to define the “dense” abstract quasi-uniformities.

The paper is organized as follows. Chapter 2 is a brief recall on uniform and quasi-uniform spaces. Chapter 3 is a brief recall of well-known facts from enriched category theory. Chapter 4 presents new categorical developments. Super monoidal functors are defined. They allow the definition of a 2-category Enr of enrichments over different bases. A general notion of Cauchy-completion holds in Enr (4.8 and 4.9). Chapter 5 introduces quasi-uniform triples. They correspond to an axiomatisation of the lattice of quasi-uniformities, they may be seen as quasi-uniformities “without points”. We show how to build monoidal bi-closed categories from quasi-uniform triples. Chapter 6 introduces abstract quasi-uniformities and dense abstract quasi-uniformities. They consist in sets of points with a “general distance” valued in a quasi-uniform triple. Then abstract quasiuniformities are related to enrichments. In Chapter 7, we prove the existence of completion for abstract quasi-uniformities. We relate this completion process to the categorical notion of Cauchycompletion. Chapter 8 relates (usual) quasi-uniformities to abstract ones. We show how to retrieve the usual completion of quasi-uniformities [FL82] from the more general completion of abstract quasi-uniformities. Chapter 9 treats reversibility. Abstract uniformities are introduced to generalise uniform spaces. We show that all the previous results may be specialised to the “reversible” case.

2 Uniformities and quasi-uniformities

Recall that a quasi-uniformity U on a set X, denoted (X, U), is a set of binary relations U ⊆ ℘(X × X) that satisfy:

- ∀u ∈ U, Δ ⊆ u,

- U is filtered (non-empty, stable by finite intersection and arbitrary union),

- ∀u ∈ U, ∃v ∈ U, v^{2} ⊆ u. A quasi-uniformity (X, U) is a uniformity when ∀u ∈ U, u^{−1} ∈ U. Given quasi-uniformities (X, U) and (Y, V), a map f : X → Y is uniformly continuous when ∀v ∈ V, ∃u ∈ U, ∀x, y ∈ X, (x, y) ∈ u ⇒ (fx, fy) ∈ v. Further on, QUnif will denote the category of quasi-uniform spaces and uniformly continuous maps. Unif will denote the full subcategory of uniformities.

Let (X, U) be a uniformity. A filter F on X is Cauchy (w.r.t U) when it satisfies: ∀u ∈ U, ∃f ∈ F, f × f ⊆ u. The neighbourhood filter of some x ∈ X is the filter consisting in the subsets {y/(x, y) ∈ u}, u ranging in U. Then (X, U) is complete when any minimal Cauchy filter occurs as the neighbourhood filter of some element of X. (X, U) is separated when the map sending elements of X to their neighbourhood filter is injective. Recall,

Theorem 2.1 [Bou] Let X be a uniformity. There is a separated and complete uniformity \overline{(X, U)}, and a uniformly continuous map i : (X, U) → \overline{(X, U)} such that for any uniformly continuous map f from (X, U) to a separated and complete uniform space (Y, V), there exists one an only one uniformly continuous map f̄: \overline{(X, U)} → (Y, V) such that f = f̄∘ i.

A generalization of this theorem to quasi-uniform spaces may be found in [FL82], (chapter 3).

Theorem 2.2 For any quasi-uniform space (X, U) there is a separated and complete quasi-uniform space \overline{(X, U)} with a uniformly continuous map i_{(X,U)} : (X, U) → \overline{(X, U)} such that for any uniformly continuous map f : (X, U) → (Y, V) with (Y, V) separated and complete there is a unique uniformly continuous f̄: \overline{(X, U)} → (Y, V) such that f̄∘ i_{(X,U)} = f.

3 Enrichments over monoidal partial orders

All the results in this chapter are well-known and reproduced here for completness. We refer the reader to [Kel82] for a general presentation of enriched category theory.

We call a monoidal category that is also a partial order, a monoidal partial order. In this section, V will denote a monoidal partial order.

Definition 3.1 (V-enrichment) An enrichment A over V or a V-category is a set Obj(A) — the objects of A — with a mapping A(−, −) : Obj(A) × Obj(A) ⟶ Obj(V) satisfying:

- for any object a of A, I ≤ A(a, a);

- for any objects a, b, c of A, A(b, c) ⊗ A(a, b) ≤ A(a, c). Given V-categories A and B, a V-functor f from A to B is a map f : Obj(A) ⟶ Obj(B) such that for any objects a, a′ of A, A(a, a′) ≤ B(fa, fa′). Given two V-functors f, g : A → B there is a V-natural transformation from f to g, — which is denoted f ⇒ g, when for any object a of A, I ≤ A(fa, ga).

The usual definition of enrichment does not require Obj(A) above to be a set. In this paper, we shall consider only “small” enrichments i.e. such that their sets of objects is small. The base categories will be also always small. This makes sense since the examples of enrichments that we will look at, are very “concrete”, and the fact that Cauchy-completion (3.9) preserves smallness is essential.

Definition 3.2 A V-category A is called skeletal when for any objects a, b of A, if I ≤ A(a, b) and I ≤ A(b, a) then a = b.

Fact 3.3 V-categories, V-functors and, V-natural transformations constitute a 2-category V −Cat.

Further on, V will be supposed cocomplete and bi-closed.

Definition 3.4 (V-modules) Given two V-categories A and B, a V-module φ from A to B — denoted φ : \xymatrix{A \ar[r]|{∘} & B} — is a map φ : Obj(B) × Obj(A) → Obj(V) such that:

- for any objects a, a′ of A, and b of B, A(a, a′) ⊗ φ(b, a) ≤ φ(b, a′);

- for any objects a of A and, b, b′ of B, φ(b, a) ⊗ B(b′, b) ≤ φ(b′, a). For any two V-modules φ : \xymatrix{A \ar[r]|{∘} & B} and ψ : \xymatrix{B \ar[r]|{∘} & C} , their composite ψ • φ : \xymatrix{A \ar[r]|{∘} & C} is defined by: ∀a ∈ Obj(A), c ∈ Obj(C), (ψ • φ)(c, a) = ⋁_{b∈Obj(B)}φ(b, a) ⊗ ψ(c, b). The set V − Mod(A, B) of V-modules from A to B is partially ordered by: φ ≤ φ′ ⇔ ∀(b, a) ∈ Obj(B) × Obj(A), φ(b, a) ≤ φ′(b, a).

Fact 3.5 V-categories and V-modules with partial orders defined above, constitute a locally posetal 2-category denoted V − Mod.

In V − Mod, the identity in A is the module with underlying map A(−, −) : Obj(A) × Obj(A) → Obj(V) sending any (a, a′) to A(a, a′). For any V-categories A and B, a V-module φ : \xymatrix{A \ar[r]|{∘} & B} has right adjoint ψ : \xymatrix{B \ar[r]|{∘} & A} if and only if A(−, −) ≤ ψ • φ and φ • ψ ≤ B(−, −). For any left adjoint module φ, φ̃ will denote its (unique!) right adjoint. Any V-functor f : A → B corresponds to a pair of adjoint modules f_{⋄} ⊣ f^{⋄}, f_{⋄} : \xymatrix{A \ar[r]|{∘} & B} , f^{⋄} : \xymatrix{B \ar[r]|{∘} & A} , as follows: for any objects a of A, and b of B, f_{⋄}(b, a) = B(b, fa) and f^{⋄}(a, b) = B(fa, b). ˆ1 denotes the enrichment on V, with one object, say ∗, and ˆ1(∗, ∗) = I. If B is a V-category, for any of its object b, b : ˆ1 → B denotes the V-functor sending ∗ to b.

Further on V − AMod will denote the locally posetal 2-category with:

- objects V-categories,

- arrows: left adjoint V-modules;

- 2-cells: partial orders inherited from V − Mod.

Fact 3.6 There is a 2-functor J_{V} : V − Cat → V − AMod as follows. It is the identity on objects, and is the map (−)_{⋄} on arrows sending any V-functor f to the V-module f_{⋄}. For any V-categories A and B:

(i) (−)_{⋄} : V − Cat(A, B) → V − Mod(A, B) is injective if and only if B is skeletal; (ii) For any V-functors f, g : A → B, f ⇒ g if and only if f_{⋄} ≤ g_{⋄}.

Definition 3.7 (Cauchy-completness) A V-category B is Cauchy-complete when one the following two equivalent assertions holds:

(i) For any left adjoint V-module φ : \xymatrix{A \ar[r]|{∘} & B} , φ = f_{⋄} for some V-functor f : A → B; (ii) For any left adjoint V-module φ : \xymatrix{ˆ1 \ar[r]|{∘} & B} , φ = b_{⋄} for some object b of B.

We define the following 2-categories.

- V − SkCcCat :

- objects: skeletal and Cauchy-complete V-categories,

- arrows and 2-cells: inherited from V − Cat.

- V − SkCcAMod :

- objects: skeletal and Cauchy-complete V-categories,

- arrows and 2-cells: inherited from V − AMod.

Fact 3.8 The restriction of J_{V} on V − SkCcCat is an isomorphism onto V − SkCcAMod.

Definition 3.9 (Cauchy-completion) Let A be a V-category. The Cauchy-completion of A is the V-category Ā defined as follows. Its objects are the left-adjoint V-modules of the form φ : \xymatrix{ˆ1 \ar[r]|{∘} & A} . For convenience we will consider modules in Obj(Ā) as well as their adjoints, as maps with domains A. For any φ, ψ ∈ Obj(Ā), Ā(φ, ψ) = (φ̃ • ψ)(∗, ∗) = ⋁_{a∈Obj(A)} ψ(a) ⊗ φ̃(a). The map Obj(A) → Obj(Ā) sending any a to the V-module a_{⋄}, defines a V-functor j_{A} : A → Ā.

Fact 3.10 For any V-category A, Ā is skeletal and Cauchy-complete.

Lemma 3.11 For any V-category A, (j_{A})_{⋄} : \xymatrix{A \ar[r]|{∘} & Ā} and (j_{A})^{⋄} : \xymatrix{Ā \ar[r]|{∘} & A} are inverse modules.

According to 3.11 and 3.10

Fact 3.12 There is a 2-functor S : V − AMod → V − SkCcCat which is an equivalence of 2categories. S is defined on objects by S(A) = Ā and on arrows by S(φ) = f_{φ} where for any left adjoint V-module φ : \xymatrix{A \ar[r]|{∘} & B} , f_{φ} : Ā → B̄ is the unique V-functor f satisfying f_{⋄} • j_{A}_{⋄} = j_{B}_{⋄} • φ.

Fact 3.13 The inclusion 2-functor V − SkCcCat → V − Cat has a left 2-adjoint.

3.13 implies that skeletal and Cauchy-complete V-categories form a reflective subcategory of the underlying category of V − Cat. Indeed, the left adjoint of the inclusion sends a V-category A to Ā and the unit takes value j_{A} : A → Ā in A.

According to 3.10 and 3.11, the following is coherent

Definition 3.14 (Morita-equivalence) Two V-categories A and B are Morita-equivalent when one the following equivalent assertions is satisfied:

(i) They are isomorphic in V − Mod; (ii) Their Cauchy-completions are isomorphic in V − Cat.

4 More on enrichments

4.1 About the change of base

The change of base for enrichments (over bicategories) has recently known some new developments [KLSS99]. Nevertheless we shall only use an old formulation of a classical result [Eil-Kel66]. Restricting ourselves to the case where bases are partial orders, we get:

Fact 4.1 There is a 2-functor (−)_{@} : Mon → 2 − Cat where Mon is the 2-category with:

- objects: monoidal partial orders,

- arrows: monoidal functors,

- 2-cells: natural transformations ^{2}, and 2 − Cat is the 2-category of 2-categories, 2-functors and 2-natural transformations. It sends any category V to the 2-category V − Cat. For any functor F : V → W, F_{@} : V − Cat →: W − Cat, is as follows. For any V-category A, F_{@}A is the W-category with the same objects as A and such that for any objects a, b of A, F_{@}A(a, b) = F A(a, b). For any V-functor f : A → B, F_{@}f : F_{@}A → F_{@}B is the W-functor with the same underlying map as f. For any monoidal transformation F ≤ G, there is a natural transformation with value in A the W-functor F_{@}A → G_{@}A, with underlying map the identity of Obj(A).

Further on we consider a monoidal functor between monoidal partial orders F : V → W.

Fact 4.2 There is a lax normal ^{3} functor F_{♯} : V − Mod → W − Mod that extends F_{@}, i.e. the diagram below commutes:

2here, monoidal transformations are just natural transformations.

3A lax functor is normal when it preserves identities.

\xymatrix{V − Cat \ar[d]_{J_{V}} \ar[r]^{F_{@}} & W − Cat \ar[d]^{J_{W}} \\ V − Mod \ar[r]_{F_{♯}} & W − Mod}

For any V-module φ : \xymatrix{A \ar[r]|{∘} & B} , F_{♯}(φ) : \xymatrix{F_{@}A \ar[r]|{∘} & F_{@}B} is the map

\{ \begin{matrix}Obj(B) × Obj(A) → Obj(V), \\ ∀a ∈ Obj(A), b ∈ Obj(B), F_{♯}(φ)(b, a) = F φ(b, a). \end{matrix}

proof: Let φ : \xymatrix{A \ar[r]|{∘} & B} . For any objects a, a′ of A and b of B,

F_{@}A(a, a′) ⊗ F_{♯}φ(b, a) = F A(a, a′) ⊗ F φ(b, a)

≤ F (A(a, a′) ⊗ φ(b, a))

≤ F (φ(b, a′)).

Similarly, for any object a of A and b, b′ of B, F_{♯}φ(b, a) ⊗ F_{@}B(b′, b) ≤ F_{♯}φ(b′, a).

Now given φ : \xymatrix{A \ar[r]|{∘} & B} and ψ : \xymatrix{B \ar[r]|{∘} & C} , for any objects a of A and c of C,

F_{♯}(ψ • φ)(c, a) = F (ψ • φ)(c, a), = F (⋁_{b∈Obj(B)} φ(b, a) ⊗ ψ(c, b)),

≥ ⋁_{b∈Obj(B)} F (φ(b, a) ⊗ ψ(c, b))
≥ ⋁_{b∈Obj(B)} F φ(b, a) ⊗ F ψ(c, b)
= (F_{♯}ψ • F_{♯}φ)(c, a).

By definition F_{♯}(f_{⋄}) = (F_{@}f)_{⋄} for any V-functor f, thus for any V-category A, F_{♯}(1_{A}_{⋄}) = (F_{@}A)_{⋄} = (1_{F_{@}A})_{⋄}.

Lemma 4.3 Let f : A → B be a V − functor. Then:

(i) F_{♯}(f_{⋄}) ⊣ F_{♯}(f^{⋄}) in W − Mod; (ii) If f_{⋄} is an isomorphism in V − Mod then F_{♯}(f^{⋄}) • F_{♯}(f_{⋄}) = 1.

proof: (i): Results from the facts that for any V-functor f, F_{♯}(f_{⋄}) = (F_{@}(f))_{⋄} and also F_{♯}(f^{⋄}) = (F_{@}(f))^{⋄}. (ii): Suppose f_{⋄} • f^{⋄} = 1 and f^{⋄} • f_{⋄} = 1. Composing the second inequality by F_{♯} which is lax and normal, one gets F_{♯}(f^{⋄}) • F_{♯}(f_{⋄}) ≤ F_{♯}(f^{⋄} • f_{⋄}) = F_{♯}(1) ≤ 1. Thus F_{♯}(f^{⋄}) • F_{♯}(f_{⋄}) = 1 since F_{♯}(f_{⋄}) ⊣ F_{♯}(f^{⋄}).

This motivates the definition

Definition 4.4 A lax functor G : V → W is super when for any family of pairs of objects (V_{i}, W_{i}) ∈ Obj(V) × Obj(V), i ranging in I, if I_{V} ≤ ⋁_{i∈I} V_{i} ⊗ W_{i} then I_{W} ≤ ⋁_{i∈I} G(V_{i}) ⊗ G(W_{i}).

Note that the composition of super functors and the identity functors are super.

Lemma 4.5 If F is super then F Ā(−, −) ≤ F_{♯}(j_{A}_{⋄}) • F_{♯}(j_{A}^{⋄}).

proof: For any objects φ, ψ of Ā,

F_{♯}(j_{A}_{⋄}) • F_{♯}(j_{A}^{⋄})(φ, ψ) = ⋁_{a∈Obj(A)} F j_{A}^{⋄}(a, ψ) ⊗ F j_{A}_{⋄}(φ, a)

= ⋁_{a∈Obj(A)} F ψ(a) ⊗ F φ̃(a)

≥ ⋁_{a∈Obj(A)}(F (Ā(φ, ψ) ⊗ φ(a)) ⊗ F φ̃(a))
= F Ā(φ, ψ) ⊗ ⋁_{a∈Obj(A)}(F φ(a) ⊗ F φ̃(a))
≥ F Ā(φ, ψ) ⊗ I_{W}
≥ F Ā(φ, ψ).

According to this and 4.3,

Lemma 4.6 If F is super then F_{♯}(j_{A}_{⋄}) is an isomorphism, in this case F_{♯} preserves the Moritaequivalence, i.e. if the V-categories A and B are Morita equivalent then also are F_{@}A and F_{@}B.

One has also

Lemma 4.7 If F is super then F_{♯} : V − Mod → W − Mod preserves adjoints.

proof: Suppose that F is super. For any left adjoint φ : \xymatrix{A \ar[r]|{∘} & B} in V − Mod, and any objects a, a′ of A, I_{V} ≤ A(a, a) ≤ ⋁_{b∈Obj(B)}(φ(b, a) ⊗ φ̃(a, b)) thus I_{W} ≤ ⋁_{b∈Obj(B)} F φ(b, a) ⊗ F φ̃(a, b) and

F A(a, a′) ≤ F A(a, a′) ⊗ (⋁_{b∈Obj(B)} F φ(b, a) ⊗ F φ̃(a, b))

= ⋁_{b∈Obj(B)}((F A(a, a′) ⊗ F φ(b, a)) ⊗ F φ̃(a, b))

≤ ⋁_{b∈Obj(B)}(F φ(b, a′)) ⊗ F φ̃(a, b))
= (F_{♯}φ̃ • F_{♯}φ)(a, a′).

That F_{♯}φ • F_{♯}φ̃ ≤ 1 results from the facts that F_{♯} is lax and normal and φ • φ̃ ≤ 1: F_{♯}(φ) • F_{♯}(φ̃) ≤ F_{♯}(φ • φ̃) ≤ F_{♯}(1) = 1.

4.2 The 2-categories Enr and SC-Enr

For convenience we write the “enrichment (A, V)” for the “V-category A”. Given a monoidal functor F : V → W, enrichments (A, V) and (B, W), and a map f : Obj(A) → Obj(B), we say that F is compatible with the triple (f, A, B) when f defines a W-functor F_{@}A → B.

In this section C is the locally preordered 2-category with:

- objects: enrichments;

- arrows from (A, V) to (B, W): ordered pairs (f, F) where f is a map Obj(A) → Obj(B) and F : V → W is a monoidal functor compatible with (f, A, B);

- composition: the composite ( \xymatrix{(B, W) \ar[r]^{(g,G)} & (C, X} )) ∘ ( \xymatrix{(A, V) \ar[r]^{(f,F)} & (B, W)} ) is the pair (GF : V → X , g ∘ f);

- 2-cells : the 2-cell (f, F) ⇒ (g, G) : A → B exists when there is a (monoidal) natural transformation F ≤ G and for any object a of A, there exists the arrow I → B(fa, ga) in W.

Enr, the locally preordered 2-category of enrichments has objects enrichments, arrows those (f, F) of C with F super and, local preorders inherited from C.

It is straightforward to check that both C and Enr are well-defined 2-categories.

SC −Enr will stand for the locally preordered 2-category with objects the skeletal and Cauchycomplete enrichments and with arrows and local preorders inherited from Enr.

Proposition 4.8 The inclusion 2-functor SC − Enr ↪ Enr has a 2-left-adjoint. Precisely, for

any enrichment A, the arrow \xymatrix{A \ar[r]^{(j_{A},1)} & Ā} (that lies in Enr) is such that:

- For any arrow \xymatrix{A \ar[r]^{(f,F)} & B} in Enr with B skeletal and Cauchy-complete, there is an unique arrow (f̄, F̄) that renders commutative the diagram in C below.

\xymatrix{A \ar[r]^{(j_{A},1)} \ar[dr]_{(f,F)} & Ā \ar[d]^{(f̄,F̄)} \\  & B}

This arrow (f̄, F̄) is in Enr. It is given by F̄ = F and f̄_{⋄} = f_{⋄} • F_{♯}(j_{A}_{⋄}).

- There is a 2-cell (f, F) ≤ (g, G) : A → B if and only if there is one (f̄, F) ≤ (ḡ, G) : Ā → B.

proof: Let (f, F) : A → B be in Enr, Ā be the Cauchy-completion of A and, B be skeletal and Cauchy-complete. We are looking for some arrow (f̄, F̄) : Ā → B in Enr such that (f̄, F̄) ∘ (j_{A}, 1_{A}) = (f, F). Necessarely F̄ = F. The composition on the right by the isomorphism of W − Mod, F_{♯}(j_{A}^{⋄}) (see below),

\xymatrix{F_{@}(A) \ar[r]|{∘}^{F_{♯}(j_{A}_{⋄})} \ar[dr]_{f} & F_{@}(Ā) \ar[d]^{f̄} \\  & B}

yields an isomorphism of categories W − Mod(F_{@}Ā, B) ≅ W − Mod(F_{@}A, B). Thus by Cauchycompletness and skeletality of B, composition on the right by F_{@}(j_{A}) is an isomorphism W − Cat(F_{@}Ā, B) ≅ W − Cat(F_{@}A, B). Therefore such an f̄ exists and is unique, it satisfies f̄_{⋄} = f_{⋄} • F_{♯}(j_{A}_{⋄}). Given another functor (g, G) : A → B. We have to show that there is a natural transformation (f, F) ⇒ (g, G) if and only if there is one (f̄, F) ⇒ (ḡ, G) where ḡ is such that ḡ∘G_{@}(j_{A}) = g. For any V-category B, let θ_{B} : F_{@}B → G_{@}B denote the V-functor with underlying map the identity of Obj(B). Then the situation in W − Mod is depicted below

B F_{@}A G_{@}Ā G_{@}A  F_{@}Ā 1  B θĀ θ_{A}  F_{♯}(j_{A}_{⋄}) F_{♯}(j_{A}^{⋄}) G_{♯}(j_{A}_{⋄}) G_{♯}(j_{A}^{⋄})  f_{⋄} g_{⋄}  ḡ_{⋄}  f̄_{⋄} ≅  ≅ (1)  (2)

It is now straightforward that 2-cell (1) exists if and only 2-cell (2) exists. This is equivalent to the desired result since B is skeletal and Cauchy-complete.

4.3 Enr^{∘} and derivatives

The category Enr^{∘} is as follows. It has the same objects as Enr. A morphism (A, V) → (B, W), is a map f : Obj(A) → Obj(B) such that there is a super functor F : V → W compatible with (f, A, B). There is an obvious forgetfull 2-functor Enr → Enr^{∘} where Enr^{∘} is seen as a locally discrete 2-category. We define also the category SC − Enr^{∘} as the full subcategory of Enr^{∘} with objects skeletal and Cauchy-complete enrichments. A question is to know whether the inclusion functor SC − Enr^{∘} ↪ Enr^{∘} admits some left adjoint. Indeed we could not answer it but the following related result appears relevant for our case of study.

Futher on F denotes a locally partially ordered 2-category with objects some monoidal partial orders, with arrows some super functors and, with 2-cells inherited from Mon. We will note Enr(F) the locally preordered 2-category with objects enrichments over objects of F, with arrows the (f, F) of Enr where F is an arrow of F, and with local preorders inherited from Enr. SC − Enr(F) will be the 2-category with objects skeletal and Cauchy-complete enrichments of Enr(F) and, with arrows and 2-cells inherited from Enr(F). Finally we extend the notation (−)^{∘} to define obviously the categories Enr(F)^{∘} and SC − Enr(F)^{∘}. Then,

Theorem 4.9 If F is locally filtered then the inclusion functor SC − Enr(F)^{∘} ↪ Enr(F)^{∘} has a left adjoint.

To see this, we will check that given any enrichment (A, V) with V in F, the morphism (j_{A}, 1) : (A, V) → (Ā, V) in Enr(F), is such that for any morphism (f, F) : (A, V) → (B, W) in Enr(F) with (B, W) skeletal and Cauchy-complete, there is a unique map g : Obj(Ā) → Obj(B) such that:

- g ∘ j_{A} = f;

- there exists a G in F compatible with (g, Ā, B) . We already know that such a map exists, namely f̄ as defined in 4.8 works. Indeed, it is the unique such map. Given a map g (with some G) that satisfies the two conditions above, we shall prove g = f̄.

By assumption there is an H in F with H ≤ F, H ≤ G. Note that this H is compatible with (f̄, Ā, B) and (g, Ā, B)

At this point, we need to introduce notations to avoid ambiguities. Mainly we shall distinguish enriched functors from their underlying maps. j^{F} : F_{@}A → F_{@}Ā, j^{G} : G_{@}A → G_{@}Ā, and j^{H} : H_{@}A → H_{@}Ā will denote the W-functors with underlying map j_{A} : A → Ā. So we define the W-functors f^{F} : F_{@}A → B, g^{G} : G_{@}Ā → B, g^{H} : H_{@}Ā → B, f̄^{F} : F_{@}Ā → B and, f̄^{H} : H_{@}Ā → B with respective underlying maps f, g, g, f̄ and, f̄.

Since F, G and H are super, the W-modules (j^{F})_{⋄}, (j^{G})_{⋄} and (j^{H})_{⋄} are isomorphims with respective inverse (j^{F})^{⋄}, (j^{G})^{⋄} and (j^{H})^{⋄}. Now since H ≤ F and H ≤ G, the identity map 1_{A} : Obj(A) → Obj(A) defines W-functors 1_{A}^{F} : H_{@}A → F_{@}A and 1_{A}^{G} : H_{@}A → G_{@}A. As well, the identity map 1Ā : Obj(Ā) → Obj(Ā) defines W-functors 1F ; Ā : H_{@}Ā → F_{@}Ā, 1G ; Ā : H_{@}Ā → G_{@}Ā. The situation in W − Mod is depicted below.

F_{@}Ā  H_{@}Ā F_{@}A  G_{@}Ā B H_{@}A  G_{@}A H_{@}Ā  G_{@}Ā F_{@}Ā 1  1^{(g^{G})_{⋄}} ≅  ≅  ≅ ≅ ≅ ≅  ^{(1F ; Ā)_{⋄}}  ^{(1G ; Ā)_{⋄}}^{(1_{A}^{G})_{⋄}}^{(1G ; Ā)_{⋄}}^{(1_{A}^{F})_{⋄}} (III)  (IV) (I)  (II)^{(1F ; Ā)_{⋄}}^{(j^{F})^{⋄}}^{(j^{F})_{⋄}}  ^{(j^{H})^{⋄}}^{(j^{H})_{⋄}}  ^{(j^{G})_{⋄}}^{(j^{G})^{⋄}}^{(f̄^{F})_{⋄}}

Since j^{F} ∘ 1_{A}^{F} = 1F ; Ā ∘ j^{H}, (j^{F})_{⋄} ∘ (1_{A}^{F})_{⋄} = (1F ; Ā)_{⋄} ∘ (j^{H})_{⋄} — this is the commutation of diagram (I). Multiplying on the left (j^{F})^{⋄} and on the right by (j^{H})_{⋄} the two sides of the previous equation, one gets (j^{F})^{⋄} ∘ (1F ; Ā)_{⋄} = (1_{A}^{F})_{⋄} ∘ (j^{H})^{⋄} which is the commutation of (III). Similarly (II) and (IV) commute. This shows that (f̄^{F})_{⋄} ∘ (1F ; Ā)_{⋄} = (g^{G})_{⋄} ∘ (1G ; Ā)_{⋄}, thus (f̄^{H})_{⋄} = (g^{H})_{⋄} and f̄ = g since B is skeletal and Cauchy-complete.

5 Quasi-uniform triples

5.1 Definition

Definition 5.1 (Quasi-uniform triples) A quasi-uniform triple (S, ·, ≤) consists of a semi-group (S, ·) with base S, composition operation “·”, with S partially ordered by ≤. It satisfies morever the following axioms: (A1) ∀s, t ∈ S, (s · t ≥ s) ∧ (s · t ≥ t); (A2) ∀s, t, u ∈ S, t ≤ u ⇒ (s · t ≤ s · u) ∧ (t · s ≤ u · s); (A3) ∀s ∈ S, ∃t ∈ S, t^{2} ≤ s; (A4) S is filtered, i.e. ∀s, t ∈ S, ∃u ∈ S, u ≤ s ∧ u ≤ t.

For the quasi-uniform triple (S, ·, ≤), we define the following. The operation “·” is extended to ℘(S) by V · W = {s · t | s ∈ V, t ∈ W }. Further on, for some V ⊆ S, [V] will denote the upperclosure of V with respect to ≤. I_{S} will stand for the set of upper-closed subsets of S. The binary operation ⊗ in I_{S} is given by V ⊗ W = [V · W]. We define the operation ρ by ρ(V) = S ⊗ V ⊗ S. Note that according to (A3), S ⊗ S = S. From this, one checks that ρ is indeed a lower closure operation (it is monotonous, decreasing with respect to the inclusion and idempotent). O_{S} will stand for the sets of fixpoints for ρ. It is stable for ⊗ since for any V, W ∈ O_{S},

V ⊗ W = S ⊗ V ⊗ S ⊗ S ⊗ W ⊗ S
= S ⊗ S ⊗ V ⊗ S ⊗ S ⊗ W ⊗ S ⊗ S
= S ⊗ V ⊗ W ⊗ S.

Therefore O_{S} provided with the restriction of ⊗ – still denoted ⊗, is a monoid with S as neutral element.

Observation 5.2 Given a quasi-uniform triple (S, ·, ≤), W and a family (V_{i})_{i∈I} both in I_{S}, W ⊗ (⋃_{i∈I} V_{i}) = ⋃_{i∈I}(W ⊗ V_{i}) and (⋃_{i∈I} V_{i}) ⊗ W = ⋃_{i∈I}(V_{i} ⊗ W).

Fact 5.3 Given a quasi-uniform triple (S, ·, ≤), a monoidal partial order C_{S} is defined by the following data:

- objects: elements of O_{S};

- order: inclusion;

- identity: S;

- tensor product: ⊗. C_{S} is moreover cocomplete and biclosed. For any family (V_{i})_{i∈I}, its least upper bound is ⋁_{i∈I} V_{i} = ⋃_{i∈I} V_{i}, its greatest lower bound is ⋀_{i∈I} V_{i} = ρ(⋂_{i∈I} V_{i}).

proof: Checking that C_{S} is a posetal monoidal category is straightforward. Let us see that it is cocomplete. For any family (V_{i})_{i∈I} of elements O_{S}, ⋃_{i∈I} V_{i} lies in O_{S}, since it is upper closed and, according to 5.2, S ⊗ (⋃_{i∈I} V_{i}) ⊗ S = ⋃_{i∈I} S ⊗ V_{i} ⊗ S = ⋃_{i∈I} V_{i}. Therefore ⋃_{i∈I} V_{i} is the least upper bound of the V_{i}’s in O_{S}. Since ρ is a right adjoint, it preserves greatest lower bounds. Thus C_{S} is cocomplete with greatest lower bounds as above. C_{S} with inclusion ordering has a least upper bound S and greatest lower bound ∅. C_{S} is biclosed: for any V, V′, W ∈ O_{S},

V′ ⊗ V ⊆ W if and only if
V′ ⊆ {s ∈ S | ∀t ∈ V, s · t ∈ W } if and only if
V′ ⊆ ρ({s ∈ S | ∀t ∈ V, s · t ∈ W }).

Analogously, V ⊗ V′ ⊆ W if and only if V′ ⊆ ρ({s ∈ S | ∀t ∈ V, t · s ∈ W }).

5.2 Morphisms of quasi-uniform triples

Recall that given a binary relation R ⊆ S × T. The maps

\{ \begin{matrix}∃_{R} : ℘(S) → ℘(T), \\ V ↦ \{u′ | ∃u ∈ V, R(u, u′)\} \end{matrix}

and

\{ \begin{matrix}∀_{R} : ℘(T) → ℘(S), \\ V′ ↦ \{u | ∀u′ ∈ T, R(u, u′) ⇒ u′ ∈ V′\} \end{matrix}

defines an adjoint pair ∃_{R} ⊣ ∀_{R} : ℘(S) → ℘(T), the powersets ℘(S) and ℘(T) being partially ordered with inclusion. (For any V ⊆ S and V′ ⊆ T, ∃_{R}(V) ⊆ V′ ⇔ V ⊆ ∀_{R}(V′)).

Definition 5.4 Given two quasi-uniform triples S and T, a morphism from S to T is a relation R ⊆ S × T such that:

- (i) ∃_{R}(S) = T;

- (ii) ∀s_{1}, s_{2} ∈ S, ∀t ∈ T, R(s_{2}, t) ∧ (s_{1} ≤ s_{2}) ⇒ R(s_{1}, t);

- (iii) ∀s ∈ S, ∀t_{1}, t_{2} ∈ T, R(s, t_{1}) ∧ (t_{1} ≤ t_{2}) ⇒ R(s, t_{2});

- (iv) ∀s_{1}, s_{2} ∈ S, ∀t_{1}, t_{2} ∈ T, R(s_{1}, t_{1}) ∧ R(s_{2}, t_{2}) ⇒ R(s_{1}s_{2}, t_{1}t_{2}).

For any quasi-uniform triple S the partial order ≤⊆ S × S is a morphism of quasi-uniform triples. It is straightforward to check that quasi-uniform triples and their morphisms form a locally posetal 2-category QUT, the local partial orders being the inclusion relations.

Fact 5.5 QUT is locally finitely complete. Precisely, given morphisms of quasi-uniform triples P, Q : S → T, the relation P ∩ Q defines a morphism S → T.

proof: For any t ∈ T, since P and Q satisfies (i), there are some s_{1}, s_{2} ∈ S such that P (s_{1}, t) and Q(s_{2}, t). Since S is filtered there is some s ∈ S such that s ≤ s_{1} and s ≤ s_{2}. Thus since P, Q satisfy (ii), P (s, t) and Q(s, t). That P ∩ Q satisfies axioms (ii), (iii) (iv) is straightforward.

Further on R : S → T is a morphism of quasi-uniform triples. Since R satisfies (ii) for any subsets P ⊆ S, ∃_{R}(P) is upper closed, and dually since R satifies (iii), for any Q ⊆ T, ∀_{R}(Q) is downward-closed. Also, for any subsets P, Q of S,

∃_{R}(P) ⊗ ∃_{R}(Q) = ↑ {v_{1} · v_{2} | (∃u_{1} ∈ P, R(u_{1}, v_{1})) ∧ (∃u_{2} ∈ Q, R(u_{2}, v_{2}))}
⊆ ↑ {v | ∃u_{1} ∈ P, u_{2} ∈ Q, (u_{1} · u_{2}, v) ∈ R} (according to (vi))
= ∃_{R}(P ⊗ Q).

Definition 5.6 The morphism R is said dense when for any V ∈ I_{S}, ∃_{R}(V) belongs to O_{T}.

Note that R is dense when one of the two equivalent assertions hold:

- for all s ∈ S, the upper set {t ∈ T | R(s, t)} belongs to O_{T};

- ∀s ∈ S, t ∈ T, R(s, t) ⇒ ∃t_{1}, t_{2}, t_{3} ∈ T, t ≥ t_{1}t_{2}t_{3} ∧ R(s, t_{2}).

Fact 5.7 There is larger dense morphism amongst subrelations of R, namely R̄ ⊆ R. It is defined for all s ∈ S ans t ∈ T by: R̄(s, t) ⇔ ∃t_{1}, t_{2}, t_{3} ∈ T, t ≥ t_{1}t_{2}t_{3} ∧ R(s, t_{2}).

proof: To check that R̄ satifies axioms (i), (ii) and (iii), is straightforward. Let us see that it satisfies (iv). Suppose s_{1}, s_{2} ∈ S, t_{1}, t_{2} ∈ T, R̄(s_{1}, t_{1}) and R̄(s_{2}, t_{2}). Then there are u_{1}, u_{2}, u_{3}, v_{1}, v_{2}, v_{3} ∈ T such that t_{1} ≥ u_{1}u_{2}u_{3}, R(s_{1}, u_{2}), t_{2} ≥ v_{1}v_{2}v_{3} and R(s_{2}, v_{2}). Then t_{1}t_{2} ≥ u_{1}u_{2}u_{3}v_{1}v_{2}v_{3} ≥ u_{1}u_{2}v_{2}u_{3} and R(s_{1}s_{2}, u_{2}v_{2}).

Note also that ¯ R̄ = R̄ and the composition of dense morphism is also dense. Consider the partial order ≤_{S} ⊆ S × S, we will write ⊑_{S} for the relation ¯≤_{S}. ⊑_{S} is the relation given for all s, t ∈ S by: s⊑_{S}t ⇔ ∃t_{1}, t_{2}, t_{3} ∈ S, t ≥ t_{1}t_{2}t_{3} ∧ s ≤ t_{2}. Then R̄ is just the composite ⊑_{T} ∘R. One has also

Fact 5.8 If R is dense then R∘ ⊑_{S}= R.

proof: Since ⊑_{S} is a subrelation of ≤_{S} then (R∘ ⊑_{S}) is a subrelation of (R∘ ≤_{S}) = R. Conversely, suppose R(s, t). Since R is dense, there are t_{1}, t_{2}, t_{3} ∈ T such that t ≥ t_{1}t_{2}t_{3} and R(s, t_{2}). Now there are some s_{1}, s_{3} ∈ S such that R(s_{1}, t_{1}) and R(s_{3}, t_{3}). Thus R(s_{1}ss_{3}, t) showing (R∘ ⊑_{S})(s, t).

5.3 Dense morphims and super functors

For any morphism of quasi-uniform triples R : S → T, we define maps F′(R) : I_{S} → I_{T} and F (R) : O_{S} → O_{T} as follows. F′(R) is the restriction of ∃_{R} on I_{S} and and F (R) = ρ ∘ F′(R) ∘ i where i denotes there the embedding O_{S} → I_{S} (see below)

\xymatrix{I_{S} \ar[r]^{F′(R)} & I_{T} \ar[d]^{ρ} \\ O_{S} \ar[u]^{i} \ar[r]_{F (R)} & O_{T}}

Then R̄ is the unique dense morphism P such that F (P) = F (R).

Facts 5.9 Given morphisms of abstract quasi-uniform uples R : S → T and P : T → U, one has:

- F′(R) preserves least upper bounds.

- F (R) preserves least upper bounds.

- F′(P ∘ R) = F′(P) ∘ F′(R)

- F (P ∘ R) ⊇ F (P) ∘ F (R).

proof: Since ∃_{R} is a left adjoint and any union of upper-closed subsets is upper-closed, F′ preserves least upper bound. Since i and ρ also preserves least upper bounds, F preserves least upper bounds. It is immediate that F′(P ∘ R) = F′(P) ∘ F′(R). Thus

F (P) ∘ F (R) = ρ ∘ F′(P) ∘ i ∘ ρ ∘ F′(R) ∘ i
⊆ ρ ∘ F′(P) ∘ F′(R) ∘ i
= ρ ∘ F′(P ∘ R) ∘ i
= F (P ∘ R).

Fact 5.10 For any morphism of quasi-uniform triples R : S → T, the above map F (R) defines a normal super monoidal functor C_{S} → C_{T}

proof: Since F′(R)(S) = T, F (R)(S) = ρ ∘ F′(R)(T) = ρ(T) = T. For any P, Q ∈ O_{S},

F (R)(P ⊗ Q) = ρ ∘ F′(R) ∘ i(P ⊗ Q)

⊇ ρ ∘ (F′(R) ∘ i(P) ⊗ F′(R) ∘ i(Q))

⊇ (ρ ∘ F′(R) ∘ i)(P) ⊗ (ρ ∘ F′(R) ∘ i)(Q)
= F (R)(P) ⊗ F (R)(Q)

Let us see that F (R) is super. Let (V_{i}, W_{i})_{i∈I} be a family of pairs of objects in C_{S} such that S ⊆ ⋃_{i∈I} V_{i} ⊗ W_{i}. Given s′ ∈ T there is a s′′ ∈ T such that s′′^{2} ≤ s′. Since T = F (R)(S) = F (R)(⋃_{i∈I} V_{i} ⊗ W_{i}) = ⋃_{i∈I} F (R)(V_{i} ⊗ W_{i}), for this s′′ there is some i ∈ I, such that s′′ ∈ F (R)(V_{i} ⊗ W_{i}). Thus s′′ ∈ F (R)(V_{i}) ∩ F (R)(W_{i}) and s′ ∈ F (R)(V_{i}) ⊗ F (R)(W_{i}).

6 Abstract Quasi-uniformities

6.1 Definition

Definition 6.1 An abstract quasi-uniformities A over a quasi-uniform triple S denoted also (A, S), is a set Obj(A) together with a map: A(−, −) : Obj(A) × Obj(A) → I_{S} that satisfies:

(i) ∀x, y, z ∈ Obj(A), A(x, y) ⊗ A(y, z) ⊆ A(x, z), (ii) ∀x ∈ Obj(A), S ⊆ A(x, x).

Let S, T, U be quasi-uniform triples, and A, B and, C be abstract quasi-uniformities respectively over S, T and U.

Definition 6.2 Let f be a map Obj(A) → Obj(B). A relation R ⊆ S × T is said compatible with the triple (f, A, B) — or simply “with f”, when there is no ambiguity — when (∗) for all s, s′ ∈ S, for all x, y ∈ Obj(A), s ∈ A(x, y) ∧ R(s, s′) ⇒ s′ ∈ B(fx, fy).

The set of compatible relations with some f is stable under arbitrary unions and intersections. The largest relation compatible with (f, A, B) is denoted R_{f,A,B} (or simply R_{f}), it is defined for all s ∈ S and t ∈ T by R_{f}(s, t) if and only if ∀x, y ∈ Obj(A), s ∈ A(x, y) ⇒ t ∈ B(fx, fy) R_{f} satisfies the conditions (ii) and (iii) of 5.4. If the relation R is compatible with (f, A, B) and Q is compatible with (g, B, C) then R ∘ Q is compatible (g ∘ f, A, C).

A morphism of quasi-uniformities (A, S) → (B, T) consists in a pair (f, R) where f is a map Obj(A) → Obj(B) and R is a morphism S → T of quasi-uniform triples that is compatible with (f, A, B). Note that if R : S → T is a morphism then it is compatible with (f, A, B) if and only if for all objects x, y of A, F′(R)A(x, y) ⊆ B(fx, fy).

Abstract quasi-uniformities and their morphisms form a locally preordered 2-category AQUnif. The preorder on the homsets AQUnif((A, S), (B, T)) is given by (f, R) ≤ (g, P) if and only if R ⊆ P and for all object x of A, T ⊆ B(fx, gx). The identity morphism in (A, S) is the pair (1_{A}, ≤_{S}⊆ S × S).

The category AQUnif^{∘} is as follows. It has objects abstract quasi-uniformities and arrows (A, S) → (B, T), the maps f : Obj(A) → Obj(B) for which there is a morphism R : S → T such that the pair (f, R) becomes a morphism of abstract quasi uniformities (A, S) → (B, T). There is an evident “forgetfull” 2-functor U : AQUnif → AQUnif^{∘} where AQUnif^{∘} is a considered as a locally discrete category.

Observation 6.3 Given a morphism of abstract quasi-uniformities (f, R) : (A, S) → (B, T), since R satisfies (i) of 5.4 and is compatible with f, then

∀t ∈ T, ∃s ∈ S, ∀x, y ∈ Obj(A), s ∈ A(x, y) ⇒ t ∈ B(fx, fy)

6.2 Dense abstract quasi-uniformities

Let S be some quasi-uniform triple. An enrichment of the form (A, C_{S}) corresponds exactly to an abstract quasi-uniformity (A, S) where the map A(−, −) takes its values in O_{S}. Such an abstract quasi-uniformity will be called dense. A morphism between dense abstract quasi-uniformities (f, R) : (A, S) → (B, T) is said dense when R is dense. DAQUnif will denote the locally preordered 2-category with:

- objects: dense abstract quasi-uniformities,

- arrows: dense morphisms,

- local partial orders inherited from AQUnif. Note that in DAQUnif, the unit at (A, S) is the pair (1_{A}, ⊑_{S}) where 1_{A} denotes the identity map on Obj(A).

For any abstract quasi-uniformity A over S, the dense abstract quasi-uniformities Int(A) over S, is as follows. It has the same objects as A and for any x, y ∈ Obj(A), Int(A)(x, y) = ρ(A(x, y)). For any morphism of abstract quasi-uniformities (f, R) : A → B, since for all x, y ∈ Obj(A),

F′(R̄) ∘ ρ(A(x, y)) = ρ ∘ F′(R) ∘ ρ(A(x, y))

⊆ ρ ∘ F′(R)(A(x, y))

⊆ ρ(B(fx, fy)),

we let Int(f, R) denote the dense morphism (f, R̄) : Int(A) → Int(B).

Fact 6.4 The above data defines a normal lax functor Int : AQUnif → DAQUnif.

Let (A, S) be an abstract quasi-uniformity. Since for any objects x, y of A, ρ(A(x, y)) ⊆ A(x, y), there is a morphism of abstract quasi-uniformities Int(A) → A defined by the pair (1_{A}, ≤_{S}). Now ⊑_{S} is compatible with (1_{A}, A, Int(A)). Thus the pair (1_{A}, ⊑_{S}) defines a morphism A → Int(A). This shows that

Proposition 6.5 There is an equivalence Int^{∘} : AQUnif^{∘} ≅ DAQUnif^{∘}.

where DAQUnif^{∘} has an obvious meaning.

Given abstract quasi-uniformities (A, S), we write E(A) for the enrichment over C_{S} that corresponds to Int(A). For any arrow (f, R) : (A, S) → (B, T) in AQUnif, one gets the arrow E(f, R) = (f, F (R)) : E(A) → E(B). These assignments define a lax functor E : AQUnif → Enr. It factorises as E = E_{D} ∘ Int, where E_{D} is a lax functor DAQUnif → Enr, that is such that the E_{D} : DAQUnif(A, B) → Enr(E(A), E(B)) are full embeddings. From this, one deduces also two functors E^{∘} : AQUnif^{∘} → Enr^{∘} and E_{D}^{∘} : DAQUnif^{∘} → Enr^{∘}, with E^{∘} = E_{D}^{∘} ∘ Int^{∘}. Since E_{D}^{∘} is faithfull then also is E^{∘}.

7 Completions

7.1 Cauchy filters / adjoint modules

Let (A, S) be an abstract quasi-uniformity.

Definition 7.1 (Cauchy filters) A filter F on Obj(A) is Cauchy when it satisfies

(C) ∀u ∈ U, ∃f ∈ F, ∀x, y ∈ f, u ∈ A(x, y)

A filter basis is Cauchy when the filter that it generates is Cauchy.

In the definition above we could have replaced the condition (C) by:

(C′) ∀u ∈ U, ∃f ∈ F, ∀x, y ∈ f, u ∈ A(x, y) ⊗ A(y, x).

Fact 7.2 Let (f, R) : (A, S) → (B, T) be a morphism of abstract quasi-uniformities. If F is a Cauchy filter on (A, S) then f(F), the direct image by of F by f is a Cauchy basis on (B, T).

proof: Since the direct image of any filter is a filter basis, it remains to check that f(F) is Cauchy. Let t ∈ T. According to 6.3, we may find some s ∈ S such that ∀x, y ∈ Obj(A), s ∈ A(x, y) ⇒ t ∈ B(fx, fy). Since F is Cauchy, there is p ∈ F such that ∀x, y ∈ Obj(A), x, y ∈ p ⇒ s ∈ A(x, y) Then ∀x, y ∈ Obj(A), x, y ∈ p ⇒ t ∈ B(fx, fy).

According to this and since for any abstract quasi-uniformities A, there are two morphisms (1_{A}, ≤) : A → Int(A) and (1_{A}, ⊑) : A → Int(A), Cauchy filters on A and on Int(A) are the same and A is complete (resp. separated) if and only if Int(A) is.

Further on we suppose that A is dense. We shall denote by A the corresponding C_{S}-category. We shall relate adjoint pairs φ ⊣ φ̃ : \xymatrix{1 \ar[r]|{∘} & A} and Cauchy-filters on A (7.5). A Cauchy filter F is said minimal when it is minimal for the inclusion relation.

Fact 7.3 To any adjoint pair of C_{S}-modules φ ⊣ φ̃ with φ : \xymatrix{ˆ1 \ar[r]|{∘} & A} corresponds a Cauchy filter Γ̄(φ) on Obj(A) with basis the family Γ(φ) of sets Γ(φ)(u) = {x ∈ Obj(A) | u ∈ φ(x) ⊗ φ̃(x)}, u ranging in S. Γ̄(φ) is a minimal Cauchy filter. Precisely any Cauchy filter compatible with Γ(φ) is finer than Γ(φ). proof: Consider an adjoint pair as above. By definition, the maps φ : Obj(A) → O_{S}, φ̃ : Obj(A) → O_{S} are such that

- ∀x, y ∈ Obj(A), φ(x) ⊗ A(y, x) ⊆ φ(y);
- ∀x, y ∈ Obj(A), A(x, y) ⊗ φ̃(x) ⊆ φ̃(y).

and

(1) S ⊆ ⋃_{x∈Obj(A)}φ(x) ⊗ φ̃(x);
(2) ∀x, y ∈ Obj(A), φ̃(y) ⊗ φ(x) ⊆ A(x, y).

According to (1), for all u, Γ(φ)(u) is non empty. Since S is filtered, for all u, v ∈ S, there is some w ∈ S with w ≤ u, w ≤ v and Γ(φ)(w) ⊆ Γ(φ)(u) ∩ Γ(φ)(v). Therefore the family Γ(φ) is a filter basis. Let us see now that this basis is Cauchy. Let u ∈ S. Consider v ∈ S such that v^{2} ≤ u and let x, y ∈ Γ(φ)(v). v^{2} ∈ φ(y)⊗φ̃(y)⊗φ(x)⊗φ̃(x) ⊆ φ̃(y)⊗φ(x). From (2) we have u ∈ A(x, y).

Suppose now that F is a Cauchy filter compatible with Γ(φ). We show that F is finer than Γ(φ). Consider u ∈ S, we have to find some f ∈ F such that f ⊆ Γ(φ)(u). Let v ∈ S, such that v^{4} ≤ u. Choose some f ∈ F such that ∀x, y ∈ f, v ∈ A(x, y). Since F is compatible with Γ(φ), Γ(φ)(v)∩f ≠ ∅. Let x̄ ∈ Γ(φ)(v)∩f. Then for all y ∈ f, u ≥ v^{4} ∈ φ(x̄)⊗A(y, x̄)⊗A(x̄, y)⊗φ̃(x̄) ⊆ φ(y) ⊗ φ̃(y).

Fact 7.4 If F ∈ ℘(℘(Obj(A))) satisfies:

∀u ∈ S, ∃f ∈ F, ∀x, y ∈ f, u ∈ A(x, y)

then the maps

\{ \begin{matrix}φ : X → O_{S}, \\ x ↦ ⋃_{f∈F} ⋀_{y∈f} A(x, y) \end{matrix} and,

\{ \begin{matrix}φ̃ : X → O_{S}, \\ x ↦ ⋃_{f∈F} ⋀_{y∈f} A(y, x) \end{matrix}

define respectively modules φ : \xymatrix{ˆ1 \ar[r]|{∘} & A} and, φ̃ : \xymatrix{A \ar[r]|{∘} & ˆ1}. If F is moreover a filter basis then φ ⊣ φ̃ and the Cauchy filter Γ̄(φ) (see 7.3) is less fine than F.

proof: φ is a module from ˆ1 to A since for all objects x, y of A,

φ(x) ⊗ A(y, x) = (⋃_{f∈F} ⋀_{z∈f} A(x, z)) ⊗ A(y, x)
= ⋃_{f∈F} ( (⋀_{z∈f} A(x, z)) ⊗ A(y, x) )
⊆ ⋃_{f∈F} ⋀_{z∈f}(A(x, z) ⊗ A(y, x))
⊆ φ(y).

Analogously φ̃ is a module \xymatrix{A \ar[r]|{∘} & 1}.

Now suppose that F is moreover a filter basis. Let us see that φ ⊣ φ̃. Let us check first S ⊆ ⋃_{x∈Obj(A)} φ(x) ⊗ φ̃(x).

⋃_{x∈Obj(A)} φ(x) ⊗ φ̃(x) = ⋃_{x∈Obj(A)}((⋃_{f∈F} ⋀_{y∈f} A(x, y)) ⊗ (⋃_{f∈F} ⋀_{y∈f} A(y, x)))
= ⋃_{x∈Obj(A)} ⋃_{f_{1},f_{2}∈F} (⋀_{y∈f_{1}} A(x, y) ⊗ ⋀_{y∈f_{2}} A(y, x))
= ⋃_{x∈Obj(A)} ⋃_{f∈F} (⋀_{y∈f} A(x, y) ⊗ ⋀_{y∈f} A(y, x)) (F is filtered)

Let u ∈ S. There is some v ∈ S such that u ≥ v^{8}. By assumption there is some non-empty f(v) ∈ F such that x, y ∈ f(v) ⇒ v ∈ A(x, y). Then for some x ∈ f(v), v ∈ A(x, y) ∩ A(y, x) for any y ∈ f(v), v^{3} ∈ ⋀_{y∈f(v)} A(x, y) ∩ ⋀_{y∈f(v)} A(y, x) and u ∈ ⋀_{y∈f(v)} A(x, y) ⊗ ⋀_{y∈f(v)} A(y, x).

Let us check that for any objects x, y of A, φ̃(x) ⊗ φ(y) ⊆ A(y, x). Let x, y ∈ Obj(A),

φ̃(x) ⊗ φ(y) = (⋃_{f∈F} ⋀_{z∈f} A(z, x)) ⊗ (⋃_{f∈F} ⋀_{z∈f} A(y, z))
= ⋃_{f_{1},f_{2}∈F} (⋀_{z∈f_{1}} A(z, x) ⊗ ⋀_{z∈f_{2}} A(y, z))
= ⋃_{f∈F} (⋀_{z∈f} A(z, x) ⊗ ⋀_{z∈f} A(y, z)).

Let f ∈ F. Since f is non empty then for some z_{0} ∈ f, ⋀_{z∈f} A(z, x) ⊆ A(z_{0}, x), ⋀_{z∈f} A(y, z) ⊆ A(y, z_{0}) and, ⋀_{z∈f} A(z, x) ⊗ ⋀_{z∈f} A(y, z) ⊆ A(z_{0}, x) ⊗ A(y, z_{0}). Thus φ̃(x) ⊗ φ(y) ⊆ A(y, x).

Now we prove that F is finer than Γ(φ). Let u ∈ S. There is some v ∈ S such that v^{2} ≤ u. Since Γ(φ)(v) ≠ ∅, we choose some x̄ ∈ Γ(φ)(v) i.e. v ∈ φ(x̄) ⊗ φ̃(x̄), i.e. v ≥ v_{1} · v_{2} for some v_{1} ∈ φ(x̄) and v_{2} ∈ φ̃(x̄). Thus there are some f_{1}, f_{2} ∈ F such that

v_{1} ∈ ⋀_{y∈f_{1}} A(x̄, y) ⊆ ⋂_{y∈f_{1}} A(x̄, y) and,
v_{2} ∈ ⋀_{y∈f_{2}} A(y, x̄) ⊆ ⋂_{y∈f_{2}} A(y, x̄).

Thus given y ∈ f_{1} ∩ f_{2}, v_{1} ∈ A(x̄, y) and v_{2} ∈ A(y, x̄) and u ≥ v^{2} ≥ v_{1}v_{2}v_{1}v_{2} ∈ φ(x̄) ⊗ A(y, x̄) ⊗ A(x̄, y) ⊗ φ̃(x̄) ⊆ φ(y) ⊗ φ̃(y). Thus f_{1} ∩ f_{2} ⊆ Γ(φ)(u).

Fact 7.5 Γ̄ defines a bijection between adjoint pairs of modules of the form φ ⊣ φ̃ : \xymatrix{1 \ar[r]|{∘} & A} and minimal Cauchy-filters on A.

This is a consequence of

Fact 7.6 Given an adjoint pair φ ⊣ φ̃ : \xymatrix{1 \ar[r]|{∘} & A} then it is the pair corresponding to the Cauchy filter Γ̄(φ) as in 7.4, — i.e. for any object x of A, φ(x) = ⋃_{u∈S} ⋀_{y∈Γ(φ)(u)} A(x, y)

proof: It is enough to show that for any object x of A, φ(x) ⊆ ⋃_{u∈S} ⋀_{y∈Γ(φ)(u)} A(x, y) since one should get analogously φ̃(x) ⊆ ⋃_{u∈S} ⋀_{y∈Γ(φ)(u)} A(y, x) and we are considering two adjoint pairs. Let x ∈ Obj(A) and u ∈ φ(x). u ≥ v_{1}v_{2}u′v_{3} for some v_{1}, v_{2}, v_{3} ∈ S and u′ ∈ φ(x). For any y ∈ Γ(φ)(v_{2}), by definition v_{2} ∈ φ(y) ⊗ φ̃(y) thus v_{2} ∈ φ̃(y) and v_{2}u′ ∈ φ̃(y) ⊗ φ(x) ⊆ A(x, y). Therefore u ≥ v_{1}v_{2}u′v_{3} ∈ ⋀_{y∈Γ(φ)(v_{2})} A(x, y).

If F is Cauchy-filter on A, we shall write φ_{F} for the corresponding left adjoint module \xymatrix{1 \ar[r]|{∘} & A} define in 7.4. 7.3 and 7.4 show that for a given Cauchy filter F there is one and only minimal Cauchy filter that is less fine than F. We will denote it F̄. F̄ = Γ̄(φ_{F}).

7.2 Completion and separation / Cauchy-completion and skeletality

Let (A, S) be some abstract quasi-uniformities. Given some object x of A, let B^{1}(x), B^{2}(x), B^{3}(x), B^{4}(x) denote respectively the set of subsets,

- {y ∈ Obj(A) | u ∈ A(y, x) ∩ A(x, y)}, u ranging in S;

- {y ∈ Obj(A) | u ∈ A(y, x) ⊗ A(x, y)}, u ranging in S;

- {y ∈ Obj(A) | u ∈ ρ ∘ A(y, x) ∩ ρ ∘ A(x, y)} u ranging in S;

- {y ∈ Obj(A) | u ∈ ρ ∘ A(y, x) ⊗ ρ ∘ A(x, y)}, u ranging in S. Then B_{1}(x), B_{2}(x), B_{3}(x), B_{4}(x) are Cauchy bases. They are equivalent in the sense that they generate the same filter on Obj(A). The filter that they generate is denoted V (x). An element of V (x) is called a neigbourhood of x (in (A, S)). If (A, S) is dense then B_{1}(x) = B_{3}(x) and B_{2}(x) = B_{4}(x). In this case, for any object x of A, V (x) is just Γ̄(x_{⋄}). According to 7.3 it is a minimal Cauchy filter on (A, S).

This motivates the definitions:

Definition 7.7 (A, S) is said complete when one of the following equivalent assertion holds:

- any Cauchy filter is finer than the filter of neighbourhoods for some x;

- any minimal Cauchy-filter occurs as some V (x) for some object x of A.

Fact 7.8 The following assertions are equivalent:

- (A, S) is complete;

- Int(A, S) is complete;

- E(A, S) (= E_{D} ∘ Int(A)) is Cauchy-complete.

We define (dually)

Definition 7.9 (A, S) is said separated when one the following equivalent assertion holds:

- (i) the map assigning any object x to the filter V (x) is injective;

- (ii) For all objects x, y of Obj(A), if x ≠ y then there is a u ∈ S such that u ∉ A(x, y) ∩ A(y, x).

The assertion (ii) is equivalent to the existence of a u ∈ S such that y ∉ Γ(x)(u) or x ∉ Γ(y)(u). So clearly (ii) implies (i). That (i) implies (ii) results from the following observations. Since any refinement of a Cauchy filter is Cauchy, two distinct minimal Cauchy filters could not have a common refinement. Thus given objects x and y of A, if V (x) ≠ V (y) then there are v and u such that Γ(x)(u) ∩ Γ(y)(v) = ∅.

Fact 7.10 The following assertions are equivalent:

- (A, S) is separated;

- Int(A, S) is separated;

- E(A, S) is skeletal.

7.3 A completion for abstract quasi-uniformities

SC − QAUnif is the locally preordered 2-category with:

- objects: separated and complete abstract quasi-uniformities;

- arrows and local preorders inherited from QAUnif. SC − DQAUnif is the locally preordered 2-category with:

- objects: separated and complete dense abstract quasi-uniformities;

- arrows and local preorders inherited from DQAUnif. We extend the notation “(−)^{∘}” to define the categories SC − QAUnif^{∘} and SC − DQAUnif^{∘}.

Theorem 7.11 The inclusion functor SC − QAUnif^{∘} ↪ QAUnif^{∘} has a left adjoint.

According to 6.5, this is equivalent to

Theorem 7.12 The inclusion functor SC − DQAUnif^{∘} ↪ DQAUnif^{∘} has a left adjoint.

Since E_{D}^{∘} is faithfull, this holds due to 5.5 and 4.9.

Further on we shall write \overline{(A, S)} for the completion of the abstract quasi-uniformity (A, S) (which is its image by the left adjoint in 7.11).

8 Quasi-uniformities

8.1 Quasi-uniformities/ abstract ones

An abstract uniformities (A, S) is said concrete when it satisfies: ∀s, t ∈ S, ∀x, y ∈ Obj(A), t ∘ s ∈ A(x, y) ⇔ ∃z ∈ Obj(A), s ∈ A(x, z) ∧ t ∈ A(z, y).

Given two abstract uniformities (A, S) and (B, T) with A concrete and a map f : Obj(A) → Obj(B), the relation R_{f,A,B} (see section 6.1) satisfies conditions (iv) of 5.4, as shown below. For all s, s′ ∈ S and t, t′ ∈ T, such that R_{f}(s, t) and R_{f}(s′, t′), one has:

∀x, y ∈ Obj(A), s′ ∘ s ∈ A(x, y) ⇒ ∃z ∈ Obj(A), s ∈ A(x, z) ∧ s′ ∈ A(z, y)

⇒ t ∈ B(fx, fz) ∧ t′ ∈ B(fz, fy)

⇒ t′ ∘ t ∈ B(fx, fy).

Let (X, U) be a quasi-uniformity. One builts a quasi-uniform triple U = (U, ∘, ⊆), where ∘ denotes the composition of relations, ⊆ is the inclusion ordering. For any pair (x, y) in X, we let (x, y)^{∈} = {u ∈ U | (x, y) ∈ U}. From (X, U), one can built P (X, U), the abstract quasi-uniformity over U as follows. P (X, U) has objects those of X, and for any x, y ∈ X, P (X, U)(x, y) = (x, y)^{∈}.

Let (X, U), (Y, T) be quasi-uniformities, A = P (X, U), B = P (Y, T) and let f be a map X → Y. The relation R_{f,A,B} ⊆ U × T (section 6.1) is for all u ∈ U and v ∈ T

(u, v) ∈ R_{f} ⇔ ∀x, y ∈ X, (x, y) ∈ u ⇒ (fx, fy) ∈ v.

Now f is uniformly continuous exactly when R_{f}(U) = T, that is condition (i) of 5.4. Thus if f is uniformly continuous, we let:

Definition 8.1 P (f) stands for the morphism (f, R_{f}) : P (X, U) → P (Y, T) in AUnif.

Finally, if QUnif is seen as a locally discrete 2-category denoted QUnif^{2} then P defines a normal lax functor QUnif^{2} → AQUnif. We write P^{∘} for the corresponding functor QUnif → AQUnif^{∘}. P^{∘} is faithfull.

Let (A, S) be some abstract quasi-uniformity, one may built a quasi-uniformity denoted Q(A) over Obj(A) as follows. Q(A) is the quasi-uniformity with basis B(A) ⊆ ℘(Obj(A) × Obj(A)), that is the set of sets û = {(x, y) | u ∈ A(x, y)}, u ranging in U. Observe that (A, S) is concrete exactly when for all s, t ∈ S, ŝ ∘ t̂ ⊆ ŝ ∘ t. Let us check that B(A) is a basis of a quasi-uniformity on Obj(A). For any s ∈ S, Δ ⊆ ŝ since for any s ∈ S, s ∈ A(x, x). The assignment s ↦ ŝ defines a monotonous map S → ℘(Obj(A) × Obj(A)), thus B(A) is filtered since S is. Note that for any s, t ∈ S, ŝ ∘ t̂ ⊆ ŝ ∘ t. So, for any p ∈ B(A), since p = ŝ for some s ∈ S there is some t ∈ U such that t^{2} ≤ u and (t̂)^{2} ⊆ t̂^{2} ⊆ p.

Let (f, R) : (A, S) → (B, T) be a morphism of abstract quasi-uniformities. According to 6.3, f is a uniformely continuous map Q(A) → Q(B). We let Q(f) denote this arrow in QUnif. Q defines a functor AQUnif^{∘} → QUnif.

Trivially the Q ∘ P^{∘} ≅ 1.

Let (A, S) be an abstract quasi-uniformities. One may define a morphism R_{S} : Ŝ → S by: ∀s ∈ S, ∀p ∈ Ŝ, R_{S}(p, s) ⇔ p ⊆ ŝ. R_{S} is also compatible with (1_{A}, P ∘ Q(A), A). Suppose that the abstract quasi-uniformity (A, S) is moreover concrete. Then there is a morphism of quasi-uniform triples R′_{S} : S → Ŝ defined for all s ∈ S and p ∈ P by R′_{S}(s, p) ⇔ ŝ ⊆ p. It is straightforward that R_{S} satisfies conditions (i), (ii), (iii), of 5.4. It satifies (iv) since for all s, t ∈ S, p, q ∈ Ŝ if ŝ ⊆ p and t̂⊆ q then ŝ ∘ t = ŝ ∘ t̂ ⊆ p ∘ q. This R′_{S} is compatible with (1_{A}, A, P ∘ Q(A)). This shows

Fact 8.2 There is a coreflection P^{∘} ⊣ Q : QUnif → AQUnif^{∘}. QUnif is equivalent to the full subcategory of AQUnif^{∘} of concrete abstract uniformities.

8.2 Completions of quasi-uniformities

We show now how to infer 2.2 from 7.11.

It is straightforward to check that both P and Q preserve separation and completion. That is:

- For any quasi-uniformity (X, U), (X, U) is separated (resp. complete) if and only if P (X, U) is.

- For any abstract quasi-uniformity (A, S), (A, S) is separated (resp. complete) if and only if Q(A, S) is.

Let (X, U) be a quasi-abstract uniformity. Since the components of unit of the adjunction P ⊣ Q are isomorphisms, ϵ_{P (X,U)} : P ∘ Q ∘ P (X, U) → P (X, U), the counit of the co-reflection 8.2 in P (X, U) is an isomorphism. The situation is depicted by the commuting diagram in QAUnif below.

\xymatrix{P ∘ Q ∘ P (X, U) \ar[d]_{P ∘Q(i_{P (X,U)})} \ar[r]^{ϵ_{P (X,U)}}_{≅} & P (X, U) \ar[d]^{i_{P (X,U)}} \\ P ∘ Q ∘ \overline{P (X, U)} \ar[r]_{ϵ\overline{P (X,U)}} & \overline{P (X, U)}}

Since P ∘ Q ∘ \overline{P (X, U)} is separated and complete, there is a unique morphism of quasi-uniformities r such that r ∘ i_{P (X,U)} = P ∘ Q(i_{P (X,U)}) ∘ ϵ_{P (X,U)}^{−1}. It satisfies ϵ\overline{P (X,U)} ∘ r = 1. Since ϵ\overline{P (X,U)} is a monomorphism, it is an isomorphism. From this, the fact that P is full and faithfull and, 7.11, one gets that for any uniformly continuous map f : (X, U) → (Y, V) where (Y, V) is separated and complete there is one and only one uniformly continuous map f̄: Q ∘ \overline{P (X, U)} → (Y, V) that makes the diagram below in QUnif commute

\xymatrix{(X, U) \ar[drr]_{f} \ar[r]^{≅} & Q ∘ P (X, U) \ar[r]^{Q(i_{P (X,U)})} & Q ∘ \overline{P (X, U)} \ar[d]^{f̄} \\  &  & (Y, V)}

9 Reversibility

9.1 Reversible enrichments

Definition 9.1 A monoidal partial order V is reversible when there is an automorphism rev of the category V such that:

- rev is involutive i.e. rev^{2} = 1;

- for any objects a,b of V, rev(a ⊗ b) = rev(b) ⊗ rev(a);

- rev(I) = I.

Further on, V denotes a reversible closed monoidal partial order with automorphism rev.

Definition 9.2 A V-category A is reversible when the diagram of maps, below, commutes.

\xymatrix{Obj(A) × Obj(A) \ar[dd]_{rev×rev} \ar[dr]^{A(−,−)} \\  & Obj(V) \\ Obj(A) × Obj(A) \ar[ur]_{A(−,−)}}

Observation 9.3 Let A, B be V-categories. If B is reversible then the preoder on V − Cat(A, B) is an equivalence, if B is moreover skeletal then this equivalence is the identity.

A monoidal functor F : V → W between reversible categories preserves reversibility when the diagram in Cat, below, commutes:

\xymatrix{V \ar[d]_{rev} \ar[r]^{F} & W \ar[d]^{rev} \\ V \ar[r]_{F} & W}

Note that if the monoidal functor F : V → W preserves reversibility and the enrichment (A, V) is reversible then F_{@}(A, V) is reversible. So we define REnr as the locally preordered 2-category with:

- objects: reversible enrichments;

- arrows: the (f, F) : (A, V) → (B, W) of Enr such that F preserves the reversibility.

- local pre-orders inherited from Enr.

9.2 Uniform 4-uples

A uniform 4-uple (S, ·, ≤, rev) is a quasi-uniform triple (S, ·, ≤) with a bijective map rev : S → S such that:

- rev^{2} = rev,

- ∀u, v ∈ S, rev(uv) = rev(v) ∘ rev(u),

- ∀u, v ∈ S, u ≤ v ⇒ rev(u) ≤ rev(v).

Observation 9.4 If (S, ·, ≤, rev) is a quasi-uniform 4-uple, then C_{S} is reversible with the direct image of rev as automorphism.

Given uniform 4-uples S and T, we say that a relation R ⊆ S × T preserves reversibility when ∀s ∈ S, ∀t ∈ T, R(s, t) ⇔ R(rev(s), rev(t)). The composition of reversibility-preserving relations preserves reversibility. For any uniform 4-uple S, ≤_{S} and ⊑_{S} preserve reversibility. We define the morphisms of uniform 4-uples as the morphisms of uniform triples that preserve reversibility. For such a morphism R : S → T, R̄ also preserves reversibility, and F (R) : C_{S} → C_{T} preserves reversibility.

9.3 Abstract uniformities

An abstract uniformity (A, S) is a quasi-abstract uniformity A on a uniform 4-uple S, such that the diagram of maps below commutes

\xymatrix{Obj(A) × Obj(A) \ar[dd]_{rev×rev} \ar[dr]^{A(−,−)} \\  & I_{S} \\ Obj(A) × Obj(A) \ar[ur]_{A(−,−)}}

The definitions and the various observations above allow us to define of the following 2-categories. AUnif is the local preordered 2-category of abstract uniformities. It consists in the following data:

- objects: abstract uniformities,

- arrows A → B: the (f, R) : (A, S) → (B, T) in QAUnif with R preserving reversibility,

- local pre-orders inherited from QAUnif. DAUnif is the locally preordered 2-category with:

- objects: dense abstract uniformities,

- arrows A → B: the (f, R) : A → B of QAUnif with R dense and preserving reversibility,

- local pre-orders inherited from QAUnif.

There are inclusion 2-functors AUnif ↪ QAUnif and DAUnif ↪ DQAUnif. We define the 2-functor Int_{R} and the lax functors E_{R}, E_{RD}, as the unique lax-functors making the diagrams below commute.

AQUnif  Enr AUnif  REnr DAQUnif E Int_{R}  E_{RD} E_{D} Int  E_{R} DAUnif

Again we will obviously extend the notation (−)^{∘}. We get an equivalence of categories Int_{R}^{∘} : AUnif^{∘} ≅ DAUnif^{∘}. The functors E_{R}^{∘} : AUnif^{∘} → REnr^{∘} and E_{RD}^{∘} : DAUnif^{∘} → REnr^{∘} satisfy E_{R}^{∘} = E_{RD}^{∘} ∘ Int_{R}^{∘}. Since E_{RD}^{∘} is faithfull then also is E_{R}^{∘}.

Recall that the Cauchy-completion does not generally preserve reversibility [Be-Wal82]. But according to 7.5, one sees that for any uniform 4-uple S, the Cauchy-completion of any reversible enrichment over C_{S} is also reversible. Thus from 7.11, one deduces

Theorem 9.5 The inclusion functor SC − AUnif^{∘} ↪ AUnif^{∘} has a left adjoint.

Precisely the restriction to the category AUnif^{∘} of Comp, the left adjoint to SC − AQUnif^{∘} ↪ SC − AQUnif^{∘}, defines a functor Comp_{R} : AUnif^{∘} → SC − AUnif^{∘} that is left adjoint to the inclusion SC − AUnif^{∘} ↪ AUnif^{∘}. The situation is depicted below by the commuting diagrams in Cat.

AUnif^{∘}^{SC − AQUnif^{∘}}^{AQUnif^{∘}}^{AUnif^{∘}}  ^{AQUnif^{∘}}^{AQUnif^{∘}} Comp^{R} Comp

9.4 Abstract uniformities/ uniform spaces

Any uniformity (X, U) obviously defines a uniform 4-uple (U, ⊆, ∘, (−)^{−1}) where (U, ⊆, ∘) is the quasi-uniform triple defined in 8.1 and (−)^{−1} is the map sending u ∈ U to its inverse u^{−1}. Now the abstract-quasi uniformity P (X, U) is easily seen to be an abstract uniformity, and one may define a full and faithfull P_{R} : Unif → AUnif^{∘} by considering the restriction of P on Unif. Obviously P_{R} has a right adjoint Q_{R} deduced from Q by restriction on AUnif^{∘}. Analogous arguments to those used in 8.2 prove 2.1.

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