Theory and Applications of Categories, Vol. 10, No. 10, 2002, pp. 187–219.

CHANGE OF BASE, CAUCHY COMPLETENESS AND REVERSIBILITY

ANNA LABELLA AND VINCENT SCHMITT

ABSTRACT. We investigate the effect on Cauchy complete objects of the change of base 2-functor V-Cat → W-Cat induced by a two-sided enrichment V → W. We restrict our study to the case of locally partially ordered bases. The reversibility notion introduced in [Wal82] is extended to two-sided enrichments and Cauchy completion. We show that a reversible left adjoint two-sided enrichment F : V → W between locally partially ordered reversible bicategories induces an adjunction F_{∼} ⊣ F^{∼} : V-SkCRcCat ⇀ W-SkCRcCat between sub-categories of skeletal and Cauchy-reversible complete enrichments. We give two applications: sheaves over locales and group actions.

1. Introduction

Our motivation for the study of the change of base for enrichments over bicategories originated with the two following facts. Sheaves over a locale have been characterised as Cauchy complete enrichments over bicategories [Wal81]. Any continuous map f : X → Y yields an adjunction between categories of sheaves f_{∗} ⊣ f^{∗} : Sh(Y) ⇀ Sh(X). From this we hoped to find a categorical generalisation in terms of enrichments of the later adjunction.

Our first problem was to define a good notion of base morphism F : V → W. We had in mind that such a morphism should induce a 2-adjunction between 2-categories of enrichments, say F_{@} ⊣ F^{@} : V-Cat ⇀ W-Cat. This question was largely answered in [KLSS99] with the introduction of the so-called two-sided enrichments. To explain partly these results, we should start from the definition of MonCat, the category of monoidal functors between monoidal categories [Ben63], and enrichments over them [EiKe66], [Law73]. A monoidal functor F : V → W induces a 2-functor F_{@} : V-Cat → W-Cat. MonCat is equipped with a 2-categorical structure by defining 2-cells in it as monoidal natural transformations ([EiKe66]). The process (−)_{@} of sending V to V-Cat and F to F_{@} extends to a 2-functor from MonCat to 2-Cat. Adjunctions in MonCat were characterised in [Kel74]. Moving to the case of enrichments over bicategories, several notions of ”morphism” between bicategories were proposed in the literature, but no analogous 2-functor (−)_{@} was known in this situation.

Two-sided enrichments are a slight generalisation of monoidal functors and Bénabou’s lax functors [Ben67]. With these new morphisms of bicategories one obtained a bicategory Base together with the expected pseudo-functor (−)_{@} : Base → 2-Cat. Introducing some 3-cells on Base, one eventually gets a tricategory Caten such that the 2-categories of enrichments are representable V-Cat ≅ Caten(1, V) (1 is the unit bicategory). These new morphisms of bicategories could be seen as a two-sided version of the usual enrichments over bicategories, hence their name. Adjunctions in Caten — of particular interest in this paper — are as follows: left adjoint two-sided enrichments are exactly the pseudo functors with “local right adjoints”. This notion of local adjunction is slightly more general than the one in [BePo88]. To complete the picture, let us recall the following results. The cartesian product of bicategories extends to a pseudo-functor Caten×Caten → Caten that makes Caten into a monoidal tricategory. Caten has a closed structure Caten(V×W, Z) ≅ Caten(V, Conv(W, Z)). Eventually two-sided enrichments are also just enrichments since thus Caten(V, W) ≅ Conv(V, W)-Cat.

Received by the editors 2000 October 12 and, in revised form, 2002 April 8.

Transmitted by R.J Wood. Published on 2002 April 20.

2000 Mathematics Subject Classification: 18D20,18D99.

Key words and phrases: Enriched categories, two-sided enrichments, change of base, reversibility,

Cauchy completion, sheaves.

\begin{matrix}c \\ ◯ \end{matrix} Anna Labella and Vincent Schmitt, 2002. Permission to copy for private use granted.

Our second problem was to understand the effect of change of base F_{@} for a two-sided enrichment F, on the skeletality and Cauchy completeness of objects. This is the object of this paper. The idea is that the adjunction V-Cat → W-Cat given by an adjoint 2sided enrichment F : V → W should also “behave well” with respect to the skeletality and Cauchy completeness of objects. One may hope for some (categorical) connection between the sub-2-categories of skeletal and Cauchy complete objects respectively of V-Cat and W-Cat.

Our investigation may be summarised as follows. We restrict our study to the case when the base bicategories are locally partially ordered. This is both for convenience and because our purpose was to give a satisfying categorical generalisation of the adjunction between categories of sheaves. First we look for conditions on a two-sided enrichment F that ensure that it preserves skeletality, Cauchy completion, Morita equivalence and so on... We introduce the “super” two-sided enrichments. They generalise the super monoidal functors used by one of the authors to code uniformly continuous maps from enriched functors [Sc01]. They are more general than left adjoint two-sided enrichments and enjoy nice properties. It was known from [KLSS99] that a two-sided enrichment F induces a normal lax functor between bicategories of modules F_{♯} : V-Mod → W-Mod. For any super F, F_{♯} : V-Mod → W-Mod preserves adjoints and F_{@} : V-Cat → W-Cat preserves the Morita equivalence. Eventually we extend the notion of reversibility defined for enrichments in [Wal82] to two-sided enrichments and to the Cauchy completion. Most familiar examples of enrichments over locally partially ordered bicategories (sheaves and group actions) enjoy a nice property of symmetry, called reversibility. As pointed out by Kasangian [BeWal82] a Cauchy completion of a reversible enrichment fails in general to be reversible. This fact forces us to define a completion for reversible enrichments that preserves reversibility. This is the so-called Cauchy-reversible completion. Thus we get the desired generalisation: an adjoint pair in Caten F ⊣ G : V ⇀ W where F is reversible induces some adjunction between sub-categories of skeletal and Cauchyreversible complete objects V-SkCRcCat ⇀ W-SkCRcCat. An instance of this theorem is also the classical result in group theory that a group morphism f : H → G induces an adjunction between categories of group actions f_{∗} ⊣ f^{∗} : H-Set ⇀ G-Set.

This paper is organised as follows. Section 2 recalls basics of the theory of enrichments over locally partially ordered bicategories. This makes the paper self-contained and allows later comparisons with the reversible case. (Anyway, we refer the reader to [Kel82] for the fundamentals of enriched category theory). Section 3 treats the change of base for enrichments and the connection with two-sided enrichments in the case when the base categories are locally partially ordered. We begin by recalling some elements of the theory of two-sided enrichments developed in [KLSS99]: the definition of the tricategory Caten, the connection with enrichments, the characterisation of adjoints in Caten, and the change of base. Then we present the various new results of our investigation into the effect of the change of base bicategories on skeletality, and Cauchy completeness of enrichments. The super two-sided enrichments are introduced. Section 4 is devoted to the development of the theory in the reversible case. The notion of reversibility is defined for bicategories, (two-sided) enrichments, enriched modules, completion, etc. We retrace all the steps of the theory. Two detailed examples are given, namely sheaves on locales and group actions.

2. Basic enriched category theory

2.1. Definition. A bicategory V is locally partially ordered (respectively locally preordered), when for any objects a and b, the homset V(a, b) is a partial order (respectively a preorder).

Further on in this section, V will denote a locally partially ordered bicategory that is moreover:

- biclosed: i.e. for any arrow f : u → v and any object w of V, the functors f ∘ − : V(w, u) → V(w, v) and − ∘ f : V(v, w) → V(u, w) are adjoints.

- locally cocomplete: i.e. each homset V(u, v) is cocomplete. For any object v, I_{v} stands for the identity in v.

2.2. Definition. [V-categories] An enrichment A over V, also called a V-category, is a set Obj(A) — the objects of A, with two mappings (−)_{+}^{A} : Obj(A) ⟶ Obj(V) (sometimes just written (−)_{+}) and A(−, −) : Obj(A) × Obj(A) ⟶ Arrows(V) that satisfy:

- (enr − i) for any objects a,b of A, A(a, b) : a_{+} ⟶ b_{+};

- (enr − ii) for any object a of A, I_{a_{+}} ≤ A(a, a);

- (enr − iii) 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:

- (fun − i) (−)_{+}^{A} = (−)_{+}^{B} ∘ f;

- (fun − ii) for any objects a, b of A, A(a, b) ≤ B(fa, fb). Given V-functors f, g : A → B there is a unique V-natural transformation from f to g, when for any object a of A, I_{a_{+}} ≤ B(fa, ga).

2.3. Proposition. V-categories, V-functors and V-natural transformations form a locally preordered 2-category denoted V-Cat.

For any object u in V, û will denote the enrichment on V, with one object say ∗, with ∗_{+} = u and û(∗, ∗) = I_{u}. If B is a V-category, for any object b in B, b : b̂_{+} → B will denote the V-functor sending ∗ to b.

2.4. Definition. [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) → Arrows(V) such that:

- (mod − i) for all objects a of A, and b of B, φ(b, a) is an arrow b_{+} → a_{+} in V.

- (mod − ii) for all objects a, a′ of A, and b of B, A(a, a′) ∘ φ(b, a) ≤ φ(b, a′);

- (mod − iii) for all objects a ∈ Obj(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 ψ •φ(c, a) = ⋁_{b∈Obj(B)}φ(b, a)∘ψ(c, b). The set V-Mod(A, B) of Vmodules from A to B is partially ordered by φ ≤ ψ ⇔ ∀(b, a) ∈ Obj(B)×Obj(A), φ(b, a) ≤ ψ(b, a).

2.5. Proposition. V-categories and V-modules with partial orders defined above, constitute a locally partially ordered bicategory denoted V-Mod.

In V-Mod, the identity in A is the module with underlying map A(−, −) : Obj(A) × Obj(A) → Arrows(V) sending (y, x) to A(y, x). For any V-categories A and B, a Vmodule 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 all objects a of A, and b of B, f_{⋄}(b, a) = B(b, fa), f^{⋄}(a, b) = B(fa, b).

Further on V-AMod will denote the locally partially ordered bicategory with:

- objects V-categories,

- arrows: left adjoint V-modules;

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

There is a 2-functor J_{V} : V-Cat → V-Mod as follows. It is the identity on objects, and is the map (−)_{⋄} on arrows sending any V-functor f to the V-module f_{⋄}. Note that for any V-categories A and B, for any V-functors f, g : A → B, f ⇒ g if and only if f_{⋄} ≤ g_{⋄}.

2.6. Definition. [Skeletality, Cauchy completeness] Let B be a V-category. B is called skeletal if and only if for any V-category A, the map (−)_{⋄} : V-Cat(A, B) → V-AMod(A, B) is injective. B is called Cauchy complete if and only if for any V-category A, the map (−)_{⋄} : V-Cat(A, B) → V-AMod(A, B) is surjective.

2.7. Observation. For any V-category B, the following assertions are equivalent:

- B is skeletal;

- For any object u of V, the map (−)_{⋄} : V-Cat(û, B) → V-AMod(û, B) is injective;

- For any objects a and b of B, if a_{+} = b_{+}, I_{a_{+}} ≤ B(a, b) and, I_{a_{+}} ≤ B(b, a) then a = b.

2.8. Proposition. A V-category B is Cauchy complete when for any object u of V, the map (−)_{⋄} : V-Cat(û, B) → V-AMod(û, B) is surjective (i.e. for any object u of V and any left adjoint V-module φ : \xymatrix{û \ar[r]|{∘} & B} , φ = b_{⋄} for some object b of B with b_{+} = u).

Proof. Suppose that φ : \xymatrix{A \ar[r]|{∘} & B} has right adjoint φ̃. Then for any object a of A, φ • a_{⋄} : \xymatrix{â_{+} \ar[r]|{∘} & B} and a^{⋄} • φ̃ : \xymatrix{B \ar[r]|{∘} & â_{+}} are adjoints. For any objects a of A and b of B,

φ • a_{⋄}(b, ∗) = ⋁_{a′∈Obj(A)} A(a′, a) ∘ φ(b, a′) = φ(b, a)

and

a^{⋄} • φ̃(∗, b) = ⋁_{a′∈Obj(A)} φ̃(a′, b) ∘ A(a, a′) = φ̃(a, b).

By assumption, for any object a of A, there is a V-functor f_{a} : â_{+} → B such that (f_{a})_{⋄} = φ • a_{⋄} and (f_{a})^{⋄} = a^{⋄} • φ̃. This means that for any objects a of A and b of B, φ(b, a) = B(b, f_{a}(∗)) and φ̃(a, b) = B(f_{a}(∗), b). Let f denote the map from Obj(A) to Obj(B) sending a to f_{a}(∗). f defines a V-functor from A to B since for any objects a,a′ of A,

B(fa, fa′) = ⋁_{b∈Obj(B)} B(b, fa′) ∘ B(fa, b)

= ⋁_{b∈Obj(B)} φ(b, a′) ∘ φ̃(a, b)

= φ̃ • φ(a, a′)
≥ A(a, a′).

We define the following locally partially ordered bicategories.

- V-SkCcCat, with:

- objects: skeletal and Cauchy complete V-categories,

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

- V-SkCcAMod, with:

- objects: skeletal and Cauchy complete V-categories,

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

2.9. Proposition. The restriction of J_{V} on V-SkCcCat is an isomorphism onto V-SkCcAMod.

2.10. Definition. [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{û \ar[r]|{∘} & A} , u ranging in Obj(V). For any object φ of Ā with φ : \xymatrix{û \ar[r]|{∘} & A} , φ_{+} = u. For convenience we will consider modules in Obj(Ā) as well as their adjoints, as maps with domains A. For any objects φ,ψ of Ā, Ā(φ, ψ) = (φ̃ • ψ)(∗, ∗).

Let us check that the Ā of 2.10 is a well defined V-category. Ā trivially satisfies (enr − i). Let us show that it satisfies (enr − ii). Let φ : \xymatrix{û \ar[r]|{∘} & A} be an object of Ā, then

Ā(φ, φ) = ⋁_{a∈Obj(A)} φ(a) ∘ φ̃(a)

= φ̃ • φ(∗, ∗)

≥ I_{u}.

Let us show that Ā satisfies (enr − iii). Let φ, ψ, γ be objects of Ā. Then

Ā(γ, ψ) ∘ Ā(φ, γ) = (⋁_{a∈Obj(A)} ψ(a) ∘ γ̃(a)) ∘ (⋁_{a′∈Obj(A)} γ(a′) ∘ φ̃(a′))

= ⋁_{a,a′∈Obj(A)} ψ(a) ∘ γ̃(a) ∘ γ(a′) ∘ φ̃(a′)

≤ ⋁_{a,a′∈Obj(A)} ψ(a) ∘ A(a′, a) ∘ φ̃(a′)
= ⋁_{a∈Obj(A)} ψ(a) ∘ φ̃(a)
= Ā(φ, ψ).

Let A be a V-category with Cauchy completion Ā. For any objects a of A and φ of Ā, Ā(a_{⋄}, φ) = φ(a) and Ā(φ, a_{⋄}) = φ̃(a). The map i_{A} : Obj(A) → Obj(Ā), sending a to a_{⋄} is a V-functor from A to Ā since for any object a of A, (a_{⋄})_{+} = a_{+}, and for any objects a,b of A, Ā(a_{⋄}, b_{⋄}) = ⋁_{c∈Obj(A)} A(c, b) ∘ A(a, c) = A(a, b).

2.11. Remark. For any V-category A,

- A is skeletal if and only if the map i_{A} : Obj(A) → Obj(Ā) is injective;

- A is Cauchy complete if and only the map i_{A} is onto.

2.12. Proposition. For any V-category A, Ā is skeletal.

Proof. Suppose that φ, ψ are some objects of Ā such that φ_{+} = ψ_{+} = u and, (i) I_{u} ≤ Ā(φ, ψ) and, (ii) I_{u} ≤ Ā(ψ, φ). The inequation (i) is equivalent to (1û)_{⋄} ≤ φ̃ • ψ which is equivalent to φ ≤ ψ since φ ⊣ φ̃. Similarly, (ii) is equivalent to ψ ≤ φ. Therefore φ = ψ.

2.13. Lemma. Let A be a V-category and, Γ : \xymatrix{û \ar[r]|{∘} & Ā} be some V-module where u is an object of V. For any object φ of Ā, φ̃ denoting its adjoint, Γ(φ) = ⋁_{a∈Obj(A)}Γ(a_{⋄}) ∘ φ̃(a).

Proof. Let a be some object of A,

Γ(a_{⋄}) = ⋁_{ψ∈Obj(Ā)} Γ(ψ) ∘ Ā(a_{⋄}, ψ), = ⋁_{ψ∈Obj(Ā)} Γ(ψ) ∘ ψ(a).

Therefore, for any object φ of Ā,

⋁_{a∈Obj(A)} Γ(a_{⋄}) ∘ φ̃(a) = ⋁_{a∈Obj(A)}(⋁_{ψ∈Obj(Ā)} Γ(ψ) ∘ ψ(a)) ∘ φ̃(a)

= ⋁_{ψ∈Obj(Ā)} Γ(ψ) ∘ Ā(φ, ψ)

= Γ(φ).

2.14. Proposition. For any V-category A, Ā is Cauchy complete.

Proof. Given a left adjoint module Γ : \xymatrix{û \ar[r]|{∘} & Ā} with u an object of V. We define the maps γ, γ′ : Obj(A) → Arrows(V) respectively by γ(a) = Γ(a_{⋄}) and γ′(a) = Γ̃(a_{⋄}). We are going to show that γ is an object of Ā and Γ = Ā(−, γ). It is immediate that γ and γ′ are modules respectively from û to A and from A to û. Let us show that γ ⊣ γ′. According to 2.13 (and its dual!), ⋁_{φ∈Obj(Ā)} Γ(φ) ∘ Γ̃(φ) = ⋁_{φ∈Obj(Ā)} ⋁_{a,b∈Obj(A)} Γ(a_{⋄}) ∘ φ̃(a) ∘ φ(b) ∘ Γ̃(b_{⋄})

≤ ⋁_{a,b∈Obj(A)} Γ(a_{⋄}) ∘ A(b, a) ∘ Γ̃(b_{⋄})

= ⋁_{a,b∈Obj(A)} Γ(a_{⋄}) ∘ Ā(b_{⋄}, a_{⋄}) ∘ Γ̃(b_{⋄})
= ⋁_{a∈Obj(A)} Γ(a_{⋄}) ∘ Γ̃(a_{⋄}).

Since Γ ⊣ Γ̃,

I_{u} ≤ ⋁_{φ∈Obj(Ā)} Γ(φ) ∘ Γ̃(φ)

= ⋁_{a∈Obj(A)} Γ(a_{⋄}) ∘ Γ̃(a_{⋄})

= ⋁_{a∈Obj(A)} γ(a) ∘ γ′(a)

and for any objects a,b of A,

γ • γ′(a, b)= γ′(b) ∘ γ(a)

= Γ̃(b_{⋄}) ∘ Γ(a_{⋄})

= Γ ∘ Γ̃(a_{⋄}, b_{⋄})
≤ Ā(a_{⋄}, b_{⋄})
= A(a, b).

Now, since γ is an object of Ā, 2.13 shows that Γ = Ā(−, γ).

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

Proof. For any object φ of Ā, the maps Ā(−, φ) and Ā(φ, −) — sending respectively an object ψ of Ā to Ā(ψ, φ) and to Ā(φ, ψ) — define modules. Therefore, for any objects a, b of A,

(i_{A}^{⋄} • i_{A}_{⋄})(a, b) = ⋁_{φ∈Obj(Ā)} Ā(φ, b_{⋄}) ∘ Ā(a_{⋄}, φ)

= Ā(a_{⋄}, b_{⋄})

= A(a, b).

For any objects φ, ψ of Ā,

(i_{A}_{⋄} • i_{A}^{⋄})(φ, ψ) = ⋁_{a∈Obj(A)} Ā(a_{⋄}, ψ) ∘ Ā(φ, a_{⋄})

= ⋁_{a∈Obj(A)} ψ(a) ∘ φ̃(a)

= Ā(φ, ψ).

According to this,

2.16. Proposition. V-AMod and V-SkCcAMod are equivalent. There is a 2-functor S : V-AMod → V-SkCcCat which is an equivalence of 2-categories. S is defined on objects by S(A) = Ā and on arrow by S(φ) = f_{φ} where for any left adjoint V-module φ : \xymatrix{A \ar[r]|{∘} & B} , f_{φ} is the unique V-functor f : Ā → B̄ satisfying f_{⋄} • i_{A}_{⋄} = i_{B}_{⋄} • φ. The inclusion 2-functor V-SkCcCat → V-Cat is a right 2-adjoint.

The left 2-adjoint of the inclusion V-SkCcCat → V-Cat sends a V-category A to Ā and the unit takes value i_{A} : A → Ā in A.

According to 2.12 and 2.15, the following is coherent 2.17. Definition. [Morita equivalence] Two V-categories A and B are Morita equivalent when one of the following equivalent assertions is satisfied:

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

3. Change of base

The study of the change of base for enrichments over bicategories yielded the concept of two-sided enrichments (see [KLSS99] for the general treatment). We will specify the theory in the particular and simpler case when the base bicategories for enrichments are locally partially ordered (3.1-3.5). We keep on using the original terminology of [KLSS99]. Section 3.8 presents a study of the effect on the change of base bicategories on skeletality and Cauchy completeness of enrichments. Super two-sided enrichments are introduced.

3.1. Two-sided enrichments. Caten is the tricategory as follows. Its objects are locally partially ordered bicategories. An arrow A : V → W in Caten is a two-sided enrichment. It consists in:

- a span in Set

\xymatrix{ & Obj(A) \ar[dl]_{_{(−){A \atop −}}} \ar[dr]^{_{(−){A \atop +}}} \\ Obj(V) &  & Obj(W)}

where Obj(A) is the set of “objects” of A,

- a collection of functors A_{a,b} : V(a_{−}, b_{−}) → W(a_{+}, b_{+}), (a, b) ranging in Obj(A)^{2}, that satisfy the coherence conditions (c),(u) below.

(c) For any objects a,b,c of A, A_{a,b}(−) ∘ A_{b,c}(?) ≤ A_{a,c}(−∘?), i.e there exists a 2-cell in P o, the locally partially ordered bicategory of partial orders, monotonous maps with pointwise ordering, as follows

\xymatrix{V(a_{−}, b_{−}) × V(b_{−}, c_{−}) \ar[d]_{A_{a,b}×A_{b,c}} \ar[r]^{∘^{V}} & V(a_{−}, c_{−}) \ar[d]^{A_{a,c}} \\ W(a_{+}, b_{+}) × W(b_{+}, c_{+}) \ar@{=>}[ur] \ar[r]_{∘^{W}} & W(a_{+}, c_{+})}

(u) For any object a of A, I_{a_{+}}^{W} ≤ A_{a,a}(I_{a_{−}}^{V}) which corresponds to the 2-cell in P o

[figure: 1; I_{a_{−}}^{V}; V(a_{−}, a_{−}); A_{a,a}; 1; I_{a_{+}}^{W}; W(a_{+}, a_{+})]

For any two-sided enrichments V and W, the bicategory Caten(V, W) is locally partially ordered, as follows. A 2-cell f : A → B : V → W in Caten is called a functor, it consists in a map f : Obj(A) → Obj(B) such that:

- (−)_{−}^{A} = (−)_{−}^{B} ∘ f;

- (−)_{+}^{A} = (−)_{+}^{B} ∘ f;

\xymatrix{ & Obj(A) \ar[dd]^{f} \ar[dl]_{_{(−){A \atop −}}} \ar[dr]^{_{(−){A \atop +}}} \\ Obj(V) &  & Obj(W) \\  & Obj(B) \ar[ul]^{_{(−){B \atop −}}} \ar[ur]_{_{(−){B \atop +}}}}

- for any objects a,b of A, A_{a,b} ≤ B_{fa,fb} : V(a_{−}, b_{−}) → W(a_{+}, b_{+}), i.e. there is a 2-cell in P o

\xymatrix{V(a_{−}, b_{−}) \ar[r]_{⇓}^{A_{a,b}} \ar[r]_{B_{fa,fb}} & W(a_{+}, b_{+})}

2-cells in Caten are ordered as follows. f ≤ g : A → B : V → W when for any object a of A, I_{a_{+}}^{W} ≤ B_{fa,ga}(I_{a_{−}}^{V}) : a_{+} → a_{+}, which corresponds to the 2-cell in P o

[figure: 1; I_{a_{−}}^{V}; V(a_{−}, a_{−}); B_{fa,ga}; 1; I_{a_{+}}^{W}; W(a_{+}, a_{+})]

The (vertical) composition of the 2-cells f : A → B, g : B → C : V → W is the composite of the maps on objects f and g. The identity 1_{A} of A : V → V has underlying map the identity map on Obj(A).

Let A : U → V and B : V → W be two-sided enrichments. Their composite B ∘ A : U → W has for span the composition of the spans of A and B: its objects are the pairs (a, b) in Obj(A) × Obj(B) such that a_{+} = b_{−} and the maps (−)_{−}^{B∘A} and (−)_{+}^{B∘A} are given by (a, b)_{−} = a_{−} and (a, b)_{+} = b_{+}. The functors B ∘ A_{(a,b),(a′,b′)} are the composites

\xymatrix{U(a_{−}, a′_{−}) \ar[r]^{A_{(a,a′)}} & V(a_{+}, a′_{−}) = V(b_{−}, b′_{−}) \ar[r]^{B_{(b,b′)}} & W(b_{+}, b′_{−})} .

The horizontal composition of the 2-cells f : A → B : U → V and g : C → D : V → W is the map sending (a, c) ∈ Obj(C ∘ A) to (fa, gc).

3.2. A closed structure on Caten. The cartesian product of locally partially ordered bicategories extends straightforwardly to a pseudo functor Caten ×Caten → Caten that makes Caten into a monoidal tricategory. Caten has a biclosed structure:

Caten(U × V, W) ≅ Caten(U, Conv(V, W)).

where Conv(V, W) is defined as follows:

3.3. Definition. [The bicategory Conv(V,W)] Given two locally partially ordered bicategories V, W the locally partially ordered bicategory Conv(V, W) is defined as follows. Its objects are ordered pairs (v, w) ∈ Obj(V) × Obj(W). Its arrows (v, w) → (v′, w′) are functors V(v, v′) → W(w, w′), (i.e. monotonous maps). The partial order on arrows is pointwise: For any F, G : (v, w) → (v′, w′), F ≤ G when for any h : v → v′ in V, F (h) ≤ G(h) : w → w′ in W. For any F : (v, w) → (v′, w′) and G : (v′, w′) → (v′′, w′′), the (horizontal) composite G¯∘F : (v, w) → (v′′, w′′) is the left Kan extension of

\xymatrix{V(v, v′) × V(v′, v′′) \ar[r]^{F ×G} & W(w, w′) × W(w′, w′′) \ar[r]^{∘^{W}} & W(w, w′′)}

along \xymatrix{V(v, v′) × V(v′, v′′) \ar[r]^{∘^{V}} & U(v, v′′)} , where ∘^{V}, ∘^{W} are the horizontal compositions respectively of V and W. This means that for any arrow h in V(v, v′′), G¯∘F (h) = ⋁{G(g) ∘ F (f) | g ∘ f ≤ h}. The identity I_{(v,w)} in (v, w) is the left Kan extension of I_{w}^{W} : 1 → W(w, w) along I_{v}^{V} : 1 → V(v, v), thus for any f in V(v, v), I_{(v,w)}(f) =

\{ \begin{matrix}I_{w}^{W} if f ≥ I_{v}^{V}, \\ ⊥, the initial element of W(w, w), otherwise. \end{matrix}

For any bicategory V, one has an isomorphism in 2-Cat

V ≅ Conv(1, V).

Two-sided enrichments are also just enrichments since for any V, W

Caten(V, W) ≅ Conv(V, W)-Cat.

Thus one also gets that for any V, the 2-category V-Cat is representable:

V-Cat ≅ Caten(1, V).

3.4. Adjoints in Caten. Adjoints F ⊣ G : V ⇀ W in Caten are (up to isomorphism) as follows. Obj(F) = Obj(G) = Obj(V), the span of F is a left adjoint in the bicategory of spans on Set

\xymatrix{ & Obj(V) \ar[dl]_{1} \ar[dr]^{F} \\ Obj(V) &  & Obj(W)}

the span of G is its right adjoint

\xymatrix{ & Obj(V) \ar[dl]_{F} \ar[dr]^{1} \\ Obj(W) &  & Obj(V)}

and the functors F_{u,v} : V(u, v) → W(F u, F v) have right adjoints G_{u,v} : W(F u, F v) → V(u, v). Actually, a left adjoint F in Caten may be seen as a 2-functor V → W with “local (right) adjoints”. In this case, the above family of right adjoints satisfies some coherence properties. It can be shown that for any lax functor F : V → W such that for any objects u, v in V, the component F_{u,v} : V(u, v) → W(F u, F v) has right adjoint G_{u,v}, these adjoints in their totality satisfy the coherence conditions, on composition:

(1) for all objects u,v,t of V, for all f : F u → F v and g : F v → F t,
G_{v,t}(g) ∘ G_{u,v}(f) ≤ G_{u,t}(g ∘ f);

and on identities:

(2) for all objects u of V, I_{u} ≤ G_{u,u}(I_{F u}).

if and only if F is a 2-functor.

3.5. The change of base for enrichments. Using the representability for V-cat-egories and the fact that Caten is a tricategory, one has the change of base theorem for enrichments over bicategories. The composition on the left by any two-sided enrichment F : V → W, yields a 2-functor F ∘ − : Caten(1, V) → Caten(1, W), and by the representability result, it corresponds to a 2-functor F_{@} : V-Cat → W-Cat. In the same way, with any 2-cells σ : F → G of Caten yields a 2-natural transformation σ_{@} : F_{@} → G_{@} : V-Cat → W-Cat. Indeed,

3.6. Theorem. There is a pseudo-functor (−)_{@} from Caten to 2-Cat (the 2-category of 2-categories, 2-functors and 2-natural transformations). It sends a bicategory V to the 2-category V-Cat, its actions on two-sided enrichments and 2-cells are the ones described above.

The above F_{@} : V-Cat → W-Cat is now described in terms of enrichments. For any V-category A, F_{@}A is given by:

- Obj(F_{@}A) = {(a, x) | a ∈ Obj(A), x ∈ Obj(F), a_{+} = x_{−}};

- (a, x)_{+}^{F_{@}A} = x_{+};

- F_{@}A((a, x), (b, y)) = F_{x,y}(A(a, b)). For any V-functor f : A → B, F_{@}(f) is the W-functor with underlying map (a, x) ↦ (fa, x).

Now if F is moreover a 2-functor, then for any V-category A, F_{@}A is given by:

- Obj(F_{@}A) = Obj(A);

- (a)_{+}^{F_{@}A} = F (a_{+}); - (F_{@}A)(a, b) = F A(a, b); and for any V-functor f : A → B, F_{@}(f) is the W-functor with the same underlying map as f.

From an adjoint pair F ⊣ G in Caten, one gets the adjoint pair F_{@} ⊣ G_{@} in 2-Cat. Precisely, 3.7. Theorem. If the 2-functor between locally partially ordered bicategories F : V → W admits local adjoints G_{u,v}, (u, v) ranging in Obj(V)^{2} then the 2-functor F_{@} admits a right 2-adjoint, denoted F^{@} and defined on objects as follows. For any object B of W-Cat, F^{@}B is given by:

- Obj(F^{@}B) = {(b, v) | b ∈ Obj(B), v ∈ Obj(V), b_{+} = F v};

- (b, v)_{+} = v;

- F^{@}B((b, v), (b′, v′)) = G_{v,v′}(B(b, b′)).

In the case above F^{@} is just G_{@}, the collection of local adjoints G_{u,v} defining the right adjoint G to F in Caten. 3.8. Change of base, skeletality and Cauchy completion. Further on we consider a two-sided enrichment F : V → W where V and W are locally cocomplete and biclosed. 3.9. Lemma. If F is left adjoint in Caten then for any W-category B, if B is skeletal then also is F^{@}B. Proof. Suppose that (b, v), (b′, v) ∈ Obj(F^{@}B) are such that F^{@}B((b, v), (b′, v)) ≥ I_{v}, i.e. G_{v,v}B(b, b′) ≥ I_{v}. Since F_{v,v} ⊣ G_{v,v}, B(b, b′) ≥ F I_{v} = I_{F v}. By skeletality of B, b = b′. 3.10. Proposition. There is a lax normal ^{1} functor F_{♯} : V-Mod → W-Mod that extends F_{@} in the sense the diagram below commutes:

\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 defined by: for all objects (a, x) of F_{@}A and (b, y) of F_{@}B, F_{♯}(φ)((b, y), (a, x)) = F_{y,x}(φ(b, a)). This is depicted by the diagram in Cat below:

[figure: 1; φ(b,a); F_{♯}φ(b,a); V(b_{+}, a_{+}); W((b, y)_{+}, (a, x)_{+}); V(y_{−}, x_{−}); F_{y,x}; W(y_{+}, x_{+})]

1A lax functor is normal when it strictly preserves identities.

Proof. Let φ : \xymatrix{A \ar[r]|{∘} & B} . For any objects (a, x), (a′, x′) of F_{@}A and (b, y) of F_{@}B,

F_{@}A((a, x), (a′, x′)) ∘ F_{♯}φ((b, y), (a, x)) = F_{x,x′}A(a, a′) ∘ F_{y,x}φ(b, a)

≤ F_{y,x′}(A(a, a′) ∘ φ(b, a))

≤ F_{y,x′}(φ(b, a′))
= F_{♯}φ((y, b), (a′, x′)).

Similarly, for any objects (a, x) of F_{@}A and (b, y), (b′, y′) of F_{@}B, F_{♯}φ((b, y), (a, x)) ∘ F_{@}B((b′, y′), (b, y)) ≤ F_{♯}φ((b′, y′), (a, x)).

Now given V-module, φ : \xymatrix{A \ar[r]|{∘} & B} and ψ : \xymatrix{B \ar[r]|{∘} & C} , for any objects (a, x) of F_{@}A and (c, z) of F_{@}C,

(F_{♯}(ψ • φ))((c, z), (a, x)) = F_{z,x}(ψ • φ)(c, a), = F_{z,x}(⋁_{b∈Obj(B)} φ(b, a) ∘ ψ(c, b)),

≥ ⋁_{b∈Obj(B)} F_{z,x}(φ(b, a) ∘ ψ(c, b))
≥ ⋁_{(b,y)∈Obj(F_{@}B)} F_{y,x}φ(b, a) ∘ F_{z,y}ψ(c, b)
= (F_{♯}ψ • F_{♯}φ)((c, z), (a, x)).

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

3.11. Lemma. 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 equality 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^{⋄}).

These later facts motivated the somewhat technical definition below, as we were looking for sufficient conditions on a two-sided enrichment F that ensures that F_{@} preserves the Morita equivalence.

3.12. Definition. [Super two-sided enrichments] A two-sided enrichment G : V → W is super when for any object x of G, and any family of pairs of arrows

\xymatrix{x_{−} \ar[r]^{f_{i}} & u_{i} \ar[r]^{g_{i}} & x_{−}}

in V, i ranging in I, if I_{x_{−}} ≤ ⋁_{i∈I} g_{i} ∘ f_{i} then

I_{x_{+}} ≤ ⋁_{{y∈Obj(G)|∃i∈I,u_{i}=y_{−}}} G_{y,x}(g_{i}) ∘ G_{x,y}(f_{i}).

It is straightforward to check that the composition of super two-sided enrichments is super. Trivially identities are super.

3.13. Lemma. If F is super then F_{♯} : V-Mod → W-Mod preserves adjoints, i.e. for any φ ⊣ φ̃ in V-Mod, one has the ajoint pair F_{♯}(φ) ⊣ F_{♯}(φ̃) in W-Mod.

Proof. Suppose that F is super. Consider a left adjoint φ : \xymatrix{A \ar[r]|{∘} & B} in V-Mod. First let us show that 1 ≤ F_{♯}(φ̃) • F_{♯}(φ). Note that for any object (a, x) of F_{@}A, since I_{a_{+}}^{V} ≤ ⋁_{b∈Obj(B)}(φ(b, a) ∘ φ̃(a, b)) one has I_{x_{+}}^{W} ≤ ⋁_{(b,z)∈Obj(F_{@}B)} F_{z,x}φ(b, a) ∘ F_{x,z}φ̃(a, b). Thus for any objects (a, x), (a′, y) of F_{@}A.

F_{@}A((a, x), (a′, y)) = F_{x,y}A(a, a′)

≤ F_{x,y}A(a, a′) ∘ (⋁_{(b,z)∈Obj(F_{@}B)} F_{z,x}φ(b, a) ∘ F_{x,z}φ̃(a, b))

= ⋁_{(b,z)∈Obj(F_{@}B)}((F_{x,y}A(a, a′) ∘ F_{z,x}φ(b, a)) ∘ F_{x,z}φ̃(a, b))
≤ ⋁_{(b,z)∈Obj(F_{@}B)}(F_{z,y}φ(b, a′)) ∘ F_{x,z}φ̃(a, b))
= (F_{♯}φ̃ • F_{♯}φ)((a, x), (a′, y)).

Now since F_{♯} is lax and normal and φ • φ̃ ≤ 1, F_{♯}(φ) • F_{♯}(φ̃) ≤ F_{♯}(φ • φ̃) ≤ F_{♯}(1) = 1.

In particular the above F_{♯} : V-Mod → W-Mod will preserve inverse pairs of modules, thus isomorphisms.

3.14. Corollary. If F is super then F_{♯} preserves the Morita equivalence, i.e. if the V-categories A and B are Morita equivalent then also are F_{@}A and F_{@}B.

Further on, we suppose that F has right adjoint G in Caten, i.e. F : V → W is a 2-functor with local adjoints G_{a,b}, (a, b) ranging in Obj(V)^{2}.

3.15. Proposition. F_{♯} : V-Mod → W-Mod is a 2-functor.

Proof. We already know that F_{♯} is lax and normal. It remains to show that F_{♯} preserves composition. Let φ : \xymatrix{A \ar[r]|{∘} & B} and ψ : \xymatrix{B \ar[r]|{∘} & C} be V-modules. Since F has local right adjoints, it “preserves local least upper bounds”, therefore for any objects a of A and c of C,

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).

We shall finish with technical lemmas. Their relevance will appear in the next section treating reversibility.

3.16. Lemma. Let η be the unit of the adjunction F_{@} ⊣ F^{@}. For any left adjoint module φ : \xymatrix{A \ar[r]|{∘} & B} there is a 2-cell in V-Mod:

\xymatrix{A \ar[d]_{_{η_{A}_{⋄}}} \ar[r]^{φ} & B \ar@{=>}[dl] \ar[d]^{_{η_{B}_{⋄}}} \\ F^{@}F_{@}A \ar[r]_{G_{♯}F_{♯}φ} & F^{@}F_{@}B}

Proof. Let φ : \xymatrix{A \ar[r]|{∘} & B} be a right adjoint in V-Mod. Consider some objects a of A and (v, b) ∈ F^{@}F_{@}B. Then

(η_{B}_{⋄} • φ)((v, b), a) = ⋁_{b′∈Obj(B)} φ(b′, a) ∘ G_{v,b′_{+}}F B(b, b′)

≤ ⋁_{b′∈Obj(B)} G_{b′_{+},a_{+}}F φ(b′, a) ∘ G_{v,b′_{+}}F B(b, b′)

≤ G_{v,a_{+}}F φ(b, a);

On the other hand,

(G_{♯}F_{♯}φ • η_{A}_{⋄})((v, b), a) = ⋁_{(a′,v′)∈Obj(F^{@}F_{@}A)} G_{v′,a_{+}}F A(a′, a) ∘ G_{v,v′}F φ(b, a′)

≤ ⋁_{(a′,v′)∈Obj(F^{@}F_{@}A)} G_{v,a_{+}}F (A(a′, a) ∘ φ(b, a′))

≤ G_{v,a_{+}}F φ(b, a)

And since (a, a_{+}) ∈ Obj(F^{@}F_{@}A), one has also

G_{v,a_{+}}F φ(b, a) ≤ G_{a_{+},a_{+}}F I_{a_{+}} ∘ G_{v,a_{+}}F φ(b, a)

≤ G_{a_{+},a_{+}}F A(a, a) ∘ G_{v,a_{+}}F φ(b, a)

≤ (G_{♯}F_{♯}φ • η_{A}_{⋄})((v, b), a)

3.17. Lemma. Let A be a Cauchy complete W-category. For any object u of V and any left adjoint V-module φ : \xymatrix{û \ar[r]|{∘} & F^{@}A} , there is a V-functor f : û → F^{@}A with f_{⋄} ≥ φ.

Proof. Let u be an object of V and φ : \xymatrix{û \ar[r]|{∘} & F^{@}A} be a left adjoint V-module. According to Fact 3.15, F_{♯}φ : \xymatrix{F_{@}û \ar[r]|{∘} & F_{@}F^{@}A} is a left adjoint W-module, therefore also is ϵ_{A}_{⋄} • F_{♯}φ : \xymatrix{F_{@}û \ar[r]|{∘} & A} , ϵ denoting the co-unit of the adjunction F_{@} ⊣ F^{@} (see diagram below).

\xymatrix{F^{@}A & F_{@}F^{@}A \ar[r]^{^{ϵ_{A}^{⋄}}} & A \\ û \ar[u]_{φ} & F_{@}û \ar[u]^{F_{♯}φ} \ar@{.>}[ur]_{g_{⋄}}}

Since A is Cauchy complete there is some V-functor g : F_{@}û → A such that g_{⋄} = ϵ_{A}_{⋄}•F_{♯}φ. η denoting the unit of the adjunction F_{@} ⊣ F^{@}, let us see that the W-functor f = F^{@}g∘ηû : û → F^{@}A — see diagram below — satisfies f_{⋄} ≥ φ.

\xymatrix{û \ar[dr]_{f} \ar[r]^{ηû} & F^{@}F_{@}û \ar[d]^{F^{@}g} & F_{@}û \ar[d]^{g} \\  & F^{@}A & A}

The required 2-cell results from the pasting of diagrams in V-Mod:

[figure: û; (I); F^{@}F_{@}û; φ; F^{@}F_{@}F^{@}A; G_{♯}F_{♯}φ; (η_{F^{@}A})_{⋄}; ηû_{⋄}; (II); F^{@}A; (F^{@}ϵ_{A})_{⋄}; (F^{@}g)_{⋄}; F^{@}A]

2-cell (I) is given by 3.16, and 2-cell (II) is given by the laxity of G_{♯}.

4. Reversibility

In this section we will restrict the theory of two-sided enrichments over locally partially ordered bicategories to the symmetric case. We extend the notion of reversibility of [Wal82] to two-sided enrichments, modules, and completion. Definitions will allow to rebuild the complete theory in this special case. They will be also illustrated by two examples: sheaves over locales and action groups. Eventually we study the effect of the change of bases on the completeness and skeletality of enrichments in the reversible context. The main result is theorem 4.34.

4.1. Reversible enrichments.

4.2. Definition. A 2-category V is reversible when there are isomorphisms of categories s_{u,v} : V(u, v) ≅ V(v, u) for all objects u and v of V, and these isomorphisms satisfy the conditions:

- rev − (i) : s_{u,v} ∘ s_{v,u} = 1, for any objects u and v;

- rev − (ii) : for any objects u,v,w the diagram (in Cat) below commutes:

\xymatrix{V(v, w) × V(u, v) \ar[d]_{≅} \ar[r]^{^{∘^{u,v,w}}} & V(u, w) \ar[dd]^{s_{u,w}} \\ V(u, v) × V(v, w) \ar[d]_{s_{u,v}×s_{v,w}} \\ V(v, u) × V(w, v) \ar[r]_{_{∘_{w,v,u}}} & V(w, u)}

where ∘ is the horizontal composition in V;

- rev − (iii) : for any object v the diagram below commutes

[figure: 1; I_{v}; V(v, v); s_{v,v}; 1; I_{v}; V(v, v)]

Further on V is a reversible locally partially ordered biclosed and cocomplete bicategory.

4.3. Definition. [reversible enrichments] A V-category A is reversible when for any objects a, b of A, A(b, a) = s_{a_{+},b_{+}}(A(a, b)).

Let A and B be V-categories. If B is reversible then the preorder V-Cat(A, B), denoted here ⇒, is an equivalence since for any V-functors f, g : A → B,

f ⇒ g if and only if for all object a of A, I_{a_{+}} ≤ B(fa, ga), - - - - - - - - - - - - - - - - - - - - - , s(I_{a_{+}}) ≤ s(B(fa, ga))

- - - - - - - - - - - - - - - - - - - - - , I_{a_{+}} ≤ B(ga, fa)

if and only if g ⇒ f.

4.4. Observation. If B above is moreover skeletal then ⇒ is the identity on V-Cat(A, B).

4.5. Proposition. Let A and B be reversible V-categories.

(1) If φ : \xymatrix{A \ar[r]|{∘} & B} is a V-module then the map defined on Obj(A) × Obj(B) sending (a, b) to the arrow s_{b_{+},a_{+}}(φ(b, a)) defines a V-module φ^{s} : \xymatrix{B \ar[r]|{∘} & A} .

(2) If φ is left adjoint then φ^{s} is right adjoint with (φ̃)^{s} ⊣ φ^{s}.

Proof. Assertion (1) is straightforward. Let us see (2) : Suppose that φ is left adjoint. From 1 ≤ φ̃ • φ, i.e. for any objects a, a′ of A,

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

one gets by applying s_{(a_{+},a′_{+})} on both sides of the inequality that for any a, a′,

A(a′, a) ≤ ⋁_{b∈Obj(A)} (φ̃)^{s}(b, a) ∘ φ^{s}(a′, b),

i.e. 1 ≤ φ^{s} • (φ̃)^{s}. Similarly one gets from (φ̃)^{s} • φ^{s} ≤ 1 that φ • φ̃ ≤ 1.

There are reversible enrichments with non-reversible Cauchy completions. A very simple example (due to S. Kasangian) may be found in [BeWal82]. This motivates the notion of Cauchy-reversible completeness defined further.

4.6. Definition. [Reversible modules] A V-module φ between reversible V-categories is reversible when it has right adjoint φ^{s}.

4.7. Remark. If f is a V-functor between reversible V-categories then the V-module f_{⋄} is reversible.

4.8. Definition. [Cauchy-reversible completeness] A reversible V-category B is Cauchyreversible complete when one of the following two equivalent assertions holds:

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

Let us prove the equivalence of assertions (i) and (ii) of the previous definition.

(i) ⇒ (ii): immediate. (ii) ⇒ (i): Note that for some reversible module φ : \xymatrix{A \ar[r]|{∘} & B} , for any object a of A, φ • a_{⋄} ⊣ a^{⋄} • φ^{s} = (φ • a_{⋄})^{s}. Then one concludes analogously as for 2.8.

Clearly, any reversible Cauchy complete V-category is also a Cauchy-reversible complete one.

4.9. Definition. [Cauchy-reversible completion] Let A be a reversible V-category with Cauchy completion Ā. A^{s}, the Cauchy-reversible completion of A is the reversible Vcategory where Obj(A^{s}) ⊆ Obj(Ā), is the set of reversible V-modules of the form φ : \xymatrix{û \ar[r]|{∘} & A} , u ranging in Obj(V). The maps A^{s}(−, −) and (−)_{+} of A^{s} are the restrictions of the maps Ā(−, −) and (−)_{+} of Ā.

From 2.10, it is immediate that the above data defines a V-category A^{s}. A^{s} is moreover reversible as shown below. Let φ : \xymatrix{û \ar[r]|{∘} & A} , ψ : \xymatrix{v̂ \ar[r]|{∘} & A} be reversible adjoints, then

Ā(φ, ψ) = ⋁_{a∈Obj(A)} ψ(a) ∘ φ^{s}(a)

= ⋁_{a∈Obj(A)} s_{v,u}(φ(a) ∘ ψ^{s}(a))

= s_{v,u}(⋁_{a∈Obj(A)} φ(a) ∘ ψ^{s}(a))
= s_{v,u}(Ā(ψ, φ)).

Let A, Ā and, A^{s} be as in 4.8. The map i_{A} : Obj(A) → Obj(Ā) takes values in Obj(A^{s}). Thus it defines a V-functor from A to A^{s} further on denoted j_{A}.

An immediate consequence of 2.12 is

4.10. Proposition. For any reversible V-category A, A^{s} is skeletal.

Adapting the proof of 2.13 one shows

4.11. Lemma. Let A be a reversible V-category and, Γ : \xymatrix{û \ar[r]|{∘} & A^{s}} be some V-module where u is an object of V. For any object φ of A^{s}, Γ(φ) = ⋁_{a∈Obj(A)}Γ(a_{⋄}) ∘ φ^{s}(a).

4.12. Proposition. For any reversible V-category A, A^{s} is Cauchy-reversible complete.

Proof. Merely an adaptation of the proof of 2.14. Given Γ : \xymatrix{û \ar[r]|{∘} & A^{s}} a reversible Vmodule. One checks that the maps γ : Obj(A) → Arrows(V) given by for any a ∈ Obj(A),

γ(a) = Γ(a_{⋄})

defines a reversible module from û to A with adjoint γ′ given by for any object a of A,

γ′(a) = Γ^{s}(a_{⋄}).

Then according to 4.11, Γ = A^{s}(−, γ).

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

Proof. The proof that (j_{A})_{⋄} and (j_{A})^{⋄} are inverse modules is the one for 2.15. Since A^{s} is reversible, j_{A}^{⋄} = (j_{A}_{⋄})^{s}.

We define the following 2-categories.

- V-RCat with:

- objects: reversible V-categories,

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

- V-SkCRcCat with:

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

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

- V-RMod with:

- objects: reversible V-categories,

- arrows: reversible V-modules,

- 2-cells: inherited from V-Mod.

According to 4.13,

4.14. Proposition. V-SkCRcCat and V-RMod are 2-equivalent. The 2-equivalence S′ : V-RMod → V-SkCRcCat is defined on objects by S′(A) = A^{s} and on arrows by S′(φ) = f_{φ} where for any reversible V-module φ : \xymatrix{A \ar[r]|{∘} & B} , f_{φ} : A^{s} → B^{s} is the unique V-functor f satisfying (f_{φ})_{⋄} • j_{A}^{⋄} = j_{B}_{⋄} • φ. The inclusion 2-functor V-SkCRcCat → V-RCat has a left 2-adjoint.

According to 4.4, V-SkCRcCat and V-RMod are locally discrete, i.e. the only 2-cells are identies. Therefore we shall sometimes consider them just as categories. Skeletal and Cauchy-reversible complete V-categories form a reflective subcategory of the underlying category of V-RCat. The left adjoint of the inclusion sends a reversible V-category A to A^{s} and, the unit takes value j_{A} : A → A^{s} in A.

We may define,

4.15. Definition. [Morita reversible equivalence] Two reversible V-categories A and B are Morita reversible equivalent when one of the following equivalent assertions is satisfied:

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

Note that if two reversible V-categories are Morita reversible equivalent then they are Morita equivalent.

4.16. Examples. Further on we give two examples of reversible enrichments on locally partially ordered bicategories. Note that for the second example, the reversibility isomorphisms are not identities.

4.17. Example. [Sheaves][Wal81]

To any locale L corresponds a locally partially ordered bicategory C_{L} given by:

- objects are the u ∈ L;

- arrows from u to v are the w ∈ L with w ≤ u ∧ v;

- the partial order on C_{L}(u, v) is that of L;

- the composition of arrows is the intersection;

- the unit in u is u. Such a C_{L} is locally cocomplete and biclosed. It is also reversible where the isomorphisms s_{u,v} are the identity maps C_{L}(u, v) → C_{L}(v, u).

If L is a locale, Sh(L) will denote the category of sheaves on L. In [Wal81], the following result is proved.

4.18. Theorem. Given a locale L, Sh(L) is equivalent to the category of reversible skeletal Cauchy complete C_{L}-enrichments.

The equivalence J above is defined as follows. For any sheaf F on L, J(F) is the enrichment over C_{L} where:

- Obj(J(F)) is the set of partial sections of F;

- For any section s, s_{+} is its domain dom(s);

- For any sections s, s′, J(F)(s, s′) = ⋁{u ≤ dom(s) ∧ dom(s′) | s⌊u = s′⌊u} — s⌊u denotes the restriction of s to u ≤ dom(s). If h : F → G is a morphism of Sh(L), J(h) is the map that sends any partial section s ∈ F u to h(s) ∈ Gu.

In this case it happens that Cauchy-reversible complete enrichments are Cauchy complete. This is due to

4.19. Proposition. For any locale L, any left adjoint C_{L}-module is reversible.

Proof. Let u ∈ L and φ : \xymatrix{û \ar[r]|{∘} & A} be some left adjoint module with φ ⊣ φ̃. Then the map ψ given by: ψ(s) = φ(s) ∧ φ̃(s) defines a module ψ : \xymatrix{û \ar[r]|{∘} & A} with itself as adjoint. Since ψ ≤ φ and ψ ≤ φ̃, φ = ψ = φ̃.

So we may reformulate 4.18

4.20. Theorem. Given a locale L, Sh(L) is equivalent to the category of skeletal Cauchyreversible complete C_{L}-enrichments.

4.21. Example. [Group actions]

Let α be an action of the group G on the set X. We note for such an action:

- ⟨σ, x⟩_{α} (or simply ⟨σ, x⟩), the image of x ∈ X by α(σ) for σ ∈ G,

- “·”, the composition law of G,

- u, the neutral element of G. Such an action will be denoted (G, X, α) or simply (X, α).

Given a group G, G-Set denotes the category with objects actions of G on sets and with arrows action preserving maps i.e. f : (Z, β) → (X, α) is an arrow of G-Set when the underlying map f : Z → X makes the following diagram commute:

\xymatrix{G × Z \ar[r]^{β} \ar[d]_{1×f} & Z \ar[d]^{f} \\ G × X \ar[r]_{α} & X}

If α is an action of G on X, the locally partially ordered bicategory C_{α} is defined as follows:

- Objects are the x ∈ X;

- Arrows from x to x′ are the subsets of {σ ∈ G | ⟨σ, x⟩ = x′};

- the partial order on C_{α}(x, x′) is the inclusion;

- the composition L′ ∘ L of L : x → x′ and L′ : x′ → x′′ is {τ · σ | σ ∈ L and τ ∈ L′};

- the unit at x is {u}. Clearly C_{α} is locally cocomplete and biclosed. C_{α} is reversible with isomorphisms s: s_{x,x′}(L) = {σ^{−1} | σ ∈ L}, for any L : x → x′.

4.22. Lemma. Let α be an action of a group G on a set X and A be a reversible enrichment on C_{α}.

(1) For any a_{0} ∈ Obj(A) and any σ_{0} ∈ G, the map φ given by

(∗) ∀a ∈ Obj(A), φ(a) = {σ_{0}} ∘ A(a, a_{0})

defines a left adjoint module φ : \xymatrix{x̂ \ar[r]|{∘} & A} where x = ⟨σ_{0}, (a_{0})_{+}⟩, with right adjoint φ̃ given by

∀a ∈ Obj(A), φ̃(a) = A(a_{0}, a) ∘ {σ_{0}^{−1}}.

Conversely for any x ∈ X and any left adjoint module φ : \xymatrix{x̂ \ar[r]|{∘} & A} there is some a_{0} ∈ Obj(A) and some σ_{0} ∈ G such that x = ⟨σ_{0}, (a_{0})_{+}⟩ and the map φ is given by formulae (∗) above.

(2) If φ, ψ : \xymatrix{x̂ \ar[r]|{∘} & A} are left adjoint modules given by:

∀a ∈ Obj(A), φ(a) = {σ_{0}} ∘ A(a, a_{0})

and

∀a ∈ Obj(A), ψ(a) = {τ_{0}} ∘ A(a, a′_{0})

where σ_{0}, τ_{0} ∈ G and a_{0}, a′_{0} ∈ Obj(A), then φ = ψ if and only if (σ_{0}, a_{0}) ∼ (τ_{0}, a′_{0}), ∼ being the equivalence relation on G × X:

∀σ, σ′ ∈ G, ∀a, a′ ∈ Obj(A), (σ, a) ∼ (σ′, a′) ⇔ (σ′)^{−1} · σ ∈ A(a, a′).

Proof. (1) : Let a_{0} ∈ Obj(A), σ_{0} ∈ G, and φ be the map on Obj(A) defined by φ(a) = {σ_{0}} ∘ A(a, a_{0}). For short, let x = ⟨σ_{0}, (a_{0})_{+}⟩. Actually φ defines a module \xymatrix{x̂ \ar[r]|{∘} & A} . φ satisfies (mod − i) since for any a ∈ Obj(A), φ(a) is an object of C_{α}(a_{+}, x). φ satisfies (mod − ii) since for any a ∈ Obj(A), {u} ∘ φ(a) ⊆ φ(a). φ satisfies (mod − iii), since for any a, a′ ∈ Obj(A),

φ(a) ∘ A(a′, a) = {σ_{0}} ∘ A(a, a_{0}) ∘ A(a′, a)
⊆ {σ_{0}} ∘ A(a′, a_{0})
= φ(a′).

Analogously one shows that the map sending any a ∈ Obj(A) to A(a_{0}, a) ∘ {σ_{0}^{−1}} defines a module φ̃ : \xymatrix{A \ar[r]|{∘} & x̂}. φ̃ is right adjoint to φ as shown below. Since u ∈ ⋃_{a∈Obj(A)} A(a, a_{0}) ∘ A(a_{0}, a),

{u} ⊆{a_{0}, a_{0}}
⊆⋃_{a∈Obj(A)}{σ_{0}} ∘ A(a, a_{0}) ∘ A(a_{0}, a) ∘ {σ_{0}^{−1}}
=⋃_{a∈Obj(A)} φ(a) ∘ φ̃(a).

For any a, a′ ∈ Obj(A),

φ̃(a) ∘ φ(a′)= A(a_{0}, a) ∘ {σ_{0}^{−1}} ∘ {σ_{0}} ∘ A(a′, a_{0})
⊆ A(a′, a).

Now consider some x ∈ X, and some left adjoint module φ : \xymatrix{x̂ \ar[r]|{∘} & A} with right adjoint φ̃. Then {u} ⊆ ⋃_{a∈Obj(A)} φ(a)∘φ̃(a). This implies that there are some a_{0} ∈ Obj(A) and some σ_{0} ∈ φ(a_{0}) with σ_{0}^{−1} ∈ φ̃(a_{0}). For this a_{0}, ⟨σ_{0}, (a_{0})_{+}⟩ = x. Let ψ : \xymatrix{x̂ \ar[r]|{∘} & A} be the left adjoint module defined by ∀a ∈ Obj(A), ψ(a) = {σ_{0}} ∘ A(a, a_{0}). Then for any a ∈ Obj(A),

ψ(a) = {σ_{0}} ∘ A(a, a_{0})
⊆ φ(a_{0}) ∘ A(a, a_{0})
⊆ φ(a).

Thus ψ ≤ φ. Analogously for any a ∈ Obj(A), A(a_{0}, a) ∘ {σ_{0}^{−1}} ⊆ φ̃(a) i.e. ψ̃ ≤ φ̃. Therefore ψ = φ.

(2) : First let us check that ∼ is an equivalence relation. It is reflexive since for any object a of A, u^{−1} · u = u ∈ A(a, a). It is symmetric. If (σ, a) ∼ (σ′, a′) then σ′^{−1}.σ ∈ A(a, a′), thus σ^{−1}.σ′ ∈ A(a′, a) since A is reversible. It is transitive. Suppose (σ, a) ∼ (σ′, a′) and (σ′, a′) ∼ (σ′′, a′′). Then (σ′)^{−1}·σ ∈ A(a, a′) and (σ′′)^{−1}·σ′ ∈ A(a′, a′′). Thus σ′′^{−1} · σ ∈ A(a′, a′′) ∘ A(a, a′) ⊆ A(a, a′′).

Consider further on φ, ψ : \xymatrix{x̂ \ar[r]|{∘} & A} , σ_{0}, σ′_{0} ∈ G and, a_{0}, a′_{0} ∈ Obj(A) such that ∀a ∈ Obj(A), φ(a) = {σ_{0}} ∘ A(a, a_{0}) and ψ(a) = {σ′_{0}} ∘ A(a, a′_{0}). First, suppose (σ_{0}, a_{0}) ∼ (σ′_{0}, a′_{0}) i.e., σ′_{0}^{−1} · σ_{0} ∈ A(a_{0}, a′_{0}). Then for any a ∈ Obj(A),

φ(a) = {σ_{0}} ∘ A(a, a_{0})
= {σ′_{0}} ∘ {σ′_{0}^{−1} · σ_{0}} ∘ A(a, a_{0})
⊆ {σ′_{0}} ∘ A(a_{0}, a′_{0}) ∘ A(a, a_{0})
⊆ {σ′_{0}} ∘ A(a, a′_{0})
= ψ(a),

then φ ≤ ψ. Since σ′_{0} = σ_{0} ·γ^{−1} and then γ^{−1} ∈ A(a′_{0}, a_{0}), one obtains analogously ψ ≤ φ.

Now suppose φ = ψ. Then for any a ∈ Obj(A), {σ_{0}} ∘ A(a, a_{0}) = {σ′_{0}} ∘ A(a, a′_{0}). For a = a_{0}, we get {σ_{0}} ⊆ {σ′_{0}} ∘ A(a_{0}, a′_{0}), i.e. there is γ ∈ A(a_{0}, a′_{0}), such that σ_{0} = σ′_{0} · γ.

4.23. Corollary. Let α be an action of the group G on a set X and A be a reversible enrichment on C_{α}. Any left adjoint module φ : \xymatrix{x̂ \ar[r]|{∘} & A} has right adjoint φ^{s}.

4.24. Proposition. Let (X, α) be an action of the group G. The category of skeletal and Cauchy-reversible complete enrichments on C_{α} is equivalent to the category G-Set ↓ α of objects over α. Precisely the equivalence J : G-Set ↓ α ≅ C_{α}-SkCRcCat is given by:

- For any f : (Z, β) → (X, α) in G-Set, J(f) is the enrichment with set of objects Z, and such that for any z ∈ Z, z_{+} = f(z), and for any z, z′ ∈ Z, J(f)(z, z′) is the set {σ ∈ G | ⟨σ, z⟩_{β} = z′};

- For any arrow ⟨f_{1}, h, f_{2}⟩ of G-Set ↓ α with f_{1} : (Z_{1}, β_{1}) → (X, α) and f_{2} : (Z_{2}, β_{2}) → (X, α), J(h) is the C_{α}-functor from J(f_{1}) to J(f_{2}) defined by the map h : Z_{1} → Z_{2}. J has adjoint J′ defined on objects as follows. For any skeletal Cauchy-reversible complete enrichment A on C_{α}, J′(A) is the action (Obj(A), β) where for any σ ∈ G and any a ∈ Obj(A), ⟨σ, a⟩_{β} is the unique a′ such that σ ∈ A(a, a′).

Proof. Let us check that given f : (Z, β) → (X, α), J(f) is a skeletal Cauchy-reversible complete C_{α}-enrichment.

- It satisfies (enr − i): Since f is a morphism of G-Set, for any z ∈ Z and any σ ∈ G, ⟨σ, f(z)⟩_{α} = f(⟨σ, z⟩_{β}). Therefore for any z, z′ ∈ Z,

J(f)(z, z′) = {σ ∈ G | ⟨σ, z⟩_{β} = z′}
⊆ {σ ∈ G | ⟨σ, f(z)⟩_{α} = f(z′)}

i.e. J(f)(z, z′) is an object of C_{α}(z_{+}, z′_{+}).

- J(f) satisfies (enr − ii) since for any z ∈ Z, {u} ⊆ J(f)(z, z).

- It trivially satisfies (enr − iii).

- J(f) is trivially reversible.

- J(f) is skeletal since for any z, z′ ∈ Z, u ∈ J(f)(z, z′) ⇒ z = z′.

- Let us show that it is Cauchy complete. Consider a module φ : \xymatrix{x̂ \ar[r]|{∘} & J(f)} with right adjoint φ̃. According to Lemma 4.22 there are some z_{0} ∈ Z and some σ_{0} ∈ φ(z_{0}) such that φ(z) = {σ_{0}} ∘ J(f)(z, z_{0}). Let z_{1} = ⟨σ_{0}, z_{0}⟩_{β}. Then f(z_{1}) = ⟨σ_{0}, f(z_{0})⟩ = x. We are going to show that φ = z_{1}_{⋄}. z_{1}_{⋄} ≤ φ since for any z ∈ Z,

J(f)(z, z_{1}) = {σ_{0}} ∘ J(f)(z, z_{0})
⊆ φ(z_{0}) ∘ J(f)(z, z_{0})
⊆ φ(z).

Analogously z_{1}^{⋄} ≤ φ̃ since for any z ∈ Z, J(f)(z_{1}, z) = J(f)(z_{0}, z)∘{σ_{0}^{−1}}. Thus z_{1}_{⋄} = φ.

Let us check that given some arrow ⟨f_{1}, h, f_{2}⟩ of G-Set ↓ (X, α), J(h) is a well defined C_{α}-functor. Suppose f_{1} : (Z_{1}, β_{1}) → (X, α) and f_{2} : (Z_{2}, β_{2}) → (X, α). h is an arrow of G-Set that makes the following diagram in G-Set commute:

\xymatrix{(Z_{1}, β_{1}) \ar[dr]_{f_{1}} \ar[rr]^{h} &  & (Z_{2}, β_{2}) \ar[dl]^{f_{2}} \\  & (X, α)}

Since h is an arrow of G-Set, for any z, z′ ∈ Z_{1}, J(f_{1})(z, z′) ⊆ J(f_{2})(hz, hz′) and the map h : Z_{1} → Z_{2} defines a C_{α}-functor from J(f_{1}) to J(f_{2}).

Now it is straightforward to check that J as defined above is a functor from G-Set ↓ (X, α) to C_{α}-Cat.

Let us check that given some skeletal reversible Cauchy complete enrichment A on C_{α}, J′(A) = (Obj(A), β) is a well defined object of G-Set ↓ (X, α).

First we need to prove the property (∗): Given x ∈ X, a_{0} ∈ Obj(A) and σ_{0} ∈ G such that ⟨σ_{0}, (a_{0})_{+}⟩ = x, there is a unique a_{1} ∈ Obj(A) with σ_{0} ∈ A(a_{0}, a_{1}). It satisfies (a_{1})_{+} = x.

Let x, a_{0}, and σ_{0} be as above and φ : \xymatrix{x̂ \ar[r]|{∘} & A} be the left adjoint module defined by φ(a) = {σ_{0}} ∘ A(a, a_{0}). Due to the Cauchy completeness of A, there is an a_{1} ∈ Obj(A) such that φ = a_{1}_{⋄}. Such an a_{1} satisfies therefore (a_{1})_{+} = x and σ_{0} ∈ A(a_{0}, a_{1}). Due to skeletality, such an a_{1} is unique.

According to (∗), β : G × Obj(A) → Obj(A) is a well defined map. β defines indeed an action of G on Obj(A), as shown below (for short we omit the subscript β).

- For any a ∈ Obj(A), ⟨u, a⟩ = a since u ∈ A(a, a).

- For any a ∈ Obj(A) and any g_{1}, g_{2} ∈ G, {g_{1}} ⊆ A(a, ⟨g_{1}, a⟩) and {g_{2}} ⊆ A(⟨g_{1}, a⟩, ⟨g_{2}, ⟨g_{1}, a⟩⟩). Therefore

g_{2} · g_{1} ∈ A(⟨g_{1}, a⟩, ⟨g_{2}, ⟨g_{1}, a⟩⟩ ∘ A(a, ⟨g_{1}, a⟩)
⊆ A(a, ⟨g_{2}, ⟨g_{1}, a⟩⟩),

proving ⟨g_{2} · g_{1}, a⟩ = ⟨g_{2}, ⟨g_{1}, a⟩⟩.

Since by definition for any a, a′ ∈ Obj(A), A(a, a′) = {σ ∈ G | ⟨σ, a⟩_{β} = a′}, it is clear that (−)_{+} defines an arrow of G-Set from (Obj(A), β) above to (X, α).

In order to prove that J is an equivalence of category with adjoint J′ we must prove J′ ∘ J ≅ 1 and J ∘ J′ ≅ 1. It is immediate to see that the two natural isomorphisms above are identities.

We found convenient to present now the following result that we shall use later (4.38).

4.25. Lemma. If A is a reversible enrichment on C_{α} with (reversible) Cauchy completion Ā, then J(Ā) is isomorphic to the arrow of G-Set

f : ((G × Obj(A))/ ∼, δ) → (X, α)

where:

- ∼ is the equivalence on G × Obj(A) defined in Lemma 4.22-(2);

- G acts on (G × Obj(A))/ ∼ as follows:

⟨τ, [(σ, z)]⟩_{δ} = [(τ · σ, z)];

- for any a ∈ Obj(A) and any σ ∈ G,

f([(σ, a)]) = ⟨σ, a_{+}⟩_{α}

Proof. First, let us show that the maps δ and f are well defined. Suppose (σ, a) ∼ (σ′, a′) for some σ, σ′ ∈ G and some a, a′ ∈ Obj(A). By definition σ′^{−1} · σ ∈ A(a, a′). Thus for any τ ∈ G, (τ · σ′)^{−1}τ · σ = σ′^{−1} · σ ∈ A(a, a′), showing (τ · σ, a) ∼ (τ · σ′, a′). Also ⟨σ′^{−1}σ, a_{+}⟩_{α} = (a′)_{+} and ⟨σ, a_{+}⟩_{α} = ⟨σ′, (a′)_{+}⟩_{α}. Trivially δ defines an action of G on (G × Obj(A))/ ∼ that is preserved by f.

Further on, let Ā denote the Cauchy completion of A. According to Lemma 4.22- (2), the map h : (G × X)/ ∼→ Obj(Ā) sending [(σ_{0}, a_{0})] to φ such that ∀a, φ(a) = {σ_{0}} ∘ A(a, a_{0}) is well defined and is a bijection.

According to 4.24, in order to conclude, it is enough to show that h defines an isomorphism of C_{α}-Cat from J(f) to Ā. f ∘ h = (−)_{+} since for any φ : \xymatrix{x̂ \ar[r]|{∘} & A} = h([(σ_{0}, a_{0})]), φ_{+} = x = ⟨σ_{0}, (a_{0})_{+}⟩_{α} = f([(σ_{0}, a_{0})]). For any σ, σ′ ∈ G and any a, a′ ∈ Obj(A),

J(f)([σ, a], [σ′, a′]) = {τ ∈ G | ⟨τ, [σ, a]⟩_{δ} = [σ′, a′]}
= {τ ∈ G | [τ · σ, a] = [σ′, a′]}
= {τ ∈ G | σ′^{−1} · τ · σ ∈ A(a, a′)}
= {σ′} ∘ A(a, a′) ∘ {σ^{−1}}
= {σ′} ∘ ⋁_{b∈Obj(A)}(A(b, a′) ∘ A(a, b)) ∘ {σ^{−1}}
= Ā(h([σ, a]), h([σ′, a′])).

4.26. Change of base, reversibility and completions. If U and V are reversible then Conv(U, V) may be provided with a reversible structure. The isomorphism s_{(u,v),(u′,v′)} : Conv(U, V)((u, v), (u′, v′)) ≅ Conv(U, V)((u′, v′), (u, v)) sends any functor F : (u, v) → (u′, v′) to the functor F^{s} : (u′, v′) → (u, v) such that

\xymatrix{U(u, u′) \ar[r]^{F} & V(v, v′) \ar[d]^{s_{v,v′}^{V}} \\ U(u′, u) \ar[r]_{F^{s}} \ar[u]^{s_{u′,u}^{U}} & V(v′, v)}

and any σ : F → G : (u, v) → (u′, v′) to σ^{s} : F^{s} → G^{s} = s_{v,v′}^{V} ∘ σ ∘ s_{u′,u}^{U}.

The s_{(u,v),(u′,v′)} are trivially idempotent. To see that Conv(U, V) satisfies rev − (ii) and rev − (iii), we will need the following lemma. Its proof is left to the reader.

4.27. Lemma. Given functors M : A → B, T : A → C, M′ : A′ → B′, T′ : A′ → C′ and isomorphisms s_{A,A′} : A → A′, s_{B,B′} : B → B′ and, s_{C,C′} : C → C′ such that:

- the left Kan extension L of T along M and the left Kan extension L′ of T′ along M′ exist,

- s_{B,B′} ∘ M = M′ ∘ s_{A,A′},

- s_{C,C′} ∘ T = T′ ∘ s_{A,A′}, then L′ ∘ s_{B,B′} ≅ s_{C,C′} ∘ L.

[figure: .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; A; C; A′; C′; B′; B; M′; T; T′; M; s_{A,A′}; s_{B,B′}; L′; L; s_{C,C′}]

That Conv(U, V) satisfies rev − (ii) is equivalent to the fact that for any functors F : U(u, u′) → V(v, v′) and G : U(u′, u′′) → V(v′, v′′) the diagram below (in Cat) commutes

\xymatrix{U(u, u′′) \ar[r]^{G¯∘F} \ar[d]_{s_{u,u′′}^{U}} & V(v, v′′) \ar[d]^{s_{v,v′′}^{V}} \\ U(u′′, u) \ar[r]_{F^{s}¯∘G^{s}} & V(v′′, v)}

To see that it holds, just apply 4.27 with:

- M = \xymatrix{U(u, u′) × U(u′, u′′) \ar[r]^{^{∘^{U}}} & U(u, u′′)} ,

- T = \xymatrix{U(u, u′) × U(u′, u′′) \ar[r]^{F ×G} & V(v, v′) × V(v′, v′′) \ar[r]^{^{∘^{V}}} & V(v, v′′)} ,

- M′ = \xymatrix{U(u′′, u′) × U(u′, u) \ar[r]^{^{∘^{U}}} & U(u′′, u)} ,

- T′ = \xymatrix{U(u′′, u′) × U(u′, u) \ar[r]^{G^{s}×F^{s}} & V(v′′, v′) × V(v′, v) \ar[r]^{^{∘^{V}}} & V(v′′, v)} ,

and isomorphisms:

- \xymatrix{U(u, u′) × U(u′, u′′) \ar[r]^{≅} & U(u′, u′′) × U(u, u′) \ar[r]^{s_{u′,u′′}×s_{u,u′}} & U(u′′, u′) × U(u′, u)} ,

- \xymatrix{U(u, u′′) \ar[r]^{s_{u,u′′}} & U(u′′, u)} ,

- \xymatrix{V(v, v′′) \ar[r]^{s_{v,v′′}} & V(v′′, v)} .

That Conv(U, V) satisfies rev − (iii) is equivalent to the fact that for any objects u of U and v of V the following diagram commutes.

\xymatrix{U(u, u) \ar[d]_{s_{u,u}^{U}} \ar[r]^{I_{(u,v)}} & V(v, v) \ar[d]^{s_{v,v}^{V}} \\ U(u, u) \ar[r]_{I_{(u,v)}} & V(v, v)} .

This results from applying 4.27 with:

- M = \xymatrix{1 \ar[r]^{I_{u}^{U}} & U(u, u)} ,

- T = \xymatrix{1 \ar[r]^{I_{v}^{V}} & V(v, v)} ,

- M′ = \xymatrix{1 \ar[r]^{I_{u}^{U}} & U(u, u)} ,

- T′ = \xymatrix{1 \ar[r]^{I_{v}^{V}} & V(v, v)} ,

and isomorphisms 1 : 1 → 1, s_{u,u} and, s_{v,v}.

4.28. Definition. A two-sided enrichment F : V → W between reversible locally partially ordered bicategories is reversible when for any objects x, y of F the following diagram commutes

\xymatrix{V(x_{−}, y_{−}) \ar[r]^{F_{x,y}} \ar[d]_{s_{x_{−},y_{−}}^{V}} & W(x_{+}, y_{+}) \ar[d]^{s_{x_{+},y_{+}}^{W}} \\ V(y_{−}, x_{−}) \ar[r]_{F_{y,x}} & W(y_{+}, x_{+})}

By definition for reversible locally partially ordered bicategories U and V, the isomorphism

Caten(U, V) ≅ Conv(U, V)-Cat

sends exactly reversible two-sided enrichments to reversible enrichments. Also, if V is reversible the composite isomorphism

V-Cat ≅ Conv(1, V)-Cat ≅ Caten(1, V)

sends bijectively reversible enrichments onto reversible two-sided enrichments. Note that reversible enrichments form a subcategory of Caten. Thus for any reversible two-sided enrichment F : V → W, the corresponding change of base F_{@} : V → W preserves the reversibility of objects.

Further on V and W will denote two locally partially ordered reversible cocomplete and biclosed bicategories.

Using 3.13, it is straightforward to check that

4.29. Observation. Given a reversible super two-sided enrichment F : V → W, the lax functor F_{♯} : V-Mod → W-Mod defined in 3.15 preserves the reversibility of arrows: for any reversible V-module φ, F_{♯}(φ) is reversible.

According to this and 4.15

4.30. Proposition. For any super reversible two-sided enrichment F : V → W, F_{@} preserves the Morita reversible equivalence: if the reversible V-categories A and B are Morita reversible equivalent then also are F_{@}A and F_{@}B.

Further on, we consider some adjunction F ⊣ G : V ⇀ W in Caten, say F is a 2-functor with local right adjoints G_{a,b} : W(F a, F b) → V(a, b).

4.31. Lemma. F is reversible if and only if G is.

To prove this, we need 4.32. Lemma. Given objects a, b of V, if s : V(a, b) → V(b, a) and s′ : W(F a, F b) → W(F b, F a) are functors then

F_{b,a} ∘ s ≤ s′ ∘ F_{a,b} ⇔ s ∘ G_{a,b} ≤ G_{b,a} ∘ s′.

Proof. (⇒) Consider some arrow of W, α : F a → F b. Since F_{a,b} ⊣ G_{a,b}, F_{a,b}G_{a,b}(α) ≤ α. Therefore s′F_{a,b}G_{a,b}(α) ≤ s′(α). By assumption, F_{b,a} ∘ s ≤ s′ ∘ F_{a,b} thus F_{b,a}sG_{a,b}(α) ≤ s′F_{a,b}G_{a,b}(α) ≤ s′(α). Then, since F_{b,a} ⊣ G_{F b,F a}, sG_{a,b}(α) ≤ G_{b,a}s′(α) (see the diagram in W(F b, F a) below where subscripts a,b are omitted).

\xymatrix{s′(α) & F Gs′(α) \ar[l] \\ s′F G(α) \ar[u] & F sG(α) \ar[l] \ar@{.>}[u]}

(⇐) Consider some arrow of V, α : a → b. Since F_{a,b} ⊣ G_{a,b}, α ≤ G_{a,b}F_{a,b}(α). Therefore s(α) ≤ sG_{a,b}F_{a,b}(α). By assumption s ∘ G_{a,b} ≤ G_{b,a} ∘ s′, thus sG_{a,b}F_{a,b}(α) ≤ G_{b,a}s′F_{a,b}(α). Then since F_{b,a} ⊣ G_{b,a}, F_{b,a}s(α) ≤ s′F_{a,b}(α) (see the diagram in V(b, a) below).

\xymatrix{s(α) \ar[d] \ar[r] & GF s(α) \ar@{.>}[d] \\ sGF (α) \ar[r] & Gs′F (α)}

Proof of 4.31. As consequence of 4.32, one has that for any objects a, b of V, the following diagram (in Cat) commutes

\xymatrix{W(F a, F b) \ar[d]_{s_{F a,F b}^{W}} \ar[r]^{G_{a,b}} & V(a, b) \ar[d]^{s_{a,b}^{V}} \\ W(F b, F a) \ar[r]_{G_{b,a}} & V(b, a)}

if and only the diagram

\xymatrix{V(a, b) \ar[d]_{s_{a,b}^{V}} \ar[r]^{F_{a,b}} & W(F a, F b) \ar[d]^{s_{F a,F b}^{W}} \\ V(b, a) \ar[r]_{F_{b,a}} & W(F b, F a)}

commutes.

Further on, we suppose that the left adjoint F is reversible.

4.33. Lemma. If A is a Cauchy-reversible complete W-category then F^{@}A is Cauchyreversible complete.

Proof. First we can show that for any object v of V and any reversible V-module φ : \xymatrix{v̂ \ar[r]|{∘} & F^{@}A} , there is a V-functor f : v̂ → F^{@}A with f_{⋄} ≥ φ. The proof of this merely an adaptation of the proof of 3.17. Let v be an object of V and φ : \xymatrix{v̂ \ar[r]|{∘} & F^{@}A} be a reversible V-module. According to 4.29, F_{♯}φ : \xymatrix{F_{@}v̂ \ar[r]|{∘} & F_{@}F^{@}A} is reversible. Then also is ϵ_{A}_{⋄} • F_{@}φ : \xymatrix{F_{@}v̂ \ar[r]|{∘} & A} . Due to the Cauchy-reversible completeness of A, there is a V-functor g : F_{@}v̂ → A such that g_{⋄} = ϵ_{A}_{⋄} • F_{♯}φ. Then one concludes as for 3.17. Indeed for the f determined above f_{⋄} = φ, since the category V-RMod(v̂, F^{@}A) is discrete.

Since F_{@} and F^{@} preserves the reversibility of objects and F^{@} preserves the skeletality and the Cauchy-reversible completeness of objects (3.9, 4.33), one deduces from 3.7 and 4.14 the existence for any skeletal and Cauchy-reversible complete V-category A of a natural isomorphism in B:

V-SkCRcCat(A, F^{@}B) ≅_{B} W-SkCRcCat((F_{@}A)^{s}, B).

Let us sum up

4.34. Theorem. For any reversible left adjoint F : V → W in Caten, the functor F^{∼} : W-SkCRcCat → V-SkCRcCat

\{ \begin{matrix}B ↦ F^{@}B, \\ f : B → B′ ↦ F^{@}f : F^{@}B → F^{@}B′ \end{matrix}

has a left adjoint F_{∼} defined on objects by F_{∼}A = (F_{@}A)^{s}.

4.35. Examples (continued). Now we give two applications of theorem 4.34.

4.36. Example. [Geometric morphisms]

Let f : L → H be a continuous map of locales, with corresponding frame morphisms f^{−} : H → L. It corresponds to a 2-functor F : C_{H} → C_{L} that preserves reversibility. F sends any object u to f^{−}u; F_{u,v} sends any morphisms w : u → v (i.e. w ≤ u ∧ v) to f^{−}w : f^{−}u → f^{−}v. F_{u,v} has a right adjoint G_{u,v}, defined for any w ≤ f^{−}u ∧ f^{−}v by

G_{u,v}(w) = ⋁{w′ ≤ u ∧ v | f^{−}w′ ≤ w}.

According to this and 4.20, we may apply 4.34 to retrieve the following well known result.

4.37. Theorem. Any continuous map of locales f : L → H yields an adjunction f_{∗} ⊣ f^{∗} : Sh(H) ⇀ Sh(L), where f_{∗} and f^{∗} are respectively the inverse and direct image functors. f^{∗} is the following. Let f^{−} denote the frame morphism corresponding to f. For any sheaf A : L^{op} → Set, and any open v of H,

(f^{∗}A)(v) = A(f^{−}(v)).

For any sheaves A, B : L^{op} → Set and any natural transformation h : A → B, f^{∗}h is the natural transformation defined in each open v of H by

(f^{∗}h)_{v} = h_{f^{−}v} : A(f^{−}(v)) → B(f^{−}(v)).

To see this, it is enough to check that given such an f : L → H, if F^{∼} denotes the right adjoint of 4.34 for the F defined above, then f^{∗} is F^{∼} up to the equivalence of 4.20. Which is immediate.

4.38. Example. [Group actions]

Recall that any group morphism φ : G → H induces an adjoint pair φ_{∗} ⊣ φ^{∗} : G-Set ⇀ H-Set. φ_{∗} is named the extension functor) and φ^{∗}, the restriction functor. Actually φ^{∗} has also a right adjoint named the induction functor.

For any object (Y, β) of H-Set, φ^{∗}((Y, β)) is the set Y with action

\{ \begin{matrix}G × Y → Y, \\ (σ, y) ↦ ⟨φ(σ), y⟩_{β}. \end{matrix}

An arrow from (Y_{1}, β_{1}) to (Y_{2}, β_{2}) of H-Set with underlying map p : Y_{1} → Y_{2} has image by φ^{∗}, the arrow of G-Set (from φ^{∗}((Y_{1}, α_{1})) to φ^{∗}((Y_{2}, α_{2})) ) with underlying map p.

For any object (X, α) of G-Set, its image by φ_{∗} is the contracted product H ∧^{G} (X, α) defined as follows. Its underlying set is

(H × X)/ ∼

with ∼ the equivalence defined by

∀τ, τ′ ∈ H, ∀x, x′ ∈ X, (τ, x) ∼ (τ′, x′) ⇔ ∃σ ∈ G, ⟨σ, x⟩_{α} = x′, τ = τ′ · φ(σ).

Our purpose is to show the later adjunction may be retrieved as an instance of 4.34. Precisely, considering a category Grp-Set of group actions, described below, we will show that any arrow of Grp-Set, (φ, f) : (G, X, α) → (H, Y, β) yields an adjunction (φ, f)_{∼} ⊣ (φ, f)^{∼} with (φ, f)_{∼} : G-Set ↓ (G, X, α) → H-Set ↓ (H, Y, β). The previous pair φ_{∗} ⊣ φ^{∗} corresponding then to the adjunct pair (φ, 1_{{∗}})_{∼} ⊣ (φ, 1_{{∗}})^{∼} with (φ, 1_{{∗}})_{∼} : (G, {∗},!_{G}) → (H, {∗},!_{H}), {∗} denoting the one-point set, 1_{{∗}}, the identity map on this set, and !_{G} and !_{H}, the respective unique actions of G and H on this set.

The category Grp-Set of group actions is as follows. Objects are group actions. Arrows from (G, X, α) to (H, Y, β) are the pairs (φ, f) where φ : G → H is a group morphism, f : X → Y is a map and, the following diagram commutes:

\xymatrix{G × X \ar[r]^{α} \ar[d]_{φ×f} & X \ar[d]^{f} \\ H × Y \ar[r]_{β} & Y}

Consider an arrow (φ, f) : (G, X, α) → (H, Y, β). It corresponds to a 2-functor F : C_{α} → C_{β} that preserves reversibility. F sends any object x to f(x) and, F_{x,x′} sends any arrow L : x → x′ — i.e. L ⊆ {σ ∈ G | ⟨σ, x⟩_{α} = x′} to the arrow

φ(L) = {φ(σ) | σ ∈ L} : f(x) → f(x′).

F has local right adjoints: for any x, x′ ∈ X, G_{x,x′}, the right adjoint to F_{x,x′}, is defined for any L ⊆ {τ ∈ H | ⟨τ, f(x)⟩_{β} = f(x′)} by

G_{x,x′}(L) = φ^{−1}(L) ∩ {σ ∈ G | ⟨σ, x⟩_{α} = x′}.

Therefore we may apply 4.34 together with 4.24 and 4.25 to obtain the following result.

4.39. Theorem. Any arrow (φ, f) : (G, X, α) → (H, Y, β) of Grp-Set yields some adjunction (φ, f)_{∼} ⊣ (φ, f)^{∼} : G-Set ↓ α ⇀ H-Set ↓ β, described below.

- If h : (Z, γ) → (Y, β) is an arrow of H-Set, its image by (φ, f)^{∼} is the arrow of G-Set, π^{2} : (Z×_{h,f}X, δ) → (X, α), with

Z×_{h,f}X = {(z, x) | h(z) = f(x)},

for any (z, x) ∈ Z×_{h,f}X,

π^{2}((z, x)) = x,

and for any σ ∈ G and any (z, x) ∈ Z×_{h,f}X,

⟨σ, (z, x)⟩_{δ} = (⟨φ(σ), z⟩_{γ}, ⟨σ, x⟩_{α}).

- (φ, f)^{∼} is defined on arrows as follows. Considering the following commuting diagram in H-Set:

\xymatrix{(Z_{1}, γ_{1}) \ar[dr]_{h_{1}} \ar[rr]^{p} & & (Z_{2}, γ_{2}) \ar[dl]^{h_{2}} \\ & (Y, β)} .

the image by (φ, f)^{∼} of the arrow ⟨h_{1}, p, h_{2}⟩ of H-Set ↓ (Y, β) has underlying arrow in G-Set

p′ : (Z_{1} ×_{(h_{1},f)} X, δ_{1}) → (Z_{2} ×_{(h_{2},f)} X, δ_{2})

where for any (z, x) ∈ Z_{1} ×_{(h_{1},f)} X,

p′((z, x)) = (p(z), x)

— see the commuting diagram in G-Set below where for i = 1, 2, (φ, f)^{∼}(h_{i}) = π^{2} : (Z_{i} ×_{(h_{i},f)} X, δ_{1}) → (X, α).

\xymatrix{(Z_{1} ×_{(h_{1},f)} X, δ_{1}) \ar[dr]_{π^{2}} \ar[rr]^{p′} &  & (Z_{2} ×_{(h_{2},f)} X, δ_{2}) \ar[dl]^{π^{2}} \\  & (X, α)}

- (φ, f)_{∼} is defined on objects of G-Set ↓ (X, α) as follows. For any arrow g : (Z, γ) → (X, α) of G-Set, (φ, f)_{∼}(g) is the arrow

h : ((H × Z)/ ∼, δ) → (Y, β)

of H-Set where ∼ is the equivalence on H × Z defined by

∀τ, τ′ ∈ H, ∀z, z′ ∈ Z, ( (τ, z) ∼ (τ′, z′) ⇔ ∃σ ∈ G, ⟨σ, z⟩ = z′, τ = τ′ · φ(σ) )

for any τ, κ ∈ H and z ∈ Z,

⟨κ, [(τ, z)]⟩_{δ} = [(κ · τ, z)]

for any τ ∈ H and z ∈ Z,

h([(τ, z)]) = ⟨τ, f ∘ g(z)⟩_{β}.

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Dip. Scienze dell’Informazione, Universitá di Roma “La Sapienza”, via Salaria, 113, 00198 Roma, Italy, and Department of Mathematics and Computer Science, University of Leicester, University Road, Leicester LE1 7RH, England

Email: labella@dsi.uniroma1.it
      vs27@mcs.le.ac.uk

This article may be accessed via WWW at http://www.tac.mta.ca/tac/ or by anony-

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