Change of base, Cauchy-completness and reversibility

Anna Labella ^{∗}, Vincent Schmitt, ^{†}

20/09/00

Abstract: We investigate the effect of the change of base 2-functors V − Cat → W − Cat induced by two-sided enrichments V → W on Cauchy-complete objects. We restrict our study to the case of locally posetal bases. Reversibility is defined for locally posetal bicategories, two-sided enrichments and, Cauchy-completion. We show that a reversible left adjoint two-sided enrichment F : V → W between locally posetal bicategories induces some adjunction F_{∼} ⊣ F^{∼} : V − SkCRcCat → W − SkCRcCat between sub-categories of skeletal and Cauchy-reversible-complete enrichments. We give two examples of application, sheaves and group actions.

keywords: Enriched categories, two-sided enrichments, change of base, reversibility, Cauchy-completion, sheaves.

1 Introduction

Our motivation for the study of the change of base for enrichments over bicategories originated with the characterisation of sheaves as skeletal and Cauchycomplete enrichments over bicategories [Wal81], together with the fact a continuous map f : X → Y yields an adjunction between the categories of sheaves f_{∗} ⊣ f^{∗} : Sh(Y) ⇀ Sh(X). The problem was first to define a good notion of base morphism F : V → W. We had in mind that such a morphism should first induce a 2-adjunction at the level of enrichments, say F_{@} ⊣ F^{@} : V − Cat → W − Cat. The secund point is that this adjunction should also “behave well” with respect to the skeletality and Cauchy-completness of objects: there should exist a categorical connection between the sub-2-categories of skeletal and Cauchy-complete objects respectively of V − Cat and W − Cat. The first point of our problem was answered in [KLSS99]. Let us start from the definition of monoidal functor 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 where proposed in the literature, but adjoints were not preserved by (−)_{@}. A two-sided enrichment F : V → W between two bicategories was proposed in [KLSS99] as a slight generalisation of monoidal functors and Benabou’s lax functors [Ben67]. With these new morphisms 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 they were named “two-sided enrichments”. Adjunctions in Caten were characterised: left adjoint two-sided enrichments are exactly the pseudofunctors with “local right adjoints”. This notion of local adjointness 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)). The two-sided enrichments are also just enrichments since thus Caten(V, W) ≅ Conv(V, W)−Cat.

∗Dip. Scienze dell’Informazione, Università di Roma “La Sapienza”, via Salaria, 113, 00198 Roma, Italy, e-mail: labella@dsi.uniroma1.it

†Department of Mathematics and Computer science, University of Leicester, University road, Leicester LE1 7RH, England e-mail: vs27@mcs.le.ac.uk

This paper treats the second point of our original problem: the effect of change of base F_{@} for a two-sided enrichment F, on skeletal and Cauchy-complete objects. For convenience, since our purpose was to give a satisfying categorical generalisation of the above adjunction between categories of sheaves and also for convenience, we restricted our study to the case when the base bicategories are locally posetal.

First we investigate conditions on two-sided enrichments F such that F_{@} preserves skeletality, Cauchy-completion, Morita-equivalence, and so on. We introduce the super two-sided enrichments. They generalise the super monoidal functors used to code uniformly continuous maps as some enriched functors in [Sc00]. They are more general than left adjoint two-sided enrichments and have 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 need to introduce the notion of reversibility for bicategories, twosided-enrichments and Cauchy-completion. Most familiar examples of enrichments over locally posetal bicategories (sheaves and group actions) enjoy a nice property of symmetry that we call 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 Cauchy-reversible-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 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 posetal bicategories to make the paper self-contained and to allow later comparisons with the reversible case. 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 posetal. We first recall some elements of the theory of two-sided enrichments developped [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-completness 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 and group actions.

2 Basic enriched category theory

Definition 2.1 A bicategory V is locally posetal (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 posetal bicategory.

Definition 2.2 (Enrichments over a locally posetal bicategory) 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).

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

Further on, we suppose that:

- V is biclosed i.e. for any arrow f : a → b and any object c of V, the functors f ∘ − : V(c, a) → V(c, b) and − ∘ f : V(c, b) → V(c, a) are adjoints.

- V is locally cocomplete i.e. each homset V(a, b) is cocomplete.

Definition 2.3 (V-module) 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:

- (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 V-modules from A to B is partially ordered by φ ≤ ψ ⇔ ∀(b, a) ∈ Obj(B) × Obj(A), φ(b, a) ≤ ψ(b, a).

Fact 2.4 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 (y, x) to A(y, x). 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 all objects a of A, and b of B, f_{⋄}(b, a) = B(b, fa), f^{⋄}(a, b) = B(fa, b). 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.

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.

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_{⋄}.

Definition 2.5 (Skeletality, Cauchy-completness) Let B be a V-category. B is called skeletal when for any V-category A, the map (−)_{⋄} : V − Cat(A, B) → V − AMod(A, B) is injective. B is called Cauchy-complete when for any V-category A, the map (−)_{⋄} : V − Cat(A, B) → V − AMod(A, B) is surjective.

Observation 2.6 A V-category B is skeletal when for any of its objects a and b, if a_{+} = b_{+}, I_{a_{+}} ≤ B(a, b) and, I_{a_{+}} ≤ B(b, a) then a = b.

Fact 2.7 A V-category B is Cauchy-complete when 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. 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) = ⋁_{b′∈Obj(B)} 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 posetal 2-categories.

- 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.

Fact 2.8 The restriction of J_{V} on V − SkCcCat is an isomorphism onto V − SkCcAMod.

Definition 2.9 (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 Vmodules 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 2.9 is a well defined V-category. Ā trivially satisfies (enr − i). Let us show that it satifies (enr − ii). Let φ 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(x) ∘ 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).

Remark 2.10 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.

Fact 2.11 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.

Lemma 2.12 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)

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

= ⋁_{ψ∈Obj(Ā)} Γ(ψ) ∘ Ā(φ, ψ)
= Γ(φ).

Fact 2.13 For any V-category A, Ā is Cauchy-complete. proof: Given a right 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 2.12 (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, 2.12 shows that Γ = Ā(−, γ).

Lemma 2.14 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 adjoint modules. Therefore, for any objects a, b of A,

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

= Ā(b_{⋄}, a_{⋄})

= 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,

Fact 2.15 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.

According to 2.15, skeletal and Cauchy-complete V-categories form a reflective subcategory of the underlying category of V − Cat. The left adjoint of the inclusion sends a V-category A to Ā and the unit takes value i_{A} : A → Ā in A.

According to 2.11 and 2.14, the following is coherent

Definition 2.16 (Morita-equivalence) Two V-categories A and B are Moritaequivalent when one 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 general 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 posetal (3.1-3.4). We keep on using the original terminology of [KLSS99]. Section 3.5 presents a study of the effect on the change of base bicategories on skeletality and Cauchy-completness of enrichments. Super two-sided enrichments are introduced.

3.1 Two-sided enrichments

Caten is the tricategory as follows. Its objects are locally posetal 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 posetal 2-category 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 posetal, 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_{+}}^{V} ≤ B(fa, ga)(I_{a_{−}}^{W}) : 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

For any (locally posetal) bicategory V, the 2-category V − Cat is representable

V − Cat ≅ Caten(1, V).

The cartesian product of locally posetal bicategories extends straightforwardly to a pseudofunctor 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:

Definition 3.1 (The bicategory Conv(V,W)) Given two locally posetal bicategories V, W the locally posetal bicategory Conv(V, W) is defined as follows. Its objects are ordered pairs (v, w) ∈ Ob(V) × Ob(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′′), F ¯∘G(h) = ⋁{G(g) ∘ F (f) | g ∘ f ≤ h}. The identity in (u, v) is the left Kan extension of I_{w}^{W} : 1 → W(w, w) along 1_{v}^{V} : 1 → V(v, v), thus for any arrow f in V(v, v), 1_{(v,w)}(f) = V(v, v)(I_{v}, f) · 1_{v}.

There is a further connection between enrichments over bicategories and two-sided enrichments: two-sided enrichments are also just enrichments since

Caten(V, W) ≅ Caten(1, Conv(V, W)) ≅ Conv(V, W) − Cat.

3.3 Adjoints in Caten

Adjoints F ⊣ G : V → W in Caten are (up to isorphism) 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_{a,b} : V(a, b) → W(F a, F b) have right adjoints G_{a,b} : W(F a, F b) → V(a, b). Indeed the left adjoint F may be seen as a 2-functor V → W with “local (right) adjoints”. In this case, the above family of right adjoints satisfy some coherence properties. It can be shown that for any lax functor F : V → W such that for any objects a, b in V, the restriction F : V[a, b] → W[F a, F b] admits a right adjoint G_{a,b} : W[F a, F b] → V[a, b], these adjoints satisfy the coherence conditions, on composition:

(1) for all objects a,b,c of V, for all β : F a → F b and γ : F b → F c,
G_{b,c}γ ∘ G_{a,b}β ≤ G_{a,c}(γ ∘ β);

and on identities:

(2) for all objects a of A, I_{a} ≤ G_{a,a}I_{F a}.

if and only if F is a 2-functor.

3.4 The change of base for enrichments

Using the representability for V-categories 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,

Theorem 3.2 There is a pseudo-functor (−)_{@} from Caten to 2 − Cat (the 2category 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(V), 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).

Note that 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)(x, y) = F A(x, y); 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,

Theorem 3.3 If the 2-functor between locally posetal 2-categories 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 collections of local adjoints G_{u,v} defining the right adjoint G to F in Caten.

3.5 Change of base, skeletality and Cauchy-completion

Further on we consider a two-sided enrichment F : V → W.

Lemma 3.4 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′.

Further on we suppose moreover that V and V are locally cocomplete and biclosed.

Fact 3.5 There is a lax normal ^{1} functor F_{♯} : V − Mod → W − Mod that extends F_{@} in the sense the diagram above commutes:

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

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

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_{+})]

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})_{⋄}.

Lemma 3.6 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 quality 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 in order F_{@} to preserve the Morita-equivalence.

Definition 3.7 (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}} & a_{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,a_{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.

Lemma 3.8 If F is super then 1 ≤ F_{♯}(i_{A}_{⋄}) • F_{♯}(i_{A}^{⋄}) : F_{@}A → F_{@}A in W − Mod.

proof: Consider some objects φ, ψ of Ā, and x, y of F, with φ : \xymatrix{ˆ (x_{−}) \ar[r]|{∘} & A} ,

ψ : \xymatrix{ˆ (y_{−}) \ar[r]|{∘} & A} . Since I_{x_{−}}^{V} ≤ ⋁_{a∈Obj(A)} φ(a) ∘ φ̃(a),

(F_{♯}(i_{A}_{⋄}) • F_{♯}(i_{A}^{⋄}))((φ, x), (ψ, y)) = ⋁_{(a,z)∈Obj(F_{@}A)}( F_{♯}(i_{A}^{⋄})((a, z), (ψ, y))∘

F_{♯}(i_{A}_{⋄})((φ, x), (a, z)) )

= ⋁_{(a,z)∈Obj(F_{@}A)} F_{z,y}ψ(a) ∘ F_{x,z}φ̃(a)

≥ ⋁_{(a,z)∈Obj(F_{@}A)}(F_{z,y}(Ā(φ, ψ) ∘ φ(a)) ∘ F_{x,z}φ̃(a)) ≥ ⋁_{(a,z)∈Obj(F_{@}A)}((F_{x,y}Ā(φ, ψ) ∘ F_{z,x}φ(a)) ∘ F_{x,z}φ̃(a)) = F_{x,y}Ā(φ, ψ) ∘ ⋁_{(a,z)∈Obj(F_{@}A)}(F_{z,x}φ(a) ∘ F_{x,z}φ̃(a))

≥ F_{x,y}Ā(φ, ψ) ∘ I_{x_{+}}
≥ F_{x,y}Ā(φ, ψ).

According to this and 3.6,

Lemma 3.9 If F is super then F_{♯}(i_{A}_{⋄}) is an isomorphism, in this case 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.

One has also

Lemma 3.10 If F is super then F_{♯} : V − Mod → W − Mod preserves adjoints.

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.

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}.

Fact 3.11 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 which relevance will appear in the next section treating reversibility.

Lemma 3.12 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);

Also, (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)

But since (a, a_{+}) ∈ Obj(F^{@}F_{@}A) and 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)

Lemma 3.13 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 some object of V and φ : \xymatrix{û \ar[r]|{∘} & F^{@}A} be a left adjoint V-module. According to Fact 3.11, F_{♯}φ : \xymatrix{F_{@}â \ar[r]|{∘} & F_{@}F^{@}A} is some 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 arrow 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 3.12, and 2-cells (II) is given by the laxity of G_{♯}.

4 Reversibility

In this section we will restrict the theory of two-sided enrichments over locally posetal bicategories to the symmetric case. We will define the notion of reversibility for bicategories, two-sided enrichments, modules, and completion. Definitions will be suggested by the examples and will allow to rebuild the complete theory in this special case. Eventually we study the effect of the change of bases on the completness and skeletality of enrichments in the reversible context. The main result is theorem 4.31.

4.1 Reversible enrichments

Definition 4.1 A 2-category V is reversible when there are isomorphisms of categories s_{u,v} : V(u, v) ≅ V(v, u) for 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; u_{v}; V(v, v); s_{v,v}; 1; u_{v}; V(v, v)]

Further on V is a reversible locally posetal bicategory.

Definition 4.2 (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)
- - - - - - - - - - - - - - - - - - - - - , g ⇒ f.

Observation 4.3 If B above is moreover skeletal then ⇒ is the identity on V − Cat(A, B).

Further on we assume that the bicategory V is biclosed and locally cocomplete.

Fact 4.4 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 further notion of Cauchy-reversible-completion.

Definition 4.5 (Reversible modules) A V-module φ between reversible V-categories is reversible when it has right adjoint φ^{s}.

Remark 4.6 If f is a V-functor between reversible V-categories then the V-module f_{⋄} is reversible.

Definition 4.7 (Cauchy-reversible-completness) A reversible V-category B is Cauchy-reversible-complete when one 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 2.7.

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

Definition 4.8 (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 V-category 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.

Similarly to 2.9, one checks that the data above 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 4.7. 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 2.11 is

Fact 4.9 For any reversible V-category A, A^{s} is skeletal.

Adapting the proof of 2.12 one shows

Lemma 4.10 Let A be a reversible V-category and, Γ : \xymatrix{û \ar[r]|{∘} & A^{s}} be some Vmodule where u is an object of V. For any object φ of A^{s}, Γ(φ) = ⋁_{a∈Obj(A)}Γ(a_{⋄}) ∘ φ^{s}(a).

Fact 4.11 For any reversible V-category A, A^{s} is Cauchy-reversible-complete.

proof: Merely an adaptation of the proof of 2.13. Given Γ : \xymatrix{û \ar[r]|{∘} & A^{s}} a reversible V-module. 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 4.10, Γ = A^{s}(−, γ).

Lemma 4.12 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 2.14. 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 4.12,

Proposition 4.13 V − SkCRcCat and V − RMod are 2-equivalent. There is a 2-functor S′ : V − RMod → V − SkCRcCat which is an equivalence of categories. S′ 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 −SkCsycCat → V −RCat has a left 2-adjoint.

According to 4.3, 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,

Definition 4.14 (Morita-reversible-equivalence) Two reversible V-categories A and B are Morita-reversible-equivalent when one 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.

4.2 Examples

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

Example 4.15 (Sheaves) [Wal81]

To any locale L corresponds a locally posetal 2-category 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.

Theorem 4.16 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.

It happens in this case that Cauchy-reversible-complete enrichments are Cauchycomplete. This is due to

Fact 4.17 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 4.16

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

Example 4.19 (Group actions)

Let (G, X, α) denote the action α of the group G on the set X. We note for such an action:

- < g, x >_{α} (or simply < g, x >), the action of g ∈ G on x ∈ X,

- “·”, the composition law of G,

- u, the unit of G. (G, X, α) may be also written (X, α).

Given a group G, G − Set denotes the category with objects actions of G 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, a locally posetal 2-category C_{(G,X,α)} is defined as follows:

- Objects are the x ∈ X;

- Arrows from x to x′ are the subsets of {g ∈ G |< g, x >= x′};

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

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

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

Lemma 4.20 Let (G, X, α) be a group action and A be a reversible enrichment on C_{(G,X,α)}.

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

(∗) ∀z ∈ Obj(A), φ(z) = {σ_{0}} ∘ A(z, z_{0})

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

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

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

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

∀z ∈ Obj(A), φ(z) = {σ_{0}} ∘ A(z, z_{0})

and

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

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

∀σ, σ′ ∈ G, ∀z, z′ ∈ Obj(A), (σ, z) ∼ (σ′, z′) ⇔ (∃γ ∈ A(z, z′), σ = σ′ ∘ γ).

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

φ(z) ∘ A(z′, z) = {σ_{0}} ∘ A(z, z_{0}) ∘ A(z′, z)
⊆ {σ_{0}} ∘ A(z′, z_{0})
= φ(z′)

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

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

For any z, z′ ∈ Obj(A),

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

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

ψ(z) = {σ_{0}} ∘ A(z, z_{0})
⊆ φ(z_{0}) ∘ A(z, z_{0})
⊆ φ(z).

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

(2) : First let us check that ∼ is an equivalence relation. It is reflexive since for any object z of A, u ∈ A(z, z). It is symmetric since if (σ, z) ∼ (σ′, z′) then there is a γ ∈ A(z, z′) such that σ = σ′ ∘ γ, then, γ^{−1} ∈ A(z′, z) and σ′ = σ ∘ γ^{−1}. To check transitivity, suppose (σ, z) ∼ (σ′, z′) and (σ′, z′) ∼ (σ′′, z′′). Then there are some γ ∈ A(z, z′) and some γ′ ∈ A(z′, z′′) such that σ = σ′ ∘ γ and σ′ = σ′′ ∘ γ′. Thus γ′ ∘ γ ∈ A(z′, z′′) ∘ A(z, z′) ⊆ A(z, z′′) and σ = σ′′ ∘ γ′ ∘ γ.

Consider further on φ, ψ : \xymatrix{x̂ \ar[r]|{∘} & A} , σ_{0}, σ′_{0} ∈ G and, z_{0}, z′_{0} ∈ Obj(A) such that ∀z ∈ Obj(A), φ(z) = {σ_{0}} ∘ A(z, z_{0}) and ψ(z) = {σ′_{0}} ∘ A(z, z′_{0}). First, suppose (σ_{0}, z_{0}) ∼ (σ′_{0}, z′_{0}) i.e., there is some γ ∈ A(z_{0}, z′_{0}) such that σ_{0} = σ′_{0} ∘ γ. Then for any z ∈ Obj(A),

φ(z) = {σ_{0}} ∘ A(z, z_{0})
= {σ′_{0}} ∘ {γ} ∘ A(z, z_{0})
= {σ′_{0}} ∘ A(z_{0}, z′_{0}) ∘ A(z, z_{0})
⊆ {σ′_{0}} ∘ A(z, z′_{0})
= ψ(z),

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

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

Corollary 4.21 Let (G, X, α) be a group action and A be a reversible enrichment on C_{(G,X,α)}. Any left adjoint module φ : \xymatrix{x̂ \ar[r]|{∘} & A} has right adjoint φ^{s}.

Proposition 4.22 Let (G, X, α) be a group action. The category of skeletal and Cauchy-reversible-complete enrichments on C_{(G,X,α)} is equivalent to the category G− Set ↓ (X, α) of objects over (X, α). Precisely the equivalence J : G − Set ↓ (X, α) ≅ C_{(G,X,α)} − SkCcCat 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 ↓ (X, α) with f_{1} : (Z_{1}, β_{1}) → (X, α) and f_{2} : (Z_{2}, β_{2}) → (X, α), J(h) is the C_{(G,X,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_{(G,X,α)}, J′(A) = (G, Obj(A), β) where for any σ ∈ G and any z ∈ Obj(A), < σ, z >_{β} is the unique z′ such that σ ∈ A(z, z′).

proof: Let us check that given f : (Z, β) → (X, α), J(f) is a skeletal Cauchyreversible-complete C_{(G,X,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_{(G,X,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′ ∈ G, 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 4.20 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_{(G,X,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_{(G,X,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_{(G,X,α)} − Cat.

Let us check that given some skeletal reversible Cauchy-complete enrichment A on C_{(G,X,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, z_{0} ∈ Obj(A) and σ_{0} ∈ G such that < σ_{0}, (z_{0})_{+} >= x, there is a unique z_{1} ∈ Obj(A) with (z_{1})_{+} = x and {σ_{0}} ⊆ A(z_{0}, z_{1}).

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

- For any z ∈ Obj(A) and any g_{1}, g_{2} ∈ G, {g_{1}} ⊆ A(z, < g_{1}, z >) and {g_{2}} ⊆ A(< g_{1}, z >, < g_{2}, < g_{1}, z >>). Therefore

g_{2} · g_{1} ∈ A(< g_{1}, z >, < g_{2}, < g_{1}, z >> ∘A(z, < g_{1}, z >)
⊆ A(z, < g_{2}, < g_{1}, z >>),

proving < g_{2} · g_{1}, z >=< g_{2}, < g_{1}, z >>.

Since by definition for any z, z′ ∈ Obj(A), A(z, z′) = {σ ∈ G | < σ, z >_{β} = z′}, 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 isomorphims above are identities.

We found convenient to present now the following result that we shall use later (4.34).

Lemma 4.23 If A is a reversible enrichment on C_{(G,X,α)} with (reversible) Cauchycompletion 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 4.20-(2);

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

< τ, [(σ, z)] >_{δ} = [(τ ∘ σ, z)];

- for any z ∈ Obj(A) and any σ ∈ G,

f([(σ, z)]) = < σ, z_{+} >_{α}

proof: First, let us show that the maps δ and f are well defined. Suppose (σ, z) ∼ (σ′, z′) for some σ, σ′ ∈ G and some z, z′ ∈ Obj(A). By definition there is a γ ∈ A(z, z′) such that σ = σ′∘γ. For any τ ∈ G, τ ∘σ = τ ∘σ′∘γ, therefore (τ ∘σ, z) ∼ (τ ∘ σ′, z′). Since γ ∈ A(z, z′), < γ, z_{+} >_{α} = (z′)_{+} and < σ, z_{+} >_{α} = < σ′ ∘ γ, z_{+} >_{α} = < σ′, (z′)_{+} >_{α}. δ trivially defines an action of G on (G × Obj(A))/ ∼.

Further on, let Ā denote the Cauchy-completion of A. According to Lemma 4.20-(2), the map h : Obj(Ā) → (G × X)/ ∼ sending any φ to [(σ_{0}, z_{0})] such that ∀z, φ(z) = {σ_{0}} ∘ A(z, z_{0}) is well defined and is a bijection.

According to 4.22, in order to conclude, it is enough to show that h defines an isomorphism of C_{(G,X,α)} − Cat from Ā to J′(f) = ((G × Obj(A))/ ∼, f, d). f ∘ h = (−)_{+} since for any φ : \xymatrix{x̂ \ar[r]|{∘} & A} = h([(σ_{0}, z_{0})]), φ_{+} = x = < σ_{0}, (z_{0})_{+} >_{α} = f([(σ_{0}, z_{0})]). For any σ, σ′ ∈ G and any z, z′ ∈ Obj(A),

A([σ, z], [σ′, z′]) = {τ ∈ G | < τ, [σ, z] >_{δ} = [σ′, z′]}
= {τ ∈ G | [τ ∘ σ, z] = [σ′, z′]}
= {τ ∈ G | ∃γ ∈ A(z, z′), τ ∘ σ = σ′ ∘ γ}
= {σ′} ∘ A(z, z′) ∘ {σ^{−1}}
= {σ′} ∘ ⋁_{y∈Obj(A)}(A(y, z′) ∘ A(z, y)) ∘ {σ^{−1}}
= Ā(h([σ, z]), h([σ′, z′])).

4.3 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 which proof is left to the reader.

Lemma 4.24 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}; s_{B}; s_{C}; L′; L]

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 4.24 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]^{^{∘^{V}}} & 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]^{1_{(u,v)}} & V(v, v) \ar[d]^{s_{v,v}^{V}} \\ U(u, u) \ar[r]_{1_{(u,v)}} & V(v, v)} .

This results from applying 4.24 with:

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

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

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

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

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

Definition 4.25 A two-sided enrichment F : V → W between reversible locally posetal 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 posetal bicategories U and V, the isomorphism

Caten(U, V) ≅ Conv(U, V) − Cat

sends exactly reversible two-sided enrichments to reversible enrichments. Conversely, 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. It is immediate 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 posetal reversible bicategories.

Using 3.10, it is straightforward to check that

Observation 4.26 Given a reversible super two-sided enrichment F : V → W, the lax functor F_{♯} : V − Mod → W − Mod defined in 3.11 preserves the reversibility of arrows: for any reversible V-module φ, F_{♯}(φ) is reversible.

According to this and 4.12

Fact 4.27 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 Moritareversible-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).

Lemma 4.28 F is reversible if and only if G is.

To prove this, we need

Lemma 4.29 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 omited).

\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 4.28: As consequence of 4.29, 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.

Lemma 4.30 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 3.13. Let v be an object of V and φ : \xymatrix{v̂ \ar[r]|{∘} & F^{@}A} be a reversible V-module. According to 4.26, 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-completness of A, there is a V-functor g : F_{@}v̂ → A such that g_{⋄} = ϵ_{A}_{⋄} • F_{♯}φ. Then one concludes as for 3.13. 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-completness of objects (3.4, 4.30), one deduces from 3.3 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

Theorem 4.31 For any reversible left adjoint F : V → W in Caten, the functor F^{∼} : W − SkCRcCat → W − 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}.

4.4 Examples (continued)

Now we give two applications of theorem 4.31.

Example 4.32 (Geometric morphisms)

Let f : L → H be a continuous map of locales, with corresponding frame morphims 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}.

Therefore and according to 4.18, we may apply 4.31 to retrieve the following well known result.

Theorem 4.33 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 a f : L → H, if F_{∼} denotes the left adjoint of 4.31 for the F defined above, then f^{∗} is F_{∼} up to the equivalence of 4.18. Which is immediate.

Example 4.34 (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 H-set (Y, β), 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 G-set (X, α), 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 4.31. 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_{(G,X,α)} → C_{(H,Y,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 4.31 together with 4.22 and 4.23 to obtain the following result.

Theorem 4.35 Any arrow (φ, f) : (G, X, α) → (H, Y, β) of Grp − Set yields some adjunction (φ, f)^{∼} ⊣ (φ, f)_{∼}, with (φ, f)^{∼} : G − Set ↓ (X, α) → H − Set ↓ (Y, β), given 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, φ ∘ g(z) >_{β} .

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