Theory and Applications of Categories, Vol. 26, No. 14, 2012, pp. 349–384.

BIEQUIVALENCES IN TRICATEGORIES

NICK GURSKI

Abstract. We show that every internal biequivalence in a tricategory T is part of a biadjoint biequivalence. We give two applications of this result, one for transporting monoidal structures and one for equipping a monoidal bicategory with invertible objects with a coherent choice of those inverses.

Introduction

It is common in mathematics to regard two objects X and Y as being the same if there is an isomorphism between them, that is a pair of maps f : X → Y and g : Y → X such that fg = 1_{Y} and gf = 1_{X}. More importantly, any specific isomorphism f : X → Y gives an explicit means for transporting information about X to information about Y. The choice of ambient category plays a very important role in this process, especially when we examine two categories with the same objects but different morphisms. For example, first take the category of CW-complexes and continuous maps. In this category, an isomorphism between X and Y gives a formula for transporting cell structures from X to Y. If we now take the category of CW-complexes and homotopy classes of continuous maps, then isomorphisms are now homotopy equivalences. An isomorphism then no longer gives a recipe for transporting cell structures, but instead only gives a recipe for transporting homotopical information like homotopy and homology groups.

An important property of an isomorphism f : X → Y is that it uniquely determines the inverse g : Y → X by the formulas fg = 1_{Y} and gf = 1_{X}; the proof is exactly the same as that showing that any group element has a unique inverse. Thus we have two concepts which are a priori different:

• the property that f : X → Y is an isomorphism, and

• the structure consisting of a pair (f, g) of morphisms f : X → Y, g : Y → X such that fg = 1_{Y} and gf = 1_{X}.

We have then uncovered that a morphism f has the property of being an isomorphism if and only if there is a pair (f, g) with the isomorphism structure, and that moreover the pair (f, g) is uniquely determined by f alone. This is an example of a structure (being part of an isomorphism pair) that is determined in a unique way by a property (the existence of an inverse).

Received by the editors 2009-12-22 and, in revised form, 2012-08-06.

Transmitted by Ezra Getzler. Published on 2012-08-07.

2000 Mathematics Subject Classification: Primary 18D05, 18D10.

Key words and phrases: biequivalence, biadjoint biequivalence, tricategory, monoidal bicategory.

\begin{matrix}c \\ ◯ \end{matrix} Nick Gurski, 2012. Permission to copy for private use granted.

Moving up to the case in which the objects of study are now the 0-cells of some 2category (or more generally, some bicategory), there are now more possible notions of sameness. While we can ask if two 0-cells of a 2-category are isomorphic, it is much more common to ask if they are equivalent. The canonical example of a 2-category is that of categories, functors, and natural transformations, and in this 2-category we see from experience that equivalence is the natural notion of sameness.

A functor F : X → Y is often defined to be an equivalence if it is essentially surjective, full, and faithful. It is then shown that a functor F is an equivalence if and only if there exists a functor G : Y → X such that the composites F G, GF are naturally isomorphic to the identity functors 1_{Y} , 1_{X}, respectively. This definition identifies a property of a functor F that allows us to conclude that two categories X and Y are, in some sense, the same.

On the other hand, we can make the definition of the structure of adjoint equivalence F ⊣_{eq} G which consists of a functor F : X → Y, a functor G : Y → X, and two natural isomorphisms η : 1_{X} ⇒ GF, ε : F G ⇒ 1_{Y} such that the following two diagrams commute.

[figure: F; F GF; F η; F GF; F; εF; F; F; 1; G; GF G; ηG; GF G; G; Gε; G; G; 1]

It is clear that if F ⊣_{eq} G is an adjoint equivalence, then F is an equivalence. It is also well-known [Mac] that every equivalence F can be completed to an adjoint equivalence. Now F no longer determines G, η, ε uniquely, but instead only determines them up to a unique isomorphism preserving the adjoint equivalence structure. We refer the reader to the paper [KL] for a general discussion of this phenomenon in the setting of algebras for a 2-monad.

The aim of this paper is to establish an analogous result in three-dimensional category theory. We define two different notions of sameness internal to a tricategory, one a property of a 1-cell and the other a structure involving 1-, 2-, and 3-cells satisfying certain axioms. These two notions are that of a 1-cell having the property “is a biequivalence” on the one hand, and the structure of a biadjoint biequivalence on the other hand. Our main result is that, in any tricategory T, every 1-cell which is a biequivalence is part of a biadjoint biequivalence.

The proof of this result proceeds in three steps. First, we show that it is true for the special case when T = Bicat, the tricategory of bicategories, functors, pseudo-natural transformations, and modifications. Second, we prove a result about transporting biadjoint biequivalences; more precisely, we show that if a functor F : S → T satisfies a kind of local embedding condition and T has the property that every biequivalence is part of a biadjoint biequivalence, then S has that property as well. Finally, we prove that this property is inherited by functor tricategories from the target, so that if T is a tricategory in which every biequivalence is part of a biadjoint biequivalence then the same holds for the functor tricategory Tricat(S, T) for any S. The main result then follows from these theorems and the Yoneda embedding for cubical tricategories.

It should be noted that the Gray-categorical analogue of this result appears in [L2], and that the full result can then be deduced by coherence for tricategories. The purpose of this paper, though, is not merely to prove this theorem, but to show that its proof can be deduced from the Yoneda Lemma together with some computations in the most obvious case. When considering similar theorems for higher dimensional categories, it might be possible to give a Yoneda embedding without necessarily having a good coherence theorem; such a scenario appears entirely plausible in the case of tetracategories.

We also give two applications of this result. The first is a transport-of-structure result, showing how biequivalences F : X → Y between bicategories can be used to transport monoidal structures from X to Y. This relies on choosing a weak inverse G : Y → X for the definition of the tensor product on Y, and then requires the rest of the biadjoint biequivalence structure in order to define the higher cells that are part of the definition of a monoidal bicategory and to check that they satisfy the necessary axioms. We also indicate how to prove similar results when the monoidal structure is replaced with a braided monoidal, sylleptic monoidal, or symmetric monoidal one.

The second application of the main result is to an elucidation of those monoidal bicategories in which every object is weakly invertible, called Picard 2-categories here. We show that every Picard 2-category is monoidally biequivalent to one in which a coherent choice of inverses has been made. In fact, the result is much stronger in that we show that the forgetful functor from coherent Picard 2-categories (those with a choice of inverses) to Picard 2-categories (those monoidal bicategories which merely have the property that every object is invertible) is a triequivalence. We go on to improve this result by defining functors that preserve a given coherent structure up to equivalence, and show that every monoidal functor between Picard 2-categories can be given the structure of such.

The paper is organized as follows. The first section is a warm-up in which we give a proof of the fact that every equivalence in a bicategory is part of an adjoint equivalence. We do this to give the reader a taste of the strategy that will be used later in the tricategorical case so as to clearly indicate the crucial points. This material is well-known although I am unaware of a reference that presents this result in full detail using the argument we give below.

The second section gives the definitions of biequivalence and biadjoint biequivalence that are at the heart of this paper. Both of these we express in the completely general case, working in an arbitrary tricategory T. The definition of biadjoint biequivalence has two forms, with and without the “horizontal cusp” axioms, and we discuss briefly why these two definitions are logically equivalent using the calculus of mates.

Section 3 gives a proof of our main result in the special case where T = Bicat. This proof is largely calculation, much as the proof that every equivalence in Cat is part of an adjoint equivalence is done by straightforward calculation in Section 1. We use that a biequivalence in Bicat can be characterized in two different ways: as a functor having a weak inverse or as a functor which is biessentially surjective and a local equivalence. The proof largely consists of using this alternate characterization of biequivalences, results from section 1, and the biadjoint biequivalence axioms to construct the other cells of the biadjoint biequivalence. This is done by constructing the components on objects first, and then building up the rest of the structure afterwards. The reader will note that the technical difficulties lie in two places: in constructing these cells once the components on objects are given, and in checking that our constructions satisfy all of the biadjoint biequivalence axioms since we only need a subset of them in order to define all the needed cells.

The fourth section provides the proof of our main result, that every biequivalence in a tricategory is part of a biadjoint biequivalence. The proofs in this section rely very heavily on coherence for tricategories in the form “every diagram in a free tricategory commutes” to simplify the pasting diagrams required. We often leave the particulars of checking axioms to the reader as the diagrams are very large, but we state exactly which axioms are needed in each case. The bulk of the technical work goes into showing that the components we construct satisfy the axioms required to be transformations, modifications, or perturbations, while the biadjoint biequivalence axioms are immediate.

Sections 5 and 6 give the two applications mentioned above. Both of these results should be thought of in the form “a certain 3-dimensional monad has property P” in each case. In the case of lifting monoidal structures, the monad would be the free monoidal bicategory monad on the tricategory Bicat, and the property P would be a 3-dimensional version of flexibility [BKP]. In the case of Picard 2-categories, the monad would be the free Picard 2-category monad on the tricategory of monoidal bicategories, and the property P would be a kind of 3-dimensional idempotency [KL]. This is the proper way to view these results, although we do not pursue the details here because of the lack of groundwork on 3-dimensional monads on tricategories.

The author would like to thank Bruce Bartlett for his interest in these results, and John Baez and Eugenia Cheng for enlightening discussions.

1. Equivalences and adjoint equivalences in bicategories

We begin by reviewing the relevant results for bicategories that we will later generalize to tricategories. We assume that the reader is familiar with bicategories and the coherence theorem for bicategories (see [MP] for coherence for bicategories, or [JS] for a discussion of coherence, including functors, for the case of monoidal categories instead of bicategories). We begin with some basic definitions.

1.1. Definition. Let B be a bicategory, and let f : x → y and g : y → x be 1-cells in B. An adjunction f ⊣ g consists of a 2-cell ε : fg ⇒ 1_{y} and a 2-cell η : 1_{x} ⇒ gf such that the following two diagrams (the triangle identities) commute.

\xymatrix{g \ar[r]^{l_{g}^{−1}} \ar@{-->}[drrrr]_{1_{g}} & 1_{x}g \ar[r]^{η∗1_{g}} & (gf)g \ar[r]^{a} & g(fg) \ar[r]^{1_{g}∗ε} & g1_{y} \ar[d]^{r} \\  &  &  &  & g}

\xymatrix{f \ar[r]^{r_{f}^{−1}} \ar@{-->}[drrrr]_{1_{f}} & f1_{x} \ar[r]^{1_{f} ∗η} & f(gf) \ar[r]^{a^{−1}} & (fg)f \ar[r]^{ε∗1_{f}} & 1_{y}f \ar[d]^{l} \\  &  &  &  & f}

We then say that f is left adjoint to g, or that g is right adjoint to f.

1.2. Remark. In the bicategory Cat, the associativity and unit isomorphisms are all identities. In that case, this definition reduces to the usual definition of an adjunction between functors.

1.3. Definition. An adjunction f ⊣ g is an adjoint equivalence if ε and η are invertible. In this case, we write f ⊣_{eq} g.

1.4. Theorem. Let F : X → Y and G : Y → X be functors, and let α : F G ⇒ 1_{Y} and β : 1_{X} ⇒ GF be natural isomorphisms. Then there is a unique adjoint equivalence (F, G, ε, η) in Cat such that ε = α.

Proof. Let ε = α. The second triangle identity states that εF ∘ F η = 1_{F}. By the invertibility of ε, this equation is the same as F η = (εF)^{−1}. The righthand side of this equation is well-defined and F is full and faithful since it is an equivalence of categories, so we define η_{x} : x → GF x to be the unique arrow such that F η_{x} = (ε_{F x})^{−1}.

We must now check that η is natural and that the first triangle identity holds. For naturality, we consider the square below.

\xymatrix{x \ar[r]^{f} \ar[d]_{η_{x}} & y \ar[d]^{η_{y}} \\ GF x \ar[r]_{GF f} & GF y}

Applying F to the diagram and using functoriality gives this square.

\xymatrix{F x \ar[r]^{F f} \ar[d]_{F η_{x}} & F y \ar[d]^{F η_{y}} \\ F GF x \ar[r]_{F GF f} & F GF y}

By the definition of η, this square is the naturality square of (εF)^{−1} and thus must commute. By the faithfulness of F, the original square commutes as well and so η is natural.

For the first triangle identity, we consider the composite

\xymatrix{Gy \ar[r]^{η_{Gy}} & GF Gy \ar[r]^{Gε_{y}} & Gy.}

Applying F to this yields F Gε_{y} ∘ F η_{Gy}, which is by definition F Gε_{y} ∘ ε_{F Gy}^{−1}. Now the following square commutes by the naturality of ε.

\xymatrix{F GF Gy \ar[r]^{ε_{F Gy}} \ar[d]_{F Gε_{y}} & F Gy \ar[d]^{ε_{y}} \\ F Gy \ar[r]_{ε_{y}} & y}

By the invertibility of ε_{y}, we get that ε_{F Gy} = F Gε_{y}. Therefore F Gε_{y} ∘ ε_{F Gy}^{−1} = 1_{F Gy}. Once again by the faithfulness of F, Gε_{y} ∘ η_{Gy} = 1_{y} and thus the first triangle identity is satisfied.

1.5. Remark. We could have just as easily constructed an adjoint equivalence with η = β instead of ε = α. In general it is not possible to require both of these conditions, though, as the choice of either η or ϵ fixes the other.

It would be possible at this point to prove, by a series of calculations, an analogous result with Cat replaced by any bicategory, but instead we choose a different approach that generalizes more easily to the case of tricategories.

1.6. Definition. Let B be a bicategory, and let f : x → y be a 1-cell in B. Then f is an equivalence if there exists a g : y → x such that fg ≅ 1_{y} and gf ≅ 1_{x}.

1.7. Lemma. Let B, C be bicategories, and assume that every equivalence f in C is part of an adjoint equivalence f ⊣_{eq} g. Then every equivalence in the functor bicategory [B, C] is part of an adjoint equivalence.

Proof. Let α : F ⇒ G be an equivalence in the functor category. Then each 1-cell α_{x} : F x → Gx is an equivalence in C, thus we can produce adjoint equivalences α_{x} ⊣_{eq} β_{x} in C for every object x in B. We will now define a transformation β : F ⇒ G using the β_{x} constructed above as the components on objects. Now given a morphism f : x → y in B, we must also produce an invertible 2-cell β_{f} : β_{y} ∘ Gf ⇒ F f ∘ β_{x} in C, subject to the transformation axioms. We define β_{f} by the requirement that it provides the equality of pasting diagrams given below.

[figure: Gx; F x; β_{x}; F x; Gx; α_{x}; Gx; Gx; 1; Gx; Gy; Gf; Gx; Gy; Gf; Gy; F y; β_{y}; F y; Gy; α_{y}; F x; F y; F f; β_{f}; α_{f}; ε_{x}; =; Gx; Gy; Gf; Gy; F y; β_{y}; F y; Gy; α_{y}; Gy; Gy; 1; Gx; Gx; 1; Gx; Gy; Gf; ε_{y}; ≅]

Here, ε_{x} is the counit of the adjoint equivalence α_{x} ⊣_{eq} β_{x}. This gives a well-defined β_{f} as follows. Since all the 2-cells in this pasting diagram are invertible, this equality determines 1_{α_{y}} ∗ β_{f}. But α_{y} is an equivalence 1-cell, hence the functor α_{y} ∘ − is an equivalence of categories, so that 1_{α_{y}} ∗ β_{f} determines β_{f}.

In a similar fashion, we can also construct an invertible 2-cell β′_{f} with the same source and target as β_{f} by requiring it provides the equality of pasting diagrams given below.

[figure: F x; Gx; α_{x}; Gx; F x; β_{x}; F x; F y; F f; F x; F y; F f; F y; Gy; α_{y}; Gy; F y; β_{y}; Gx; Gy; Gf; F y; F y; 1; α_{f}; β′_{f}; η_{y}; =; F x; F y; F f; F x; Gx; α_{x}; Gx; F x; β_{x}; F x; F x; 1; F y; F y; 1; F x; F y; F f; η_{x}; ≅]

After applying left and right unit isomorphisms, the pasting diagram below can be shown to be equal to both β_{f} and β′_{f} using the triangle identities, so β_{f} = β′_{f}.

[figure: Gx; F x; β_{x}; F x; Gx; α_{x}; Gx; Gx; 1; Gx; Gy; Gf; Gx; Gy; Gf; Gy; F y; β_{y}; F y; Gy; α_{y}; F x; F y; F f; β_{f}; α_{f}; ε_{x}; Gx; F x; β_{x}; F x; F y; F f; Gy; F y; β_{y}; β′_{f}; F y; F y; 1; η_{y}]

This defines the components of β on both objects and morphisms.

Now we check that these components satisfy the axioms for a transformation. First, we show that β_{f} is natural in f. Given a 2-cell δ : f ⇒ g in B, we must show that

β_{g} ∘ (1_{β_{y}} ∗ Gδ) = (F δ ∗ 1_{β_{x}}) ∘ β_{f}.

This follows from the naturality of both α_{f} and the coherence isomorphisms used in the definition of β_{f}.

Second, we must show that β_{1_{x}} is the composite

β_{x} ∘ G1 ≅ β_{x}1 ≅ β_{x} ≅ 1β_{x} ≅ F 1 ∘ β_{x}

where every isomorphism is given by a unique coherence isomorphism. To do this, we need only show that the composite above gives the equality of pasting diagrams we used to define β_{1_{x}}. This is trivial using the unit axiom for the transformation α and the fact that

[figure: Gx; F x; β_{x}; F x; Gx; α_{x}; Gx; Gx; G1; Gx; Gx; G1; Gx; F x; β_{x}; F x; Gx; α_{x}; F x; F x; F 1; β_{1x}; α_{1x}]

is the unique coherence isomorphism

(α_{x} ∘ β_{x}) ∘ G1 ≅ G1 ∘ (α_{x} ∘ β_{x})

by the definition of β_{1_{x}}.

The third and final transformation axiom follows from a similar proof.

1.8. Lemma. Assume that every equivalence f in C is part of an adjoint equivalence f ⊣_{eq} g, and let F : B → C be a functor which is locally an equivalence. Then every equivalence r in B is part of an adjoint equivalence r ⊣_{eq} s.

Proof. If r : x → y is an equivalence in B, then there is an s : y → x such that rs ≅ 1_{y} and sr ≅ 1_{x}. Then F r is an equivalence in C with F s as a pseudoinverse. By hypothesis, we can find an adjoint equivalence F r ⊣_{eq} t. By the uniqueness of pseudo-inverses, we must have t ≅ F s, so we have an adjoint equivalence F r ⊣_{eq} F s. Since F is locally full, this means we can find a 2-cell in B which maps to the following composite.

F 1 ≅ 1 ⟶^{η′} F sF r ≅ F (sr)

(Here η′ denotes the unit of the adjoint equivalence F r ⊣_{eq} F s.) This 2-cell will be the unit of our adjoint equivalence, and the counit is constructed similarly; the triangle identities follow from coherence for functors and the fact that F is locally faithful.

1.9. Theorem. Let B be a bicategory, and let f be an equivalence in B. Then f is part of an adjoint equivalence f ⊣_{eq} g.

Proof. Since every equivalence in Cat is part of an adjoint equivalence, the same is true for [B^{op}, Cat]. Let Y : B → [B^{op}, Cat] be the Yoneda embedding. The functor Y satisfies the hypotheses of the above lemma, hence there is an adjoint equivalence f ⊣_{eq} g in B.

1.10. Remark. We have actually shown something stronger than the fact that every equivalence is part of an adjoint equivalence. We have actually shown that given any equivalence f, a pseudo-inverse g, and an isomorphism α : fg ≅ 1, there is a unique adjoint equivalence f ⊣ g with α as its counit. This is true in Cat, hence in any functor bicategory into Cat. Therefore any two adjoint equivalences f ⊣_{eq} g with the same counit α : fg ⇒ 1 will necessarily have the same unit after applying the Yoneda embedding, therefore must have the same unit before applying Y since it is a local equivalence.

2. Definitions

This section will provide the definition of a biadjoint biequivalence in an arbitrary tricategory T. This proceeds in two steps: first we define a biadjunction in a tricategory, and then equip it with extra structure to define a biadjoint biequivalence. There are two possible options for the definition of a biadjoint biequivalence. We provide the concise definition first (omitting the “horizontal cusp” axioms), and then explain how it is equivalent to a definition with additional data and axioms.

Before giving the definition of a biadjunction in a tricategory T, we note that our definition is merely the weakening of previous definitions of biadjunctions in Gray-categories. This weakening is done in the most straightforward manner, and is done because the most natural and concise definition of a biadjunction in a Gray-category uses the Graycategory axioms implicitly. A different approach to these structures might, for instance, involve constructing the “free living biadjunction” - this would be a tricategory 𝔹 with the property that biadjunctions in an arbitrary tricategory T would correspond to maps 𝔹 → T. This is the approach taken by Lack in [L1] in the context of Gray-categories in order to discuss the relationship between biadjunctions and pseudomonads. Since our focus is on biadjoint biequivalences, and not the more general biadjunctions, we do not proceed in this fashion.

2.1. Definition. Let T be a tricategory. Then a biadjunction f ⊣_{bi} g consists of

• 1-cells f : x → y, g : y → x,

• 2-cells α : f ⊗ g ⇒ I_{y}, β : I_{x} ⇒ g ⊗ f, and

• invertible 3-cells Φ, Ψ below,

[figure: f; f ⊗ I; r^{▪}; f ⊗ I; f ⊗ (g ⊗ f); 1⊗β; f ⊗ (g ⊗ f); (f ⊗ g) ⊗ f; a^{▪}; (f ⊗ g) ⊗ f; I ⊗ f; α⊗1; I ⊗ f; f; l; f; f; 1; Φ; g; I ⊗ g; l^{▪}; I ⊗ g; (g ⊗ f) ⊗ g; β⊗1; (g ⊗ f) ⊗ g; g ⊗ (f ⊗ g); a; g ⊗ (f ⊗ g); g ⊗ I; 1⊗α; g ⊗ I; g; r; g; g; 1; Ψ]

such that the pasting diagrams in Figures 1 and 2 are both the identity.

2.2. Remark. In the presence of the simplifying assumption that the tricategory T is actually a strict, cubical tricategory (i.e., a Gray-category), the axioms simplify to the equality of pasting diagrams below.

[figure: fg; fgfg; fβg; fgfg; fg; αfg; fg; I; α; fg; fg; 1; fg; I; α; fgfg; fg; fgα; ⇓fΨ; ≅; fg; fgfg; fβg; fgfg; fg; αfg; fg; I; α; fg; fg; 1; fg; I; α; fg; fg; 1; ⇓Φg; =]

[figure: I; gf; β; gf; gfgf; βgf; gfgf; gf; gαf; I; gf; β; gf; gf; 1; gf; gfgf; gfβ; ≅; ⇓gΦ; I; gf; β; gf; gfgf; βgf; gfgf; gf; gαf; I; gf; β; gf; gf; 1; gf; gf; 1; ⇓Ψf; =]

See [St], [Ver], or [L1] for earlier definitions.

358  NICK GURSKI

[figure: fg; (fI)g; r^{▪}1; (fI)g; (f(gf))g; (1β)1; (f(gf))g; ((fg)f)g; a^{▪}1; ((fg)f)g; (If)g; (α1)1; (If)g; fg; l1; fg; I(fg); l^{▪}; I(fg); II; 1α; fg; f(Ig); 1l^{▪}; f(Ig); f((gf)g); 1(β1); f((gf)g); f(g(fg)); 1a; f(g(fg)); f(gI); 1(1α); f(gI); fg; 1r; fg; (fg)I; r^{▪}; (fg)I; II; α1; II; I; l; II; I; r; (fI)g; f(Ig); a; (f(gf))g; f((gf)g); a; ((fg)f)g; (fg)(fg); a; f(g(fg)); (fg)(fg); a^{▪}; f(gI); (fg)I; a^{▪}; (If)g; I(fg); a; (fg)(fg); I(fg); α1; (fg)(fg); (fg)I; 1α; fg; fg; 1; fg; fg; 1; fg; I; α; fg; I; α; ⇓μ; ≅; ⇓π; ≅; ≅; ⇓λ; ⇓ρ; ≅; ≅; ≅; ≅; ⇓1Ψ; ⇓Φ^{−1}1]

Figure 1: First pasting

BIEQUIVALENCES IN TRICATEGORIES  359

[figure: I; gf; β; I; gf; β; I; II; l^{▪}; I; II; r^{▪}; II; I(gf); 1β; II; (gf)I; β1; I(gf); gf; l; (gf)I; gf; r; I(gf); (gf)(gf); β1; (gf)I; (gf)(gf); 1β; gf; (Ig)f; l^{▪}1; (Ig)f; ((gf)g)f; (β^{▪}1)1; ((gf)g)f; (g(fg))f; a1; (g(fg))f; (gI)f; (1α)1; (gI)f; gf; r1; gf; g(fI); 1r^{▪}; g(fI); g(f(gf)); 1(1β); g(f(gf)); g((fg)f); 1a^{▪}; g((fg)f); g(If); 1(α1); g(If); gf; 1l; (gI)f; g(If); a; (g(fg))f; g((fg)f); a; (gf)(gf); ((gf)g)f; a^{▪}; (gf)(gf); g(f(gf)); a; I(gf); (Ig)f; a^{▪}; (gf)I; g(fI); a; gf; gf; 1; gf; gf; 1; ≅; ⇓μ; ≅; ⇓π; ≅; ≅; ⇓λ; ⇓ρ; ≅; ≅; ≅; ⇓1Φ; ⇓Ψ^{−1}1]

Figure 2: Second pasting 2.3. Definition. Let T be a tricategory. Then a biadjoint biequivalence f ⊣_{bieq} g consists of

• a biadjunction f ⊣_{bi} g and

• adjoint equivalences α ⊣_{eq} α^{▪}, β ⊣_{eq} β^{▪} in the respective hom-bicategories.

It is also possible to give a longer version of a biadjoint biequivalence which includes extra data satisfying the so-called horizontal cusp axioms. Such a definition is equivalent to the one given above by using the calculus of mates, as we explain below.

The extra data needed to express the horizontal cusp axioms are a pair of invertible 3-cells \underline{Φ}, \underline{Ψ}.

[figure: f; I ⊗ f; l^{▪}; I ⊗ f; (f ⊗ g) ⊗ f; α^{▪}⊗1; (f ⊗ g) ⊗ f; f ⊗ (g ⊗ f); a; f ⊗ (g ⊗ f); f ⊗ I; 1⊗β^{▪}; f ⊗ I; f; r; f; f; 1; \underline{Φ}; g; g ⊗ I; r^{▪}; g ⊗ I; g ⊗ (f ⊗ g); 1⊗α^{▪}; g ⊗ (f ⊗ g); (g ⊗ f) ⊗ g; a^{▪}; (g ⊗ f) ⊗ g; I ⊗ g; β^{▪}⊗1; I ⊗ g; g; l; g; g; 1; \underline{Ψ}]

These additional 3-cells are then required to satisfy the horizontal cusp axioms, named for their relationship with certain “braid movie moves” between braided surfaces in ℝ^{4}. One such axiom, written categorically, is given below.

[figure: f(gf); f1; 1β^{▪}; f1; f(gf); 1β; f(gf); f(gf); 1; f(gf); (fg)f; a^{▪}; (fg)f; 1f; α1; 1f; f; l; f1; f; r; ⇓Δ; ⇓ Φ^{−1}; f(gf); (fg)f; a^{▪}; (fg)f; 1f; α1; (fg)f; (fg)f; 1; 1f; (fg)f; α^{▪}1; (fg)f; f(gf); a; f(gf); f1; 1β^{▪}; f1; f; r; 1f; f; l; f(gf); f(gf); 1; ⇓ η_{a}; ⇓\underline{Γ}1; ⇓ \underline{Φ}]

It is then clear that this axiom merely says that \underline{Φ} is the mate of Φ, and similarly for \underline{Ψ} and Ψ. All of the horizontal cusp-type axioms can be expressed in this fashion.

3. Biequivalences in Bicat

This section presents a computational proof that every biequivalence in the tricategory Bicat is part of a biadjoint biequivalence. The proof here will proceed much as the proof in Cat did, by relying on an alternate description of biequivalences. Thus we begin with a simple lemma.

3.1. Lemma. Let F : B → C be a functor between bicategories. Then F is biessentially surjective and locally an equivalence of categories if and only if there is a functor G : C → B such that F G ≃ 1_{C} and GF ≃ 1_{B} in the respective functor bicategories.

Proof. Since F is biessentially surjective, for every object c in C we can find an object b in B and an adjoint equivalence f_{c} ⊣_{eq} g_{c} between F b and c by Theorem 1.9; here we choose f_{c} to have source F b and target c. We choose such an adjoint equivalence for every c, and define the functor G on objects by Gc = b. Now F_{b,b′} : B(b, b′) → C(F b, F b′) is an equivalence of categories for every pair b, b′, and we choose an adjoint equivalence F_{b,b′} ⊣_{eq} G̃_{b,b′}. We define the functor G on hom-categories G_{b,b′} : C(c, c′) → B(Gc, Gc′) to be the composite

C(c, c′) →^{f_{c}^{∗}} C(F b, c′) →^{g_{c∗}} C(F b, F b′) →^{G̃_{b,b′}} B(b, b′).

We must now construct isomorphisms 1_{Gc} ≅ G(1_{c}), Gf ∘ Gg ≅ G(f ∘ g) and check the axioms for a functor. For the first of these, we compute that

G(1_{c}) = G̃_{b,b}(g_{c} ∘ (1_{c} ∘ f_{c})),

while 1_{Gc} = 1_{b}. Now note that the adjoint equivalence F_{b,b′} ⊣_{eq} G̃_{b,b′} has a unit isomorphism 1 ⇒ G̃_{b,b′} ∘ F_{b,b′}, and when specialized to the case b = b′ and then evaluated at 1_{b} yields

1_{b} ≅ G̃_{b,b}(F_{b,b}(1_{b})).

Since F is a functor, we have an isomorphism φ_{0}^{F} : 1_{F b} ≅ F (1_{b}) which we can compose with the previous isomorphism to get

1_{b} ≅ G̃_{b,b}(F_{b,b}(1_{b})) ⟶^{G̃_{b,b}(φ_{0}^{F})} G̃_{b,b}(1_{F b})

which we denote by \overline{φ}_{0}^{G}. Writing η_{c} : 1_{F b} ⇒ g_{c} ∘ f_{c} for the unit of the adjoint equivalence f_{c} ⊣_{eq} g_{c}, we obtain the isomorphism

φ_{0}^{G} : 1_{b} ≅ G(1_{c})

as the following composite.

1_{b} ⟶^{\overline{φ}_{0}^{G}} G̃_{b,b}(1_{F b}) ⟶^{G̃_{b,b}(η_{c})} G̃_{b,b}(g_{c} ∘ f_{c}) ⟶^{G̃_{b,b}(1∗l^{−1})} G̃_{b,b}(g_{c} ∘ (1 ∘ f_{c}))

The isomorphism φ_{2}^{G} : Gf ∘ Gg ≅ G(f ∘ g) is obtained in a similar fashion, and the functor axioms for G follow from those for F and the adjoint equivalence axioms for both f_{c} ⊣_{eq} g_{c} and F_{b,b′} ⊣_{eq} G̃_{b,b′}.

We must finally check that F G ≃ 1_{C} and GF ≃ 1_{B} in the relevant functor bicategories. For the first of these, note that F G(c) = F b by construction. We already have equivalence 1-cells f_{c} : F G(c) → 1_{C}(c) that we now need to complete to a natural transformation. This requires that we give natural isomorphisms

f_{r} : f_{c′}g_{c′}rf_{c} ≅ rf_{c},

one for each r, which we define to be the obvious whiskering of the counit isomorphism f_{c′}g_{c′} ≅ 1_{c′} composed with the left unit isomorphism. The adjoint equivalence axioms for f_{c} ⊣_{eq} g_{c} and coherence for bicategories imply all of the transformation axioms. This shows that F G ≃ 1_{C}, and we leave it to the reader to prove GF ≃ 1_{B}.

3.2. Theorem. Let F : B → C be a biequivalence between bicategories. Then there is a biadjoint biequivalence F ⊣_{bieq} G.

Proof. Since F is a biequivalence, choose a functor G : C → B such that F G is equivalent to 1_{C} in the bicategory Bicat(C, C) and GF is equivalent to 1_{B} in Bicat(B, B). Taking any equivalence α : F G ⇒ 1_{C} exhibiting this fact, we can construct an adjoint equivalence α ⊣_{eq} α^{▪} in Bicat(C, C) by Theorem 1.9. We will write Γ : αα^{▪} ⇛ 1 and \underline{Γ} : 1 ⇛ α^{▪}α for the counit and unit of this adjoint equivalence, respectively.

Now we construct the adjoint equivalence β ⊣_{eq} β^{▪} (between 1_{B} and GF) and the invertible modification Φ simultaneously. The component of β at an object b ∈ B is a 1-cell β_{b} : b → GF b. The component of Φ at b ∈ B is an invertible 2-cell in C

[figure: F b; F b; 1; F b; F GF b; F β_{b}; F GF b; F GF b; 1; F GF b; F b; α_{F b}; F b; F b; 1; F b; F b; 1; Φ]

since the associativity and unit 2-cells in Bicat have identities as their components. By coherence, such an invertible 2-cells determines and is determined by an invertible 2-cell Φ̃ : α_{F b} ∘F (β_{b}) ⇒ 1_{F b}. Since α is an equivalence, giving such an isomorphism is equivalent to giving an isomorphism α_{F b}^{▪} ≅ F (β_{b}). Now F is locally an equivalence of categories, and in particular essentially surjective, so there exists a morphism β_{b} : b → GF b such that F (β_{b}) ≅ α_{F b}^{▪}. For every object b ∈ B, choose such a β_{b} and a specified isomorphism δ_{b} : F (β_{b}) ≅ α_{F b}^{▪}.

For the component of β at f : b → c, consider the following composite.

F (β_{c} ∘ f) ≅ F β_{c} ∘ F f →^{δ_{c}∗1} α_{F c}^{▪} ∘ F f
→^{α_{f}^{▪}} F GF f ∘ α_{F b}^{▪} →^{1∗δ_{b}^{−1}} F GF f ∘ F β_{b}
≅ F (GF f ∘ β_{b})

Since F is locally an equivalence, there is a unique isomorphism

β_{f} : β_{c} ∘ f ⇒ GF f ∘ β_{b}

that maps to the composite above. It is then simple to check that β is a transformation 1 ⇒ GF, and that it is an equivalence. This construction also immediately implies that δ is an invertible modification

δ : 1_{F} ⊗ β ⇛ α^{▪}.

We then define the adjoint equivalence β ⊣_{eq} β^{▪} to be any adjoint equivalence containing β.

The 2-cell Φ̃ : α_{F b} ∘ F (β_{b}) ⇒ 1_{F b} is defined to be the following composite.

α_{F b} ∘ F (β_{b}) ⇒^{1∗δ_{b}} α_{F b} ∘ α_{F b}^{▪} ⇒^{Γ} 1_{F b}

By coherence for bicategories, this determines the 2-cell Φ_{b} uniquely. These 2-cells Φ_{b} then give the data for an invertible modification Φ since all of the cells used to construct the Φ_{b} are either components of modifications or are appropriately natural.

All that remains is to construct the invertible 3-cell Ψ and to check the two biadjunction axioms. Before doing so, we remind the reader that, for 3-cells in a tricategory, ∘ denotes the composition along 2-cell boundaries, ∗ denotes composition along 1-cell boundaries, and ⊗ denotes composition along 0-cell boundaries. Now the second axiom determines the 3-cell (1_{G} ⊗ Ψ) ∗ 1_{β}. Since the 2-cell β is an equivalence, the functor − ∘ β is an equivalence of categories, and in particular the cell (1_{G} ⊗Ψ)∗1_{β} uniquely determines the cell 1_{G} ⊗ Ψ. Similarly, since the functor F is a biequivalence, G is also, so the functor G ∘ − is a biequivalence of bicategories; thus 1_{G} ⊗ Ψ uniquely determines the invertible modification Ψ. By construction, the second biadjoint biequivalence axiom is satisfied.

Now we show that this choice of Ψ satisfies the first biadjoint biequivalence axiom. First, note that, while Bicat is not a Gray-category, it does have a strictly associative and unital composition law for 1-cells in the following sense. The composite H(GF) equals the composite (HG)F, and similarly F 1 = F = 1F, but we still have associativity and unit equivalences for this composition law. These are the 1-cells in the biadjoint biequivalence axioms labeled a, l, r, and they have components on objects given by identities and components on morphisms given by unique coherence 2-cells. For examples, the transformation l : 1F ⇒ F has its component at an object x the identity 1_{F x} : F x → F x and its component at a 1-cell f : x → y the unique coherence cell

1_{F y} ∘ F f ≅ F f ∘ 1_{F x}.

Similarly, the modifications π, μ, λ, and ρ in Bicat all have unique coherence 2-cells as their components. Thus coherence for bicategories reduces the first biadjoint biequivalence to checking that the pasting

[figure: F Gx; F Gx; 1; F Gx; x; α_{x}; F Gx; F Gx; 1; F Gx; x; α_{x}; F Gx; F GF Gx; F β_{Gx}; F GF Gx; F Gx; F Gα_{x}; F GF Gx; F Gx; α_{F Gx}; ⇓Φ̃_{Gx}^{−1}; ⇓F Ψ̃_{x}; ≅_{α}]

is equal to the identity on α ∘ 1_{F G}. (Here we use the same convention that Ψ̃ is derived from Ψ via unique coherence isomorphisms.) From this point on, we mark our naturality isomorphisms with a subscript to indicate which transformation they are naturality isomorphisms for to avoid confusion, and we refer to all instances of the above pasting diagram as “Axiom 1”, perhaps with some descriptor to indicate which object x is being used.

First, note that Axiom 1 is the identity if and only if it is the identity when x is of the form F y for some y in B. Indeed, consider the following pasting diagram.

[figure: F Gx; F Gx; 1; F Gx; x; α_{x}; F Gx; F Gx; 1; F Gx; x; α_{x}; F Gx; F GF Gx; F β_{Gx}; F GF Gx; F Gx; F Gα_{x}; F GF Gx; F Gx; α_{F Gx}; ⇓Φ̃_{Gx}^{−1}; ⇓F Ψ̃_{x}; ≅_{α}; F GF Gx; F Gx; F Gα_{x}; F GF Gx; F GF Gx; 1; F GF Gx; F Gx; α_{F Gx}; F Gx; x; α_{x}; F GF Gx; F Gx; F Gα_{x}; ^{≅^{1}}; ^{≅^{α}}]

Using the modification and transformation axioms, it is equal to the pasting below.

[figure: F GF Gx; F GF Gx; 1; F GF Gx; F Gx; α_{F Gx}; F GF Gx; F GF Gx; 1; F GF Gx; F Gx; α_{F Gx}; F GF Gx; F GF GF Gx; F β_{GF Gx}; F GF GF Gx; F GF Gx; F Gα_{F Gx}; F GF GF Gx; F GF Gx; α_{F GF Gx}; ⇓Φ̃_{GF Gx}^{−1}; ⇓F Ψ̃_{F Gx}; ≅_{α}; F GF Gx; F Gx; F Gα_{x}; F Gx; F Gx; 1; F GF Gx; F Gx; F Gα_{x}; F Gx; x; α_{x}; F Gx; x; α_{x}; _{≅_{1}}; _{≅_{α}}]

Thus the pasting Axiom 1 for x is the identity if and only if Axiom 1 for F Gx is the identity, so taking y = Gx proves the claim. From this point, we replace x with F y.

Now Axiom 1 for F y is the identity if and only if the following pasting diagram is the identity since F β_{y} is an equivalence 1-cell.

[figure: F GF y; F GF y; 1; F GF y; F y; α_{F y}; F GF y; F GF y; 1; F GF y; F y; α_{F y}; F GF y; F GF GF y; F β_{GF y}; F GF GF y; F GF y; F Gα_{F y}; F GF GF y; F GF y; α_{F GF y}; ⇓Φ̃_{GF y}^{−1}; ⇓F Ψ̃_{F y}; ≅_{α}; F y; F GF y; F β_{y}]

Applying F to the second biadjoint biequivalence axiom and rewriting, we see that the above pasting diagram is the identity if and only if the following one is.

[figure: F y; F GF y; F β_{y}; F GF y; F GF y; 1; F GF y; F y; α_{F y}; F y; F GF y; F β_{y}; F GF y; F GF y; 1; F GF y; F y; α_{F y}; F GF y; F GF GF y; F β_{GF y}; F GF GF y; F GF y; α_{F GF y}; F GF y; F GF GF y; F GF β_{y}; F GF GF y; F GF y; F Gα_{F y}; ≅_{F β}; ⇓ Φ̃_{GF y}^{−1}; ⇓ F GΦ̃_{y}; ≅_{α}]

Recall now that β was constructed together with an invertible modification δ with components δ_{y} : F β_{y} ≅ α_{F y}^{▪} such that the composite

α_{F y} ∘ α_{F y}^{▪} ⟹^{1∗δ_{y}^{−1}} α_{F y} ∘ F β_{y} ⟹^{Φ_{y}} 1_{F y}

is the counit of the adjoint equivalence α ⊣_{eq} α^{▪}. The previous pasting diagram is then the identity if and only if we pre-compose it with δ_{y}^{−1} and post-compose it with δ_{y}. Using the naturality axiom for α^{▪}, the modification axiom for δ, and the equality relating δ to the counit of the adjoint equivalence α ⊣_{eq} α^{▪}, the pre- and post-composed pasting is the identity if and only if the one displayed below is the identity.

[figure: F y; F GF y; α_{F y}^{▪}; F GF y; F GF y; 1; F GF y; F y; α_{F y}; F y; F GF y; α_{F y}^{▪}; F GF y; F GF y; 1; F GF y; F y; α_{F y}; F GF y; F GF GF y; α_{F GF y}^{▪}; F GF GF y; F GF y; α_{F GF y}; F GF y; F GF GF y; F Gα_{F y}^{▪}; F GF GF y; F GF y; F Gα_{F y}; ≅_{α_{F}^{▪}}; ^{≅^{c^{−1}}}; _{≅_{c}}; ≅_{α_{F}}]

Here we have written ≅_{c} for the counit isomorphism of the adjoint equivalence α ⊣ α^{▪}, and ≅_{c^{−1}} for the inverse of the counit. This diagram is the identity following from a general lemma on mates that we give below.

3.3. Lemma. Let B be a bicategory, and T : B → B a functor. Let α : T ⇒ 1, α^{▪} : 1 ⇒ T be part of an adjoint equivalence α ⊣_{eq} α^{▪}. Then for any object a in B, the pasting diagram below is the identity.

[figure: a; T a; α_{a}^{▪}; T a; T a; 1; T a; a; α_{a}; a; T a; α_{a}^{▪}; T a; T a; 1; T a; a; α_{a}; T a; T^{2}a; α_{T a}^{▪}; T^{2}a; T a; α_{T a}; T a; T^{2}a; T α_{a}^{▪}; T^{2}a; T a; T α_{a}; ≅_{α^{▪}}; ^{≅^{c^{−1}}}; _{≅_{c}}; ≅_{α}]

Proof. First, the above pasting diagram is the identity if and only if the one below is.

[figure: a; T a; α_{a}^{▪}; T a; T a; 1; T a; a; α_{a}; a; T a; α_{a}^{▪}; T a; T a; 1; T a; a; α_{a}; T a; T^{2}a; α_{T a}^{▪}; T^{2}a; T a; α_{T a}; T a; T^{2}a; T α_{a}^{▪}; T^{2}a; T a; T α_{a}; ≅_{α^{▪}}; ^{≅^{c^{−1}}}; _{≅_{c}}; ≅_{α}; T a; a; α_{a}; T a; T a; 1; T a; T a; 1; ^{≅^{u}}; _{≅_{u^{−1}}}]

Here we have written ≅_{u} to denote a unit isomorphism, and ≅_{u^{−1}} the inverse of a unit isomorphism.

It is now a simple calculation using the adjoint equivalence axioms for transformations to show that the pasting below (modulo unit isomorphisms to alter the 1-cell source and target, which we ignore for now but record the presence of for later)

[figure: T a; a; α_{a}; a; T a; α_{a}^{▪}; T a; T a; 1; T a; T a; 1; T a; T^{2}a; T α_{a}^{▪}; T^{2}a; T a; α_{T a}; a; T a; α_{a}^{▪}; T a; T^{2}a; α_{T a}^{▪}; _{≅_{α^{▪}}}; _{≅_{u^{−1}}}; _{≅_{c^{−1}}}]

is equal to the naturality square below.

Thus we have shown that the pasting diagram in the previous paragraph is equal (modulo units) to the pasting diagram displayed below.

[figure: T a; a; α_{a}; a; T a; α_{a}^{▪}; T a; T a; 1; T a; T^{2}a; T α_{a}^{▪}; T^{2}a; T a; α_{T a}; T a; a; α_{a}; T^{2}a; T a; T α_{a}; T a; a; α_{a}; T a; T a; 1; ^{≅^{u}}; ^{≅^{α}}; ^{≅^{α}}; _{≅_{c}}]

Using the triangle identities and the naturality axioms for α, this is now equal to the pasting below (once again modulo units).

[figure: T a; a; α_{a}; a; a; α_{a}α_{a}^{▪}; T a; T a; T (α_{a}α_{a}^{▪}); T a; a; α_{a}; a; a; 1; T a; T a; 1; ^{≅^{α}}; ^{≅^{c^{−1}}}; _{≅_{c}}]

By the naturality axioms for α and coherence for functors, this is equal to a composite of left and right unit isomorphisms so the original diagram is a composite of coherence isomorphisms, hence is the identity by coherence for bicategories.

4. Biequivalences in general tricategories

This section will establish the general result that every biequivalence in a tricategory T is part of a biadjoint biequivalence in T. Our proof will proceed largely as did the general case for equivalences in bicategories by first examining the case of functor tricategories and then using a Yoneda embedding. Since a Yoneda embedding is only known for cubical tricategories rather than the general case, the proof for tricategories is slightly longer although essentially the same. We refer the reader to [GPS], [Gur], or the upcoming [Gur2] for the relevant tricategorical results.

Recall that if S is any tricategory and T is a Gray-category, then there is a Graycategory Tricat(S, T) with objects functors S → T, 1-cells transformations, 2-cells modifications, and 3-cells perturbations.

4.1. Proposition. Let S be any tricategory and T be a Gray -category. Assume that every biequivalence 1-cell f in T is part of a biadjoint biequivalence f ⊣_{bieq} g. Then every biequivalence 1-cell α in Tricat (S, T) is part of a biadjoint biequivalence α ⊣_{bieq} β.

4.2. Remark. We have written the proof of this proposition out so that it should be obvious to the reader that it remains true when T is merely a tricategory and not a Graycategory. By this we mean the following: the assumption that T is a Gray-category is only present to use the results of [Gur] to give a concrete construction of the tricategory Tricat(S, T). Using coherence for tricategories, it is possible to construct a tricategory Tricat(S, T) when T is any tricategory, and then the proof below applies verbatim to the analogous proposition.

Proof. Let α : F ⇒ G be a biequivalence in Tricat(S, T). Then α_{a} : F a → Ga is a biequivalence in T for every object a, so we choose biadjoint biequivalences α_{a} ⊣_{bieq} β_{a} for every object a of S. To complete the proof, we must do the following:

1. equip the components β_{a} with the structure of a transformation;

2. equip the componentwise adjoint equivalences

ε_{a} : α_{a}β_{a} ⊣_{eq} 1_{Ga}, η_{b} : 1_{F b} ⊣_{eq} β_{b}α_{b}

with the structure of adjoint equivalences in the hom-bicategories Tricat(S, T)(G, G), Tricat(S, T)(F, F); and

3. check that Φ, Ψ are perturbations.

Since equations between perturbations are checked componentwise, the fact we have biadjoint biequivalences α_{a} ⊣_{bieq} β_{a} will then imply that there is a global biadjoint biequivalence α ⊣_{bieq} β in Tricat(S, T).

We begin by defining a transformation β : G ⇒ F with components given by these β_{a}. Since we have already given the components on objects, there are three pieces of data left to define. The first is an adjoint equivalence

[figure: S(a, b); T (Ga, Gb); G; T (Ga, Gb); T (Ga, F b); T (1,β_{b}); S(a, b); T (F a, F b); F; T (F a, F b); T (Ga, F b); T (β_{a},1); β]

in the bicategory Bicat(S(a, b), T (Ga, F b)). We write down the component β_{f} of β at an object f and leave it to the reader to construct the rest of the adjoint equivalence in the obvious fashion.

β_{b} ⊗ Gf ⟶^{1⊗r^{▪}} β_{b} ⊗ (Gf ⊗ 1)
⟶^{1⊗(1⊗ε_{a}^{▪})} β_{b} ⊗ (Gf ⊗ (α_{a} ⊗ β_{a}))
⟶^{1⊗a^{▪}} β_{b} ⊗ ((Gf ⊗ α_{a}) ⊗ β_{a})
⟶^{1⊗(α_{f}^{▪} ⊗1)} β_{b} ⊗ ((α_{b} ⊗ F f) ⊗ β_{a})
⟶^{a^{▪}} (β_{b} ⊗ (α_{b} ⊗ F f)) ⊗ β_{a}
⟶^{a^{▪}⊗1} ((β_{b} ⊗ α_{b}) ⊗ F f) ⊗ β_{a}
⟶^{η_{b}^{▪}⊗1} (1 ⊗ F f) ⊗ β_{a}
⟶^{l⊗1} F f ⊗ β_{a}

We have written this out as if if were a 2-cell an arbitrary tricategory, not necessarily a Gray-category. In the case that T is Gray, this cell is as below.

β_{b}Gf ⟶^{11ϵ_{a}^{▪}} β_{a}Gfα_{a}β_{a} ⟶^{1α_{f}^{▪} 1} β_{b}α_{b}F fβ_{a} ⟶^{η_{b}^{▪}1} F fβ_{a}

We now must produce a pair of invertible modifications Π, M to complete the definition of the data for the transformation β. The component of the modification M^{β} at the object a is given by the isomorphism shown below where the unmarked isomorphisms are unique by coherence and the two marked 3-cells are both appropriate mates.

[figure: β_{a}; β_{a}GI; 1ι^{G}; β_{a}GI; β_{a}GIα_{a}β_{a}; 11ε^{▪}; β_{a}GIα_{a}β_{a}; β_{a}α_{a}F Iβ_{a}; 1α_{I}^{▪}1; β_{a}α_{a}F Iβ_{a}; F Iβ_{a}; η^{▪}11; β_{a}; β_{a}α_{a}β_{a}; 1ε^{▪}; β_{a}α_{a}β_{a}; β_{a}GIα_{a}β_{a}; 1ι^{G}11; β_{a}α_{a}β_{a}; β_{a}α_{a}F Iβ_{a}; 11ι^{F} 1; β_{a}α_{a}β_{a}; β_{a}; η^{▪}1; β_{a}; β_{a}; 1; β_{a}; F Iβ_{a}; ι^{F} 1; ⇓Ψ^{−1}; ≅; ⇓1(M_{a}^{α})^{−1}1; ≅]

The component of the modification Π^{β} at the composable pair (g, f) is given by the pasting below, once again following the same conventions. To conserve space, we omit the subscripts for the components of α and β given that they can be deduced from the other 1-cells in any given term.

[figure: βGgGf; βGgαβGf; 11ϵ^{▪}1; βGgαβGf; βαF gβGf; 1α^{▪}; βαF gβGf; F gβGf; η^{▪}111; F gβGf; F gβGfαβ; 111ϵ^{▪}; F gβGfαβ; F gβαF fβ; 11α^{▪}1; F gβαF fβ; F gF fβ; 1η^{▪}11; βGgGf; βGgGfαβ; 111ϵ^{▪}; βGgGfαβ; βGgαβGfαβ; 11ϵ^{▪}111; βGgαβGfαβ; βαF gβGfαβ; 1α^{▪}1111; βαF gβGfαβ; F gβGfαβ; η^{▪}11111; βGgGfαβ; βGgαF fβ; 11α^{▪}1; βGgαF fβ; βGgαβαF fβ; 11ϵ^{▪}111; βGgαβαF fβ; βαF gβαF fβ; 1α^{▪}1111; βαF gβαF fβ; F gβαF fβ; η^{▪}11111; βGgαβαF fβ; βGgαF fβ; 111η^{▪}11; βGgαF fβ; βαF gF fβ; 1α^{▪}11; βαF gF fβ; F gF fβ; η^{▪}111; βGgαF fβ; βGgαF fβ; 1; βGgGf; βG(gf); 1χ^{G}; βG(gf); βG(gf)αβ; 11ϵ^{▪}; βGgGfαβ; βG(gf)αβ; 1χ^{G}11; βG(gf)αβ; βαF (gf)β; 1α^{▪}1; βαF (gf)β; F (gf)β; η^{▪}11; βαF gF fβ; βαF (gf)β; 11χ^{F} 1; F gF fβ; F (gf)β; χ^{F} 1; ≅; ≅; ≅; ≅; ≅; ⇓11Ψ^{−1}11; ⇓1(Π_{gf}^{α})^{−1}1]

All of the unmarked isomorphisms are isomorphisms of the form

(β ⊗ 1) ∘ (1 ⊗ α) ≅ (1 ⊗ α) ∘ (β ⊗ 1)

arising from the functoriality of the horizontal composition

⊗ : T (y, z) × T (x, y) → T (x, z).

The transformation axioms for β then follow from those for α, thus completing the construction of a weak inverse for α.

The next step is to construct adjoint equivalences

ε : αβ ⊣_{eq} 1_{G}, η : 1_{F} ⊣_{eq} βα.

We already have the adjoint equivalences

ε_{a} : α_{a}β_{a} ⊣_{eq} 1_{Ga}, η_{b} : 1_{F b} ⊣_{eq} β_{b}α_{b}

on components in the tricategory T, we need only lift these to adjoint equivalences in the hom-bicategories Tricat(S, T)(G, G), Tricat(S, T)(F, F), respectively. Thus we need to equip the collection ε_{a} : α_{a}β_{a} ⇒ I_{Ga} with the structure of a trimodification. The 2-cell ε_{a} is the 2-cell in T of the same name, so we must give the invertible 3-cell displayed below.

[figure: α_{b}β_{b}Gf; Gfα_{a}β_{a}; (αβ)_{f}; Gfα_{a}β_{a}; Gf1_{Ga}; 1ε_{a}; α_{b}β_{b}Gf; 1_{Gb}Gf; ε_{b}1; 1_{Gb}Gf; Gf1_{Ga}; (1_{G})_{f}; ε_{f}]

The component (1_{G})_{f} is given by r^{▪} ⊗ l, so once again using coherence we write this as the identity and can define ε_{f} as shown below.

[figure: α_{b}β_{b}Gf; α_{b}β_{b}Gfα_{a}β_{a}; 111ϵ^{▪}; α_{b}β_{b}Gfα_{a}β_{a}; α_{b}β_{b}α_{b}F fβ_{a}; 11α_{f}^{▪} 1; α_{b}β_{b}α_{b}F fβ_{a}; α_{b}F fβ_{a}; 1η^{▪}11; α_{b}F fβ_{a}; Gfα_{a}β_{a}; α_{f} 1; Gfα_{a}β_{a}; Gf; 1ϵ; α_{b}β_{b}Gf; α_{b}β_{b}Gf; 1; α_{b}β_{b}Gf; Gf; ϵ1; α_{b}β_{b}Gfα_{a}β_{a}; α_{b}β_{b}Gfα_{a}β_{a}; 1; α_{b}β_{b}Gfα_{a}β_{a}; α_{b}β_{b}Gf; 111ϵ; Gfα_{a}β_{a}; α_{b}β_{b}Gfα_{a}β_{a}; ϵ^{▪}11; α_{b}β_{b}α_{b}F fβ_{a}; α_{b}β_{b}α_{b}F fβ_{a}; 1; α_{b}F fβ_{a}; α_{b}β_{b}α_{b}F fβ_{a}; ϵ^{▪}111; α_{b}β_{b}α_{b}F fβ_{a}; α_{b}β_{b}Gfα_{a}β_{a}; 11α_{f} 1; ≅; ≅; ≅; ≅; ⇐Ψ^{−1}11]

The unmarked isomorphisms in this pasting diagram are (up to coherence) counits for adjoint equivalences for the two left squares, functoriality of the tensor for the top right square, and a counit together with functoriality of the tensor for the bottom right square. There are now two trimodification axioms to check, one for composition and one for units. Both of these axioms follows from coherence, the definitions of Π^{β} and M^{β}, and the definitions of Π^{βα} and M^{βα}; they do not require the biadjunction axioms for α_{a} ⊣_{bieq} β_{a}.

To complete the proof that we have an adjoint equivalence

ε : αβ ⊣_{eq} 1_{G}

in Tricat(S, T)(G, G), we need only note that a modification m : θ → ϕ in this bicategory is an equivalence if and only if each component m_{x} : θ_{x} ⇒ ϕ_{x} is an equivalence in the appropriate hom-bicategory of T. In this case, the modification m is ε, and each component ε_{a} is an equivalence in T (Ga, Ga) by construction. We note in passing that the components of the specified pseudo-inverse for ε can be taken to be the 2-cells ε_{a}^{▪} that appear in the pointwise adjoint equivalences

ε : α_{a}β_{a} ⊣_{eq} 1_{Ga}.

The construction of the adjoint equivalence

η : 1_{F} ⊣_{eq} βα

follows exactly the same pattern as for ε, so we omit most of the details here. In order to check that Φ and Ψ are perturbations, we will need to know the components

η_{f} : 1η_{a} ∘ (1_{F})_{f} ⇛ (βα)_{f} ∘ η_{b}1

for each 1-cell f : a → b. Using coherence, we write η_{f} as the pasting below.

[figure: F f; F fβα; 1η; F fβα; F fβα; 1; F f; βαF f; η1; βαF f; βGfα; 1α_{f}; βGfα; βGfαβα; 11ε^{▪}1; βGfαβα; βαF fβα; 1α_{f}^{▪} 11; βαF fβα; F fβα; η^{▪}111; F fβα; βαF fβα; η111; βαF fβα; βαF fβα; 1; βαF fβα; βαF f; 111η^{▪}; βαF fβα; βGfαβα; 1α_{f} 11; βGfαβα; βGfαβα; 1; βGfαβα; βGfα; 111η^{▪}; ≅; ≅; ≅; ≅; ⇐11Ψ]

The isomorphisms are all obtained from units of adjunctions and functoriality of the tensor.

Finally, we must check that the cells Φ_{a}, Ψ_{a} constitute a pair of invertible perturbations. This involves checking a single axiom for each 1-cell f : a → b which we leave to the reader, but it follows from coherence and the biadjunction axioms for both α_{a} ⊣_{bieq} β_{a} and α_{b} ⊣_{bieq} β_{b}.

To prove our main result, we require two lemmas.

4.3. Lemma. Let f : a → b and g, h : b → a be 1-cells in a tricategory T. If both gf is equivalent to 1_{a} in T (a, a) and fh is equivalent to 1_{b} in T (b, b), then h is equivalent to g in T (b, a). In particular, if f is a biequivalence in T and both g_{1}, g_{2} are 1-cells such that fg_{i} ≃ 1 and g_{i}f ≃ 1 for i = 1, 2, then g_{1} ≃ g_{2}.

Proof. This follows in the standard way by considering the 1-cell g_{1}fg_{2} in T (b, a).

4.4. Lemma. Let S, T be tricategories, and assume that every biequivalence f in T is part of a biadjoint biequivalence f ⊣_{bieq} g. Assume that F : S → T is 2-locally an equivalence, i.e., every functor

F : S(a, b)(f, g) → T (F a, F b)(F f, F g)

is an equivalence of categories. Then every biequivalence h in S is part of a biadjoint biequivalence.

Proof. This lemma follows in the same way that Lemma 1.7 did; the proof requires using the previous lemma in exactly the same way that Lemma 1.7 required the uniqueness of weak inverses in a bicategory.

We now present our main result.

4.5. Theorem. Let T be a tricategory, and let f be a biequivalence in T. Then there is a 1-cell g in T and a biadjoint biequivalence f ⊣_{bieq} g.

Proof. First, recall that for every tricategory T there is a cubical tricategory (i.e., a tricategory in which the hom-bicategories are 2-categories and the unit and composition functors are cubical) st(T) and a triequivalence T → st(T). Since triequivalences satisfy the hypotheses of the lemma above, we are left proving the theorem in the case that T is a cubical tricategory. If T is cubical, then it has a Yoneda embedding T ↪ Tricat(T^{op}, Gray) which also satisfies the hypotheses of the above lemma. Thus if we show that every biequivalence in Gray is part of a biadjoint biequivalence, we will have proven the theorem for arbitrary T. But the inclusion Gray ↪ Bicat satisfies the hypotheses of the lemma and we have already proven the claim directly for Bicat.

5. Application: lifting monoidal structures

This section will show how to lift monoidal structures on bicategories using the results of Section 4. We refer the reader to [DS] for the definitions of the morphisms and higher cells between monoidal bicategories in the case of Gray-monoids, and [GPS] or [Gur] for the definitions of higher cells between tricategories from which these are derived. From this point forward, most calculations will only be described as they are generally straightforward but the pastings used can be very large.

5.1. Theorem. Let B be a monoidal bicategory and let C be any bicategory. If F : C → B is a biequivalence from C to the underlying bicategory of B, then C can be equipped with the structure of a monoidal bicategory and F can be compatibly equipped with the structure of a monoidal functor.

Proof. First choose a biadjoint biequivalence F ⊣_{bieq} G in Bicat. We now define a monoidal structure on C as follows. The tensor ⊠ : C × C → C is the composite

C × C ⟶^{F ×F} B × B →^{⊗} B →^{G} C.

The unit I^{C} : ∗ → C is the composite

∗ →^{I^{B}} B →^{G} C.

The associativity adjoint equivalence is given by the pasting below. Here we have marked identity adjoint equivalences with equal signs.

[figure: (C × C) × C; (B × B) × C; (F ×F)×1; (B × B) × C; B × C; ⊗×1; B × C; C × C; G×1; C × C; B × B; F ×F; B × B; B; ⊗; B; C; G; (C × C) × C; C × (C × C); a; C × (C × C); C × (B × B); 1×(F ×F); C × (B × B); C × B; 1×⊗; C × B; C × C; 1×G; C × C; B × B; F ×F; B × B; B; ⊗; B; C; G; (B × B) × C; (B × B) × B; 1×F; B × C; B × B; 1×F; (C × C) × C; (B × B) × B; (F ×F)×F; (B × B) × B; B × B; ⊗×1; (B × B) × B; B × (B × B); a; C × (C × C); B × (B × B); F ×(F ×F); C × (B × B); B × (B × B); F ×1; C × B; B × B; F ×1; B × (B × B); B × B; 1×⊗; B; B; ⇓α×1; =; =; =; =; =; ⇓1×α^{▪}; ⇓a^{B}]

The left unit adjoint equivalence is given by the pasting below, following the same conventions as above.

[figure: ∗ × C; B × C; I^{B}×1; B × C; C × C; G×1; C × C; B × B; F ×F; B × B; B; ⊗; B; C; G; B × C; B × B; 1×F; ∗ × C; ∗ × B; 1×F; ∗ × B; B × B; I^{B}×1; ∗ × B; B; _{proj_{2}}; ∗ × B; ∗ × C; 1×G; ∗ × C; C; _{proj_{2}}; ∗ × C; ∗ × C; 1; ⇓α×1; =; ⇓l^{B}; =; 1×β^{▪}^{⇐}]

The right unit adjoint equivalence is constructed in an analogous fashion.

In order to construct the remaining data, it is useful to compute the components of the above transformations. The component of a^{C} at (x, y, z) is given as the following composite which we write omitting all the identity components from adjoint equivalences marked with an equal sign.

G(F G(F x ⊗ F y) ⊗ F z) \underrightarrow{G(α⊗1)} G((F x ⊗ F y) ⊗ F z)
⟶^{Ga^{B}} G(F x ⊗ (F y ⊗ F z))
\underrightarrow{G(1⊗α^{▪})} G(F x ⊗ F G(F y ⊗ F z))

The component of l^{C} at x is given by the following composite.

G(F GI ⊗ F x) ⟶^{G(α⊗1)} G(I ⊗ F x) →^{Gl} GF x →^{β^{▪}} x

374  NICK GURSKI

The component of r^{C} at x is given by the following composite.

G(F x ⊗ F GI) ⟶^{G(1⊗α)} G(F x ⊗ I) →^{Gr} GF x →^{β^{▪}} x

Now we define the invertible modifications π, μ, λ, ρ. We explicitly define the unit modifications μ, λ, ρ, but only describe the construction of the modification π due to the size of the pasting diagram.

The unit modification μ is the pasting below (composed with the unique coherence isomorphism 1∘1∘1 ≅ 1), where all unmarked isomorphisms are naturality isomorphisms and Φ̃ is the mate of Φ.

[figure: G(F xF y); G(F GF xF y); G(F β1); G(F GF xF y); G(F G(F xI)F y); G(F G(r^{▪})1); G(F G(F xI)F y); G(F G(F xF GI)F y); G(F G(1α^{▪})1); G(F G(F xF GI)F y); G((F xF GI)F y); G(α1); G((F xF GI)F y); G(F x(F GIF y)); Ga; G(F x(F GIF y)); G(F xF G(F GIF y)); G(1α^{▪}); G(F xF G(F GIF y)); G(F xF G(IF y)); G(1F G(α1)); G(F xF G(IF y)); G(F xF GF y); G(1F G(l)); G(F xF GF y); G(F xF y); G(1F β^{▪}); G(F GF xF y); G(F xF y); G(α1); G(F xF y); G((F xI)F y); G(r^{▪}1); G(F G(F xI)F y); G((F xI)F y); G(α1); G((F xI)F y); G((F xF GI)F y); G((1α^{▪})1); G(F xF y); G(F xF y); 1; G((F xI)F y); G(F x(IF y)); Ga; G(F x(IF y)); G(F x(F GIF y)); G(1(α^{▪}1)); G(F x(IF y)); G(F xF G(IF y)); G(1α^{▪}); G(F x(IF y)); G(F xF y); G(1l); G(F xF y); G(F xF y); 1; G(F xF y); G(F xF GF y); G(1α^{▪}); G(F xF y); G(F xF y); 1; ⇓Φ; ≅; ≅; ≅; ≅; ≅; ⇓Φ̃; ⇓Gμ]

The unit modification λ is the pasting below.

[figure: G(F G(F GIF x)F y); G(F G(IF x)F y); G(F G(α1)1); G(F G(IF x)F y); G(F GF xF y); G(F G(l)1); G(F GF xF y); G(F xF y); G(F β^{▪}1); G(F G(F GIF x)F y); G((F GIF x)F y); G(α1); G((F GIF x)F y); G(F GI(F xF y)); Ga; G(F GI(F xF y)); G(F GIF G(F xF y)); G(1α^{▪}); G(F GIF G(F xF y)); G(IF G(F xF y)); G(α1); G(IF G(F xF y)); GF G(F xF y); Gl; GF G(F xF y); G(F xF y); β^{▪}; G(F G(IF x)F y); G((IF x)F y); G(α1); G((F GIF x)F y); G((IF x)F y); G((α1)1); G((IF x)F y); G(I(F xF y)); Ga; G(F GI(F xF y)); G(I(F xF y)); G(α(11)); G(I(F xF y)); G(IF G(F xF y)); G(1α^{▪}); G(F GF xF y); G(F xF y); G(α1); G((IF x)F y); G(F xF y); G(l1); G(I(F xF y)); G(F xF y); Gl; G(F xF y); G(F xF y); 1; G(F xF y); GF G(F xF y); Gα^{▪}; ≅; ≅; ≅; ≅; ⇓Gλ; ≅; ⇓G(Φ̃1); ⇓Ψ̃]

For the unit modification ρ : 1 ⊗ r^{▪} ⇒ a ∘ r^{▪}, we give the following non-standard presentation. Let ρ̃ denote the mate of ρ^{−1} with source (1 ⊗ r) ∘ a and target r. We thus define ρ̃ as the pasting below and leave it to the reader to construct the usual modification ρ as necessary.

[figure: G(F G(F xF y)F GI); G((F xF y)F GI); G(α1); G((F xF y)F GI); G(F x(F yF GI)); Ga; G(F x(F yF GI)); G(F xF G(F yF GI)); G(1α^{▪}); G(F xF G(F yF GI)); G(F xF G(F yI)); G(1F G(1α)); G(F xF G(F yI)); G(F xF GF y); G(1F Gr); G(F xF GF y); G(F xF y); G(1F β^{▪}); G(F G(F xF y)F GI); G(F G(F xF y)I); G(1α); G(F G(F xF y)I); GF G(F xF y); Gr; GF G(F xF y); G(F xF y); β^{▪}; G((F xF y)F GI); G((F xF y)I); G((11)α); G((F xF y)I); G(F x(F yI)); Ga; G(F x(F yF GI)); G(F x(F yI)); G(1(1α)); G(F x(F yI)); G(F xF G(F yI)); G(1α^{▪}); G((F xF y)I); G(F xF y); Gr; G(F G(F xF y)I); G((F xF y)I); G(α1); G(F x(F yI)); G(F xF y); G(1r); G(F xF y); G(F xF GF y); G(1α^{▪}); GF G(F xF y); G(F xF y); Gα; G(F xF y); G(F xF y); 1; ≅; ≅; ≅; ≅; ⇓Gρ̃; ≅; ⇓G(1Φ̃); ⇓Ψ̃]

We now describe the construction of the modification π for the monoidal structure on C. This invertible modification is constructed like the unit modifications by pasting together

• naturality isomorphisms for α, α^{▪}, and the associator a;

• a single counit coming from α ⊣_{eq} α^{▪}; and

• the cell G(π^{B}).

We leave it to the reader to construct the appropriate pasting from these cells.

There are now three monoidal bicategory axioms to check. We leave these to the reader as the diagrams are large but the computations simple – the associativity axiom follows by coherence, naturality, and the associativity axiom in B, while both unit axioms follow by coherence, naturality axioms, the corresponding unit axioms in B, and the biadjoint biequivalence axioms.

Now we show that F can be equipped with the structure of a monoidal functor. The adjoint equivalence

χ : ⊗_{B} ∘ (F × F) → F ∘ ⊗_{C}

is α^{▪}, and the adjoint equivalence

ι : I_{B} → F ∘ I_{C}

is α_{I}^{▪}. The invertible modification ω

[figure: (F xF y)F z; F (xy)F z; χ⊗1; F (xy)F z; F ((xy)z); χ; F ((xy)z); F (x(yz)); F a; (F xF y)F z; F x(F yF z); a; F x(F yF z); F xF (yz); 1⊗χ; F xF (yz); F (x(yz)); χ; ⇓ ω]

is given by the pasting diagram below in which all the cells are naturality isomorphisms or a counit for F G(α ⊗ 1) ⊣ F G(α^{▪} ⊗ 1).

[figure: (F xF y)F z; F G(F xF y)F z; α^{▪}1; F G(F xF y)F z; F G(F G(F xF y)F z); α^{▪}; F G(F G(F xF y)F z); F G((F xF y)F z); F G(α1); F G((F xF y)F z); F G(F x(F yF z)); F Ga; F G(F x(F yF z)); F G(F xF G(F yF z)); F G(1α^{▪}); (F xF y)F z; F x(F yF z); a; F x(F yF z); F xF G(F yF z); 1α^{▪}; F xF G(F yF z); F G(F xF G(F yF z)); α^{▪}; (F xF y)F z; F G((F xF y)F z); α^{▪}; F G((F xF y)F z); F G(F G(F xF y)F z); F G(α^{▪}1); F G((F xF y)F z); F G((F xF y)F z); 1; F x(F yF z); F G(F x(F yF z)); α^{▪}; ≅; ≅; ≅; ≅]

The invertible modification γ

[figure: IF x; F IF x; ι⊗1; F IF x; F (Ix); χ; F (Ix); F x; F l; IF x; F x; l; ⇓ γ]

is the pasting diagram shown below where both cells are mates of naturality isomorphisms.

[figure: IF x; F GIF x; α^{▪}1; F GIF x; F G(F GIF x); α^{▪}; F G(F GIF x); F G(IF x); F G(α1); F G(IF x); F GF x; F Gl; F GF x; F x; α; IF x; F x; l; IF x; F G(IF x); α^{▪}; ≅; ≅]

The invertible modification δ : χ ∘ (1 ⊗ ι) ∘ r^{▪} ⇒ F r^{▪} is defined similarly. There are now two axioms to check to show that these data give a monoidal functor between monoidal bicategories, and once again we leave these simple albeit long computations to the reader. The associativity axiom follows from the transformation axioms for α while the unit axiom requires using the biadjoint biequivalence axioms.

5.2. Remark. It would be possible at this point to prove that the functor G chosen in the proof above can also be given the structure of a monoidal functor. For instance, the transformation χ is given by the transformation β used in constructing the biadjoint biequivalence F ⊣_{bieq} G. We could go further, and even show that the entire biadjoint biequivalence F ⊣_{bieq} G can be lifted from Bicat to MonBicat, showing that the forgetful functor

MonBicat → Bicat is the tricategorical analogue of an isofibration. Put another way, the free monoidal bicategory construction is an example of a “flexible 3-monad.” The proofs of all of these statements follow in exactly the same fashion as the construction of the monoidal structure on C and F given above.

5.3. Remark. We could also lift braided monoidal, sylleptic monoidal, or symmetric monoidal structures along biequivalences in a similar fashion. For instance, if B is braided with braiding R_{x,y} : xy → yx then C can be given a braided structure with braiding

F (GxGy) ⟶^{F R_{Gx,Gy}} F (GyGx).

In these cases, as above, the entire biadjoint biequivalence could be lifted from Bicat to the relevant tricategory of monoidal bicategories of the kind considered.

6. Application: Picard 2-categories

This section will present an application of the main result to the study of Picard 2categories. This is the analogue, for monoidal bicategories, of the result of Baez and Lauda that the 2-category of 2-groups (or Picard groupoids) is 2-equivalent to the 2category of coherent 2-groups [BL]. It should be noted that all of the results of this section remain true when we add braided, sylleptic, or symmetric structures. This result could be considered the beginning of the coherence theory for Picard 2-categories, which itself is likely a crucial ingredient in understanding the relationship between Picard 2-categories and stable homotopy 2-types.

6.1. Definition. A Picard 2-category X is a monoidal bicategory such that for every object x, there exists an object y such that

x ⊗ y ≃ I ≃ y ⊗ x.

6.2. Remark. The reader should note that we call these Picard 2-categories even though the underlying object is a mere bicategory. We have also not assumed that all the 1- and 2-cells are invertible, nor that the monoidal structure is braided, sylleptic, or symmetric. All of these additional features (strictness, invertible higher cells, and symmetry) can be added as desired to produce the notion of Picard 2-category appropriate to a particular application. Analogous results to those we present below can then be proven.

6.3. Definition. The tricategory Pic2Cat is the full sub-tricategory of MonBicat consisting of those monoidal bicategories which are Picard 2-categories.

For the next definition, recall that every monoidal bicategory X gives rise to a tricategory ΣX with a single object ∗ and single hom-bicategory given by

ΣX(∗, ∗) = X;

horizontal composition is then given by the tensor product, and all of the coherence constraints for the tricategory are given by those for the monoidal structure on X. Thus a biadjoint biequivalence x ⊣_{bieq} y between objects of a monoidal bicategory is defined to be a biadjoint biequivalence x ⊣_{bieq} y in ΣX where now x, y are treated as 1-cells of the tricategory ΣX.

6.4. Definition. A coherent Picard 2-category (X, inv) is a monoidal bicategory X, a function inv : ob X → ob X, and for each object x a biadjoint biequivalence x ⊣_{bieq} inv(x).

6.5. Definition. The tricategory CohPic2Cat has as its 0-cells coherent Picard 2categories X, hom-bicategories defined as

CohPic2Cat((X, inv_{X}), (Y, inv_{Y})) = MonBicat(X, Y),

and all coherence constraints those inherited from the tricategory MonBicat.

6.6. Theorem. The underlying monoidal bicategory functor U factors (as a strict functor between tricategories) through the inclusion of Pic2Cat into MonBicat.

[figure: CohPic2Cat; MonBicat; U; CohPic2Cat; Pic2Cat; U′; Pic2Cat; MonBicat]

The strict functor U′ : CohPic2Cat → Pic2Cat is a triequivalence.

Proof. The first statement is clear, as the underlying monoidal bicategory of a coherent Picard 2-category (X, inv) is obviously a Picard 2-category, and all of the higher dimensional structure involved in the definitions of these two tricategories agrees. For the second statement, we must prove that U′ is locally a biequivalence and triessentially surjective. Now U′ is the identity functor on hom-bicategories, so is locally a biequivalence. To show that U′ is triessentially surjective, note that Theorem 4.5 actually implies that U′ is surjective on objects as we can always choose a biadjoint biequivalence x ⊣_{bieq} y for any object x with the property that x ⊗ y ≃ I ≃ y ⊗ x.

This theorem produces the most basic kind of equivalence between the theory of Picard 2-categories and its coherent version. For the rest of this paper, we will sketch an improvement to this equivalence by explaining how one might go about proving that not only can Picard 2-categories be replaced by coherent ones, but also that monoidal functors can be replaced by ones that preserve the choice of inverses up to equivalence.

6.7. Definition. 1. Let X be a bicategory. Define the bicategory X^{op} to be the one with

• the same objects as X,

• X^{op}(a, b) = X(b, a),

• composition given by g ∘^{op} f = f ∘ g, and

BIEQUIVALENCES IN TRICATEGORIES  379

• constraints given by a_{h,g,f}^{op} = a_{f,g,h}^{−1}, l_{f}^{op} = r_{f}, and r_{f}^{op} = l_{f}.

2. Let Y be a monoidal bicategory. Define the monoidal bicategory Y^{rev} to be the one with

• underlying bicategory the same as Y,

• a ⊗^{rev} b = b ⊗ a,

• I^{rev} = I,

• all adjoint equivalences given by the opposites of the appropriate adjoint equivalences for the monoidal structure on Y, and

• all invertible 2-cell data the same as that for Y.

3. Let Z be a monoidal bicategory. Define Z^{r} to be (Z^{op})^{rev} = (Z^{rev})^{op}. 6.8. Proposition. Let (X, inv) be a coherent Picard 2-category. The function on objects

inv : ob X → ob X

extends to a monoidal functor of the same name,

inv : X → X^{r}.

Proof. We have already defined inv on objects, so now it is time to define it on 1- and 2-cells. To do this, we must fix notation. For an object x, the biadjoint biequivalence x ⊣_{bieq} inv(x) consists of

• an adjoint equivalence ϵ_{x} ⊣ ϵ_{x}^{▪} between the objects x ⊗ inv(x) and I,

• an adjoint equivalence η_{x} ⊣ η_{x}^{▪} between the objects I and inv(x) ⊗ x,

• and invertible 2-cells Ψ_{x}, Φ_{x}, satisfying the necessary axioms. Thus for a 1-cell f : x → y, we define inv(f) : inv(y) → inv(x) as the following composite (ignoring associativity and unit constraints by coherence).

inv(y) ⟶^{η_{x}1} inv(x) ⊗ x ⊗ inv(y) ⟶^{1f1} inv(x) ⊗ y ⊗ inv(y) ⟶^{1ϵ_{y}} inv(x)

We then define inv(α) to be 1 ∗ (1 ⊗ α ⊗ 1) ∗ 1.

Next we must define structure constraints

inv(g) ∘^{r} inv(f) ≅ inv(g ∘ f)

1_{x} ≅ inv(1_{x})

and check that these give a functor of bicategories. Now inv(g) ∘^{r} inv(f) in X^{r} is defined to be the composite inv(f) ∘ inv(g), so we in fact require an isomorphism of the form

inv(f) ∘ inv(g) ≅ inv(g ∘ f).

It is given by the pasting diagram below, in which we have written inv(x) as x^{−} and all the unmarked isomorphisms are functoriality of the tensor product.

[figure: z^{−}; y^{−}yz^{−}; η_{y}1; y^{−}yz^{−}; y^{−}zz^{−}; 1g1; y^{−}zz^{−}; y^{−}; 1ϵ_{z}; y^{−}; x^{−}xy^{−}; η_{x}1; x^{−}xy^{−}; x^{−}yy^{−}; 1f1; x^{−}yy^{−}; x^{−}; 1ϵ_{y}; z^{−}; x^{−}xz^{−}; η_{x}1; x^{−}xz^{−}; x^{−}xy^{−}yz^{−}; 11η_{y}1; x^{−}xy^{−}yz^{−}; x^{−}xy^{−}zz^{−}; 111g1; x^{−}xy^{−}zz^{−}; x^{−}xy^{−}; 111ϵ_{z}; x^{−}xz^{−}; x^{−}yz^{−}; 1f1; x^{−}yz^{−}; x^{−}yy^{−}yz^{−}; 11η_{y}1; x^{−}yy^{−}yz^{−}; x^{−}yy^{−}zz^{−}; 111g1; x^{−}yy^{−}zz^{−}; x^{−}yy^{−}; 111ϵ_{z}; x^{−}yz^{−}; x^{−}yz^{−}; 1; x^{−}yz^{−}; x^{−}zz^{−}; 1g1; x^{−}zz^{−}; x^{−}; 1ϵ_{z}; x^{−}yy^{−}yz^{−}; x^{−}yz^{−}; 1ϵ_{y}11; x^{−}yy^{−}zz^{−}; x^{−}zz^{−}; 1ϵ_{y}11; ≅; ≅; ≅; ≅; ⇓1Φ_{y}1]

The isomorphism 1_{inv(x)} ≅ inv(1_{x}) is (modulo coherence) Φ_{x}^{−1}.

There are now three axioms to check, one for associativity and two for units. All three of these axioms follow by using the functoriality of the tensor product and then invoking coherence for monoidal bicategories (i.e., coherence for tricategories in the single-object case). The unit axioms then require the equation Φ ∘ Φ^{−1} = 1, while the associativity axiom only uses naturality. Thus we have shown that inv is a functor of bicategories X → X^{op}. Now we turn to showing that it is monoidal.

The first step in showing that inv is a monoidal functor X → X^{r} is to construct an adjoint equivalence

χ : ⊗^{r} ∘ (inv × inv) ⇒ inv ∘ ⊗.

On components, this gives adjoint equivalences

inv(x) ⊗^{r} inv(y) = inv(y) ⊗ inv(x) → inv(x ⊗ y)

in X^{r}, hence adjoint equivalences in X with the source and target reversed. We define the component χ_{x,y} below, and the rest of the adjoint equivalence will be defined in the obvious fashion. We retain the same convention as above for writing inv(x) as x^{−}.

(xy)^{−} ⟶^{η_{y}1} y^{−}y(xy)^{−} ⟶^{1η_{x}11} y^{−}x^{−}xy(xy)^{−} ⟶^{11ϵ_{xy}} y^{−}x^{−}

The second step in giving inv a monoidal structure is to construct an adjoint equivalence

ι : I^{r} ⇒ inv(I)

in X^{r}. Since I^{r} = I, this is the obvious adjoint equivalence with left adjoint shown below.

I^{−} ⟶^{l^{▪}} II^{−} ⟶^{ϵ_{I}} I

The third step in giving inv a monoidal structure is to define three invertible modifications ω, γ, δ. We leave it to the reader to write down the pasting diagrams as they are quite large, but we explain here which cells will be included in each. In each case, there will be a large number of coherence cells from the monoidal bicategory structure, most of which will arise from the functoriality of the tensor product. The other cells in each case are as follows:

• for ω, the remaining cells are Φ_{xy}, Ψ_{inv(xyz)}, and Φ_{yz}^{−1};

• for γ, the remaining cells are Φ_{Ix}, Φ_{I}, and Ψ_{inv(x)};

• for δ, the remaining cell is Φ_{I}.

(The apparent asymmetry in the definitions of γ and δ is due to the fact that γ has the cell F l in the source while δ has the cell F r^{▪} in the target.)

Finally, there are two monoidal functor axioms to check. Both follow from coherence for monoidal functors (a special case of coherence for functors of tricategories) and the biadjoint biequivalence axioms.

6.9. Remark. One can go further to prove that the monoidal functor inv is actually a monoidal biequivalence. First we can equip X^{r} with a canonical coherent Picard 2category structure using the one given on X. Then it is easy to show that inv^{2}(x) is equivalent to x in X as both are weak inverses for inv(x), from which it follows that inv squares to the identity.

6.10. Definition. A coherent functor (F, c, u, v) : (X, inv_{X}) → (Y, inv_{Y}) between coherent Picard 2-categories consists of the following data:

• a monoidal functor F : X → Y,

• an equivalence 1-cell c_{x} : (F x)^{−} → F (x^{−}) for each object x ∈ X, and

• a pair of invertible 2-cells u_{x}, v_{x} for each object x ∈ X as displayed below.

[figure: I; (F x)^{−}F x; η_{F x}; (F x)^{−}F x; F (x^{−})F x; c_{x}1; F (x^{−})F x; F (x^{−}x); χ; I; F I; ι; F I; F (x^{−}x); F η_{x}; ⇓ u_{x}]

[figure: F x(F x)^{−}; I; ε_{F x}; I; F I; ι; F x(F x)^{−}; F xF (x^{−}); 1c_{x}; F xF (x^{−}); F (xx^{−}); χ; F (xx^{−}); F I; F ε_{x}; ⇓ v_{x}]

These are subject to the following axiom.

[figure: F xI; F x; r; F x; IF x; l^{▪}; IF x; F IF x; ι1; F IF x; F (Ix); χ; F xI; F xF I; 1ι; F xF I; F (xI); χ; F (xI); F (xx^{−}x); F (1η_{x}); F (xx^{−}x); F (Ix); F (ε_{x}1); F xI; F x(F x)^{−}F x; 1η_{F x}; F x(F x)^{−}F x; IF x; ε_{F x}1; F x(F x)^{−}F x; F xF (x^{−})F x; 1c1; F xF (x^{−})F x; F xF (x^{−}x); 1χ; F xF (x^{−})F x; F (xx^{−})F x; χ1; F (xx^{−})F x; F (xx^{−}x); χ; F xF (x^{−}x); F (xx^{−}x); χ; F xF I; F xF (x^{−}x); 1F η_{x}; F (xx^{−})F x; F IF x; F ε_{x}1; ⇒^{Φ{−1 \atop F x}}; ⇓v_{x}1; ⇓1u_{x}; ≅; ≅; ⇓ω; F xI; F x; r; F x; IF x; l^{▪}; IF x; F IF x; ι1; F IF x; F (Ix); χ; F xI; F xF I; 1ι; F xF I; F (xI); χ; F (xI); F (xx^{−}x); F (1η_{x}); F (xx^{−}x); F (Ix); F (ε_{x}1); F xI; F x; r; IF x; F x; l; F (xI); F x; F r; F x; F (xI); F l^{▪}; ≅; ⇓δ; ⇓γ; F Φ_{x}^{−1}^{⇒}]

6.11. Remark. We could have structured this definition in a slightly different fashion in a variety of ways. First, the equivalence 1-cells c_{x} could have been the components of a transformation c as shown here.

[figure: X; Y; F; Y; Y^{r}; inv; X; X^{r}; inv; X^{r}; Y^{r}; F^{r}; ⇓ c]

Second, we could have asked that the invertible 2-cells u_{x}, v_{x} could have been the components of a pair of invertible modifications. To express the axioms above as diagrams of modifications would have required that the η_{x}, ε_{x} be the components of transformations, which in turn would require that the functor inv be covariant instead of contravariant. Thus we would have to restrict attention to those Picard 2-categories in which every 1-cell is an equivalence; in fact, we would need every 1-cell to come as part of a specified adjoint equivalence in order to prescribe inv as a covariant functor.

Third, we could have required a third axiom about how c, u, v interact with Ψ. This axiom follows from the first axiom by using the biadjoint biequivalence axioms. In addition, the pastings involved are larger than the one for the axiom above as they involve two different uses of c instead of just one, so it requires additional naturality squares.

6.12. Theorem. Let (X, inv_{X}), (Y, inv_{Y}) be coherent Picard 2-categories, and let F : X → Y be a monoidal functor. Then F underlies a coherent functor

(F, c, u, v) : (X, inv_{X}) → (Y, inv_{Y}).

Proof. The 1-cell c_{x} : F (x^{−}) → F x^{−} is given by the following composite where the last arrow is given by a composite of coherence cells and is thus unique up to unique isomorphism by coherence for functors.

F (x^{−}) ⟶^{η_{F x}1} F x^{−}F xF (x^{−}) ⟶^{1χ} F x^{−}F (xx^{−}) ⟶^{1F ε} F x^{−}F I ⟶ F x^{−}

It is immediate that c_{x} is an equivalence 1-cell.

We must now construct the invertible 2-cells u_{x}, v_{x} and check the two axioms. The cell u_{x} is a pasting of coherence cells from both Y and the functor F, together with Φ_{F x}. The cell v_{x} is constructed similarly out of coherence cells and F Φ_{x}^{−1}. The two axioms are straightforward diagram chases.

6.13. Corollary. Let X be a Picard 2-category. Then X has a coherent structure which is unique in the following sense: if (X, inv) and (X, inv′) are two coherent structures on X, then the identity functor on X lifts to a coherent functor

(1, c, u, v) : (X, inv) → (X, inv′).

6.14. Remark. We leave it to the reader to define coherent transformations and modifications. A coherent transformation will involve additional data, while a coherent modification will only involve a new axiom. Defined correctly, it is then possible to prove that the forgetful functor from the tricategory in which all cells are coherent to the tricategory Pic2Cat is a triequivalence. This shows that a coherent structure on a given Picard 2-category is unique in the strongest possible sense.

References

[BL] John C. Baez and Aaron D. Lauda, Higher dimensional algebra V: 2-groups, Theory Appl. Categories 12 (2004), 423–491.

[BKP] R. Blackwell, G. M. Kelly, and A. J. Power, Two-dimensional monad theory, J. Pure Appl. Algebra 59 (1989), 1–41.

[DS] Brian Day and Ross Street, Monoidal bicategories and Hopf algebroids, Adv. Math. 129 (1997), 99–157.

[GPS] R. Gordon, A. J. Power, and Ross Street, Coherence for tricategories, Mem. Amer. Math. Soc. 117 (1995), no. 558.

[Gur] Nick Gurski, An algebraic theory of tricategories, Ph. D. thesis, University of Chicago, 2006.

[Gur2] Nick Gurski, Coherence in three-dimensional category theory, to appear in the Tracts in Mathematics Series, Cambridge University Press.

[JS] André Joyal and Ross Street, Braided tensor categories, Adv. Math. 102 (1993), 20–78.

[KL] G. M. Kelly and Stephen Lack, On property-like structures, Theory Appl. Categories 3 (1997), 213–250.

[L1] Stephen Lack, A coherent approach to pseudomonads, Adv. Math. 152 (2000), 179–202.

[L2] Stephen Lack, A Quillen model structure for Gray-categories, J. K-theory 8 (2011), 183–221.

[Mac] Saunders Mac Lane, Categories for the working mathematician, Springer, Berlin, 1971.

[MP] Saunders Mac Lane and Robert Paré, Coherence for bicategories and indexed categories, J. Pure Appl. Algebra 37 (1985), 59–80.

[St] Ross Street, Fibrations in bicategories, Cahiers Top. Géom. Diff. Catégorie 21 (1980), 111-160.

[Ver] D. Verity, Enriched categories, internal categories, and change of base, Ph. D. thesis, University of Cambridge, 1992.

Department of Pure Mathematics, University of Sheffield, Sheffield, UK, S3 7RH

Email: nick.gurski@sheffield.ac.uk

This article may be accessed at http://www.tac.mta.ca/tac/ or by anonymous ftp at

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/26/14/26-14.{dvi,ps,pdf}

THEORY AND APPLICATIONS OF CATEGORIES (ISSN 1201-561X) will disseminate articles that significantly advance the study of categorical algebra or methods, or that make significant new contributions to mathematical science using categorical methods. The scope of the journal includes: all areas of pure category theory, including higher dimensional categories; applications of category theory to algebra, geometry and topology and other areas of mathematics; applications of category theory to computer science, physics and other mathematical sciences; contributions to scientific knowledge that make use of categorical methods. Articles appearing in the journal have been carefully and critically refereed under the responsibility of members of the Editorial Board. Only papers judged to be both significant and excellent are accepted for publication. Full text of the journal is freely available in .dvi, Postscript and PDF from the journal’s server at http://www.tac.mta.ca/tac/ and by ftp. It is archived electronically and in printed paper format.

Subscription information Individual subscribers receive abstracts of articles by e-mail as they are published. To subscribe, send e-mail to tac@mta.ca including a full name and postal address. For institutional subscription, send enquiries to the Managing Editor, Robert Rosebrugh, rrosebrugh@mta.ca.

Information for authors The typesetting language of the journal is TEX, and L^{A}TEX2e strongly encouraged. Articles should be submitted by e-mail directly to a Transmitting Editor. Please obtain detailed information on submission format and style files at http://www.tac.mta.ca/tac/.

Managing editor Robert Rosebrugh, Mount Allison University: rrosebrugh@mta.ca

TEXnical editor Michael Barr, McGill University: barr@math.mcgill.ca

Assistant TEX editor Gavin Seal, Ecole Polytechnique Fédérale de Lausanne:

gavin seal@fastmail.fm

Transmitting editors

Clemens Berger, Université de Nice-Sophia Antipolis, cberger@math.unice.fr Richard Blute, Université d’ Ottawa: rblute@uottawa.ca Lawrence Breen, Université de Paris 13: breen@math.univ-paris13.fr Ronald Brown, University of North Wales: ronnie.profbrown(at)btinternet.com

Valeria de Paiva: valeria.depaiva@gmail.com

Ezra Getzler, Northwestern University: getzler(at)northwestern(dot)edu Kathryn Hess, Ecole Polytechnique Fédérale de Lausanne : kathryn.hess@epfl.ch Martin Hyland, University of Cambridge: M.Hyland@dpmms.cam.ac.uk Anders Kock, University of Aarhus: kock@imf.au.dk Stephen Lack, Macquarie University: steve.lack@mq.edu.au F. William Lawvere, State University of New York at Buffalo: wlawvere@buffalo.edu Tom Leinster, University of Glasgow, Tom.Leinster@glasgow.ac.uk Ieke Moerdijk, University of Utrecht: moerdijk@math.uu.nl Susan Niefield, Union College: niefiels@union.edu Robert Paré, Dalhousie University: pare@mathstat.dal.ca Jiri Rosicky, Masaryk University: rosicky@math.muni.cz Giuseppe Rosolini, Università di Genova: rosolini@disi.unige.it Alex Simpson, University of Edinburgh: Alex.Simpson@ed.ac.uk James Stasheff, University of North Carolina: jds@math.upenn.edu Ross Street, Macquarie University: street@math.mq.edu.au Walter Tholen, York University: tholen@mathstat.yorku.ca Myles Tierney, Rutgers University: tierney@math.rutgers.edu Robert F. C. Walters, University of Insubria: robert.walters@uninsubria.it R. J. Wood, Dalhousie University: rjwood@mathstat.dal.ca