2. Bicategories and Some Examples

Bicategories were introduced by BÉnabou (Reference BÉnabou1967), encompassing monoidal categories, 2-categories (in particular, the 2-category of categories), and other examples. He (and later many other authors) defines bicategories in the style of “categories weakly enriched in categories.” That is, the hom-objects $\mathsf{B}_1(a,b)$ of a bicategory $\mathsf{B}$ are taken to be (1-)categories, and composition is given by a functor $\mathsf{B}_1(a,b)\times \mathsf{B}_1(b,c) \to \mathsf{B}_1(a,c)$ . This presentation of bicategories is concise and convenient for communication between mathematicians.

In this article, we use a different, more unfolded definition of bicategories, which is inspired by BÉnabou (Reference BÉnabou1967), Section 1.3 and nLab authors (2018), Section “Details”. One the one hand, it is more verbose than the definition via weak enrichment. On the other hand, it is better suited for our purposes; in particular, it is suitable for defining displayed bicategories, cf. Section 6.

1. (1) a type $\mathsf{B}_0$ of objects;

2. (2) a type $\mathsf{B}_1(a, b)$ of 1-cells for all $a, b : \mathsf{B}_0$ ;

3. (3) a type $\mathsf{B}_2(f, g)$ of 2-cells for all $a, b : \mathsf{B}_0$ and $f, g : \mathsf{B}_1(a,b)$ ;

4. (4) an identity 1-cell $\operatorname{id}_1\!(a) : \mathsf{B}_1(a,a)$ ;

5. (5) a composition $\mathsf{B}_1(a,b) \times \mathsf{B}_1(b,c) \rightarrow \mathsf{B}_1(a,c)$ , written $f \cdot g$ ;

6. (6) an identity 2-cell $\operatorname{id}_2\!(f) : \mathsf{B}_2(f,f)$ ;

7. (7) a vertical composition $\theta \bullet \gamma : \mathsf{B}_2(f,h)$ for all 1-cells $f,g,h : \mathsf{B}_1(a,b)$ and 2-cells $\theta : \mathsf{B}_2(f,g)$ and $\gamma : \mathsf{B}_2(g,h)$ ;

8. (8) a left whiskering $f \vartriangleleft \theta : \mathsf{B}_2(f \cdot g, f \cdot h)$ for all 1-cells $f : \mathsf{B}_1(a,b)$ and $g,h : \mathsf{B}_1(b,c)$ and 2-cells $\theta : \mathsf{B}_2(g,h)$ ;

9. (9) a right whiskering $\theta \vartriangleright h : \mathsf{B}_2(f \cdot h, g \cdot h)$ for all 1-cells $f, g : \mathsf{B}_1(a,b)$ and $h : \mathsf{B}_1(b,c)$ and 2-cells $\theta : \mathsf{B}_2(f,g)$ ;

10. (10) a left unitor $\lambda(f) : \mathsf{B}_2({\operatorname{id}}_1(a) \cdot f,f)$ and its inverse $\lambda(f)^{-1} : \mathsf{B}_2(f,\operatorname{id}_1\!(a) \cdot f)$ ;

11. (11) a right unitor $\rho(f) : \mathsf{B}_2(f \cdot \operatorname{id}_1\!(b),f)$ and its inverse $\rho(f)^{-1} : \mathsf{B}_2(f,f \cdot \operatorname{id}_1\!(b))$ ;

12. (12) a left associator $\alpha(f,g,h) : \mathsf{B}_2(f \cdot (g \cdot h), (f \cdot g) \cdot h)$ and a right associator $\alpha(f,g,h)^{-1} : \mathsf{B}_2((f \cdot g) \cdot h, f \cdot (g \cdot h))$ for $f : \mathsf{B}_1(a,b)$ , $g : \mathsf{B}_1(b,c)$ , and $h : \mathsf{B}_1(c,d)$

such that, for all suitable objects, 1-cells, and 2-cells,

1. (13) $\operatorname{id}_2\!(f) \bullet \theta = \theta, \quad \theta \bullet \operatorname{id}_2\!(g) = \theta, \quad \theta \bullet (\gamma \bullet \tau) = (\theta \bullet \gamma) \bullet \tau$ ;

2. (14) $f \vartriangleleft ({\operatorname{id}}_2 g) = \operatorname{id}_2\!(f \cdot g), \quad f \vartriangleleft (\theta \bullet \gamma) = (f \vartriangleleft \theta) \bullet (f \vartriangleleft \gamma)$ ;

3. (15) $({\operatorname{id}}_2 f) \vartriangleright g = \operatorname{id}_2\!(f \cdot g), \quad (\theta \bullet \gamma) \vartriangleright g = (\theta \vartriangleright g) \bullet (\gamma \vartriangleright g)$ ;

4. (16) $({\operatorname{id}}_1(a) \vartriangleleft \theta) \bullet \lambda(g) = \lambda(f) \bullet \theta$ ;

5. (17) $(\theta \vartriangleright \operatorname{id}_1\!(b)) \bullet \rho(g) = \rho(f) \bullet \theta$ ;

6. (18) $(f \vartriangleleft (g \vartriangleleft \theta)) \bullet \alpha(f, g, i) = \alpha(f,g,h) \bullet ((f \cdot g) \vartriangleleft \theta)$ ;

7. (19) $(f \vartriangleleft (\theta \vartriangleright i)) \bullet \alpha(f,h,i) = \alpha(f,g,i) \bullet ((f \vartriangleleft \theta) \vartriangleright i)$ ;

8. (20) $(\theta \vartriangleright (h \cdot i)) \bullet \alpha(g,h,i) = \alpha(f,h,i) \bullet ((\theta \vartriangleright h) \vartriangleright i)$ ;

9. (21) $(\theta \vartriangleright h) \bullet (g \vartriangleleft \gamma) = (f \vartriangleleft \gamma) \bullet (\theta \vartriangleright i)$ ;

10. (22) $\lambda(f) \bullet \lambda(f)^{-1} = \operatorname{id}_2\!({\operatorname{id}}_1(a) \cdot f), \quad \lambda(f)^{-1} \bullet \lambda(f) = \operatorname{id}_2\!(f)$ ;

11. (23) $\rho(f) \bullet \rho(f)^{-1} = \operatorname{id}_2\!(f \cdot \operatorname{id}_1\!(b)), \quad \rho(f)^{-1} \bullet \rho(f) = \operatorname{id}_2\!(f)$ ;

12. (24) $\alpha(f,g,h) \bullet \alpha(f,g,h)^{-1} = \operatorname{id}_2\!(f \cdot (g \cdot h)), \quad \alpha(f,g,h)^{-1} \bullet \alpha(f,g,h) = \operatorname{id}_2\!((f \cdot g) \cdot h)$ ;

13. (25) $\alpha(f, \operatorname{id}_1\!(b),g) \bullet (\rho(f) \vartriangleright g) = f \vartriangleleft \lambda(g)$ ;

    

14. (26) $\alpha(f,g,h \cdot i) \bullet \alpha(f \cdot g, h, i) = (f \vartriangleleft \alpha(g,h,i)) \bullet \alpha(f,g \cdot h, i) \bullet (\alpha(f,g,h) \vartriangleright i)$ .

    

A bicategory is a prebicategory whose types of 2-cells $\mathsf{B}_2(f,g)$ are sets for all $a,b : \mathsf{B}_0$ and $f,g : \mathsf{B}_1(a,b)$ .

We write $a \rightarrow b$ for $\mathsf{B}_1(a,b)$ and $f \Rightarrow g$ for $\mathsf{B}_2(f,g)$ .

Mitchell Riley formalized a definition of bicategories as “categories weakly enriched in categories” in UniMath, based on work by Peter LeFanu Lumsdaine. We do not reproduce this definition here; it is available as prebicategory. That definition is equivalent to our definition, in the following sense:

For this result, one needs to show that each $\mathsf{B}_1(a,b)$ forms a category whose morphisms are 2-cells. Let us introduce this formally.

Recall that our goal is to study univalence of bicategories, which is a property that relates equivalence and equality. For this reason, we study the two analogs of the 1-categorical notion of isomorphism. The corresponding notion for 2-cells is that of invertible 2-cells.

Since 2-cells form a set and inverses are unique, being an invertible 2-cell is a proposition. In addition, $\operatorname{id}_2\!(f)$ is invertible, and we write $\operatorname{id}_2\!(f) : \mathsf{inv2cell}(f,f)$ for this invertible 2-cell.

The bicategorical analog of isomorphisms for 1-cells is the notion of adjoint equivalence.

Definition 2.5 ((adjoint_equivalence)) An adjoint equivalence structure on a 1-cell $f: a \rightarrow b$ consists of a 1-cell $g : b \rightarrow a$ and invertible 2-cells $\eta : \operatorname{id}_1\!(a) \Rightarrow f \cdot g$ and $\varepsilon : g \cdot f \Rightarrow \operatorname{id}_1\!(b)$ such that the following two diagrams commute

An adjoint equivalence consists of a 1-cell f together with an adjoint equivalence structure on f. The type $\mathsf{AdjEquiv}({a},{b})$ consists of all adjoint equivalences from a to b.

We call $\eta$ and $\varepsilon$ the unit and counit of the adjoint equivalence, and we call g the right adjoint. The prime example of an adjoint equivalence is the identity 1-cell $\operatorname{id}_1\!(a),$ and we denote it by $\operatorname{id}_1\!(a) : \mathsf{AdjEquiv}({a},{a})$ . Sometimes, we write $a \simeq b$ for $\mathsf{AdjEquiv}({a},{b})$ .

Before we start our study of univalence, we present some examples of bicategories and preliminary notions from bicategory theory.

For 1-categories, the “correct” notion of equality is not isomorphism of categories, but equivalence of categories. Similarly, the right notion of equality for bicategories is biequivalence. To talk about biequivalences, we need to introduce pseudofunctors.

• A function $F_0 : \mathsf{B}_0 \rightarrow \mathsf{C}_0$ ;

• For all $a, b : \mathsf{B}_0$ , a function $F_1 : \mathsf{B}_1(a, b) \rightarrow \mathsf{C}_1(F_0(a), F_0(b))$ ;

• For all $f, g : \mathsf{B}_1(a, b)$ , a function $F_2 : \mathsf{B}_2(f, g) \rightarrow \mathsf{C}_2(F_1(f), F_1(g))$ ;

• For each $a : \mathsf{B}_0$ an invertible 2-cell ${{F}(a)}_i : \operatorname{id}_1\!(F_0(a)) \Rightarrow F_1({\operatorname{id}}_1(a))$ ;

• For each $f : \mathsf{B}_1(a, b)$ and $g : \mathsf{B}_1(b, c)$ , an invertible 2-cell ${F}_c(f, g) : F_1(f) \cdot F_1(g) \Rightarrow F_1(f \cdot g)$

such that

\begin{equation*}F_2({\operatorname{id}}_2(f)) = \operatorname{id}_2\!(F_1(f)) \quad \quad F_2(f \bullet g) = F_2(f) \bullet F_2(g)\end{equation*}

and such that the following diagrams commute (where all free variables should be taken to be universally quantified):

We write ${\mathsf{B}} \rightarrow {\mathsf{C}}$ for the type of pseudofunctors from $\mathsf{B}$ to $\mathsf{C}$ .

In the remainder of the paper, we sometimes write F(a) instead of $F_0(a)$ , and we use the same convention for $F_1$ and $F_2$ . We call the 2-cells ${{F}}_i$ and ${F}_c$ the identitor and compositor, respectively. From each pseudofunctor $F : \mathsf{B} \rightarrow \mathsf{C}$ we can assemble functors $F_1(a, b) : \underline{{\mathsf{B}_1}({a},{b})} \rightarrow \underline{{\mathsf{C}_1}({F(a)},{F(b)})}$ between the hom-categories.

• For each $a : \mathsf{B}_0$ a 1-cell $\eta_0(a) : F_0(a) \rightarrow G_0(a)$ ;

• For each $a, b : \mathsf{B}_0$ and $f : \mathsf{B}_1(a, b)$ , an invertible 2-cell $\eta_1(f) : \eta_0 (a) \cdot G_1(f) \Rightarrow F_1(g) \cdot \eta_0(b)$

such that the following diagrams commute

We write ${F} \Rightarrow {G}$ for the type of pseudotransformations from F to G.

Definition 2.14 ((modification)) Let $\mathsf{B}$ and $\mathsf{C}$ be bicategories, $F, G : \mathsf{B} \rightarrow \mathsf{C}$ be pseudofunctors, and $\eta, \theta : {F} \Rightarrow {G}$ be pseudotransformations. A modification $\Gamma$ from $\eta$ to $\theta$ consists of 2-cells $\Gamma(a) : \eta(a) \Rightarrow \theta(a)$ for each $a : \mathsf{B}$ such that

commutes for any $a, b : \mathsf{B}$ and $f : \mathsf{B}_1(a, b)$ . We write ${\eta} \Rrightarrow {\theta}$ for the type of modifications from $\eta$ to $\theta$ .

To illustrate these three definitions, we look at some examples.

• (ap_psfunctor) Each function $f : X \rightarrow Y$ induces a pseudofunctor $\overline{f} : {\pi({X})} \rightarrow {\pi({Y})}$ , which sends objects $x : X$ to f(x), 1-cells $p : x = y$ to $\mathsf{ap} \ {f} \ {p}$ , and 2-cells $h : p = q$ to $\mathsf{ap} \ {(\mathsf{ap} \ f)} \ {h}$ .

• (ap_pstrans) Suppose we have $f, g : X \rightarrow Y$ and $e : \prod_{x : X} f(x) = g(x)$ . Then we obtain a pseudotransformation $\overline{e} : {\overline{f}} \Rightarrow {\overline{g}}$ whose component at x is e(x), and whose actions on 1-cells are given by path induction.

• (ap_modification) Let $f, g : X \rightarrow Y$ and $e_1, e_2 : \prod_{x : X} f(x) = g(x)$ . Then each family of paths $h : \prod_{x : X} e_1(x) = e_2(x)$ gives rise to a modification $\overline{h} : {\overline{e_1}} \Rrightarrow {\overline{e_2}}$ whose component at x is h(x).

• (id_psfunctor) Given a bicategory $\mathsf{B}$ , we have the identity pseudofunctor $\operatorname{id}(\mathsf{B})$ from $\mathsf{B}$ to $\mathsf{B}$ . Its action on 0-cells, 1-cells, and 2-cells is the identity.

• (comp_psfunctor) Given bicategories $\mathsf{B}_1$ , $\mathsf{B}_2$ , and $\mathsf{B}_3$ and pseudofunctors $F : {\mathsf{B}_1} \rightarrow {\mathsf{B}_2}$ and $G : {\mathsf{B}_2} \rightarrow {\mathsf{B}_3}$ , then we have a pseudofunctor $F \cdot G$ from $\mathsf{B}_1$ to $\mathsf{B}_3$ . It sends objects a to $G_0(F_0(a))$ , 1-cells f to $G_1(F_1(f))$ , and 2-cells $\theta$ to $G_2(F_2(\theta))$ .

• (id_pstrans) Given bicategories $\mathsf{B}_1$ and $\mathsf{B}_2$ and a pseudofunctor $F : {\mathsf{B}_1} \rightarrow {\mathsf{B}_2}$ , we have a pseudotransformation $\operatorname{id}(F)$ from F to F. It sends objects a to $\operatorname{id}_1\!(F_1(a))$ , and similarly for 1-cells.

• (comp_pstrans) Given bicategories $\mathsf{B}_1$ and $\mathsf{B}_2$ , pseudofunctors $F, G, H : {\mathsf{B}_1} \rightarrow {\mathsf{B}_2}$ , and two pseudotransformations $\theta_1 : {F} \Rightarrow {G}$ and $\theta_2 : {G} \Rightarrow {H}$ , we have a pseudotransformation $\eta_1 \bullet \eta_2 : {F} \Rightarrow {H}$ . It sends objects a to $\theta_1(a) \cdot \theta_2(a)$ .

Note that we have a bicategory $\mathsf{Pseudo}(\mathsf{B}, \mathsf{C})$ of pseudofunctors, pseudotransformations, and modifications. We construct this bicategory in section 9.1 using displayed bicategories, and then, we define invertible modifications to be invertible 2-cells in this bicategory. With all this in place, we can define biequivalences.

• A pseudofunctor $L : \mathsf{B} \rightarrow \mathsf{C}$ ;

• A pseudofunctor $R : {\mathsf{C}} \rightarrow {\mathsf{B}}$ ;

• Pseudotransformations $\eta : R \cdot L \Rightarrow \operatorname{id}(\mathsf{C})$ and $\eta_i : \operatorname{id}(\mathsf{C}) \Rightarrow R \cdot L$ ;

• Pseudotransformations $\varepsilon : L \cdot R \Rightarrow \operatorname{id}(\mathsf{B})$ ad $\varepsilon_i : \operatorname{id}(\mathsf{B}) \Rightarrow L \cdot R$ ;

• Invertible modifications

  \begin{equation*} m_1 : {\eta \bullet \eta_i} \Rrightarrow \operatorname{id} \quad \quad m_2 : {\eta_i \bullet \eta} \Rrightarrow \operatorname{id} \quad m_3 : {\varepsilon \bullet \varepsilon_i} \Rrightarrow \operatorname{id} \quad \quad m_4 : {\varepsilon_i \bullet \varepsilon} \Rrightarrow \operatorname{id} \end{equation*}

Usually, the notion of biequivalence is not sufficient, and instead biadjoint biequivalences are used. The latter notion has an extra requirement, namely that L and R form a pseudoadjunction (Lack Reference Lack2000). Note that this is similar to the situation in types (Univalent Foundations Program 2013, Section 4) and categories (Mac Lane Reference Mac Lane1978, Section IV.4), where one also considers coherent notions of equivalence. However, we restrict our attention to biequivalences, because every biequivalence can be refined to a biadjoint biequivalence Gurski (Reference Gurski2012, Theorem 3.1).

As an example, we construct a biequivalence between 1-types (Example 2.7) and univalent groupoids (Example 2.11).

• (path_groupoid) Define a pseudofunctor $\mathsf{PathGrpd} : {\mbox{1-}\mathsf{Type}}\rightarrow {\mathsf{Grpd}}$ . It sends a 1-type X to the groupoid $\mathsf{PathGrpd}(X)$ whose objects are X and morphisms from x to y are paths $x = y$ .

• (objects_of_grpd) Define a pseudofunctor $\mathsf{Ob} : {\mathsf{Grpd}} \rightarrow {\mbox{1-}\mathsf{Type}}$ . It sends a groupoid G to the 1-type $\mathsf{Ob}(G)$ whose inhabitants are objects of G. Note that this is a 1-truncated type, because G is univalent.

3. Univalent Bicategories

Recall that a (1-)category $\mathsf{C}$ (called “precategory” by Ahrens et al. Reference Ahrens, Kapulkin and Shulman2015) is called univalent if, for every two objects $a,b : \mathsf{C}_0$ , the function $\mathsf{idtoiso}_{a,b} : (a = b) \to \mathsf{Iso}({a},{b})$ mapping the constant path to the identity isomorphism is an equivalence. For bicategories, where we have one more layer of structure, univalence can be imposed both locally and globally.

1. (1) Let $a, b : \mathsf{B}_0$ and $f, g : \mathsf{B}_1(a,b)$ be objects and morphisms of $\mathsf{B}$ ; by path induction we define a function $\mathsf{idtoiso}^{2,1}_{f,g} : f = g \to \mathsf{inv2cell}(f,g)$ which sends $\operatorname{\mathsf{{refl}}}(f)$ to $\operatorname{id}_2\!(f)$ . A bicategory $\mathsf{B}$ is locally univalent if, for every two objects $a,b : \mathsf{B}_0$ and two 1-cells $f,g : \mathsf{B}_1(a,b)$ , the function $\mathsf{idtoiso}^{2,1}_{f,g}$ is an equivalence.

2. (2) Let $a, b : \mathsf{B}_0$ be objects of $\mathsf{B}$ ; using path induction we define $\mathsf{idtoiso}^{2,0}_{a,b} : a = b \to \mathsf{AdjEquiv}({a},{b})$ sending $\operatorname{\mathsf{{refl}}}(a)$ to $\operatorname{id}_1\!(a)$ . A bicategory $\mathsf{B}$ is globally univalent if, for every two objects $a,b : \mathsf{B}_0$ , the canonical function $\mathsf{idtoiso}^{2,0}_{a,b}$ is an equivalence.

3. (3) (is_univalent_2) We say that $\mathsf{B}$ is univalent if $\mathsf{B}$ is both locally and globally univalent.

Local univalence can be characterized via the hom-categories. More precisely, it is equivalent to all hom-categories being univalent.

Univalent bicategories satisfy a variant of the elimination principle of path induction. More precisely, there are two such principles: a local one for invertible 2-cells and a global one for adjoint equivalences. We start with the induction principle associated with invertible 2-cells:

Proposition 3.4 (J_2_1). Let $\mathsf{B}$ be a locally univalent bicategory. Given a type family Y and a function y with types

there is a function

such that $\mathsf{J}_{2,1}(Y,y,a , b , f , f , \operatorname{id}_2\!(f)) = y(a , b , f)$ .

In particular, in order to prove a predicate over all invertible 2-cells in a given locally univalent bicategory, it suffices to prove it for all identity 2-cells.

Next, we present the induction principle associated with adjoint equivalences:

Proposition 3.5 ((J_2_0)) Let $\mathsf{B}$ be a globally univalent bicategory. Given a type family Y and a function y with types

there is a function

\begin{equation*}\mathsf{J}_{2,0} (Y,y) : \prod_{(a , b : \mathsf{B}_0)} . \prod_{(f : a \simeq b)} . Y (a,b,f)\end{equation*}

such that $\mathsf{J}_{2,0} (Y,y,a , a , \operatorname{id}_1\!(a)) = y(a)$ .

In particular, in order to prove a predicate over all adjoint equivalences in a given globally univalent bicategory, it suffices to prove it for all identity 1-cells. Notice that in both induction principles, the computation rules hold only up to propositional equality. Next, we present some usage examples of how to use Propositions 3.4 and 3.5. The constructions described in Example 3.6 and Proposition 3.9 work for arbitrary bicategories, not just globally/locally univalent ones. Nevertheless, these constructions are considerably simpler if the involved bicategories satisfy certain univalence assumptions.

Example 3.6 ((comp_adjoint_equivalence)) In a globally univalent bicategory $\mathsf{B}$ , sequential composition of adjoint equivalences can be defined in a way that resembles the construction of composition of paths. Consider the type family $Y (a, b , f) :\equiv \prod_{(c : \mathsf{B}_0)} . b \simeq c \to a \simeq c$ and the function $y(a) :\equiv \lambda\, (c : \mathsf{B}_0) (f : a \simeq c). f$ . The composition of $f : a \simeq b$ and $g : b \simeq c$ is given by

\begin{equation*}f \cdot_{\simeq} g :\equiv \mathsf{J}_{2,0}(Y,y,a,b,f,c,g).\end{equation*}

Proof. Lengthy but straightforward.

If $\mathsf{B}$ is globally univalent and $\mathsf{C}$ is locally univalent, the above statement can be proved very easily.

Proof. Applying Proposition 3.5 on f, we are left to prove that $F_1({\operatorname{id}}_1(a))$ is an adjoint equivalence. Since F is a pseudofunctor, there exists an invertible 2-cell ${{F}}_i(a) :\operatorname{id}_1\!(F_0(a)) \Rightarrow F_1({\operatorname{id}}_1(a))$ . Therefore, by Example 3.7 and the fact that $\operatorname{id}_1\!(F_0(a))$ is an adjoint equivalence, we conclude that $F_1({\operatorname{id}}_1(a))$ is an adjoint equivalence as well.

Another consequence is that biequivalences between univalent bicategories gives rise to equivalences on the level of objects.

While right adjoints are only unique up to isomorphism in general, they are unique up to identity if the bicategory is locally univalent:

As a consequence of this proposition, we get the following:

Proposition 3.11 has another important use: to prove global univalence of a bicategory, we need to show that $\mathsf{idtoiso}^{2,0}_{a,b}$ is an equivalence. Often we do that by constructing a function in the other direction and showing these two are inverses. This requires comparing adjoint equivalences, which is done with the help of Proposition 3.11.

Local univalence is also relevant when one discusses bicategorical analogues of limits and colimits. To exemplify this, we look at biinitial objects, and we note that a similar discussion can be given for bifinal objects (bifinal_unique). We start by defining biinitiality structures.

In general, adjoint equivalence structures are not necessarily unique, but they are if the bicategory is locally univalent. As such, having a biinitiality structure is not necessarily a proposition, and instead, it should be viewed as a structure on the objects. If the bicategory is locally univalent, however, then we can use Proposition 3.11 to show that biinitiality structures form a proposition.

While local univalence affects the uniqueness of biinitiality structures, global univalence affects the uniqueness of biinitial objects. Since limits and colimits are unique up adjoint equivalence, the type of biinitial objects is a proposition if the bicategory is univalent.

Before we discuss examples of biinitial objects, we give an equivalent definition of biinitiality formulated using universal mapping properties.

• for every b there is a 1-cell $a \rightarrow b$ ;

• for every two 1-cells $f, g : a \rightarrow b$ there is a unique 2-cell $f \Rightarrow g$ .

• (biinitial_1_types) The empty type is a biinitial object in $\mbox{1-}\mathsf{Type}$ .

• (biinitial_cats) The empty category is a biinitial object in $\mathsf{Cat}$ .

Now let us prove that some examples from Section 2 are univalent.

1. (1) (TwoType, Example 2.6 cont’d) The fundamental bigroupoid of each 2-type is univalent.

2. (2) (OneTypes, Example 2.7 cont’d) The bicategory of 1-types of a universe $\operatorname{\textsf{U}}$ is locally univalent; this is a consequence of function extensionality. If we assume the univalence axiom for $\operatorname{\textsf{U}}$ , then 1-types form a univalent bicategory. To show that, we factor $\mathsf{idtoiso}^{2,0}$ as follows.

   
   The left function is an equivalence by univalence, and the right function is an equivalence by the characterization of adjoint equivalences in Example 2.7. The fact that this diagram commutes follows from Proposition 3.11.

3. (3) (FullSub, If $\mathsf{B}$ is univalent and P is a predicate on $\mathsf{B}$ , then so is the full subbicategory of $\mathsf{B}$ with those objects satisfying P.

It is more difficult to prove that the bicategory of univalent categories is univalent, and we only give a brief sketch of this proof.

Local univalence follows from the fact that the functor category [C,D] is univalent if D is. For global univalence, we use that the type of identities on categories is equivalent to the type of adjoint equivalences between categories (Ahrens et al. Reference Ahrens, Kapulkin and Shulman2015, Theorem 6.17). The proof proceeds by factoring $\mathsf{idtoiso}^{2,0}$ as a chain of equivalences $(C = D) \xrightarrow{\sim} \mathsf{CatIso}({C},{D}) \xrightarrow{\sim} \mathsf{AdjEquiv}({C},{D})$ . To our knowledge, a proof of global univalence was first computer-formalized by RafaËl Bocquet. ^{Footnote 2}

In the previous examples, we proved univalence directly. However, in many complicated bicategories such proofs are not feasible. An example of such a bicategory is the bicategory $\mathsf{Pseudo}(\mathsf{B},\mathsf{C})$ of pseudofunctors from $\mathsf{B}$ to $\mathsf{C}$ , pseudotransformations, and modifications (Leinster Reference Leinster1998) (for a univalent bicategory $\mathsf{C}$ ). Even in the 1-categorical case, proving the univalence of the category [C, D] of functors from C to D, and natural transformations between them, is tedious. In Section 7, we develop some machinery to prove the following theorem.

4. Bicategories and 2-Categories

In this section, we propose a definition of 2-category and compare 2-categories to bicategories. We start by defining strict bicategories.

1. (1) for each $a, b : \mathsf{B}$ and $f : a \rightarrow b$ we have $\mathsf{p}_{\lambda}(f) : \operatorname{id}_1\!(a) \cdot f = f$ , and $\mathsf{idtoiso}^{2,1}(\mathsf{p}_{\lambda}(f)) = \lambda(f)$ ;

2. (2) for each $a, b : \mathsf{B}$ and $f : a \rightarrow b$ we have $\mathsf{p}_{\rho}(f) : f \cdot \operatorname{id}_1\!(b) = f$ , and $\mathsf{idtoiso}^{2,1}(\mathsf{p}_{\rho}(f)) = \rho(f)$ ;

3. (3) for each $a, b, c, d : \mathsf{B}$ and $f : a \rightarrow b, g : b \rightarrow c$ , and $h : c \rightarrow d$ we have $\mathsf{p}_{\alpha}(f,g,h) : f \cdot (g \cdot h) = (f \cdot g) \cdot h$ , and $\mathsf{idtoiso}^{2,1}(\mathsf{p}_{\alpha}(f,g,h)) = \alpha(f,g,h)$ .

Now let us look at an example of a 1-strict bicategory.

The bicategory $\mathsf{Cat}$ of univalent categories is not 1-strict. This is because functors between two categories do not necessarily form a set.

Next we look at 2-categories. These are defined as 1-categories with additional structure and properties.

• a category $\mathsf{C}_0$ ;

• for each $x, y : \mathsf{C}_0$ and $f, g : x \rightarrow y$ a set $\mathsf{C}_2(f, g)$ of 2-cells;

• an identity 2-cell $\operatorname{id}_2\!(f) : \mathsf{C}_2(f,f)$ ;

• a vertical composition $\theta \bullet \gamma : \mathsf{C}_2(f,h)$ for all 1-cells $f,g,h : \mathsf{C}_1(a,b)$ and 2-cells $\theta : \mathsf{C}_2(f,g)$ and $\gamma : \mathsf{C}_2(g,h)$ ;

• a left whiskering $f \vartriangleleft \theta : \mathsf{C}_2(f \cdot g, f \cdot h)$ for all 1-cells $f : \mathsf{C}_1(a,b)$ and $g,h : \mathsf{C}_1(b,c)$ and 2-cells $\theta : \mathsf{C}_2(g,h)$ ;

• a right whiskering $\theta \vartriangleright h : \mathsf{C}_2(f \cdot h, g \cdot h)$ for all 1-cells $f, g : \mathsf{C}_1(a,b)$ and $h : \mathsf{C}_1(b,c)$ and 2-cells $\theta : \mathsf{C}_2(f,g)$ ;

such that, for all suitable objects, 1-cells, and 2-cells,

• $\operatorname{id}_2\!(f) \bullet \theta = \theta, \quad \theta \bullet \operatorname{id}_2\!(g) = \theta, \quad \theta \bullet (\gamma \bullet \tau) = (\theta \bullet \gamma) \bullet \tau$ ;

• $f \vartriangleleft ({\operatorname{id}}_2 g) = \operatorname{id}_2\!(f \cdot g), \quad f \vartriangleleft (\theta \bullet \gamma) = (f \vartriangleleft \theta) \bullet (f \vartriangleleft \gamma)$ ;

• $({\operatorname{id}}_2 f) \vartriangleright g = \operatorname{id}_2\!(f \cdot g), \quad (\theta \bullet \gamma) \vartriangleright g = (\theta \vartriangleright g) \bullet (\gamma \vartriangleright g)$ ;

• $({\operatorname{id}}_1 a \vartriangleleft x) \bullet \mathsf{idto2cell} (\mathsf{idleft}(g)) = \mathsf{idto2cell} (\mathsf{idleft}(f)) \bullet x$ ;

• $(x \vartriangleright \operatorname{id}_1 b) \bullet \mathsf{idto2cell}(\mathsf{idright}(g)) = \mathsf{idto2cell}(\mathsf{idright}(f)) \bullet x$ ;

• $(f \vartriangleleft (g \vartriangleleft x)) \bullet \mathsf{idto2cell}(\mathsf{assoc}(f,g,i)) = \mathsf{idto2cell}(\mathsf{assoc}(f,g,h)) \bullet (f \cdot g \vartriangleleft x)$ ;

• $f \vartriangleleft (x \vartriangleright i) \bullet \mathsf{idto2cell}(\mathsf{assoc} (f,h,i)) = \mathsf{idto2cell} (\mathsf{assoc}(f,g,i)) \bullet ((f \vartriangleleft x) \vartriangleright i)$ ;

• $ \mathsf{idto2cell}(\mathsf{assoc}(f,h,i)) \bullet (x \vartriangleright h \vartriangleright i) = (x \vartriangleright h \cdot i) \bullet \mathsf{idto2cell}(\mathsf{assoc}(g,h,i))$ .

Here, the function $\mathsf{idto2cell}_{f,g} : (f = g) \to (f \Rightarrow g)$ is defined by path induction, sending the identity path to the identity 2-cell. The paths $\mathsf{idleft}(f)$ , $\mathsf{idright}(g)$ , and $\mathsf{assoc}(f,g,h)$ are those given by the categorical axioms for $\mathsf{C}_0$ .

We call 0-cells of a 2-category $\mathsf{C}$ the objects of $\mathsf{C}_0$ , and 1-cells the morphisms of the category $\mathsf{C}_0$ . In particular, the 1-cells between every two 0-cells of a 2-category always form a set.

The type of 1-strict bicategories is equivalent to that of 2-categories.

Construction 4.9 for Problem 4.8; (strict_bicat_to_two_cat). In one direction, suppose $\mathsf{C}$ is a 2-category. We associate with $\mathsf{C}$ the following bicategory:

1. (1) 0-cells, 1-cells, and 2-cells are those of $\mathsf{C}$ ;

2. (2) composition and identity of 1-cells and 2-cells are those of $\mathsf{C}$ , respectively;

3. (3) whiskering is given by the whiskering of $\mathsf{C}$ ;

4. (4) left and right unitors, and associators, are 2-cells induced by the corresponding equality axioms via $\mathsf{idto2cell}$ .

The bicategorical axioms are then easily shown, using compatibility, in a suitable sense, of $\mathsf{idto2cell}$ with composition of paths (which corresponds to composition of 2-cells) and functions on paths (which corresponds to whiskering). The resulting bicategory is 1-strict.

In the other direction, suppose $\mathsf{B}$ is a 1-strict bicategory. We associate the following 2-category to $\mathsf{B}$ :

1. (1) 0-cells, 1-cells, and 2-cells are those of $\mathsf{B}$ , respectively;

2. (2) composition, identities, and whiskering are given by the corresponding operations of $\mathsf{B}$ ;

3. (3) the equality axioms for composition of 1-cells are proved using the strictness properties of $\mathsf{B}$ ;

4. (4) the remaining axioms are proved using suitable compatibility results about $\mathsf{idto2cell}$ .

The two functions are easily shown to be inverse to each other, thus forming an equivalence of types.

5. The Yoneda Embedding

In this section, we show that any locally univalent bicategory naturally embeds into a univalent one, via the Yoneda embedding. This construction is similar to the Rezk completion for categories (Ahrens et al. Reference Ahrens, Kapulkin and Shulman2015, Theorem 8.5), and it makes use of the Yoneda lemma. We start by discussing representable pseudofunctors, pseudotransformations, and modifications. These are used to define the desired embedding.

• (representable) Given an object $a : \mathsf{B}$ , we define the representable pseudofunctor $\mathsf{Rep}_0(a)$ from $\mathsf{B}^{\operatorname{\mathsf{{op}}}}$ (see Example 29) to $\mathsf{Cat}$ . It sends objects b to the category $\underline{{\mathsf{B}_1}({b},{a})}$ and 1-cells $f : b_1 \rightarrow b_2$ to the functor $\mathsf{Rep}_0(a)(f) : \underline{{\mathsf{B}_1}({b_2},{a})} \to \underline{{\mathsf{B}_1}({b_1},{a})}$ given by $g \mapsto f \cdot g$ . If we have 1-cells $f, g : b_1 \rightarrow b_2$ and a 2-cell $\theta : f \Rightarrow g$ , then $\mathsf{Rep}_0(a)(\theta) : \mathsf{Rep}_0(a)(f) \Rightarrow \mathsf{Rep}_0(a)(g)$ is the natural transformation whose component for each $h : b_2 \rightarrow a$ is $\theta \vartriangleright h$ .

• (representable1) Let $a, b : \mathsf{B}$ be objects and let $f : a \rightarrow b$ be a 1-cell. Then, we define the representable pseudotransformation $\mathsf{Rep}_1(f)$ from $\mathsf{Rep}_0(a)$ to $\mathsf{Rep}_0(b)$ . Its component for each $c : \mathsf{B}$ is the functor $\mathsf{Rep}_1(f)(c) : \underline{{\mathsf{B}_1}({c},{a})} \rightarrow \underline{{\mathsf{B}_1}({c},{b})}$ sending g to $g \cdot f$ . If we have $c_1, c_2 : \mathsf{B}$ and a 1-cell $g : c_1 \rightarrow c_2$ , then the naturality 2-cell $\mathsf{Rep}_1(f)(g) : \mathsf{Rep}_1(f)(c_1) \cdot \mathsf{Rep}_0(b)(g) \Rightarrow \mathsf{Rep}_0(a)(g) \cdot \mathsf{Rep}_1(f)(c_2)$ is a natural transformation, whose component for each h is $\alpha(g, h, f) : g \cdot (h \cdot f) \Rightarrow (g \cdot h) \cdot f$ .

• (representable2) Suppose that we have 0-cells $a, b : \mathsf{B}$ , 1-cells $f, g : a \rightarrow b$ , and a 2-cell $\theta : f \Rightarrow g$ . Then, the representable modification $\mathsf{Rep}_2(\alpha)$ from $\mathsf{Rep}_1(f)$ to $\mathsf{Rep}_1(g)$ is a modification, whose component for each $c : \mathsf{B}$ is the natural transformation defined on $h : \mathsf{B}(c,a)$ by $h \vartriangleleft \theta$ .

Definition 5.2 ((y)) Let $\mathsf{B}$ be a locally univalent bicategory. Then, the Yoneda embedding $\mathsf{y} : {\mathsf{B}} \rightarrow {\mathsf{Pseudo}(\mathsf{B}^{\operatorname{\mathsf{{op}}}}, \mathsf{Cat})}$ is defined as

\begin{align*}\mathsf{y}(a) &= \mathsf{Rep}_0(a) && \mbox{for } a : \mathsf{B} \\\mathsf{y}(f) &= \mathsf{Rep}_1(f) && \mbox{for } a, b : \mathsf{B}, f : a \rightarrow b \\\mathsf{y}(\theta) &= \mathsf{Rep}_2(\theta) && \mbox{for } a, b : \mathsf{B}, f, g : a \rightarrow b, \theta : f \Rightarrow b\end{align*}

Construction 5.4 for Problem 5.3; (bicategorical_yoneda_lemma). To construct this, we provide

• (yoneda_to_presheaf) A functor F from ${\mathsf{y}(a)} \Rightarrow {P}$ to P(a);

• (presheaf_to_yoneda) A functor G from P(a) to ${\mathsf{y}(a)} \Rightarrow {P}$ ;

• (yoneda_unit) A natural isomorphism from the identity to $F \cdot G$ ;

• (yoneda_counit) A natural isomorphism from $G \cdot F$ to the identity.

We only discuss the data of the involved functors. The functor F sends pseudotransformations $\tau$ to $\tau(a)({\operatorname{id}}_1(a))$ and modifications m to $m(a)({\operatorname{id}}_1(a))(a)$ . In the other direction, G sends objects $z : P(a)$ to the pseudotransformation whose components are P(f)(z) with $b : \mathsf{B}^{\operatorname{\mathsf{{op}}}}$ and $f : b \rightarrow a$ .

Now let us use the bicategorical Yoneda lemma to construct for each locally univalent bicategory a weakly equivalent univalent bicategory. We follow the construction of the Rezk completion by Ahrens et al. (Reference Ahrens, Kapulkin and Shulman2015) and take the image of the Yoneda embedding to be the univalent completion.

First, we define weak equivalences of bicategories.

• (local_equivalence) F is a local equivalence if for each $x, y : \mathsf{B}$ the functor from $\underline{{\mathsf{B}_1}({x},{y})}$ to $\underline{{\mathsf{C}_1}({F(x)},{F(y)})}$ induced by F is an adjoint equivalence.

• (essentially_surjective) F is essentially surjective if for each $y : \mathsf{C}$ there merely exists an $x : \mathsf{B}$ and an adjoint equivalence from F(x) to y.

• (weak_equivalence) F is a weak equivalence if F is both a local equivalence and essentially surjective.

The notion of weak equivalence has already been studied in classical mathematics where, using the axiom of choice, it was shown to be equivalent to the usual notion of equivalence (Lack Reference Lack2010; Leinster Reference Leinster1998). However, these notions are generally not equivalent in a constructive setting, but we conjecture that they are for univalent bicategories.

Furthermore, the notion of weak equivalence can be weakened by requiring that the pseudofunctor only induces a weak equivalence of categories on the hom-categories. Such a weaker notion would be useful if one desires to find a univalent completion of arbitrary bicategories instead of just locally univalent ones. To do so, we anticipate a two-step process: first, a local completion, which embeds bicategories in locally univalent ones, followed by, second, the construction described in this section. More concretely, for any bicategory $\mathsf{B}$ we expect to be able to construct pseudofunctors as in the following diagram.

Here, $\mathcal{RC}_{\mathsf{loc}}(B)$ is a locally univalent bicategory, and $\mathcal{RC}(B)$ is a univalent one. While the pseudofunctor ${{{\eta_{\mathsf{glob}}}}} : \mathcal{RC}_{\mathsf{loc}}(B) \to \mathcal{RC}(B)$ would be a weak equivalence according to Definition 5.5, ${{{\eta_{\mathsf{loc}}}}} : \mathsf{B} \to \mathcal{RC}_{\mathsf{loc}}(B)$ , would not be: locally, it consists of weak equivalences of categories instead of equivalences. Hence, the more general notion would thus be applicable if one is interested in the univalent completion of arbitrary bicategories. We expect such a local completion ${{{\eta_{\mathsf{loc}}}}} : B \to \mathcal{RC}_{\mathsf{loc}}(B)$ can be constructed by taking the Rezk completion of every hom-category, and since that only yields a weak equivalence of categories, the resulting pseudofunctor is not locally an equivalence. However, we only consider the second step in this paper and leave the construction of ${{{\eta_{\mathsf{loc}}}}} : \mathsf{B} \to \mathcal{RC}_{\mathsf{loc}}(B)$ as an open problem.

Weak equivalences between univalent categories are actually equivalences (Ahrens et al., Reference Ahrens, Kapulkin and Shulman2015, Lemma 6.8).We conjecture that the same is possible for bicategories.

From the Yoneda lemma, we know that $\mathsf{y}$ is a local equivalence:

However, $\mathsf{y}$ is not essentially surjective: the bicategory $\mathsf{Pseudo}(\mathsf{B}^{\operatorname{\mathsf{{op}}}}, \mathsf{Cat})$ contains non-representable presheaves. To make $\mathsf{y}$ essentially surjective, we restrict the bicategory of presheaves to the full image of the Yoneda embedding.

Proof. Follows from Item 3 in Example 3.18.

Now everything is in place to construct the desired embedding into a univalent bicategory.

Construction 5.13 for Problem 5.12; (rezk_completion_2_0). We define $\mathcal{RC}(\mathsf{B})$ to be the image of the Yoneda embedding $\mathsf{y} : {\mathsf{B}} \rightarrow {\mathsf{Pseudo}(\mathsf{B}^{\operatorname{\mathsf{{op}}}}, \mathsf{Cat})}$ . Since the codomain of $\mathsf{y}$ is univalent by Theorem 3.20, the image is univalent as well by Proposition 5.10. Note that the corestriction gives rise to a pseudofunctor $\overline{\mathsf{y}} : {\mathsf{B}} \rightarrow {\mathcal{RC}(\mathsf{B})}$ . It is essentially surjective by construction. Furthermore, $\mathsf{y}$ is a local equivalence by Corollary 5.8, and local equivalences are preserved by corestriction. Hence, $\overline{\mathsf{y}}$ is indeed a weak equivalence.

Note that Construction 5.13 raises universe levels: the bicategory $\mathcal{RC}(\mathsf{B})$ lives in a higher universe than B itself, for the same reasons as in the 1-categorical case (Ahrens et al. Reference Ahrens, Kapulkin and Shulman2015, Remark 8.6).

6. Displayed Bicategories

Now let us study how to construct more complicated univalent bicategories. To that end, we introduce displayed bicategories, the bicategorical analog to the notion of displayed category developed by Ahrens and Lumsdaine (Reference Ahrens and Lumsdaine2019). A displayed (1-)category D over a given (base) category C consists of a family of objects over objects in C and a family of morphisms over morphisms in C together with suitable displayed operations of composition and identity. A category ${\textstyle \int_{{D}}{}}$ is then constructed, the objects and morphisms of which are pairs of objects and morphisms from C and D, respectively. Properties of ${\textstyle \int_{{D}}{}}$ , in particular univalence, can be shown from analogous, but simpler, conditions on C and D.

A prototypical example is the following displayed category over $C:\equiv \mathsf{Set}$ : an object over a set X is a group structure on X, and a morphism over a function $f : X \to X'$ from group structure G (on X) to group structure G’ (on X’) is a proof of the fact that f is compatible with G and G’. The total category is the category of groups, and its univalence follows from univalence of $\mathsf{Set}$ and a univalence property of the displayed data.

Just like in 1-category theory, many examples of bicategories are obtained by endowing previously considered bicategories with additional structure. An example is the bicategory of pointed 1-types in $\operatorname{\textsf{U}}$ . The objects in this bicategory are pairs of a 1-type A and an inhabitant $a : A$ . The morphisms are pairs of a morphism f of 1-types and a path witnessing that f preserves the selected points. Similarly, the 2-cells are pairs of a homotopy p and a proof that this p commutes with the point preservation proofs. Thus, this bicategory is obtained from $\mbox{1-}\mathsf{Type}_{\operatorname{\textsf{U}}}$ by endowing the cells on each level with additional structure.

Of course, the structure should be added in such a way that we are guaranteed to obtain a bicategory at the end. Now let us give the formal definition of displayed bicategories.

1. 1. for each $a : \mathsf{B}_0$ a type ${{{{{\mathsf{D}}}_{a}}}}$ of displayed 0-cells over a;

2. 2. for each $f : a \rightarrow b$ in $\mathsf{B}$ and $\bar{a} : {{{{\mathsf{D}}_{a}}}} , \bar{b} : {{{{\mathsf{D}}_{b}}}}$ a type ${\bar{a}} \xrightarrow{f} {\bar{b}}$ of displayed 1-cells over f;

3. 3. for each $\theta : f \Rightarrow g$ in $\mathsf{B}$ , $\bar{f} : {\bar{a}} \xrightarrow{f} {\bar{b}}$ and $\bar{g} : {\bar{a}} \xrightarrow{g} {\bar{b}}$ a set of displayed 2-cells over $\theta$ and dependent versions of operations and laws from Definition 2.1, which are

4. 4. for each $a : \mathsf{B}_0$ and $\bar{a} : {{{{\mathsf{D}}_{a}}}}$ , we have $\operatorname{id}_1(\bar{a}) : {\bar{a}} \xrightarrow{{\operatorname{id}_1\!(a)}} {\bar{a}}$ ;

5. 5. for all 1-cells $f : a \rightarrow b$ , $g : b \rightarrow c$ , and displayed 1-cells $\bar{f} : {\bar{a}} \xrightarrow{f} {\bar{b}}$ and $\bar{g} : {\bar{b}} \xrightarrow{\bar{c}} {\bar{c}}$ , we have a displayed 1-cell $\bar{f} \cdot \bar{g} : {\bar{a}} \xrightarrow{{f \cdot g}} {\bar{c}}$ ;

6. 6. for all $f : \mathsf{B}_1(a,b)$ , $\bar{a} : {{{{\mathsf{D}}_{a}}}}$ , $\bar{b} : {{{{\mathsf{D}}_{b}}}}$ , and $\bar{f} : {\bar{a}} \xrightarrow{f} {\bar{b}}$ , we have ;

7. 7. for 2-cells $\theta : f \Rightarrow g$ and $\gamma : g \Rightarrow h$ , and displayed 2-cells and $\bar{\gamma} : {\bar{g}} \xrightarrow{\gamma} {\bar{h}}$ , we have a displayed 2-cell .

8. 8. for each displayed 1-cell $\bar{f} : {\bar{a}} \xrightarrow{f} {\bar{b}}$ and each displayed 2-cell , we have a displayed 2-cell ;

9. 9. for each displayed 1-cell $\bar{h} : {\bar{b}} \xrightarrow{h} {\bar{c}}$ and each displayed 2-cell , we have a displayed 2-cell ;

10. 10. for each $\bar{f} : {\bar{a}} \xrightarrow{f} {\bar{b}}$ , we have displayed 2-cells and ;

11. 11. for each $\bar{f} : {\bar{a}} \xrightarrow{f} {\bar{b}}$ , displayed 2-cells and ;

12. 12. for each , , and , we have displayed 2-cells and .

Note that we use the same notation for the displayed and the non-displayed operations.

These operations are subject to laws, which are derived systematically from the non-displayed version. Just as for displayed 1-categories, the laws of displayed bicategories are heterogeneous, because they are transported along the analogous law in the base bicategory. For instance, the displayed left-unitary law for identity reads as $\operatorname{id}_2\!(\bar{f}) \bullet \bar{\theta} =_e \bar{\theta}$ , where e is the corresponding identity of item 13 in Definition 2.1.

1. 13. $\operatorname{id}_2\!(f) \bullet \theta =_* \theta, \quad \theta \bullet \operatorname{id}_2\!(g) =_* \theta, \quad \theta \bullet (\gamma \bullet \tau) =_* (\theta \bullet \gamma) \bullet \tau$ ;

2. 14. $f \vartriangleleft ({\operatorname{id}}_2 g) =_* \operatorname{id}_2\!(f \cdot g), \quad f \vartriangleleft (\theta \bullet \gamma) =_* (f \vartriangleleft \theta) \bullet (f \vartriangleleft \gamma)$ ;

3. 15. $({\operatorname{id}}_2 f) \vartriangleright g =_* \operatorname{id}_2\!(f \cdot g), \quad (\theta \bullet \gamma) \vartriangleright g =_* (\theta \vartriangleright g) \bullet (\gamma \vartriangleright g)$ ;

4. 16. $({\operatorname{id}}_1(a) \vartriangleleft \theta) \bullet \lambda(g) =_* \lambda(f) \bullet \theta$ ;

5. 17. $(\theta \vartriangleright \operatorname{id}_1\!(b)) \bullet \rho(g) =_* \rho(f) \bullet \theta$ ;

6. 18. $(f \vartriangleleft (g \vartriangleleft \theta)) \bullet \alpha(f, g, i) =_* \alpha(f,g,h) \bullet ((f \cdot g) \vartriangleleft \theta)$ ;

7. 19. $(f \vartriangleleft (\theta \vartriangleright i)) \bullet \alpha(f,h,i) =_* \alpha(f,g,i) \bullet ((f \vartriangleleft \theta) \vartriangleright i)$ ;

8. 20. $(\theta \vartriangleright (h \cdot i)) \bullet \alpha(g,h,i) =_* \alpha(f,h,i) \bullet ((\theta \vartriangleright h) \vartriangleright i)$ ;

9. 21. $\lambda(f) \bullet \lambda(f)^{-1} =_* \operatorname{id}_2\!({\operatorname{id}}_1(a) \cdot f), \quad \lambda(f)^{-1} \bullet \lambda(f) =_* \operatorname{id}_2\!(f)$ ;

10. 22. $\rho(f) \bullet \rho(f)^{-1} =_* \operatorname{id}_2\!(f \cdot \operatorname{id}_1\!(b)), \quad \rho(f)^{-1} \bullet \rho(f) =_* \operatorname{id}_2\!(f)$ ;

11. 23. $\alpha(f,g,h) \bullet \alpha(f,g,h)^{-1} =_* \operatorname{id}_2\!(f \cdot (g \cdot h)), \quad \alpha(f,g,h)^{-1} \bullet \alpha(f,g,h) =_* \operatorname{id}_2\!((f \cdot g) \cdot h)$ ;

12. 24. $\alpha(f, \operatorname{id}_1\!(b),g) \bullet (\rho(f) \vartriangleright g) =_* f \vartriangleleft \lambda(f)$ ;

13. 25. $\alpha(f,g,h \cdot i) \bullet \alpha(f \cdot g, h, i) =_* (f \vartriangleleft \alpha(g,h,i)) \bullet \alpha(f,g \cdot h, i) \bullet (\alpha(f,g,h) \vartriangleright i)$ .

The purpose of displayed bicategories is to give rise to a total bicategory together with a projection pseudofunctor. They are defined as follows:

1. (1) as 0-cells tuples $(a, \bar{a})$ , where $a : \mathsf{B}$ and $\bar{a} : {{{{\mathsf{D}}_{a}}}}$ ;

2. (2) as 1-cells tuples $(f, \bar{f}) : (a,\bar{a}) \rightarrow (b,\bar{b})$ , where $f : a \rightarrow b$ and $\bar{f} : {\bar{a}} \xrightarrow{f} {\bar{b}}$ ;

3. (3) as 2-cells tuples $(\theta, \bar{\theta}) : (f, \bar{f}) \Rightarrow (g, \bar{g})$ , where $\theta : f \Rightarrow g$ and .

We also have a projection pseudofunctor $\pi_D : {{\textstyle \int_{{D}}{}}} \rightarrow {\mathsf{B}}$ .

As mentioned before, the bicategory of pointed 1-types is the total bicategory of the following displayed bicategory.

• For 1-type A in $\operatorname{\textsf{U}}$ , the objects over A are inhabitants of A.

• For $f : A \rightarrow B$ with A, B 1-types in $\operatorname{\textsf{U}}$ , the displayed 1-cells over f from a to b are paths $f(a) = b$ .

• Given two functions $f, g : A \rightarrow B$ , a homotopy $p : f \sim g$ , two points $a : A$ and $b : B$ , and paths $q_f : f(a) = b$ and $q_g : g(a) = b$ , the 2-cells over p are paths $q_f = p(a) \bullet q_g$ .

The bicategory of pointed 1-types is the total bicategory of this displayed bicategory.

• For a groupoid G, the objects over G are objects of G.

• For a functor $F : G_1 \rightarrow G_2$ between groupoids $G_1$ and $G_2$ , the displayed 1-cells over F from x to y are isomorphisms $F(a) \cong b$ .

• Given two functors $F_1, F_2 : G_1 \rightarrow G_2$ , a natural transformation $n : F_1 \Rightarrow F_2$ , two points $x : G_1$ and $y : G_2$ , and isomorphisms $q_1 : F_1(x) \cong y$ and $q_2 : F_2(x) = y$ , the displayed 2-cells over n are paths $p(a) \bullet q_g = q_f$ .

The bicategory of pointed groupoids is the total bicategory of this displayed bicategory.

We end this section presenting several general constructions of displayed bicategories.

1. (1) (disp_dirprod_bicat) Given displayed bicategories $\mathsf{D}_1$ and $\mathsf{D}_2$ over a bicategory $\mathsf{B}$ , we construct the product $\mathsf{D}_1 \times \mathsf{D}_2$ over $\mathsf{B}$ . The 0-cells, 1-cells, and 2-cells are pairs of 0-cells, 1-cells, and 2-cells, respectively.

2. (2) (sigma_bicat) Given a displayed bicategory $\mathsf{D}$ over a base $\mathsf{B}$ and a displayed bicategory $\mathsf{E}$ over ${\textstyle \int_{\mathsf{D}}{}}$ , we construct the sigma displayed bicategory ${\sum_{[\mathsf{D}]}{{\mathsf{E}}}}$ over $\mathsf{B}$ as follows. The objects over $a : \mathsf{B}$ are pairs $(\bar{a}, e)$ , where $\bar{a} : {{{{\mathsf{D}}_{a}}}}$ and $e : {{{{{\mathsf{E}}}_{{(a,\bar{a})}}}}}$ , the morphisms over $f : {a} \rightarrow {b}$ from $(\bar{a}, e)$ to $(\bar{b}, e')$ are pairs $(\bar{f}, \varphi)$ , where $\bar{f} : {\bar{a}} \xrightarrow{f} {\bar{b}}$ and $\varphi : {e} \xrightarrow{{(f, \bar{f})}} {e'}$ , and similarly for 2-cells.

3. (3) (trivial_displayed_bicat) Every bicategory $\mathsf{D}$ is, in a trivial way, a displayed bicategory over any other bicategory $\mathsf{B}$ . Its total bicategory is the direct product $\mathsf{B} \times \mathsf{D}$ .

4. (4) (disp_cell_unit_bicat) We say a displayed bicategory $\mathsf{D}$ over $\mathsf{B}$ is locally chaotic if, for each $\alpha : f \Rightarrow g$ and $\bar{f} : {\bar{a}} \xrightarrow{f} {\bar{b}}$ and $\bar{g} : {\bar{a}} \xrightarrow{g} {\bar{b}}$ , the type is contractible. Let $\mathsf{B}$ be a bicategory and suppose we have

5. − for each object a in $\mathsf{B}$ a type ${{{{\mathsf{D}}_{b}}}}$ of displayed 0-cells;

6. − for each 1-cell $f : a \rightarrow b$ in $\mathsf{B}$ and for each $\bar{a} : {{{{\mathsf{D}}_{a}}}}, \bar{b} : {{{{\mathsf{D}}_{b}}}}$ a type ${\bar{a}} \xrightarrow{f} {\bar{b}}$ of displayed 1-cells;

7. − displayed 1-identities $\operatorname{id}_1$ and compositions $(\cdot)$ of displayed 1-cells as in Definition 6.1. Then we have an associated locally chaotic displayed bicategory $\hat{\mathsf{D}}$ over $\mathsf{B}$ by stipulating that the types of 2-cells are the unit type. Note that this construction essentially gives a way of obtaining a displayed bicategory from the data of a displayed category (Ahrens and Lumsdaine Reference Ahrens and Lumsdaine2019, Def. 3.1, Items 1–4).

Now let us discuss two more examples of bicategories obtained from displayed bicategories: firstly, monads internal to an arbitrary bicategory and secondly, Kleisli triples. In Construction 8.14, we construct a biequivalence between the bicategory of Kleisli triples and the bicategory of monads internal to $\mathsf{Cat}$ .

• The displayed objects over $a : \mathsf{B}$ are monad structures on a. A monad structure on a consists of a 1-cell $m_a : a \rightarrow a$ and 2-cells $\eta_a : \operatorname{id}_1\!(a) \Rightarrow m$ and $\mu_a : m \cdot m \Rightarrow m$ such that the following diagrams commute

  

• The displayed 1-cells over $f : a \rightarrow b$ from $(m_a, \eta_a, \mu_a)$ to $(m_b, \eta_b, \mu_b)$ consist of invertible 2-cells $n_f : m_a \cdot f \Rightarrow f \cdot m_b$ such that the following two diagrams commute

  
  

• The displayed 2-cells over $x : f \Rightarrow g$ from $n_f$ to $n_g$ are proofs that the following diagrams commute

  

The total bicategory of $\mathsf{M}(\mathsf{B})$ is the bicategory of monads internal to $\mathsf{B}$ .

Next, we define a bicategory of Kleisli triples, also known as extension systems (Marmolejo and Wood Reference Marmolejo and Wood2010).

• The displayed objects over C are Kleisli triples over C. These consist of a function $M : C_0 \rightarrow C_0$ , for each $a : C$ an arrow $\eta(a) : a \rightarrow M(a)$ , and for each arrow $f : a \rightarrow M(b)$ , an arrow $f^* : M(a) \rightarrow M(b)$ such that the usual laws hold.

• The displayed 1-cells over a functor $F : C \rightarrow D$ from $M_C$ to $M_D$ consists of isomorphisms $F_M$ from $M_D(F(a))$ to $F(M_C(a))$ for each $a : C_0$ such that the usual laws hold.

• The displayed 2-cells over $n : F \Rightarrow G$ from $F_M$ to $G_M$ are equalities

  \begin{equation*} F_M(a) \cdot n(M_C(a)) = M_D(n(a)) \cdot G_M(a). \end{equation*}

The total bicategory of $\mathsf{K}$ is the bicategory of Kleisli triples.

7. Displayed Univalence

Given a bicategory $\mathsf{B}$ and a displayed bicategory $\mathsf{D}$ over $\mathsf{B}$ , our goal is to prove the univalence of ${\textstyle \int_{\mathsf{D}}{}}$ from conditions on $\mathsf{B}$ and $\mathsf{D}$ . For that, we develop the notion of univalent displayed bicategories. We start by defining displayed versions of invertible 2-cells.

A displayed invertible 2-cell over $\theta$ , where $\theta$ is an invertible 2-cell, is a pair of a displayed 2-cell $\bar{\theta}$ over $\theta$ and a proof that $\bar{\theta}$ is invertible. The type of displayed invertible 2-cells from $\bar{f}$ to $\bar{g}$ over $\theta$ is denoted by ${{{ {{\bar{f}}} \cong_{{\theta}} {{\bar{g}}} }}}$ .

Being a displayed invertible 2-cell is a proposition and the displayed 2-cell $\operatorname{id}_2\!(\bar{f})$ over $\operatorname{id}_2\!(f)$ is invertible. Next, we define displayed adjoint equivalences.

• A displayed 1-cell $\bar{r} : {\bar{b}} \xrightarrow{{r}} {\bar{a}}$ ;

• An invertible displayed 2-cell ;

• An invertible displayed 2-cell .

In addition, two laws reminiscent of those in Definition 2.5 need to be satisfied.

A displayed adjoint equivalence over the adjoint equivalence A is a pair of a displayed 1-cell $\bar{f}$ over f together with a displayed adjoint equivalence structure on $\bar{f}$ . The type of displayed adjoint equivalences from $\bar{a}$ to $\bar{b}$ over f is denoted by ${{{ {{\bar{a}}} \simeq_{{f}} {{\bar{b}}} }}}$ .

The displayed 1-cell $\operatorname{id}_1\!(\bar{a})$ is a displayed adjoint equivalence over $\operatorname{id}_1\!(a)$ .

Using these definitions, we define univalence of displayed bicategories similarly to univalence for ordinary bicategories. Again we separate it in a local and global condition.

1. (1) Let $a, b : \mathsf{B}$ , and $\bar{a} : {{{{\mathsf{D}}_{a}}}}, \bar{b} : {{{{\mathsf{D}}_{b}}}}$ . Let $f, g : {a} \rightarrow {b}$ , let $p : f = g$ , and let $\bar{f}$ and $\bar{g}$ be displayed morphisms over f and g respectively. Then we define a function

   \begin{equation*} \mathsf{disp\_idtoiso}^{2,1}_{p,\bar{f},\bar{g}} : \bar{f} =_{[p]} \bar{g} \rightarrow {{{ {{\bar{f}}} \cong_{{\mathsf{idtoiso}^{2,1}_{f,g}(p)}} {{\bar{g}}} }}} \end{equation*}
   sending $\operatorname{\mathsf{{refl}}}$ to the identity displayed isomorphism. We say that $\mathsf{D}$ is locally univalent if the function $\mathsf{disp\_idtoiso}^{2,1}_{p,\bar{f},\bar{g}}$ is an equivalence for each p, $\bar{f}$ , and $\bar{g}$ .

2. (2) Let $a, b : \mathsf{B}$ , and $\bar{a} : {{{{\mathsf{D}}_{a}}}}, \bar{b} : {{{{\mathsf{D}}_{b}}}}$ . Given $p : a = b$ , we define a function

   \begin{equation*}\mathsf{disp\_idtoiso}^{2,0}_{p,\bar{a},\bar{b}} : \bar{a} =_{[p]} \bar{b} \rightarrow {{{ {{\bar{a}}} \simeq_{{\mathsf{idtoiso}^{2,0}_{a,b}(p)}} {{\bar{b}}} }}}\end{equation*}
   sending $\operatorname{\mathsf{{refl}}}$ to the identity displayed adjoint equivalence. We say that $\mathsf{D}$ is globally univalent if the function $\mathsf{disp\_idtoiso}^{2,0}_{p,\bar{a},\bar{b}}$ is an equivalence for each p, $\bar{a}$ , and $\bar{b}$ .

3. (3) (disp_univalent_2) We call $\mathsf{D}$ univalent if it is both locally and globally univalent.

The following result states that univalence of the total bicategory can be proved from univalence of the base and of the displayed part. This is the bicategorical version of the analogous result for 1-categories shown by Ahrens and Lumsdaine (Reference Ahrens and Lumsdaine2019, Theorem 7.4), which in turn generalizes the Structure Identity Principle (Univalent Foundations Program 2013, Theorem 9.8.2).

1. (1) ${\textstyle \int_{\mathsf{D}}{}}$ is locally univalent if $\mathsf{B}$ is locally univalent and $\mathsf{D}$ is locally univalent;

2. (2) ${\textstyle \int_{\mathsf{D}}{}}$ is globally univalent if $\mathsf{B}$ is globally univalent and $\mathsf{D}$ is globally univalent.

Proof The main idea behind the proof is to characterize invertible 2-cells in the total bicategory as pairs of an invertible 2-cell p in the base bicategory and a displayed invertible 2-cell over p. Concretely, for the local univalence of $\mathsf{D}$ , we factor $\mathsf{idtoiso}^{2,1}$ as a composition of the following equivalences:

The function $w_1$ is just a characterization of paths in a sigma type. The function $w_2$ turns equalities into (displayed) invertible 2-cells, and it is an equivalence by local univalence of $\mathsf{B}$ and displayed local univalence of $\mathsf{D}$ . Finally, the function $w_3$ characterizes invertible 2-cells in the total bicategory.

The proof is similar in the case of global univalence. The most important step is the characterization of adjoint equivalences in the total bicategory.

\begin{equation*}(a,\bar{a}) \simeq (b, \bar{b}) \xrightarrow{\sim}\sum_{(p : a \simeq b)} . {{{ {{\bar{a}}} \simeq_{{p}} {{\bar{b}}} }}}.\end{equation*}

To check displayed univalence, it suffices to prove the condition in the case where p is reflexivity. This step, done by path induction, simplifies some proofs of displayed univalence.

• (fiberwise_local_univalent_is_univalent_2_1)

  \begin{equation*} \mathsf{disp\_idtoiso}^{2,1}_{\operatorname{\mathsf{{refl}}}(f),\bar{f},\overline{f'}} : \bar{f} = \overline{f'} \rightarrow {{{ {{\bar{f}}} \cong_{{\operatorname{id}_2\!(f)}} {{\overline{f'}}} }}} \end{equation*}

• (fiberwise_univalent_2_0_to_disp_univalent_2_0)

  \begin{equation*} \mathsf{disp\_idtoiso}^{2,0}_{\operatorname{\mathsf{{refl}}}(a),\bar{a},\overline{a'}} : \bar{a} = \overline{a'} \rightarrow {{{ {{\bar{a}}} \simeq_{{\operatorname{id}_1\!(a)}} {{\overline{a'}}} }}} \end{equation*}

Now, we establish the univalence of several examples.

1. (1) The category of pointed 1-types (see Example 6.3) is univalent (p1types_univalent_2).

2. (2) The full subbicategory (see Definition 2.10) of a univalent bicategory is univalent (is_univalent_2_fullsubbicat).

3. (3) The product of univalent displayed bicategories (Definition 6.6, Item 1) is univalent (is_univalent_2_dirprod_bicat).

For the sigma construction, we give two conditions for the univalence of the total bicategory. If we have univalent displayed bicategories $\mathsf{D}_1$ and $\mathsf{D}_2$ over $\mathsf{B}$ and ${\textstyle \int_{{\mathsf{D}_1,}}{}}$ respectively, then we can either show the univalence of ${\textstyle \int_{{({\sum_{[\mathsf{D}_1]}{{\mathsf{D}_2}}})}}{}}$ directly or we can show the displayed univalence of ${\sum_{[\mathsf{D}_1]}{{\mathsf{D}_2}}}$ . Note that the second property could be necessary as an intermediate step for proving the univalence of a more complicated bicategory. For the proof of displayed univalence of ${\sum_{[\mathsf{D}_1]}{{\mathsf{D}_2}}}$ , we need two assumptions on both displayed bicategories.

1. (1) The bicategory ${\textstyle \int_{{({\sum_{[\mathsf{D}_1]}{{\mathsf{D}_2}}})}}{}}$ (Definition 6.6, Item 2) is univalent (sigma_is_univalent_2).

2. (2) If $\mathsf{D}_1$ and $\mathsf{D}_2$ are locally propositional and groupoidal, then ${\sum_{[\mathsf{D}_1]}{{\mathsf{D}_2}}}$ is displayed univalent (sigma_disp_univalent_2_with_props).

We are not sure whether Item 2 of Proposition 7.9 is as strong as it can be – it might be possible to weaken the assumptions of $\mathsf{D}_1$ and $\mathsf{D}_2$ being locally propositional and groupoidal. However, this would make the proof significantly more complicated. In our examples, these assumptions are satisfied, and thus, the statement of Proposition 7.9, Item 2 is sufficient for our purposes.

Lastly, we give a condition for when a locally chaotic displayed bicategory is univalent.

8. Displayed Constructions

The idea of building bicategories by layering displayed bicategories does not only allow for modular proofs of univalence but also for the modular construction of maps between bicategories, for example, pseudofunctors and biequivalences. In this section, we introduce the notions of displayed pseudofunctor and biequivalence, and use them to build biequivalences. The first example we look at extends the biequivalence between 1-types and univalent groupoids in Example 2.18 to their pointed variants (Examples 6.2 and 6.3).

To construct the desired biequivalence, we first define displayed biequivalences over a given biequivalence in the base and we show that it gives rise to a total biequivalence on the total bicategories. Since biequivalences are defined using pseudofunctors, pseudotransformations, and invertible modifications, we first need to define displayed analogues of these.

• For each $a : \mathsf{B}$ a function $\bar{F}_0 : \mathsf{D}_1(a) \rightarrow \mathsf{D}_2(F(a))$ ;

• For every 1-cell $f : a \rightarrow b$ and all displayed objects $\bar{a} : \mathsf{D}_1(a)$ and $\bar{b} : \mathsf{D}_1(b)$ , a function sending $f : {\bar{a}} \xrightarrow{f} {\bar{b}}$ to $\bar{F}_1(f) : {\bar{F}_0(\bar{a})} \xrightarrow{{F(f)}} {\bar{F}_0(\bar{b})}$ ;

• For each 2-cell $\theta : f \Rightarrow g$ and displayed 1-cells $\bar{f} : {\bar{a}} \xrightarrow{f} {\bar{b}}$ and $\bar{g} : {\bar{a}} \xrightarrow{g} {\bar{b}}$ , a function sending to ;

• For all objects $a : \mathsf{B}$ and displayed objects $\bar{a} : \mathsf{D}_1(a)$ , we have a displayed invertible 2-cell ;

• For all displayed 1-cells $\bar{f} : {\bar{a}} \xrightarrow{f} {\bar{b}}$ and $\bar{g} : {\bar{b}} \xrightarrow{\bar{c}} {\bar{c}}$ , we have a displayed invertible 2-cell .

In addition, several laws similar to those in Definition 2.12 need to hold. They are just dependent variants of them, and they hold over the corresponding non-dependent law. Since the required laws are obtained in the same way as in Definition 6.1, we do not show them here and instead refer the interested reader to the formalization. We denote the type of displayed pseudofunctors from $\mathsf{D}_1$ to $\mathsf{D}_2$ over F by ${\mathsf{D}_1} \xrightarrow{{{F}}} {\mathsf{D}_2}$ .

• For each $x : \mathsf{B}$ and $\bar{x} : \mathsf{D}_1(x)$ a displayed 1-cell $\bar{\eta}_0(\bar{x}) : {\bar{F}_0(\bar{x})} \xrightarrow{{\eta_0(x)}} {\bar{G}_0(\bar{x})}$ ;

• For all 1-cells $f : x \rightarrow y$ , displayed objects $\bar{x} : \mathsf{D}_1(x)$ and $\bar{y} : \mathsf{D}_1(y)$ and displayed 1-cells $\bar{f} : {\bar{x}} \xrightarrow{{f}} {\bar{y}}$ , a displayed invertible 2-cell .

Again laws similar to those in Definition 2.13 need to hold, and again they are derived similar to those in Definition 6.1. We denote the type of displayed pseudotransformations from $\bar{F}$ to $\bar{G}$ over $\eta$ by .

In order to formulate displayed biequivalence, we need an invertible version of Definition 8.4.

Definition 8.5 ((disp-invmodification)) A displayed invertible modification over an invertible modification $m : {\eta} \Rrightarrow {\theta}$ is a displayed modification such that

is invertible for each $x : \mathsf{B}$ and $\bar{x} : \mathsf{D}_1(x)$ .

Each of the discussed notions also has a total version. These are constructed similarly to how the total bicategory is constructed in Definition 6.2.

• (total_psfunctor) Given a displayed pseudofunctor $\bar{F} : {{\mathsf{D}_1}} \xrightarrow{{F}} {\mathsf{D}_2}$ , to construct a pseudofunctor ${\textstyle \int_{{\bar{F}}}{}} : {{\textstyle \int_{{\mathsf{D}_1}}{}}} \rightarrow {{\textstyle \int_{{\mathsf{D}_2}}{}}}$ .

• (total_pstrans) Given a displayed pseudotransformation , to construct a pseudotransformation ${\textstyle \int_{{\bar{\eta}}}{}} : {{\textstyle \int_{{\bar{F}}}{}}} \Rightarrow {{\textstyle \int_{{\bar{G}}}{}}}$ .

• (total_invmodification) Given a displayed invertible modification $\bar{m}$ from $\bar{\eta}$ to $\bar{\theta}$ , to construct an invertible modification ${\textstyle \int_{{\bar{m}}}{}} : {{\textstyle \int_{{\bar{\eta}}}{}}} \Rrightarrow {{\textstyle \int_{{\bar{\theta}}}{}}}$ .

Construction 8.7 for Problem 8.6. Each of the constructions is defined componentwise. For example, ${\textstyle \int_{{\bar{F}}}{}}$ on an object $(x, \bar{x})$ is defined to be $(F(x), \bar{F}(\bar{x}))$ .

To define displayed biequivalences, we need composition and identity of displayed pseudofunctors and pseudotransformations:

• (disp_pseudo_id) We have the identity displayed pseudofunctor $\operatorname{id}(\mathsf{D}_1) : {{\mathsf{D}_1}} \xrightarrow{{{\operatorname{id}(\mathsf{B}_1)}}} {\mathsf{D}_1}$ .

• (disp_pseudo_comp) We have a composition displayed pseudofunctor $\bar{F} \cdot \bar{G} : {{\mathsf{D}_1}} \xrightarrow{{{F \cdot G}}} {\mathsf{D}_3}$ .

• (disp_id_pstrans) We have a displayed identity pseudotransformation .

• (disp_comp_pstrans) Suppose, we also have pseudofunctors $F', F'' : {\mathsf{B}_1} \rightarrow {\mathsf{B}_2}$ and pseudotransformations $\eta : {F} \Rightarrow {F'}$ and $\theta : {F'} \Rightarrow {F''}$ . If we also have displayed pseudofunctors $\bar{F'} : {{\mathsf{D}_1}} \xrightarrow{{{F'}}} {\mathsf{D}_2}$ and $\bar{F''} : {{\mathsf{D}_1}} \xrightarrow{{{F''}}} {\mathsf{D}_2}$ and displayed pseudotransformations and , then we have a composition displayed pseudotransformation .

Now we have developed sufficient displayed machinery to define displayed biequivalences.

• A displayed pseudofunctor $\bar{L} : {{\mathsf{D}_1}} \xrightarrow{{{L}}} {\mathsf{D}_2}$ ;

• A displayed pseudofunctor $\bar{R} : {{\mathsf{D}_2}} \xrightarrow{{{R}}} {\mathsf{D}_1}$ ;

• Displayed pseudotransformations and ;

• Displayed pseudotransformation and ;

• Displayed invertible modifications

  

• Displayed invertible modifications

  

Note that the total variant of each example in Definition 8.9 is its non-displayed analogue. Displayed biequivalences give rise to total biequivalences between their associated total bicategories.

Construction 8.11 for Problem 8.10; (total_is_biequivalence). The pseudofunctors, pseudotransformations, and invertible modifications are constructed using Construction 8.7.

Note that to construct a displayed biequivalence, one must show several laws and construct multiple displayed invertible 2-cells. If the involved displayed bicategories are locally groupoidal (Definition 7.7) and locally propositional (Definition 7.8), then constructing a displayed biequivalence is simpler. This is because all the necessary laws follow immediately from local propositionality and all the involved displayed 2-cells are invertible. With all this in place, we finally show how to construct the desired biequivalence in Problem 8.1 with displayed machinery.

Construction 8.12 for Problem 8.1; disp_biequiv_data_unit_counit_path_pgroupoid)]prob:biequivpgrpds By Problem 8.10, it suffices to construct a displayed biequivalence. We only show how to construct the required displayed pseudofunctor from points on 1-types to points on groupoids.

• Given a 1-type X and a point $x : X$ , we need to give an object of $\mathsf{PathGrpd}(X)$ , for which we take x.

• If we have 1-types X and Y with points $x : X$ and $y : Y$ , and a function $f : X \rightarrow Y$ with a path $p_f : f(x) = y$ , then we need to construct an isomorphism between f(x) and y in $\mathsf{PathGrpd}(X)$ . It is given by $p_f$ .

• Suppose we have 1-types X and Y with points $x : X$ and $y : Y$ . Furthermore, suppose we have a homotopy $s : f \sim g$ between functions $f, g : X \rightarrow Y$ , paths $p_f : f(x) = y$ and $p_g : g(x) = y$ , and a path $h : p_f = s(x) \bullet p_g$ . Then, the required displayed 2-cell is the inverse of h.

The compositor and the identitor are both the reflexivity path.

As a final example, we construct a biequivalence between the bicategory of monads internal to $\mathsf{Cat}$ and the bicategory of Kleisli triples. To construct a biequivalence between monads and Kleisli triples.

Construction 8.14 for Problem 8.13; (Monad_biequiv_Ktriple). Note that the bicategory of monads and Kleisli triples are defined as the total bicategories of Definitions 6.6 and 6.7, respectively. Hence, by Problem 8.10, it is sufficient to construct a displayed biequivalence between the respective displayed bicategories. For the details on this construction, we refer the reader to the formalization.

9. Univalence of Complicated Bicategories

In this section, we demonstrate the power of displayed bicategories on a number of complicated examples. We show the univalence of the bicategory of pseudofunctors between univalent bicategories and of univalent categories with families. In addition, we give two constructions to define univalent bicategories of algebras.