2. Related Work

In this section, we review work with a similar goal to ours, as well as work we rely on. We pay particular attention to the desiderata outlined in Section 1.2 and to the difference between judgmental and typal dimensions.

3 Preliminaries

Here, we sketch some definitions used later on. Many would be very long if given in full; instead, we try to convey some intuition while pointing to the formalized definitions. As a reference for bicategory theory, see Bénabou (Reference Bénabou1967). We use here the vocabulary and notation introduced by Ahrens et al. (Reference Ahrens, Frumin, Maggesi, Veltri and Van der Weide2021).

We occasionally write $1_a$ for $\operatorname{id}_1(a)$ and $1_f$ for $\operatorname{id}_2(\,f)$ .

Sometimes it is still interesting to assume that a given (displayed) bicategory is univalent; in such cases, fibrations are better behaved, since lifts can be shown to be actually unique rather than just essentially unique (see, for instance, Proposition 26). Whenever we state such a result, the assumption on the (displayed) bicategory to be univalent is stated explicitly.

We denote by $\mathsf{Cat}$ the bicategory of categories, and by $\mathsf{Grpd}$ the bicategory of groupoids. The bicategory $\mathsf{B}^{\mathsf{op}}$ has the same objects as $\mathsf{B}$ , but 1-cells from x to y in $\mathsf{B}^{\mathsf{op}}$ are 1-cells $f : y \rightarrow x$ in $\mathsf{B}$ . The 2-cells in $\mathsf{B}^{\mathsf{op}}$ are 2-cells in $\mathsf{B}$ . The bicategory $\mathsf{B}^{\mathsf{co}}$ has the same objects and 1-cells as $\mathsf{B}$ , but 2-cells from f to g in $\mathsf{B}^{\mathsf{co}}$ are the same as 2-cells from g to f in $\mathsf{B}$ .

1. (1) for any $b : \mathsf{B}_0$ , a type ${{{{\mathsf{D}}_{b}}}}$ of objects over b;

2. (2) for any $f : a\to b$ and $x : {{{{\mathsf{D}}_{a}}}}$ and $y : {{{{\mathsf{D}}_{b}}}}$ , a type $x \xrightarrow{f} y$ of 1-cells over f;

3. (3) for any $\theta :\, f \Rightarrow g$ and $\overline{f} : x \xrightarrow{f} y$ and $\overline{g} : x \xrightarrow{g} y$ , a set $\overline{f} \buildrel {\theta} \over \Rightarrow \overline{g}$ of 2-cells over $\theta$ ;

together with suitably typed composition (over composition in $\mathsf{B}$ ) and identity (over identity in $\mathsf{B}$ ) for both 1- and 2-cells. These operations are subject to “axioms over axioms in $\mathsf{B}$ ”.

We define the projection pseudofunctor $\pi : \int\!\!{\mathsf{D}} \to \mathsf{B}$ to be given, on any cell, by $(b,\overline{b})\mapsto b$ .

• the type of displayed objects over x is $P_0(x)$ ;

• the type of displayed 1-cells over $f : x \rightarrow y$ is $P_1(\,f)$ ; and

• the type of displayed 2-cells over $\theta :\, f \Rightarrow g$ is the unit type.

The total bicategory of this bicategory selects 0-cells and 1-cells from the original bicategory $\mathsf{B}$ . Its objects are objects $x : \mathsf{B}$ such that $P_0(x)$ , its 1-cells are 1-cells $f : x \rightarrow y$ in $\mathsf{B}$ such that $P_1(\,f)$ , and its 2-cells are the same as 2-cells $\tau :\, f \Rightarrow g$ in $\mathsf{B}$ . The projection pseudofunctor $\pi : \mathsf{SubBicat}(P_0, P_1) \to \mathsf{B}$ is the inclusion.

We instantiate Definition 6 to define bicategories of categories with a certain structure.

• $\mathsf{D}_{\mathsf{Terminal}}$ where $P_0(\mathsf{C})$ says that $\mathsf{C}$ has a terminal object and $P_1(F)$ says that F preserves terminal objects. We denote its total bicategory by $\mathsf{Cat}_{\mathsf{Terminal}}$ .

• $\mathsf{D}_{\mathsf{BinProd}}$ where $P_0(\mathsf{C})$ says that $\mathsf{C}$ has binary products and $P_1(F)$ says that F preserves binary products. We denote its total bicategory by $\mathsf{Cat}_{\mathsf{BinProd}}$ .

• $\mathsf{D}_{\mathsf{Pullback}}$ where $P_0(\mathsf{C})$ says that $\mathsf{C}$ has pullbacks and $P_1(F)$ says that F preserves pullbacks. We denote its total bicategory by $\mathsf{Cat}_{\mathsf{Pullback}}$ .

• $\mathsf{D}_{\mathsf{FinLim}}$ where $P_0(\mathsf{C})$ says that $\mathsf{C}$ has finite limits and $P_1(F)$ says that F preserves finite limits. We denote its total bicategory by $\mathsf{Cat}_{\mathsf{FinLim}}$ .

• $\mathsf{D}_{\mathsf{Initial}}$ where $P_0(\mathsf{C})$ says that $\mathsf{C}$ has an initial object and $P_1(F)$ says that F preserves initial objects. We denote its total bicategory by $\mathsf{Cat}_{\mathsf{Initial}}$ .

• $\mathsf{D}_{\mathsf{BinSum}}$ where $P_0(\mathsf{C})$ says that $\mathsf{C}$ has binary coproducts and $P_1(F)$ says that F preserves binary coproducts. We denote its total bicategory by $\mathsf{Cat}_{\mathsf{BinSum}}$ .

The above examples can be defined using Definition 6 assuming that the categories involved are univalent, since then the type families $P_0$ and $P_1$ stating a choice of limits are indeed predicates. For nonunivalent categories, one could consider instead the truncated predicates stating mere existence of limits.

The total bicategories of the displayed bicategories in Example 8 are bicategories of categories with certain (co)limits. In addition, we can combine the structure of these by taking the product of the suitable displayed bicategories. Note that we could use similar methods to construct bicategories of extensive categories, regular categories, exact categories, and (pre)toposes.

The following (displayed) bicategories are used:

• The displayed 0-cells over $x : \mathsf{B}_1$ are 0-cells $y : \mathsf{B}_2$ .

• The displayed 1-cells over $f : x_1 \to x_2$ from $y_1 : \mathsf{B}_2$ to $y_2 : \mathsf{B}_2$ are 1-cells $g : y_1 \to y_2$ in $\mathsf{B}_2$ .

• The displayed 2-cells over $\theta :\, f \Rightarrow g$ from $g_1 : y_1 \rightarrow y_2$ to $g_2 : y_1 \rightarrow y_2$ are 2-cells $\tau : g_1 \Rightarrow g_2$ in $\mathsf{B}_2$ .

The total bicategory is $\int\!\!{{\mathsf{B}_1}^{+{\mathsf{B}_2}}} \simeq \mathsf{B}_1 \times \mathsf{B}_2$ with projection $\pi : \mathsf{B}_1 \times \mathsf{B}_2 \to \mathsf{B}_1$ .

• The displayed objects over $y : \mathsf{B}$ are 1-cells $x \rightarrow y$ .

• The displayed 1-cells over $g : y_1 \rightarrow y_2$ from $h_1 : x_1 \rightarrow y_1$ to $h_2 : x_2 \rightarrow y_2$ are pairs consisting of a 1-cell $f : x_1 \rightarrow x_2$ and an invertible 2-cell $\gamma : g \cdot h_2 \Rightarrow h_1 \cdot f$ .

• Given displayed 1-cells $f_1 : x_1 \rightarrow x_2$ with $\gamma_1 : g_1 \cdot h_2 \Rightarrow h_1 \cdot f_1$ , and $f_2 : x_1 \rightarrow x_2$ with $\gamma_2 : g_2 \cdot h_2 \Rightarrow h_1 \cdot f_2$ , we define the displayed 2-cells over $\theta : g_1 \Rightarrow g_2$ from $(\,f_1, \gamma_1)$ to $(\,f_2, \gamma_2)$ as 2-cells $\tau :\, f_1 \Rightarrow f_2$ such that $\gamma_1 \bullet (h_1 \vartriangleleft \tau) = (\theta \vartriangleright h_2) \bullet \gamma_2$ .

The generated total bicategory is the arrow bicategory, $\int\!\!{\mathsf{B}^{\downarrow}} = \mathsf{B}^{\rightarrow}$ with projection given by the codomain, ${{{\mathsf{cod}}}} : \mathsf{B}^{\rightarrow} \to \mathsf{B}$ .

In the next example, we write $\mathsf{StrictCat}$ for the bicategory of strict categories and $\underline{\mathsf{StrictCat}}$ for the category of strict categories.

• The displayed objects over $\mathsf{C} : \mathsf{StrictCat}$ are functors $G : \mathsf{C} \rightarrow \underline{\mathsf{StrictCat}}$ .

• The displayed 1-cells over $F : \mathsf{C}_1 \rightarrow \mathsf{C}_2$ from $G_1 : \mathsf{C}_1 \rightarrow \underline{\mathsf{StrictCat}}$ to $G_2 : \mathsf{C}_2 \rightarrow \underline{\mathsf{StrictCat}}$ are natural transformations $\gamma : G_1 \Rightarrow F \cdot G_2$ .

• The displayed 2-cells over $n : F_1 \Rightarrow F_2$ from $\gamma_1 : G_1 \Rightarrow F_1 \cdot G_2$ to $\gamma_2 : G_1 \Rightarrow F_2 \cdot G_2$ are proofs that for all $x : \mathsf{C}$ we have $\gamma_1(x) \cdot G_2(n(x)) = \gamma_2(x)$ .

The associated projection pseudofunctor $\pi : \int\!\!{\mathsf{IndexedCat}} \to \mathsf{StrictCat}$ maps functors $\mathsf{C} \rightarrow \underline{\mathsf{StrictCat}}$ to their domain.

• The displayed objects over $\mathsf{C} : \mathsf{Cat}$ are displayed categories $\mathsf{D}$ over C together with an opcleaving.

• The displayed 1-cells over $F : \mathsf{C}_1 \rightarrow \mathsf{C}_2$ from $\mathsf{D}_1$ to $\mathsf{D}_2$ are displayed functors $\overline{F}$ from $\mathsf{D}_1$ to $\mathsf{D}_2$ that preserve opcartesian morphisms.

• The displayed 2-cells over $\theta : F \Rightarrow G$ from $\overline{F} : \mathsf{D}_1 \xrightarrow{F} \mathsf{D}_2$ to $\overline{G} : \mathsf{D}_1 \xrightarrow{G} \mathsf{D}_2$ are displayed natural transformations from $\overline{F}$ to $\overline{G}$ over $\theta$ .

The associated projection pseudofunctor $\pi : \int\!\!{\mathsf{OpCleav}} \to \mathsf{Cat}$ maps any opcleaving to its codomain category.

Similarly, we define displayed bicategories $\mathsf{Cleav}$ and $\mathsf{IsoFib}$ of cleavings and isocleavings, respectively.

The idea of displayed (bi)categories transfers to functors:

• for all objects $x : \mathsf{B}$ and $\overline{x} : {{{{\mathsf{D}}_{x}}}}$ an object $\overline{F}(\overline{x}) : {{{{\mathsf{D}'}_{F(x)}}}}$ ;

• for all displayed morphisms $\overline{f} : \overline{x} \xrightarrow{f} \overline{y}$ , a displayed 1-cell $\overline{F}(\overline{f}) : \overline{F}(\overline{x}) \xrightarrow{F(\,f)} \overline{F}(\overline{y})$ ;

• for all displayed 2-cells $\overline{\theta} : \overline{f} \buildrel {\theta} \over \Rightarrow\overline{g}$ , a displayed 2-cell $\overline{F}(\overline{\theta}) : \overline{F}(\overline{f}) \buildrel {F(\theta)} \over \Rightarrow \overline{F}(\overline{g})$ ;

• coherence isomorphisms for identity and composition of displayed 1-cells, over the analogous isomorphisms in the base;

• satisfying suitable equations over the corresponding equations in the base.

We denote by $\int\!\!{\overline{F}} : \int\!\!{\mathsf{D}} \to \int\!\!{\mathsf{D}'}$ the induced total pseudofunctor.

Remark 14. The square of pseudofunctors

induced by $\overline{F}$ over F commutes up to judgmental equality.

Furthermore, we need pullbacks and products in bicategories.

• for all 1-cells $p' : z \rightarrow a$ and $q' : z \rightarrow b$ and invertible 2-cells $\gamma' : p' \cdot f \Rightarrow q' \cdot g$ , we have a 1-cell $u : z \rightarrow x$ together with invertible 2-cells $\theta : u \cdot p \Rightarrow p'$ and $\tau : u \cdot q \Rightarrow q'$ such that

  $ \alpha \bullet \theta \vartriangleright f \bullet \gamma' = u \vartriangleleft \gamma \bullet \alpha^{-1} \bullet \tau \vartriangleright g. $

• for all 1-cells $u_1, u_2 : z \rightarrow x$ and 2-cells $\theta : u_1 \bullet p \Rightarrow u_2 \bullet p$ and $\tau : u_1 \bullet q \Rightarrow u_2 \bullet q$ such that

  $ u_1 \vartriangleleft \gamma \bullet \alpha \bullet \tau \vartriangleright q \bullet \alpha^{-1} = \alpha \bullet \theta \vartriangleright f \bullet \alpha^{-1} \bullet u_2 \vartriangleleft \gamma, $
  we have a unique 2-cell $\nu : u_1 \Rightarrow u_2$ such that $\nu \vartriangleright p = \theta$ and $\nu \vartriangleright q = \tau$ .

In Definition 15, it is irrelevant if we postulate data to be given explicitly or to merely exist, provided the bicategory $\mathsf{B}$ is univalent:

The bicategory $\mathsf{Cat}$ also has pullbacks, and they are given by iso-comma categories. These are defined as follows: given categories $\mathsf{C}_1$ , $\mathsf{C}_2$ , and $\mathsf{C}_3$ and functors $F : \mathsf{C}_1 \rightarrow \mathsf{C}_3$ and $G : \mathsf{C}_2 \rightarrow \mathsf{C}_3$ , we define the iso-comma category $F \mathbin{/_\simeq} G$

• Its objects consist of objects $x : \mathsf{C}_1$ and $y : \mathsf{C}_2$ together with an isomorphism $f : F(x) \rightarrow G(y)$

• The morphisms from $(x_1, y_1, f_1)$ to $(x_2, y_2, f_2)$ consists of morphisms $g : x_1 \rightarrow x_2$ and $h : y_1 \rightarrow y_2$ such that the following diagram commutes:

We define functors $\pi_1^\simeq : F \mathbin{/_\simeq} G \rightarrow \mathsf{C}_1$ and $\pi_2^\simeq : F \mathbin{/_\simeq} G \rightarrow \mathsf{C}_2$ and a natural isomorphism $n^\simeq : \pi_1^\simeq \cdot F \Rightarrow \pi_2^\simeq \cdot G$ . The category $F \mathbin{/_\simeq} G$ together with the functors $\pi_1^\simeq,\pi_2^\simeq$ and natural isomorphism $n^\simeq$ is universal among such data, making it the pullback of F and G.

As a special case of pullbacks in the presence of terminal objects, we can define products in bicategories If $\mathsf{B}$ has chosen products, we write $x \times y$ for the product of x and y, and we denote the projections by $\pi_1 : x \times y \rightarrow x$ and $\pi_2 : x \times y \rightarrow y$ .

Notation 20. In the following, given a cloven fibration, we notate the pullback (or reindexing) of A along f by ${{{{f}^*{A}}}}$ . Given a cloven opfibration, we notate the pushforward of A along f by ${{{{f}_!{A}}}}$ .

4. Comprehension Bicategories

In this section, we define the notion of global cleaving and local (op)cleavings of bicategories. Afterward, we use these notions to define comprehension bicategories. We are guided by Buckley (Reference Buckley2014), where local and global fibrations are defined, and we add definitions for local cloven iso- and opfibrations. However, there is an important difference: while Buckley works in a set-theoretic setting, we reformulate the definitions in a type-theoretic setting using the displayed technology developed by Ahrens et al. (Reference Ahrens, Frumin, Maggesi, Veltri and Van der Weide2021) and reviewed in Section 3 – see also Remark 25.

Throughout this section, we assume that $\mathsf{B}$ is a bicategory and $\mathsf{D}$ is a displayed bicategory over $\mathsf{B}$ .

1. (1) For any $\overline{g} : \overline{c} \xrightarrow{h \cdot f} \overline{b}$ , a choice of a displayed morphism $\overline{h} : \overline{c} \xrightarrow{h} \overline{a}$ and a displayed isomorphism $\theta$ over the identity isomorphism on $h \cdot f$ in $\mathsf{B}$ .

We call $(\overline{h},\theta)$ the lift of $(h,\overline{g})$ .

1. (2) Given lifts $(\overline{h}_1,\theta_1)$ and $(\overline{h}_2, \theta_2)$ of $(h_1,\overline{g}_1)$ and $(h_2,\overline{g}_2)$ , respectively, and $\delta : h_1 \Rightarrow h_2$ , and a 2-cell $\overline{\sigma} : \overline{g}_1 \Rightarrow \overline{g}_2$ over $\delta \vartriangleright f$ , we have a unique 2-cell $\overline{\delta} : \overline{h}_1 \Rightarrow \overline{h}_2$ over $\delta$ such that $\overline{\delta}\vartriangleright \overline{f} \bullet \theta_2 = \theta_1 \bullet \overline{\sigma}$ .

Given a cartesian structure on $\overline{f}$ , we call ${{{\mathsf{F_1}(\overline{f},\overline{g})}}} := \overline{h}$ and ${{{\mathsf{F_2}(\overline{f},\overline{g})}}} := \theta$ . We also call ${{{\mathsf{E}(\delta,\overline{\sigma})}}} := \overline{\delta}$ .

• () The identity $\overline{\operatorname{id}_1} : \overline{a} \rightarrow \overline{a}$ is cartesian.

• () If $\overline{f} : \overline{a} \rightarrow \overline{b}$ and $\overline{g} : \overline{b} \rightarrow \overline{c}$ are cartesian, then so is $\overline{f} \cdot \overline{g}$ .

Construction 24 for Problem 23; (). The 1-cells of the adjoint equivalence are constructed as cartesian factorizations (first item of Definition 21). In that way, we also obtain the desired invertible 2-cell.

Buckley (Reference Buckley2014) shows, in Remark 3.1.6, that these two definitions are equivalent.

1. (1) a displayed object $\overline{a}$ over a;

2. (2) a displayed 1-cell $\overline{f} : \overline{a} \xrightarrow{f} \overline{b}$ ;

3. (3) a cartesian structure on $\overline{f}$ .

Given a global cleaving on $\mathsf{D}$ , we use the notation $\overline{b}[f] := \overline{a}$ and $L_{f}(\overline{b}) := \overline{f}$ to denote the choice given by the cleaving.

Next we look at local cleavings and opcleavings. A 2-cell is opcartesian if and only if it is opcartesian in the 1-categorical sense for the hom-functor. However, in the formalization, we give the following direct definition not relying on hom-categories and prove the characterization via hom-categories afterward Similarly, we give an unfolded definition of local opcleaving.

Being an opcartesian 2-cell is always a property. The notion of cartesian 2-cells is analogous.

The notions of local cleaving and local isocleaving are defined analogously.

(Ahrens and Lumsdaine Reference Ahrens and Lumsdaine2019, Construction 5.12).

Note that one can construct a local isocleaving from either a local cleaving or a local opcleaving.

Now let us look at some examples of these notions.

Every displayed 2-cell in $\mathsf{IndexedCat}$ is opcartesian. To find local opcartesian lifts, suppose that we have functors $F_1, F_2 : \mathsf{C}_1 \rightarrow \mathsf{C}_2$ , $G_1 : \mathsf{C}_1 \rightarrow \underline{\mathsf{StrictCat}}$ and $G_2 : \mathsf{C}_2 \rightarrow \underline{\mathsf{StrictCat}}$ , and natural transformations $n : F_1 \Rightarrow F_2$ and $\gamma : G_1 \Rightarrow F_1 \cdot G_2$ . Our goal is to construct a natural transformation $G_1 \Rightarrow F_2 \cdot G_2$ , and for the desired transformation we take $\gamma \bullet (n \vartriangleright G_2)$ . Hence, $\mathsf{IndexedCat}$ has a local opcleaving.

• The displayed objects over $x : \mathsf{C}_1$ are displayed objects in $\mathsf{D}_2$ over F(x).

• The displayed morphisms over $f : x \rightarrow y$ from $\overline{x}$ to $\overline{y}$ are displayed morphisms over $\overline{x} \xrightarrow{F(\,f)} \overline{y}$ .

Note that $F^*(\mathsf{D}_2)$ inherits any opcleaving from $\mathsf{D}_2$ . In addition, we have a displayed functor over F from $F^*(\mathsf{D}_2)$ to $\mathsf{D}_2$ that preserves cartesian morphisms.

Opcartesian 2-cells in $\mathsf{OpCleav}$ correspond to displayed natural transformations of which all components are opcartesian. We form local lifts pointwise. Since displayed 1-cells in $\mathsf{OpCleav}$ preserve cartesian morphisms, cartesian 2-cells are preserved under both left and right whiskering.

Similarly, we can show that the displayed bicategory $\mathsf{Cleav}$ has a global and a local cleaving We finish this section by defining comprehension bicategories. Examples of those are given in Section 5.

Definition 38 A weak comprehension bicategory is given by a bicategory $\mathsf{B}$ , a displayed bicategory $\mathsf{D}$ over $\mathsf{B}$ , and a displayed pseudofunctor $\chi$ over the identity on $\mathsf{B}$ as pictured below,

satisfying the following properties (see also Proposition 33):

1. (1) $\mathsf{D}$ is equipped with a cloven global fibration;

2. (2) $\mathsf{D}$ has a cloven local opfibration;

3. (3) opcartesian 2-cells of $\mathsf{D}$ are preserved under both left and right whiskering.

A comprehension bicategory is a weak comprehension bicategory such that

1. (4) $\chi$ preserves cartesian 1-cells and opcartesian 2-cells.

Remark 39. Recall from Remarks 14, 28, and 32 that the displayed pseudofunctor $\chi : \mathsf{D} \to \mathsf{B}^{\downarrow}$ of Definition 38 gives rise to a strictly commuting diagram of pseudofunctors

with the structure of a global fibration and local opfibration in the sense of

Buckley (Reference Buckley2014).

We have factored out this requirement, since for our interpretation of the language $\mathsf{BTT}$ (introduced in Section 8), we do not need this condition; that is, we can interpret $\mathsf{BTT}$ in any weak comprehension bicategory. The reason has to do with how we interpret terms of the language $\mathsf{BTT}$ in a (weak) comprehension bicategory. Traditionally, in a comprehension category, terms of $\mathsf{MLTT}$ are interpreted as sections of projections. Specifically, every type A in context $\Gamma$ gives rise, in the interpretation, to a morphism $\pi_A : \Gamma . A \rightarrow \Gamma$ in the arrow category. A term of type A in context $\Gamma$ in $\mathsf{MLTT}$ is then interpreted as a section of $\pi_A$ .

In our setting, terms of $\mathsf{BTT}$ will be interpreted differently. Instead of interpreting $\mathsf{BTT}$ terms as sections of projections, we interpret them as morphisms over the identity. More concretely, in a bicategory $\mathsf{D}$ displayed over $\mathsf{B}$ , a term in context $\Gamma$ of type B with source A of $\mathsf{BTT}$ is interpreted as a morphism from A to B over the identity of $\Gamma$ .

In the 1-categorical case, these two ways of interpreting terms are equivalent in many examples. The reason is that, in practice, most comprehension categories are full, which means that $\chi$ is fully faithful, and that every fiber has a terminal object. From these two assumptions, one can conclude that these two ways of interpreting terms are actually equivalent. However, in the bicategorical setting, these two ways are often not equivalent: for example, in the comprehension bicategory of displayed categories with opcleavings, i.e. opfibrations (Examples 12, 37, and 45). There, morphisms over the identity are required to preserve opcartesian morphisms, but such a requirement is not present for sections of the projection. If one were to include $\mathsf{MLTT}$ style terms (which are interpreted as sections of projections) in $\mathsf{BTT}$ , then to give an interpretation one would also need that $\chi$ preserves cartesian 1-cells and opcartesian 2-cells in order to interpret substitution. This is why we present here both definitions.

In short, all the examples presented in Section 5 are comprehension bicategories; the interpretation of $\mathsf{BTT}$ given in Section 8 only requires a weak comprehension bicategory.

5. Examples of Comprehension Bicategories

In this section, we give several (classes of) instances of comprehension bicategories. In some of these instances, we recognize structures studied previously in the context of higher-dimensional and directed type theory.

The displayed pseudofunctor $\chi : {{{{\mathsf{B}}^{+{\mathsf{B}}}}}} \rightarrow \mathsf{B}^{\downarrow}$ sends the object $y_2$ over $y_1$ to the product projection $y_1 \times y_2 \to y_1$ , that is, the corresponding total pseudofunctor $\mathsf{B} \times \mathsf{B} \to \mathsf{B}^{\downarrow}$ is defined by $(y_1,y_2)\mapsto \pi_1 : y_1 \times y_2 \to y_1$ .

Example 42 corresponds roughly to the semantics studied by Fiore and Saville (Reference Fiore and Saville2019), but without the type formers.

Another example of a comprehension bicategory comes from locally groupoidal bicategories, such as $\mathsf{Grpd}$ .

Since every 2-cell is opcartesian in $\mathsf{B}^{\downarrow}$ if $\mathsf{B}$ is locally groupoidal, the displayed bicategory $\mathsf{B}^{\downarrow}$ has a local opcleaving.

Example 43 models undirected reductions between terms. It is thus related to the groupoid model of type theory by Hofmann and Streicher (Reference Hofmann and Streicher1994) and to the definition of comprehension 2-category by Garner (Reference Garner2009). A more detailed comparison to Garner’s comprehension 2-categories is given in Remark 65.

We can also consider directed versions of this example by using categories instead of groupoids.

To define $\chi$ , we use the Grothendieck construction. Specifically, from a functor $G : \mathsf{C} \rightarrow \underline{\mathsf{StrictCat}}$ we construct a category $\int G$ as follows

1. (1) The objects of $\int G$ are pairs (x, y) where $x : \mathsf{C}$ and $y : G(x)$ .

2. (2) The morphisms from $(x_1, y_1)$ to $(x_2, y_2)$ are pairs $(\,f, g)$ where $f : x_1 \rightarrow x_2$ and $g : G(\,f)(y_1) \rightarrow y_2$ .

Note that we also have a projection $\pi_1 : \int G \rightarrow \mathsf{C}$ . In addition, from a displayed 1-cell $\gamma : G_1 \Rightarrow F \cdot G_2$ , we get a functor $\int \gamma : \int G_1 \rightarrow \int G_2$ and a natural isomorphism $\gamma_c : \pi_1 \cdot F \Rightarrow \int \gamma \cdot \pi_1$ . If we have $p : \gamma_1(x) \cdot G_2(n(x)) = \gamma_2(x)$ for every $x : \mathsf{C}_1$ , then we get a natural transformation $\int p$ from $\int \gamma_1$ to $\int \gamma_2$ .

Example 44 is related to the interpretations given by Licata andHarper (Reference Licata, Harper, Mislove and Ouaknine2011) and North (Reference North and König2019) in their works on directed type theories, albeit without considering type formers. However, their notion of term, in both the syntax and the interpretation, is different from ours; see Section 10 for details.

In Example 44, contexts are categories and types over a context $\mathsf{C}$ are cloven split opfibrations on $\mathsf{C}$ . We also consider a version of this interpretation where there are more types: instead of looking at only those opfibrations that are split, all opfibrations are considered.

The pseudofunctor $\chi$ sends a displayed category $\mathsf{D}$ over $\mathsf{C}$ to the functor $\pi_1 : \int\!\!{\mathsf{D}} \rightarrow \mathsf{C}$ .

Similarly, we can define a contravariant comprehension bicategory using cleavings instead of opcleavings.

6. Internal Street (Op)Fibrations

In this section, we discuss Street (op)fibrations internal to a fixed bicategory $\mathsf{B}$ . They will yield, in Section 7, many examples of comprehension bicategories, see Example 57 and Remark 59. The examples of Street opfibrations internal to bicategories of stacks are particularly interesting (see Remark 59).

Note that $\mathsf{B}^{\downarrow}$ comes equipped with a cloven global fibration if $\mathsf{B}$ has pullbacks. However, to obtain a local (op)cleaving, we used that $\mathsf{B}$ is locally groupoidal in Example 35. This assumption is avoided in Example 37 where $\mathsf{B} = \mathsf{Cat}$ : instead of looking at arbitrary functors, one only considers the opfibrations. We can generalize this idea to arbitrary bicategories by using internal Street (op)fibrations (Buckley Reference Buckley2014).

The displayed bicategory $\mathsf{OpCleav}$ has a local opcleaving where the desired lifts are constructed pointwise. To generalize this to arbitrary bicategories, we need to adjust this definition so that we can lift arbitrary 2-cells. Furthermore, the notion of cloven Grothendieck opfibration of categories is stricter than appropriate for bicategories. If we have $x \rightarrow y$ and an object over y, then a cloven Grothendieck fibration gives an object strictly living over x, while a Street fibration only gives an object living weakly over x (i.e., up to an isomorphism). More information can be found in work by Loregian and Riehl (Loregian and Riehl Reference Loregian and Riehl2020, Example 4.1.2).

• For every $x : \mathsf{B}$ , the functor $p_* :$ $\mathsf{B}(x,e)$ $\rightarrow$ $\mathsf{B}(x,b)$ of hom-categories is a Street fibration.

• For every $f : x \rightarrow y$ , the following square is a morphism of Street fibrations:

  

A 1-cell is called an internal Street opfibration if it is an internal Street fibration in $\mathsf{B}^{\mathsf{co}}$

In $\mathsf{Cat}$ , internal Street fibrations are the same as Street fibrations of categories. However, the notion of internal Street fibrations can be applied in a wider variety of settings: for example, one could also look at internal Street fibrations in the bicategories from Example 8 or in presheaves or stacks valued in $\mathsf{Cat}$ . A classical result on internal Street (op)fibrations is that they are closed under taking pullbacks, see Gray (Reference Gray, Eilenberg, Harrison, MacLane and Röhrl1966) and Street (Reference Street1980).

Proposition 47 Street (op-)fibrations are closed under pullback. Concretely, given a pullback square

where $p_2$ is a Street (op)fibration, then $p_1$ is so, too.

7. Display Map Bicategories

In this section, we introduce “display map bicategories” as a convenient way to build comprehension bicategories.

Let $\mathsf{B}$ be a bicategory with pullbacks. We aim to construct a displayed bicategory $\mathsf{SOpFib}(\mathsf{B})$ of Street opfibrations over $\mathsf{B}$ with both a global cleaving and a local opcleaving. The construction should be done in a modular and general way, since using the same techniques we aim to construct a displayed bicategory $\mathsf{SFib}(\mathsf{B})$ of Street fibrations over $\mathsf{B}$ with a global and local cleaving. One can imagine more examples, such as the intersection of Street opfibrations with discrete or fully faithful 1-cells.

The common pattern among all these examples is that the displayed bicategory actually represents a subbicategory of the arrow bicategory of $\mathsf{B}$ . In the 1-categorical case, such examples are captured by display map categories (Taylor Reference Taylor1999). We adapt that notion to the bicategorical setting.

However, there is a difference between the 1-categorical and the bicategorical case. A display map category on a 1-category $\mathsf{C}$ is a full subcategory of the arrow category on $\mathsf{C}$ satisfying some requirements. Such a notion would not be useful in the bicategorical case, since neither $\mathsf{SOpFib}(\mathsf{B})$ nor $\mathsf{SFib}(\mathsf{B})$ are full subbicategories of $\mathsf{B}^{\downarrow}$ . Indeed, the 1-cells in $\mathsf{SOpFib}(\mathsf{B})$ and $\mathsf{SFib}(\mathsf{B})$ are required to preserve (op)cartesian cells, which is not required in $\mathsf{B}^{\downarrow}$ . As such, to obtain a notion of display map bicategory that captures these two examples, some care is needed in the definition of display map bicategories.

For that reason, our notion of display map bicategory comes in three different flavors. ^{Footnote 3}

1. (1) a predicate P on the 1-cells of $\mathsf{B}$ , and

2. (2) a choice of pullbacks along 1-cells satisfying P

such that

1. (3) if a morphism satisfies P, then it is an internal Street opfibration, and

2. (4) P is closed under pullback.

Given a display map bicategory $\mathsf{B}$ , a display map of $\mathsf{B}$ is a 1-cell of $\mathsf{B}$ together with a proof of the predicate P.

We also define notions of contravariant display map bicategory and isovariant display map bicategory: for the former, condition (3) above is replaced by the condition that the display maps are required to be internal Street fibrations, while for the latter, condition (3) is omitted. The reason for this terminology is that every display map bicategory of a given variance gives rise to a comprehension bicategory with the same variance.

Analogously, one can define a contravariant display map bicategory of Street fibrations. There are other examples of display map bicategories as well, for instance discrete Street opfibrations. To define those, we first need the notion of discrete morphisms. ^{Footnote 4}

• faithful if for every object x, the functor $f_* : \underline{\mathsf{B}(x,a)} \rightarrow \underline{\mathsf{B}(x,b)}$ given by postcomposition with f is faithful;

• conservative if for every object x, the functor $f_* : \underline{\mathsf{B}(x,a)} \rightarrow \underline{\mathsf{B}(x,b)}$ is conservative;

• discrete if for every object x, the functor $f_* : \underline{\mathsf{B}(x,a)} \rightarrow \underline{\mathsf{B}(x,b)}$ is faithful and conservative.

Every display map bicategory gives rise to a displayed bicategory as follows.

• The objects over $b : B$ are pairs $e : \mathsf{B}$ and $p : e \rightarrow b$ such that p is a display map;

• The 1-cells over $f : b_1 \rightarrow b_2$ from $p_1 : e_1 \rightarrow b_1$ to $p_2 : e_2 \rightarrow b_2$ are pairs of a 1-cell $g : e_1 \rightarrow e_2$ and an invertible 2-cell $g \cdot p_2 \Rightarrow p_1 \cdot f$ such that g preserves opcartesian 2-cells;

• The 2-cells are as in Example 10.

Similarly, every contravariant display map bicategory gives rise to a displayed bicategory. However, the 1-cells are required to preserve cartesian 2-cells. For isovariant display map bicategories, we do not make any restriction on the 1-cells. This way, we obtain displayed bicategories $\mathsf{SOpFib}(\mathsf{B})$ and $\mathsf{SFib}(\mathsf{B})$ over a bicategory $\mathsf{B}$ with pullbacks. In $\mathsf{SOpFib}(\mathsf{B})$ , 2-cells are the same as 2-cells in $\mathsf{B}^{\rightarrow}$ . However, 1-cells are a bit different: while 1-cells in $\mathsf{B}^{\rightarrow}$ are squares

that commute up to invertible 2-cell, 1-cells in $\mathsf{SOpFib}(\mathsf{B})$ have the additional requirement that whiskering with $f_e$ preserves opcartesian 2-cells. Similarly, we can define the bicategory $\mathsf{SFib}(\mathsf{B})$ where the objects are internal Street fibrations and where the 1-cells preserve cartesian 2-cells.

Now we can construct the desired cleavings.

Similarly, given a contravariant display map bicategory, one obtains both a global and a local cleaving. However, from an isovariant display map bicategory, one only gets a global cleaving and a local isocleaving. One can instantiate Example 54 to internal Street opfibrations to obtain a global cleaving and a local opcleaving for $\mathsf{SOpFib}(\mathsf{B})$ if $\mathsf{B}$ has pullbacks.

Using Example 54, we can generalize the example of (op)fibrations to arbitrary display map bicategories. We discuss the interest in this generalization in Remark 59.

Construction 56 for Problem 55; From bicategory $\mathsf{B}$ and a display map bicategory $\mathsf{D}$ on $\mathsf{B}$ , we construct the comprehension bicategory

The (displayed) pseudofunctor $\chi$ is the inclusion of $\mathsf{D}$ into $\mathsf{B}^{\downarrow}$ .

Similarly, we get a contravariant comprehension bicategory from a contravariant display map bicategory, and an isovariant comprehension bicategory from an isovariant display map bicategory. We can instantiate Example 56 to internal Street opfibrations to get the following comprehension bicategory.

Example 57 For a bicategory $\mathsf{B}$ with pullbacks (see, e.g., Example 7), we construct the comprehension bicategory

The (displayed) pseudofunctor $\chi$ forgets that the morphisms in $\mathsf{SOpFib}(\mathsf{B})$ are internal Street opfibrations.

With our comprehension bicategories of stacks (not just ones valued in groupoids), one could follow Coquand et al. (Reference Coquand, Mannaa and Ruch2017) and study the validity and independence of logical principles in directed type theory.

Note that we can specialize Example 57 to each of the examples given in Example 8. This is because we showed in Example 19 that those bicategories have pullbacks. In the case of $\mathsf{Cat}_{\mathsf{Terminal}}$ , we get a comprehension bicategory in which

• the “contexts” are categories with a terminal object; and

• the “types” are cloven opfibrations of categories which preserve terminal objects.

Similarly, we can instantiate this to the other categories mentioned in Example 8.

8. The Type Theory $\mathsf{BTT}$

In this section, we extract a core syntax for two-dimensional type theory from our semantic model. We call the resulting type theory Bicategorical Type Theory ( $\mathsf{BTT}$ ). In Section 9, we prove soundness of our syntax by giving an interpretation of the syntax in any weak comprehension bicategory.

The syntax extracted here is maximally general, in the sense that it reflects the structure of a general comprehension bicategory. In Section 10, we propose several orthogonal simplifications to the syntax, along with the corresponding semantic structure and properties. As such, our syntax and semantics are to be viewed as a framework to study different semantic structures and their corresponding internal languages, rather than as one particular pair of syntax and semantics.

In Section 8.1 we present the judgment forms of $\mathsf{BTT}$ , as well as the rules extracted from the bicategories of contexts and of types, respectively. In Sections 8.2 and 8.3 we present the comprehension and substitution rules, respectively.

For reasons of space we omit here several rules, namely the naturality rules in Figs. 8 and 9, and the coherencies of trifunctors. Such rules, written out in the “linear” syntax used here, would take up several lines and would be difficult to understand. Consequently, the syntax presented here is not complete. The presentation of the complete syntax and a suitable completeness theorem is left for future work.

8.1 Judgments and basic rules

$\mathsf{BTT}$ features contexts, substitutions, types, generalized terms, and reductions between terms. As befits the bicategorical semantics, judgmental equality is only postulated between parallel reductions.

There are eight kinds of judgments in $\mathsf{BTT}$ :

1. (1) $\Gamma \ \mathsf{ctx}$ , which is read as “ $\Gamma$ is a context”;

2. (2) $\Delta \vdash s : \Gamma$ (given $\Delta, \Gamma \ \mathsf{ctx}$ ), which is read as “s is a substitution from $\Delta$ to $\Gamma$ ”;

3. (3) $\Delta \vdash r : s \mathrel{\rightsquigarrow} t : \Gamma$ (where $\Delta \vdash s,t : \Gamma$ ), which is read as “r is a reduction from s to t”;

4. (4) $\Delta \vdash r \equiv r' : s \mathrel{\rightsquigarrow} t : \Gamma$ (where $\Delta \vdash r,r' : s \mathrel{\rightsquigarrow} t : \Gamma$ ), which is read as “r is equal to r”’;

5. (5) $\Gamma \vdash T \ \mathsf{type}$ (where $\Gamma \ \mathsf{ctx}$ ), which is read as “T is a type in context $\Gamma$ ”;

6. (6) $\Gamma \hspace{0.01em} \mid \hspace{0.01em} S \vdash t : T$ (where $\Gamma \vdash S, T \ \mathsf{type}$ ), which is read as “t is a term in T depending on S in context $\Gamma$ ”;

7. (7) $\Gamma \hspace{0.01em} \mid \hspace{0.01em} S \vdash \rho : t \mathrel{\rightsquigarrow} t' : T$ (where $\Gamma \hspace{0.01em} \mid \hspace{0.01em} S \vdash t,t' : T$ ), which is read as “ $\rho$ is a reduction from t to t”’;

8. (8) $\Gamma \hspace{0.01em} \mid \hspace{0.01em} S \vdash \rho \equiv \rho' : t \mathrel{\rightsquigarrow} t' : T$ (where $\Gamma \hspace{0.01em} \mid \hspace{0.01em} S \vdash \rho , \rho' : t \mathrel{\rightsquigarrow} t' : T$ ), which is read as “ $\rho$ is equal to $\rho'$ ”.

We often abbreviate the above judgments and write, e.g., just $\rho : t \mathrel{\rightsquigarrow} t'$ instead of $\Gamma \hspace{0.01em} \mid \hspace{0.01em} S \vdash \rho : t \mathrel{\rightsquigarrow} t' : T$ . For these judgments, we have rules that express the bicategorical structure of contexts and types. Rules are given in Fig. 1 for the bicategory of contexts, and in Fig. 2 for the bicategory of types.

Figure 1.

Rules for the bicategory of contexts.

Figure 2.

Rules for the the bicategory of types.

We also introduce symbols which read like judgments but stand for several judgments, using the composition and identities introduced in Figs. 1 and 2.

1. (1) $\Delta \vdash \rho : s \mathrel{\tilde{\mathrel{\rightsquigarrow}}} t : \Gamma$ stands for the following four judgments.

   • $\Delta \vdash \rho : s \mathrel{\rightsquigarrow} t : \Gamma$ $\Delta \vdash \rho^{-1} : t \mathrel{\rightsquigarrow} s : \Gamma$

   • $\rho \bullet \rho^{-1} \equiv 1_s$ $\rho^{-1} \bullet \rho \equiv 1_t$

2. (2) $\Gamma \hspace{0.01em} \mid \hspace{0.01em} S \vdash \rho : t \mathrel{\tilde{\mathrel{\rightsquigarrow}}} t' : T$ stands for the following four judgments.

   • $\Gamma \hspace{0.01em} \mid \hspace{0.01em} S \vdash \rho : t \mathrel{\rightsquigarrow} t' : T$ $\Gamma \hspace{0.01em} \mid \hspace{0.01em} S \vdash \rho ^{-1}: t' \mathrel{\rightsquigarrow} t : T$

   • $\rho \bullet \rho^{-1} \equiv 1_t$ $\rho^{-1} \bullet \rho \equiv 1_{t'}$

3. (3) $\Delta\ \tilde \vdash\ s : \Gamma$ stands for the following four judgments.

   • $\Delta \vdash s : \Gamma$ $\Gamma \vdash s^{-1} : \Delta$

   • $s^\ell: s s ^{-1} \mathrel{\tilde{\mathrel{\rightsquigarrow}}} 1_\Delta$ $s^\rho: s^{-1} s \mathrel{\tilde{\mathrel{\rightsquigarrow}}} 1_\Gamma$

We also require that $s^\ell$ and $s^\rho$ form an adjoint equivalence. This can be specified by formulating the usual triangle equalities as equality judgments.

• (4) $\Gamma \mid S \ \tilde \vdash\ t : T$ stands for the following four judgments.

  • $\Gamma \hspace{0.01em} \mid \hspace{0.01em} S \vdash t : T$ $\Gamma \hspace{0.01em} \mid \hspace{0.01em} T \vdash t^{-1} : S$

  • $t^\ell: t t ^{-1} \mathrel{\tilde{\mathrel{\rightsquigarrow}}} 1_S$ $t^\rho : t^{-1} t \mathrel{\tilde{\mathrel{\rightsquigarrow}}} 1_T$

Remark 61. By abuse of notation, we write several topically related rules that share all the same hypotheses as one rule with several conclusions. These rules then also share the same name, e.g., When referring to a rule by name, it will be clear from the context which of the possible rules we refer to. The names of inference rules in the text are hyperlinks to the corresponding rules (e.g., ). The equality $\equiv$ is assumed to be a congruence for every other constructor and judgment. For brevity, we have not recorded here the resulting rules. When parentheses are omitted, everything is associated to the left: that is, rst stands for ((rs)t). Note also that in several rules in which it is necessary to re-associate several four or more terms or substitutions, we have written $\alpha$ instead of a long composition of whiskered associators $\alpha_{\bullet, \bullet, \bullet}$ in the interest of readability.

8.2 Comprehension structure

Comprehension, that is, context extension, is extracted from the pseudofunctor $\chi$ . The rules for comprehension are given in Fig. 3. There are some notable differences to comprehension in $\mathsf{MLTT}$ . First, the rule which forms a substitution, comes together with a reduction that expresses the commutativity of a triangle. Second, we also have a rule that extends a substitution with a reduction. Since reductions are proof-relevant, this rule comes with a coherency on the commutativity.

Figure 3.

Rules for comprehension.

8.3 Substitution structure

Substitution is given, in the semantics, by the global and local (op)cleaving structure. We reflect this into the syntax as explicit substitution, as was also used, e.g., by Fiore and Saville (Reference Fiore and Saville2019), Licata and Harper (Reference Licata, Harper, Field and Hicks2012) in their respective settings.

The rules for substitution are given in Figs. 4, 5, 6, 7, 8, and 9. We distinguish them based on whether we need the global cleaving or the local opcleaving to interpret them. There are several important observations to be made about these rules. First, in line with our truly bicategorical approach, we do not assume the comprehension bicategory is split. In particular, no equality between $T[\!\operatorname{id}]$ and T is postulated. Instead, there is an equivalence between them (see sub-id and sub-comp), and terms of these types are transported along the equivalence.

Figure 4.

Rules for global substitution.

Figure 5.

Rules for global substitution (preservation of identity).

Figure 6.

Rules for global substitution (preservation of composition).

Figure 7.

Rules for local substitution.

Figure 8.

Some rules for local substitution (preservation).

Figure 9.

Some of the rules for coherence of substitution.

The rule expresses that each type T behaves “functorially”: for each $\Delta \vdash s : \Gamma$ (i.e., object in “ $\hom(\Delta,\Gamma)$ ”) we get a type T[s] (i.e., object in “the category of types in context $\Delta$ ”) by and for each $r: s \mathrel{\rightsquigarrow} s'$ (i.e., morphism in ‘ $\hom(\Delta,\Gamma)$ ’) we get a term $\Delta \hspace{0.01em} \mid \hspace{0.01em} T[s] \vdash \mathsf{map} \ T \ \theta : T[s']$ (i.e., morphism in “the category of types in context $\Delta$ ”) by . The rules and ensure that $T[-]$ preserves identity and composition. With we can then understand terms to be “natural transformations.”

Remark 62. For contravariant comprehension bicategories (Remark 41), the rule would be in the opposite direction, while for isovariant comprehension bicategories, this rule would be restricted to isomorphisms in the base.

Remark 63. Using the Grothendieck construction, we can view the rules from the perspective of fiber bicategories. If $P : \mathsf{E} \rightarrow \mathsf{B}$ has a global cleaving and a local opcleaving, and furthermore opcartesian 2-cells are preserved under whiskering, then we obtain a trifunctor $\mathsf{B}^{\mathsf{op}} \rightarrow \mathsf{Bicat}$ , which sends every $x : \mathsf{B}$ to the fiber of x along P. The rules given in Fig. 4 say that any 1-cell in the base gives rise to a pseudofunctor. For example, the rules represent the actions on objects, 1-cells, and 2-cells, respectively, while represent the invertible 2-cells that witness the preservation of the identity and composition of 1-cells. The other rules in that table are the coherence laws of pseudofunctors: for example, is the preservation of identity 2-cells.

Figs. 5 and 6 give pseudonatural equivalences expressing that the assignment of the previously mentioned pseudofunctor is actually pseudofunctorial: the identity and composition are preserved up to pseudonatural equivalence. For example, the rules in Fig. 5 represent the action on objects and the naturality squares. On the other hand, are the usual coherencies of pseudonatural transformations.

The rules of Fig. 7 state that we obtain a pseudonatural transformation from a 2-cell in the base. The action on objects and 1-cells is given by respectively, while describe the usual coherencies of pseudonatural transformations. The remaining rules can be found in Fig. 8. In this figure, four invertible modifications are described: describe the preservation of identity and composition, respectively, while describe the preservation of both left and right whiskering. Note that we left out the naturality conditions of these modifications. In addition, two additional coherencies are required which express that all this data together forms a trifunctor. We do not write them down, but instead, we refer the reader to Definition 3.3.1 of Gurski (Reference Gurski2006).

Remark 64. A similar rule to of Fig. 7 appears in work of

Johnstone (Reference Johnstone1993). There, it is shown that a 1-cell p in a 2-category is an internal fibration in a sense similar to our Definition 46 if and only if a specific functor given by precomposition with p has a semi-oplax right adjoint satisfying some properties, assuming that pullbacks along p exist. In Lemma 2.5 of Johnstone (Reference Johnstone1993), another equivalent characterization of p being a fibration is given assuming that pullbacks of p exist. This characterization uses several operations and compatibility requirements. One of the operations is described by the following diagram:

We assume that pullbacks along p are given; in particular, we have $f^* e$ and $g^* e$ . The characterization of p being a fibration entails, in particular, that for every 2-cell $\alpha :\, f \Rightarrow g$ there is a 1-cell $\hat{p}(\alpha) :\, f^*e \rightarrow g^*e$ . This takes place in a 2-category, though Johnstone claims that such a characterization would also hold in a bicategory. Modulo this difference, this operation is expressed by rule of Fig.7.

9. Soundness: Interpretation in Comprehension Bicategories

In this section, we give an interpretation of $\mathsf{BTT}$ in any weak comprehension bicategory. To this end, we fix a comprehension bicategory ${\mathsf{D}} \mathop{\longrightarrow}\limits^{\chi}\mathsf{B}^{\downarrow}$ over ${\mathsf{B}}$ . We interpret the judgments as follows.

• $\Gamma \ \mathsf{ctx}$ is interpreted as an object ${[\![} \Gamma {]\!]}$ of $\mathsf{B}$ .

• $\Delta \vdash s : \Gamma$ is interpreted as a 1-cell ${[\![} s {]\!]} : {[\![} \Delta {]\!]} \rightarrow {[\![} \Gamma {]\!]}$ in $\mathsf{B}$ .

• $\Delta \vdash r : s \mathrel{\rightsquigarrow} t : \Gamma$ is interpreted as a 2-cell ${[\![} r {]\!]} : {[\![} s {]\!]} \Rightarrow {[\![} t {]\!]}$ in $\mathsf{B}$ .

• $\Delta \vdash r \equiv r' : s \mathrel{\rightsquigarrow} t : \Gamma$ is interpreted as an equality ${[\![} r {]\!]} = {[\![} r' {]\!]}$ .

• $\Gamma \vdash T \ \mathsf{type}$ is interpreted as an object ${[\![} T {]\!]}$ in $\mathsf{D}$ over ${[\![} \Gamma {]\!]}$ .

• $\Gamma \hspace{0.01em} \mid \hspace{0.01em} S \vdash t : T$ is interpreted as a 1-cell ${[\![} t {]\!]} : {[\![} S {]\!]} \rightarrow {[\![} T {]\!]}$ over the identity on ${[\![} \Gamma {]\!]}$ .

• $\Gamma \hspace{0.01em} \mid \hspace{0.01em} S \vdash r: t \mathrel{\rightsquigarrow} t' : T$ is interpreted as a 2-cell ${[\![} r {]\!]} : {[\![} t {]\!]} \Rightarrow {[\![} t' {]\!]}$ over the identity 2-cell.

• $\Gamma \hspace{0.01em} \mid \hspace{0.01em} S \vdash r \equiv r' : t \mathrel{\rightsquigarrow} t' : T$ is interpreted as an equality ${[\![} r {]\!]} = {[\![} r' {]\!]}$ .

Regarding the “bicategorical” rules of Figs. 1 and 2, each rule is analogous to one of the operations or laws of a bicategory – see, for instance, Definition 3.1 of Ahrens et al. (Reference Ahrens, Frumin, Maggesi, Veltri and Van der Weide2021). This also indicates how it is interpreted.

9.1 Comprehension

In this section, we interpret the rules related to comprehension of Fig. 3. Suppose that we have a context $\Gamma : \mathsf{B}$ and a type T over $\Gamma$ . Its image $\chi(T)$ in $\mathsf{B}^{\downarrow}$ gives rise to an object $\Gamma . T : \mathsf{B}$ and a 1-cell $\pi_{\Gamma.T} : \Gamma . T \rightarrow \Gamma$ which interprets For a 1-cell t from S to T over the identity, the resulting 1-cell $\chi(t)$ is part of a triangle

which commutes up to invertible 2-cell. This yields the interpretation of the rules in Furthermore, a reduction $r : t \mathrel{\rightsquigarrow} t'$ is mapped by $\chi$ to a 2-cell from $\chi(t)$ to $\chi(t')$ in $\mathsf{B}^{\downarrow}$ , and this is how we interpret are interpreted by the identitor and compositor of $\chi$ , respectively, while the remaining rules in Fig. 3 are satisfied because they translate to the laws that express that this data forms a pseudofunctor.

9.2 Global substitution

Next, we interpret the rules for global substitution of Fig. 4 (in Sections 9.2.1, 9.2.2, 9.2.3, and 9.2.4) and the rules regarding preservation of identity under global substitution of Fig. 5 (in Sections 9.2.6, 9.2.5, and 9.2.7). The interpretation of the rules regarding preservation of composition of Fig. 6 is analogous to that of Fig. 5 and is not spelled out.

The interpretation of the rule is given directly by the global cleaving. The interpretation of the other rules require more explanation.

9.2.1 Interpretation of sub-tm

To interpret , we assume that we have a substitution $s : \Delta \rightarrow \Gamma$ between contexts $\Delta$ and $\Gamma$ . We also assume that we have types S, T over $\Gamma$ and a morphism $t : S \to T$ over $1_\Gamma$ . This situation is encapsulated in the following diagram, using the notation introduced in Definition 27.

Our goal is to construct a 1-cell $t[s] : S[s] \rightarrow T[s]$ that lies over the identity on $\Delta$ . Since the morphism $L_{s}(T) : T[s] \rightarrow T$ is cartesian, it suffices to construct a 1-cell, say, $\beta : S[s] \rightarrow T$ that lies over $1_\Delta \cdot s$ . We then obtain the desired 1-cell as the factorization ${{{\mathsf{F_1}(L_{s}(T),\beta)}}}$ of that 1-cell via $L_{s}(T)$ . (Recall that ${{{\mathsf{F_1}(\_,\_)}}}$ was defined in Definition 21.) We have an invertible 2-cell $r_{s} \bullet {\ell_{s}}^{-1} : s \cdot 1_\Gamma \Rightarrow 1_\Delta \cdot s$ , and we set $\beta := {{{{(r_{s} \bullet {\ell_{s}}^{-1})}_!{(L_{s}(S) \cdot t)}}}}$ . Define

\[t[s] := {{{\mathsf{F_1}(L_{s}(T),\beta)}}}.\]

Overall, we can summarize the construction with the following diagram.

9.2.2 Interpretation of sub-red

To interpret , we assume that we have a substitution $s : \Delta \rightarrow \Gamma$ and types S and T in context $\Gamma$ . We also assume that we have 1-cells $t, t' : S \to T$ over $1_\Gamma$ and a 2-cell $\rho : t \Rightarrow t'$ . Consider the following diagram.

Our goal is to construct a 2-cell from t[s] to t[s’] over the identity on $\Delta$ . The source and target, respectively, were constructed by factoring a 1-cell via a cartesian 1-cell, as follows.

To construct the desired 2-cell $\rho[s] : t[s] \Rightarrow t[s']$ , we use Item 2 of Definition 21. More specifically, it suffices to construct

• a 2-cell $\delta$ from $1_\Delta$ to $1_\Delta$ , and

• a 2-cell $\overline{\sigma}$ from $\beta$ to $\beta'$ .

Once we have defined those, we define $\rho[s] := {{{\mathsf{E}(\delta,\overline{\sigma})}}}$ .

For the first of the two, we take the identity 2-cell $1_{1_{\Delta}}$ . For the second, we take the composition of 2-cells shown in the following diagram.

The interpretation of the rules follows from the uniqueness of factorization 2-cells.

9.2.3 Interpretation of sub-pres-id

The rules are interpreted similarly, and we show the interpretation of . Suppose that we have a substitution $s : \Delta \rightarrow \Gamma$ and a type T in context $\Gamma$ . We consider two morphisms: the identity on T[s] and $1_T[s]$ . These are illustrated in the following diagram.

Our goal is to construct an invertible 2-cell from $1_{T[s]}$ to $1_T[s]$ . To do so, we first recall the construction of $1_T[s]$ .

To construct the desired invertible 2-cell, we again use Item 2 of Definition 21. As such, we need to construct the following:

• An invertible 2-cell $\delta$ from $1_\Delta$ to $1_\Delta$ ; and

• An invertible 2-cell $\overline{\sigma}$ from $1_T[s] \cdot L_{s}(T)$ to $1_{T[s]} \cdot L_{s}(T)$ .

With those in place, we define $\mathsf{Sub_I}(s)^{-1} := {{{\mathsf{E}(\delta,\overline{\sigma})}}}$ . For $\delta$ , we take the identity, while for $\overline{\sigma}$ , we take the following composition.

$1_{T[s]} \cdot L_{s}(T) \cong L_{s}(T) \cdot 1_T \cong \beta \cong 1_T[s] \cdot L_{s}(T)$

9.2.4 Interpretation of sub-pres-lunitor

The remaining laws stating equalities of 2-cells, such as , are all proven in a similar way. Our goal is to prove an equality $\overline{\alpha} = \overline{\beta}$ with 1-cells and 2-cells as in the diagram below:

Here t and t’ live over s and s’, respectively, while $\overline{\alpha}$ and $\overline{\beta}$ live over $\alpha$ and $\beta$ , respectively. To prove the equality, we take the following steps.

• (1) We construct an invertible 2-cell $\theta$ from $t \cdot L_{s''}(T)$ to $t' \cdot L_{s''}(T)$ .

• (2) We prove that $\alpha = \beta$ .

• (3) We prove that $\overline{\alpha} \vartriangleright L_{s''}(T) = \theta = \overline{\beta} \vartriangleright L_{s''}(T)$ .

From the first item, we get that both t and t’ are cartesian factorizations of the same $S \rightarrow T$ . From the second and third item, we get that both $\overline{\alpha}$ and $\overline{\beta}$ are factorization 2-cells from t to t’, and since such 2-cells are unique, we get $\overline{\alpha} = \overline{\beta}$ .

We demonstrate this for . Suppose we have a substitution $s : \Delta \rightarrow \Gamma$ , types S and T in context $\Gamma$ , and a morphism $t : S \to T$ over $1_\Gamma$ . We are in the following situation.

Here, we define $\alpha := (1_{1_\Delta} \vartriangleright 1_\Delta) \bullet \ell_{1_\Delta}$ and $\beta := 1_{1_\Delta \cdot 1_\Delta} \bullet \ell_{1_\Delta}$ . We also define $\overline{\alpha}$ and $\overline{\beta}$ as follows.

$\overline{\alpha} := (\mathsf{Sub_I}(s)^{-1} \vartriangleright t[s]) \bullet \ell_{t[s]}\quad \quad\overline{\beta} := \mathsf{Sub_C}(1_S, t, s) \bullet \ell_t[s]$

Then by construction, both $\alpha$ and $\beta$ are equal to $\ell_{1_\Delta}$ , and hence, $\alpha = \beta$ .

Next we construct an invertible 2-cell from $1_S[s] \cdot t[s] \cdot L_{s}(T)$ to $t[s] \cdot L_{s}(T)$ . By construction of t[s], we have an invertible 2-cell $\theta_1 : t[s] \cdot L_{s}(T) \cong \cdot L_{s}(S) \cdot t$ . In addition, we have an invertible 2-cell $\theta_2 : 1_S[s] \cdot L_{s}(S) \cong L_{s}(S) \cdot 1_S$ . This is illustrated in the following diagram.

Now we define an invertible 2-cell $\theta_3 : 1_S[s] \cdot t[s] \cdot L_{s}(T) \cong L_{s}(S) \cdot t$ as follows.

$\theta_3 := \alpha^{-1} \bullet (1_S[s] \vartriangleleft \theta_2) \bullet \alpha \bullet (\theta_1 \vartriangleright t) \bullet (r_\sigma \vartriangleright t)$

The desired invertible 2-cell $\theta : 1_S[s] \cdot t[s] \cdot L_{s}(T) \cong t[s] \cdot L_{s}(T)$ is $\theta_3 \bullet \theta_2^{-1}$ .

For the last step, we only prove that $\overline{\alpha} \vartriangleright L_{s}(T) = \theta$ . We first recall how we constructed $\mathsf{Sub_I}(s)$ . For brevity, we write $\sigma := L_{s}(S)$ and $\tau := L_{s}(T)$ .

By construction, we have that $(\mathsf{Sub_I}(s)^{-1} \vartriangleright \sigma) \bullet \ell_\sigma \bullet r_\sigma^{-1} = \theta_1$ . In particular, we have $\mathsf{Sub_I}(s)^{-1} \vartriangleright \sigma = \theta_1 \bullet r_\sigma \bullet \ell_{\sigma}^{-1}$ .

To show that $((\mathsf{Sub_I}(s)^{-1} \vartriangleright t[s]) \bullet \ell_{t[s]}) \vartriangleright \tau = \theta_3 \bullet \theta_2^{-1}$ , it suffices to show that $((\mathsf{Sub_I}(s)^{-1} \vartriangleright t[s]) \bullet \ell_{t[s]}) \vartriangleright \tau \bullet \theta_2 = \theta_3$ . This follows from the following chain of equalities.

$ \begin{align*} & \quad \alpha^{-1} \bullet (1_S[s] \vartriangleleft \theta_2^{-1}) \bullet \alpha \bullet ((\mathsf{Sub_I}(s)^{-1} \vartriangleright t[s]) \bullet \ell_{t[s]}) \vartriangleright \tau \bullet \theta_2 \bullet (r_\sigma \vartriangleright t)^{-1} \\ &= \alpha^{-1} \bullet (1_S[s] \vartriangleleft \theta_2^{-1}) \bullet \alpha \bullet (\mathsf{Sub_I}(s)^{-1} \vartriangleright t[s]) \vartriangleright \tau \bullet (\ell_{t[s]} \vartriangleright \tau) \bullet \theta_2 \bullet (r_\sigma \vartriangleright t)^{-1} \\ &= \alpha^{-1} \bullet (1_S[s] \vartriangleleft \theta_2^{-1}) \bullet \mathsf{Sub_I}(s)^{-1} \vartriangleright (t[s] \cdot \tau) \bullet \alpha \bullet (\ell_{t[s]} \vartriangleright \tau) \bullet \theta_2 \bullet (r_\sigma \vartriangleright t)^{-1} \\ &= \alpha^{-1} \bullet \mathsf{Sub_I}(s)^{-1} \vartriangleright (\sigma \cdot t) \bullet (1_{S[s]} \vartriangleleft \theta_2^{-1}) \bullet \alpha \bullet (\ell_{t[s]} \vartriangleright \tau) \bullet \theta_2 \bullet (r_\sigma \vartriangleright t)^{-1} \\ &= \mathsf{Sub_I}(s)^{-1} \vartriangleright \sigma \vartriangleright t \bullet \alpha^{-1} \bullet (1_{S[s]} \vartriangleleft \theta_2^{-1}) \bullet \alpha \bullet (\ell_{t[s]} \vartriangleright \tau) \bullet \theta_2 \bullet (r_\sigma \vartriangleright t)^{-1} \\ &= (\theta_1 \bullet r_\sigma \bullet \ell_{\sigma}^{-1}) \vartriangleright t \bullet \alpha^{-1} \bullet (1_{S[s]} \vartriangleleft \theta_2^{-1}) \bullet \alpha \bullet (\ell_{t[s]} \vartriangleright \tau) \bullet \theta_2 \bullet (r_\sigma \vartriangleright t)^{-1} \\ &= \theta_1 \vartriangleright t \bullet ({r_\sigma \bullet \ell_{\sigma}^{-1}}) \vartriangleright t \bullet \alpha^{-1} \bullet (1_{S[s]} \vartriangleleft \theta_2^{-1}) \bullet \alpha \bullet (\ell_{t[s]} \vartriangleright \tau) \bullet \theta_2 \bullet (r_\sigma \vartriangleright t)^{-1} \\ &= \theta_1 \vartriangleright t \bullet ({r_\sigma \bullet \ell_{\sigma}^{-1}}) \vartriangleright t \bullet \alpha^{-1} \bullet (1_{S[s]} \vartriangleleft \theta_2^{-1}) \bullet \ell_{t[s] \tau} \bullet \theta_2 \bullet (r_\sigma \vartriangleright t)^{-1} \\ &= \theta_1 \vartriangleright t \bullet ({r_\sigma \bullet \ell_{\sigma}^{-1}}) \vartriangleright t \bullet \alpha^{-1} \bullet \ell_{\sigma \cdot t} \bullet \theta_2^{-1} \bullet \theta_2 \bullet (r_\sigma \vartriangleright t)^{-1} \\ &= \theta_1 \vartriangleright t \bullet ({r_\sigma \bullet \ell_{\sigma}^{-1}}) \vartriangleright t \bullet \alpha^{-1} \bullet \ell_{\sigma \cdot t} \bullet (r_\sigma \vartriangleright t)^{-1} \\ &= \theta_1 \vartriangleright t \bullet ({r_\sigma \bullet \ell_{\sigma}^{-1}}) \vartriangleright t \bullet \ell_{\sigma} \vartriangleright t \bullet (r_\sigma \vartriangleright t)^{-1} \\ &= \theta_1 \vartriangleright t\end{align*}$

9.2.5 Interpretation of sub-id

Next we interpret Suppose we have a context $\Gamma$ and a type T in $\Gamma$ . Note that we have a diagram as follows.

We interpret $\mathsf{sub}_{\mathsf{id}}(T)$ ^{Footnote 5} by $L_{1_\Gamma}(T)$ , so it suffices to show that $L_{1_\Gamma}(T)$ is an equivalence. To construct the inverse, we use the following diagram.

We set $\iota := {{{\mathsf{F_1}(L_{1_\Gamma}(T),1_T \cdot 1_T)}}}$ , and it remains to show that we have an invertible 2-cell from $L_{1_\Gamma}(T) \cdot \iota$ to $1_T$ . To do so, we use Item 2 of Definition 21, and we consider the following diagram.

We need to construct an invertible 2-cell $\delta$ from $1_\Gamma \cdot 1_\Gamma$ to $1_\Gamma$ and an invertible 2-cell $\overline{\sigma}$ from $L_{1_\Gamma}(T) \cdot \iota \cdot L_{1_\Gamma}(T)$ to $1_{T[s]} \cdot L_{1_\Gamma}(T)$ . For $\delta$ , we take $\lambda$ . For $\overline{\sigma}$ , we use that $\iota \cdot L_{1_\Gamma}(T) \cong 1_T \cdot 1_T$ , and thus we have the following composition of invertible 2-cells.

$ \begin{equation*}\begin{split}L_{1_\Gamma}(T) \cdot \iota \cdot L_{1_\Gamma}(T)&\cong L_{1_\Gamma}(T) \cdot 1_T \cdot 1_T\\&\cong L_{1_\Gamma}(T)\\&\cong 1_{T[s]} \cdot L_{1_\Gamma}(T)\end{split}\end{equation*}$

The desired invertible 2-cell is thus defined by ${{{\mathsf{E}(\delta,\overline{\sigma})}}}$ .

9.2.6 Interpretation of sub-comp

To interpret we suppose that we have substitutions $s' : \Theta \rightarrow \Delta$ and $s : \Delta \rightarrow \Gamma$ , and a type T in $\Gamma$ . Hence, we have the following diagram.

Note that both $L_{s \cdot s'}(T)$ and $L_{s}(T[s']) \cdot L_{s'}(T)$ are cartesian 1-cells over $s \cdot s'$ , since cartesian 1-cells are closed under composition by Proposition 22. For that reason, we get an adjoint equivalence $\mathsf{sub}_{\mathsf{comp}}(s, s') : T[s'][s] \rightarrow T[s \cdot s']$ making the diagram below commute up to invertible 2-cell by Construction 24.

9.2.7 Interpretation of tm-sub-id

Next we interpret and for that, we suppose that we have a context $\Gamma$ , types S and T over $\Gamma$ , and a 1-cell $t : S \to T$ over $1_\Gamma$ . We need to construct an invertible 2-cell from $\mathsf{sub}_{\mathsf{id}}^{-1} \ \cdot t[1_{\Gamma}] \cdot \mathsf{sub}_{\mathsf{id}}$ to t, and for that, it suffices to construct an invertible 2-cell from $t[1_{\Gamma}] \cdot \mathsf{sub}_{\mathsf{id}}$ to $\mathsf{sub}_{\mathsf{id}} \cdot t$ . Recall that we defined $\mathsf{sub}_{\mathsf{id}}$ to be $L_{1_\Gamma}(T)$ and recall that we defined $t[1_\Gamma]$ as follows.

By composing the two invertible 2-cells depicted in this diagram, we get the desired invertible 2-cell. The rule is interpreted analogously.

9.3 Local substitution

Next we interpret the rules in Fig. 7 (in Sections 9.3.1 and 9.3.2), Fig. 8 (in Sections 9.3.3 and 9.3.4), and Fig. 9 (in Section 9.3.5). For the interpretation, we use the local opcleaving.

9.3.1 Interpretation of map

We start with Suppose we have two substitutions $s, s' : \Delta \rightarrow \Gamma$ and a 2-cell $\rho : s \Rightarrow s'$ . In addition, we assume that we have a type T in context $\Gamma$ . Our goal is to construct a 1-cell $\mathsf{map} \ T \ \rho : T[s] \rightarrow T[s']$ . From the local opcleaving, we obtain $\mu_{\rho}(L_{s}(T))$ over $1_\Delta \cdot s'$ as the pushforward of $L_{s}(T)$ along $\rho \bullet \ell_{s'}^{-1}$ .

Since $L_{s'}(T)$ is cartesian, we can factor $\mu_{\rho}(L_{s}(T))$ through it. From that, we obtain $\mathsf{map} \ T \ \rho$ .

9.3.2 Interpretation of rew-tm

Next, we give the interpretation for the rule Suppose we have two 1-cells $s, s' : \Delta \rightarrow \Gamma$ and a 2-cell $\rho : s \rightarrow s'$ . In addition, we assume that we have types S, T in context $\Gamma$ and a 1-cell $\Gamma \hspace{0.01em} \mid \hspace{0.01em} S \vdash t : T$ . Our goal is to construct an invertible 2-cell between $t[s] \bullet (\mathsf{map} \ T \ \rho)$ and $(\mathsf{map} \ S \ \rho) \bullet t[s']$ . For that it suffices to construct:

1. (1) An invertible 2-cell $\delta$ from $1_\Delta \cdot 1_\Delta$ to $1_\Delta \cdot 1_\Delta$ ; and

2. (2) An invertible 2-cell $\overline{\sigma}$ from $t[s] \bullet (\mathsf{map} \ T \ \rho) \bullet L_{s'}(T)$ to $(\mathsf{map} \ S \ \rho) \bullet t[s'] \bullet L_{s'}(T)$ .

Then we define $t[\rho]$ to be ${{{\mathsf{E}(\delta,\overline{\sigma})}}}$ . For $\delta$ we take $1_{1_\Delta \cdot 1_\Delta}$ .

To construct $\overline{\sigma}$ , we first note that by construction of t[s’], we have the following invertible 2-cell.

$(\mathsf{map} \ S \ \rho) \bullet t[s'] \bullet L_{s'}(T)\cong(\mathsf{map} \ S \ \rho) \bullet L_{s'}(S) \bullet t$

Note that by construction of $\mathsf{map} \ S \ \rho$ , the following triangle commutes up to invertible 2-cell.

Hence, we get

$\gamma_1 : (\mathsf{map} \ S \ \rho) \bullet L_{s'}(S) \bullet t\cong\mu_{\rho}(L_{s}(S)) \bullet t.$

Similarly, we have an invertible 2-cell $\mathsf{map} \ T \ \rho \cdot L_{s'}(S) \cong \mu_{\rho}(L_{s}(T))$ , and thus we obtain

$\gamma_2 : t[s] \bullet (\mathsf{map} \ T \ \rho) \bullet L_{s'}(T)\cong t[s] \bullet \mu_{\rho}(L_{s}(T)).$

It thus suffices to construct an invertible 2-cell from $t[s] \bullet \mu_{\rho}(L_{s}(T))$ to $\mu_{\rho}(L_{s}(S)) \bullet t$ . We can depict the situation with the following square.

In this diagram, $\tau$ and $\tau'$ are the opcartesian 2-cells coming from the opcartesian lift. Note that by construction of t[s], the outer square of this diagram commutes. More precisely, we have an invertible 2-cell $\theta : t[s] \cdot L_{s}(T) \cong L_{s}(S) \cdot t$ .

To construct an invertible 2-cell from $\mu_{\rho}(L_{s}(S)) \cdot t$ to $t[s] \cdot \mu_{\rho}(L_{s}(T))$ , we are going to construct two opcartesian 2-cells. First, note that we have a 2-cell

$\tau \vartriangleright t : L_{s}(S) \cdot t \Rightarrow \mu_{\rho}(L_{s}(S)) \cdot t'$

that lies over $\beta_1 := (\rho \bullet l^{-1}) \vartriangleright 1_\Gamma : s \cdot 1_\Gamma \Rightarrow (1_\Delta \cdot s') \cdot 1_\Gamma$ . This 2-cell is opcartesian, because opcartesian 2-cells are closed under right whiskering.

Note that we also have the 2-cell

$\theta^{-1} \bullet t[s] \vartriangleleft \tau : L_{s}(S) \cdot t \Rightarrow t[s] \cdot \mu_{\rho}(L_{s}(T))$

which lies over $\beta_2 := r \bullet \ell^{-1} \bullet 1_{\Gamma} \vartriangleleft (\rho \bullet \ell^{-1}) : s \cdot 1_{\Gamma} \Rightarrow 1_{\Delta} \cdot (1_{\Delta} \cdot s')$ . It is opcartesian, because left whiskering also preserves opcartesian 2-cells.

We also have the following invertible 2-cell

$\beta_3 := r \bullet \ell^{-1}:(1_\Delta \cdot s') \cdot 1_\Gamma\cong1_{\Delta} \cdot (1_{\Delta} \cdot s'),$

and note that $\beta_1 \bullet \beta_3 = \beta_2$ by naturality of the unitors. From all of this we get the desired invertible 2-cell $\mu_{\rho}(L_{s}(S)) \cdot t \cong t[s] \cdot \mu_{\rho}(L_{s}(T))$ .

9.3.3 Interpretation of map-id

To interpret we assume that we have a 1-cell $s : \Delta \rightarrow \Gamma$ and a type T in context $\Gamma$ . Our goal is to construct an invertible 2-cell between the following compositions.

To do so, we need to construct:

1. (1) An invertible 2-cell $\delta$ from $1_\Delta$ to $1_\Delta$ ; and

2. (2) An invertible 2-cell $\overline{\sigma}$ from $\mathsf{map} \ T \ 1_s \cdot L_{s}(T)$ to $L_{s}(T)$ .

The desired 2-cell is then defined to be ${{{\mathsf{E}(\delta,\overline{\sigma})}}}$ . For $\delta$ we take $1_{1_\Delta}$ .

To construct $\overline{\sigma}$ , we first note that by construction of $\mathsf{map} \ T \ \rho$ , the following triangle commutes up to an invertible 2-cell $\tau : \mathsf{map} \ T \ \rho \cdot L_{s'}(T) \cong \mu_{\rho}(L_{s}(T))$ .

Next we recall that $\mu_{\rho}(L_{s}(T))$ was constructed as the following pushforward.

Since $\ell_{s}^{-1}$ is invertible, the opcartesian lift $\theta$ is invertible as well. Hence, we define $\overline{\sigma} := \tau \bullet \theta^{-1}$ . We interpret similarly.

9.3.4 Interpretation of map-lwhisker

To interpret we use the same idea, and we need to use that we assumed that opcartesian 2-cells are closed under whiskering. We give the precise details for Suppose we have a 1-cell $s : \Theta \rightarrow \Delta$ , a 2-cell $\rho : s' \Rightarrow s''$ where $s', s'' : \Delta \rightarrow \Gamma$ , and a type T in context $\Gamma$ . Our goal is to construct an invertible 2-cell from $\mathsf{map} \ T \ (s \vartriangleleft \rho)$ to $\mathsf{sub}_{\mathsf{comp}}^{-1}(s, s') \cdot (\mathsf{map} \ T \ \rho)[s] \cdot \mathsf{sub}_{\mathsf{comp}}(s, s'')$ . Since $\mathsf{map} \ T \ \rho$ was constructed as a cartesian factorization, it suffices to construct:

1. (1) An invertible 2-cell $\delta$ from $1_\Delta$ to $1_\Delta \cdot 1_\Delta \cdot 1_\Delta$ ; and

2. (2) An invertible 2-cell $\overline{\sigma}$ from $\mathsf{map} \ T \ (s \vartriangleleft \rho) \cdot L_{s \cdot s''}(T)$ to $\mathsf{sub}_{\mathsf{comp}}^{-1}(s, s') \cdot (\mathsf{map} \ T \ \rho)[s] \cdot \mathsf{sub}_{\mathsf{comp}}(s, s'') \cdot L_{s \cdot s''}(T)$ .

With those in place, the desired invertible 2-cell is defined to be ${{{\mathsf{E}(\delta,\overline{\sigma})}}}$ . We define $\delta$ to be $\ell_{1_\Delta}^{-1} \cdot \ell_{1_\Delta \cdot 1_\Delta}^{-1}$ .

For the construction of $\overline{\sigma}$ , let us start by recalling the construction of $\mathsf{map} \ T \ (s \vartriangleleft \rho)$ .

Next we inspect the definition of $\mathsf{sub}_{\mathsf{comp}}(s, s'')$ .

From this, we already get the following invertible 2-cell.

$\begin{align*}\gamma_1 : & \mathsf{sub}_{\mathsf{comp}}^{-1}(s, s') \cdot (\mathsf{map} \ T \ \rho)[s] \cdot \mathsf{sub}_{\mathsf{comp}}(s, s'') \cdot L_{s \cdot s''}(T) \\ & \cong \\ & \mathsf{sub}_{\mathsf{comp}}^{-1}(s, s') \cdot (\mathsf{map} \ T \ \rho)[s] \cdot L_{s}(T[s'']) \cdot L_{s''}(T)\end{align*}$

We also inspect the construction of $(\mathsf{map} \ T \ \rho)[s]$ , from which we get the following invertible 2-cell.

As such, we get another invertible 2-cell as follows.

$ \begin{align*}\gamma_2 : & \mathsf{sub}_{\mathsf{comp}}^{-1}(s, s') \cdot (\mathsf{map} \ T \ \rho)[s] \cdot L_{s}(T[s'']) \cdot L_{s''}(T) \\ & \cong \\ & \mathsf{sub}_{\mathsf{comp}}^{-1}(s, s') \cdot L_{s}(T[s']) \cdot \mathsf{map} \ T \ \rho \cdot L_{s''}(T).\end{align*}$

Let us recall the construction of $\mathsf{map} \ T \ \rho$ as well.

From this, we obtain yet another invertible 2-cell.

$ \begin{align*} \gamma_3 : & \mathsf{sub}_{\mathsf{comp}}^{-1}(s, s') \cdot (\mathsf{map} \ T \ \rho)[s] \cdot L_{s}(T[s'']) \cdot L_{s''}(T) \\ & \cong \\ & \mathsf{sub}_{\mathsf{comp}}^{-1}(s, s') \cdot L_{s}(T[s']) \cdot \mu_{\rho}(L_{s''}(T))\end{align*}$

Hence, we achieve our goal if we construct an invertible 2-cell of the following form.

$\gamma_4 :\mathsf{sub}_{\mathsf{comp}}^{-1}(s, s') \cdot L_{s}(T[s']) \cdot \mu_{\rho}(L_{s'}(T))\cong\mu_{s \vartriangleleft \rho}(L_{s \cdot s'}(T))$

We can depict this situation as follows.

Here $\tau$ and $\tau'$ are the 2-cells coming from the opcartesian lifts.

Since whiskering preserves opcartesian 2-cells, the following 2-cell is opcartesian.

$\begin{align*}(\mathsf{sub}_{\mathsf{comp}}^{-1}(s, s') \cdot L_{s}(T[s'])) \vartriangleleft \tau\enspace : \enspace & \mathsf{sub}_{\mathsf{comp}}^{-1}(s, s') \cdot L_{s}(T[s']) \cdot L_{s'}(T) \\ & \Rightarrow \\ & \mathsf{sub}_{\mathsf{comp}}^{-1}(s, s') \cdot L_{s}(T[s']) \cdot \mu_{\rho}(L_{s'}(T))\end{align*}$

Now we unfold the definition of $\mathsf{sub}_{\mathsf{comp}}(s, s')$ , and from that we get the following invertible 2-cell.

Since invertible 2-cells are opcartesian and opcartesian 2-cells are closed under composition, we get an opcartesian 2-cell

$\theta : L_{s \cdot s'}(T)\Rightarrow\mathsf{sub}_{\mathsf{comp}}^{-1}(s, s') \cdot L_{s}(T[s']) \cdot \mu_{\rho}(L_{s'}(T)).$

Since we already had an opcartesian 2-cell $\tau' : L_{s \cdot s'}(T) \Rightarrow \mu_{s \vartriangleleft \rho}(L_{s \cdot s'}(T))$ , we get an invertible 2-cell between the codomains of $\theta$ and $\tau'$ . Hence, we obtain an invertible 2-cell

$\gamma_5 : \mathsf{sub}_{\mathsf{comp}}^{-1}(s, s') \cdot L_{s}(T[s']) \cdot \mu_{\rho}(L_{s'}(T))\cong\mu_{s \vartriangleleft \rho}(L_{s \cdot s'}(T)).$

By chaining all the invertible 2-cells, we get the desired 2-cell $\overline{\sigma}$ .

9.3.5 Interpretation of sub-comp-map-id-r

Finally, we show how to interpret Suppose that we have objects $\Gamma$ and $\Delta$ , an object T over $\Gamma$ , and a 1-cell $s : \Delta \rightarrow \Gamma$ . Our goal is to construct an invertible 2-cell from $\mathsf{sub}_{\mathsf{comp}}(s, 1_\Delta) {(\mathsf{map} \ T \ r_{s})}$ to $\mathsf{sub}_{\mathsf{id}}$ . To do so, it suffices to construct an invertible 2-cell from $\mathsf{sub}_{\mathsf{comp}}(s, 1_\Delta) \cdot (\mathsf{map} \ T \ r_{s}) \cdot L_{s}(T)$ to $\mathsf{sub}_{\mathsf{id}} \cdot L_{s}(T)$ . Our situation is depicted in the following diagram.

Recall that $\mathsf{sub}_{\mathsf{id}} : T[s][1_\Delta] \rightarrow T[s]$ is defined to be $L_{1_\Delta}(T[s])$ . By construction, we have an invertible 2-cell

$\varphi_1 :\mathsf{map} \ T \ r_{s} \cdot L_{s}(T)\Rightarrow\mu_{r_{s}}(L_{s \cdot 1_{\Delta}}(T)) .$

In addition, since $\mathsf{map} \ T \ r_{s}$ was constructed as an opcartesian lift, we have the following invertible 2-cell.

$\varphi_2 :L_{s \cdot 1_{\Delta}}(T)\Rightarrow\mu_{r_{s}}(L_{s \cdot 1_{\Delta}}(T))$

Note that $\varphi_2$ is invertible: this is because it is the opcartesian lift of an invertible 2-cell. By construction of $\mathsf{sub}_{\mathsf{comp}}(s, 1_\Delta)$ , we have the following invertible 2-cell.

$\varphi_3 :\mathsf{sub}_{\mathsf{comp}}(s, 1_\Delta) \cdot L_{s \cdot 1_{\Delta}}(T)\Rightarrow L_{1_{\Delta}}(T[1_\Delta]) \cdot L_{s}(T) .$

The composition $\varphi_3 \bullet \varphi_2 \bullet \varphi_1^{-1}$ yields the desired 2-cell.

We omit the description of the verification of several equalities. All in all, we can state the following theorem.

Theorem (Soundness). We can interpret $\mathsf{BTT}$ in any weak comprehension bicategory.

10. Variations on Syntax

$\mathsf{BTT}$ is a very complicated type theory and might not be feasible to implement or use in practice. Its main purpose is to serve as a framework for studying specialized syntax and the corresponding semantics. Based on a user’s goal, they might adopt some of the following simplifications.

V1: Strictness. The rules in Figs. 1 and 2 are aimed at bicategories. When working with strict 2-categories instead, the unitors and associators would become equalities, and as a result, rules for inverse laws, naturality, and the pentagon and triangle equations are not needed. For this variant, an equality judgment on contexts, types, substitutions, and terms would be added to $\mathsf{BTT}$ .

V2: Splitness. We could assume the comprehension bicategory to be split in addition to being strict; thus, in particular, the rules would collapse into ordinary equalities.

V3: Undirected TT. We could add a rule postulating inverses of reductions. Semantically, this would amount to working in groupoid-enriched categories.

V4: Proof-irrelevant reductions. Our syntax and the semantics allow us to distinguish parallel reductions (2-cells). We could instead “truncate” them, by moving to proof-irrelevant reductions, making the judgmental equality on them superfluous. This would yield a directed analog to the judgments of $\mathsf{MLTT}$ ; semantically, it corresponds to working in poset-enriched categories instead of general bicategories.

We have developed comprehension bicategories with the explicit goal of encompassing previously defined interpretations of higher-dimensional and directed type theory. In the following remarks, we summarize the relationship with two previous works.

Garner Reference Garner2009). To summarize the differences between Garner’s comprehension 2-categories (Garner Reference Garner2009) and our comprehension bicategories: the former are full, strict, split, undirected (i.e., locally groupoidal), and incorporate type constructors.

Licata and Harper (Reference Licata, Harper, Mislove and Ouaknine2011) interpret a type in context $\Gamma$ as a strict 1-functor from the category interpreting $\Gamma$ into a 1-category $\mathsf{CAT}$ of categories. Formally, they thus consider the slice bicategory $\mathsf{Cat}/\mathsf{CAT}$ , where $\mathsf{Cat}$ is the bicategory of categories, in which $\mathsf{CAT}$ is assumed to be a 0-cell. The “domain” pseudofunctor ${{{\mathsf{dom}}}} : \mathsf{Cat}/\mathsf{CAT} \to \mathsf{Cat}$ carries the structure of a global cleaving and local opcleaving; this is more generally the case for any “domain” pseudofunctor ${{{\mathsf{dom}}}} : \mathsf{B}/a \to \mathsf{B}$ (). To construct the comprehension pseudofunctor, one needs to use specifics of the category of strict categories, namely the Grothendieck construction of functors into $\mathsf{CAT}$ .

11. Terms in Directed Type Theory

The notion of term in $\mathsf{BTT}$ , and the interpretation of these terms, is different from the notion of term studied in the syntax and semantics by Licata and Harper (Reference Licata, Harper, Mislove and Ouaknine2011) or North (Reference North and König2019). Specifically, our terms correspond to 1-cells in the (total) category of types and are interpreted as such. Terms in the sense of Licata and Harper, and North, instead correspond to particular 1-cells in the category of contexts, specifically, to sections of the canonical projections $\pi_{\Gamma,A} : \Gamma.A \to \Gamma$ .

One might attempt to reconcile these two, on the syntactic side, by

• (1) adding a unit type to $\mathsf{BTT}$ ; and

• (2) asking for context morphisms to be built from terms, that is, from type morphisms. (Semantically, this corresponds to asking for $\chi$ to be full.)

However, note that most of our comprehension bicategories are not full. Furthermore, this syntactic modification would still leave a difference between the interpretations of terms given by Licata and Harper, and North, on the one hand, and our interpretation on the other hand. Specifically, in Example 45, terms are interpreted as functors that preserve opcartesian cells; by the Grothendieck construction, terms can equivalently be interpreted as pseudonatural transformations. In contrast, terms are interpreted as lax natural transformations by others (Licata and Harper Reference Licata, Harper, Mislove and Ouaknine2011; North Reference North and König2019). Hence, there is a mismatch between how terms are interpreted in comprehension bicategories compared to other interpretations.

We can analyze this mismatch more closely and suggest a potential way of reconciling it, by looking again at the intended interpretation of two-dimensional and directed type theory in categories via the opcleaving model. Taking this mismatch seriously leads us in the direction of directed type theory, where there can be different flavors of terms. For example, in work by Nuyts (Reference Nuyts2015) terms can be used isovariantly, covariantly, and contravariantly. In the opcleaving model (Example 45), these different notions of terms would correspond to different kinds of natural transformations: pseudonatural transformations, lax natural transformations, and oplax natural transformations.

To reconcile this difference, one would need to modify the syntax and the definition of comprehension bicategory. In the syntax, one would need to add a term judgment for every kind of term present in directed type theory. Accordingly, one would also add the corresponding operations like substitution. The notion of comprehension bicategory would also require slight modifications for this purpose; one necessary requirement would be that $\chi$ preserves both cartesian 1-cells and opcartesian 2-cells. The reason for this change is that this would be necessary to interpret substitution of a certain class of terms. More precisely, in the opfibration model, lax transformations can be identified with sections of the projections $\int \mathsf{D} \rightarrow \mathsf{C}$ . To interpret substitution for these, we need that certain squares in the arrow bicategory are pullbacks; to guarantee that, one requires that $\chi$ preserves cartesian 1-cells.

It is an open problem to develop categorical semantics for type theories that feature terms with different variances.