3.1 Discrete fibrations

We recall some basic properties on discrete fibrations.

Proof. The implications and are immediate. To see , suppose that p is a discrete fibration. By definition, the functor $A^{\to} \to A \times_{\mathcal{S}} \mathcal{S}^{\to}$ is bijective on objects. To see that this functor is also fully faithful, let $f_{1} : a_{1} \to a_{1}'$ and $f_{2} : a_{2} \to a_{2}'$ be objects in $A^{\to}$ , let $g : a_{1}' \to a_{2}'$ be an arrow in A, let $u : p(a_{1}) \to p(a_{2})$ be an arrow in $\mathcal{S}$ , and suppose that $p(f_{2}) \circ u = p(g) \circ p(f_{1})$ . We have to show that there exists a unique arrow $\widehat{u} : a_{1} \to a_{2}$ in A such that $p(\widehat{u}) = u$ and $f_{2} \circ \widehat{u} = g \circ f_{1}$ . Such a $\widehat{u}$ must be $\overline{u}_{a_{2}} : u^{*} a_{2} \to a_{2}$ by the condition $p(\widehat{u}) = u$ , and it indeed satisfies $f_{2} \circ \overline{u}_{a_{2}} = g \circ f_{1}$ since both $f_{2} \circ \overline{u}_{a_{2}}$ and $g \circ f_{1}$ have the same codomain and are sent by p to $p(f_{2}) \circ u = p(g) \circ p(f_{1})$ .

For a small category $\mathcal{S}$ , the category $\mathbf{DFib}_{\mathcal{S}}$ is equivalent to the category of presheaves over $\mathcal{S}$ : for a presheaf P over $\mathcal{S}$ , its category of elements $\int_{\mathcal{S}}P$ together with the projection $\int_{\mathcal{S}}P \to \mathcal{S}$ is a discrete fibration over $\mathcal{S}$ ; for a discrete fibration A over $\mathcal{S}$ , we have a presheaf $I \mapsto A(I)$ where A(I) denotes the fiber $p_{A}^{-1}(I)$ . A representable presheaf $\mathcal{S}({-}, I)$ corresponds to the slice category $\mathcal{S}/I$ with domain functor $\mathcal{S}/I \to \mathcal{S}$ . We say a discrete fibration A over $\mathcal{S}$ is representable if it is isomorphic to $\mathcal{S}/I$ for some $I \in \mathcal{S}$ . We call the functor $\mathcal{S} \ni I \mapsto \mathcal{S}/I \in \mathbf{DFib}_{\mathcal{S}}$ the Yoneda embedding. The Yoneda Lemma for discrete fibrations is formulated as follows.

Theorem 3.4. The Yoneda Lemma Let $\mathcal{S}$ be a category and A a discrete fibration over $\mathcal{S}$ . For any object $I \in \mathcal{S}$ , the map

$$ \mathbf{DFib}_{\mathcal{S}}(\mathcal{S}/I, A) \ni f \mapsto f(\mathrm{id}_{I}) \in A(I) $$

is bijective.

By the Yoneda Lemma, we identify an element $a \in A(I)$ with the corresponding map $\mathcal{S}/I \to A$ of discrete fibrations over $\mathcal{S}$ . We also recall the following criterion for representability.

Discrete fibrations are stable under “base change”.

1. (1) If

   
   is a pullback of categories, then $p_{A'} : A' \to \mathcal{S}'$ is a discrete fibration called the base change of A along F and denoted by $F^{*}A$ .

2. (2) If $\sigma : F \Rightarrow G : \mathcal{S}' \to \mathcal{S}$ is a natural transformation and

   
   are pullbacks, then there exists a unique pair $(\sigma^{*}_{A}, \overline{\sigma}_{A})$ consisting of a map $\sigma^{*}_{A} : A'_{2} \to A'_{1}$ of discrete fibrations over $\mathcal{S}'$ and a natural transformation $\overline{\sigma}_{A} : \overline{F} \sigma^{*}_{A} \Rightarrow \overline{G}$ such that $p_{A}\overline{\sigma}_{A} = \sigma p_{A'_{2}}$ .
   

3.2 Representable maps of discrete fibrations

It follows from Corollary 3.13 below that if $\mathcal{S}$ has a terminal object then 2 is equivalent to that the discrete fibration A over $\mathcal{S}$ is representable by, say, I and $\mathcal{S}$ has products with I. In particular, if $\mathcal{S}$ has finite products then 1 and 2 are equivalent.

Proof. By Item 3 of Proposition 3.2, Item 2 is equivalent to that the square

is a pullback, where the left functor sends $(n : a' \to a) \in A/a$ to $(m \circ fn : fa' \to b) \in B/b$ . This is equivalent to that the map $A/a \ni (n : a' \to a) \mapsto (m \circ fn) : fa' \to b) \in f^{*}(B/b)$ of discrete fibrations over A is an isomorphism. By Proposition 3.5, this is equivalent to that $m \circ f(\mathrm{id}_{a}) = m$ is the terminal object in $f^{*}(B/b) \cong (f \downarrow b)$ .

Corollary 3.13 Let $f : A \to B$ be a map of discrete fibrations over a category $\mathcal{S}$ . Then f is representable if and only if, for any object $I \in \mathcal{S}$ and element $b : \mathcal{S}/I \to B$ , the pullback $b^{*}A$ is a representable discrete fibration over $\mathcal{S}$ . More precisely, the right adjoint $\delta : B \to A$ and the counit $\varepsilon$ of f fit into the pullback square

for any element $b : \mathcal{S}/I \to B$ .

Definition 3.14 Let

be a square of categories that commutes up to isomorphism, and suppose that f and f’ have right adjoints $\delta$ and $\delta'$ , respectively. We say this square satisfies the Beck–Chevalley condition if the canonical natural transformation $g \delta' \Rightarrow \delta h$ defined by the following diagram is an isomorphism.

Here $\varepsilon'$ is the counit of the adjunction $f' \dashv \delta'$ and $\eta$ is the unit of $f \dashv \delta$ .

Corollary 3.15. Let

be a commutative square of discrete fibrations over a category $\mathcal{S}$ where f is representable. If this square is a pullback, then f’ is representable and this square satisfies the Beck–Chevalley condition.

3.3 Modeling type theory

A representable map $f : A \to B$ of discrete fibrations over $\mathcal{S}$ is considered to be a model of dependent type theory. We think of objects $I \in \mathcal{S}$ as contexts, elements $b \in B(I)$ as types over I and elements $a \in A(I)$ as terms over I. For a term $a \in A(I)$ , the type of a is $f(a) \in B(I)$ . The representability of f is used for modeling context extensions.

Definition 3.17. Let $f : A \to B$ be a representable map of discrete fibrations over $\mathcal{S}$ . We denote by $\delta^{f} : B \to A$ the right adjoint to f and by $\varepsilon^{f}$ the counit of the adjunction $f \dashv \delta^{f}$ . For an object $I \in \mathcal{S}$ and an element $b \in B(I)$ , we write $\{b\}^{f}$ for the object $p_{A}(\delta^{f}_{b}) \in \mathcal{S}$ . Let $\pi^{f}_{b} = p_{B}(\varepsilon^{f}_{b}) : \{b\}^{f} \to I$ . We call $\{b\}^{f}$ the context extension of b with respect to f. By Corollary 3.13, these fit into the pullback

Syntactically, the context extension $B(I) \ni b \mapsto \{b\}^{f}\in \mathcal{S}$ models the rule for extending a context by a type

and $\pi_{b}^{f}$ corresponds to the context morphism $(I, x : b) \to I$ and $\delta_{b}^{f}$ corresponds to the term $I, x : b \vdash x : b$ .

We demonstrate how to model type constructors using representable maps of discrete fibrations. First, we recall the notions of a pushforward and a polynomial functor.

Definition 3.19. For an exponentiable arrow $f : A \to B$ in a cartesian category $\mathcal{C}$ , we define a functor $\mathrm{P}_{f} : \mathcal{C} \to \mathcal{C}$ called the polynomial functor associated with f to be the composite

where $A^{*}$ is the pullback functor along $A \to 1$ and $B_{!}$ is the forgetful functor.

Since the category of discrete fibrations over a category $\mathcal{S}$ is equivalent to the category of presheaves over $\mathcal{S}$ , the pushforward along an arbitrary map exists. But the pushforward along a representable map has a simple description.

Lemma 3.20. Let

be a commutative triangle of categories and suppose that p is a discrete fibration. Then f is a discrete fibration if and only if q is. Consequently, the isomorphism $(\mathfrak{Cat}/\mathcal{S})/A \cong \mathfrak{Cat}/A$ restricts to an isomorphism $(\mathbf{DFib}_{\mathcal{S}})/A \cong \mathbf{DFib}_{A}$ .

Proof. By Corollary 3.7, the adjunction $f \dashv \delta^{f}$ is mapped to the adjunction

By Lemma 3.20, $f^{*}$ is isomorphic to the pullback functor $(\mathbf{DFib}_{\mathcal{S}})/B \to (\mathbf{DFib}_{\mathcal{S}})/A$ , and thus $(\delta^{f})^{*}$ is the pushforward along f.

Let $f : A \to B$ be a representable map of discrete fibrations over a category $\mathcal{S}$ . Consider the discrete fibration $\mathrm{P}_{f}B$ over $\mathcal{S}$ . It is the pullback

by the construction of $f_{*}$ given in Proposition 3.21. Hence, an element of $\mathrm{P}_{f}B$ over $I \in \mathcal{S}$ is a pair $(b_{1}, b_{2})$ of elements $b_{1} \in B(I)$ and $b_{2} \in B(\{b_{1}\}^{f})$ . One can think of $b_{2}$ as a type family over $b_{1}$ . Then a map $g : \mathrm{P}_{f}B \to B$ is thought of as a type constructor that takes types $b_{1} \in B(I)$ and $b_{2} \in B(\{b_{1}\}^{f})$ and returns a type $g(b_{1}, b_{2}) \in B(I)$ . Syntactically, g is a type constructor of the form

where the expression $x.b_{2}$ means that the variable x is bound. Similarly, a map $h : \mathrm{P}_{f}A \to A$ is a term constructor that takes a type $b_{1} \in B(I)$ and a term $a_{2} \in A(\{b_{1}\}^{f})$ and returns a term $h(b_{1}, a_{2}) \in A(I)$ . For example, dependent products are modeled by maps $\Pi : \mathrm{P}_{f}B \to B$ and $\mathsf{abs} : \mathrm{P}_{f}A \to A$ of discrete fibrations over $\mathcal{S}$ such that the square

commutes and is a pullback (Awodey Reference Awodey2018, Section 2.1). The commutativity means that $\mathsf{abs}$ has a typing rule

and being a pullback means that $\mathsf{abs}$ induces a bijection between the set of terms $I, x : b_{1} \vdash a_{2} : b_{2}$ and the set of terms $I \vdash a : \Pi(b_{1}, x.b_{2})$ . We refer the reader to Awodey (Reference Awodey2018), Newstead (Reference Newstead2018) for further examples.

4.1 Example: Basic dependent type theory

Most examples of type theories in the sense of Definition 4.4 are constructed using a logical framework introduced in Section 5. Here we only give a simple example which naturally arises from the syntax of dependent type theory.

Let $\mathbb{G}$ denote the opposite of the category of finite generalized algebraic theories (Cartmell Reference Cartmell1978) and interpretations between them. We do not need the precise definition of a generalized algebraic theory, but remember that a generalized algebraic theory consists of sets of type constants, term constants, type equations and term equations. $\mathbb{G}$ has finite limits, or equivalently $\mathbb{G}^{\mathrm{op}}$ has finite colimits: coproducts of generalized algebraic theories are given by disjoint union; coequalizers of generalized algebraic theories are obtained by adjoining equations.

Let $U_{n}$ be the generalized algebraic theory consisting of type constants $({} \vdash A_{0}), (x_{0} : A_{0} \vdash A_{1}), \dots, (x_{0} : A_{0}, \dots,x_{n-1} : A_{n-1}(x_{0}, \dots, x_{n-2}) \vdash A_{n})$ and let $E_{n}$ be the extension of $U_{n}$ by a term constant $(x_{0} : A_{0}, \dots, x_{n-1} : A_{n-1}(x_{0}, \dots, x_{n-2}) \vdash a_{n} : A_{n}(x_{0}, \dots, x_{n-1}))$ . We denote by $\partial_{n} : U_{n} \to E_{n}$ and $\mathrm{ft}_{n} : U_{n-1} \to U_{n}$ the obvious inclusions (in $\mathbb{G}^{\mathrm{op}}$ ), where we define $U_{-1}$ to be the empty theory. We also regard $\partial_{n}$ and $\mathrm{ft}_{n}$ as arrows $\partial_{n} : E_{n} \to U_{n}$ and $\mathrm{ft}_{n} : U_{n} \to U_{n-1}$ , respectively, in $\mathbb{G}$ . The category $\mathbb{G}$ is “freely generated by an exponentiable arrow” in the following sense.

By Item 1 of Theorem 4.12, we regard $\mathbb{G}$ as a representable map category with the smallest stable class of exponentiable arrows containing $\partial_{0} : E_{0} \to U_{0}$ and call it the basic dependent type theory. Item 2 of Theorem 4.12 can be rephrased as follows.

By this universal property, a model $\mathcal{S}$ of the type theory $\mathbb{G}$ consists of the following data:

• a category $\mathcal{S}$ with a terminal object;

• a representable map $\partial_{0}^{\mathcal{S}} : E_{0}^{\mathcal{S}} \to U_{0}^{\mathcal{S}}$ of discrete fibrations over $\mathcal{S}$ .

This is precisely a natural model (category with families).

4.2 The 2-category of models of a type theory

The models of a type theory are part of a 2-category.

A 2-morphism $\sigma : F \Rightarrow G : \mathcal{S} \to \mathcal{T}$ of morphisms of models of $\mathbb{T}$ is a natural transformation $\sigma : F \Rightarrow G$ between the underlying functors such that, for any object $A \in \mathbb{T}$ , there exists a (necessarily unique) natural transformation $\sigma_{A} : F_{A} \Rightarrow G_{A}$ such that $p_{(A^{\mathcal{T}})} \sigma_{A} = \sigma p_{(A^{\mathcal{S}})}$ .

We denote by $\mathfrak{Mod}_{\mathbb{T}}$ the 2-category of models of $\mathbb{T}$ and their morphisms and 2-morphisms.

4.3 Another definition of morphisms of models

Results in this subsection are required only in proofs of some propositions, so the reader may skip this subsection until needed.

A model of a type theory is defined to be a representable map functor into a category of discrete fibrations. In this subsection, we see that morphisms and 2-morphisms of models of a type theory are also regarded as representable map functors into suitable representable map categories.

We say a map $f : A \to B$ in $(\mathbf{DFib}^{\mathfrak{I}})_{\mathcal{S}}$ is representable if every component $f_{i} : A i \to B i$ is a representable map of discrete fibrations over $\mathcal{S} i$ and, for any 1-cell $u : i \to i'$ in $\mathfrak{I}$ , the square

satisfies the Beck–Chevalley condition.

Proof. Let

be a pullback in $(\mathbf{DFib}^{\mathfrak{I}})_{\mathcal{S}}$ and suppose that f is representable. One can check that pullbacks in $(\mathbf{DFib}^{\mathfrak{I}})_{\mathcal{S}}$ are componentwise, which means that

is a pullback in $\mathbf{DFib}_{\mathcal{S}i}$ for every $i \in \mathfrak{I}$ . By Corollary 3.15 each $f'_{i}$ is representable. To see the Beck–Chevalley condition, let $u : i \to i'$ be a 1-cell in $\mathfrak{I}$ . Consider the following commutative diagram.

The front square satisfies the Beck–Chevalley condition by assumption. The left and right squares satisfy the Beck-Chevalley condition by Corollary 3.15. The functor $g_{i'} : A'i' \to Ai'$ is a discrete fibration by Lemma 3.20 and thus reflects isomorphisms by Proposition 3.3. Hence, it follows that the back square satisfies the Beck–Chevalley condition.

Let $f : A \to B$ be a representable map in $(\mathbf{DFib}^{\mathfrak{I}})_{\mathcal{S}}$ . Although the right adjoint $\delta^{f} : B \to A$ to f is only a pseudo-natural transformation, we can define the base change $(\delta^{f})^{*}C \to B$ of C along $\delta^{f}$ for a map $C \to A$ in $(\mathbf{DFib}^{\mathfrak{I}})_{\mathcal{S}}$ as follows:

• for an object $i \in \mathfrak{I}$ , we define $((\delta^{f})^{*}C)i = (\delta^{f}_{i})^{*}(C i)$ ;

• for a 1-cell $u : i \to i'$ in $\mathfrak{I}$ , we have a natural isomorphism

  
  Using Proposition 3.6, we have a unique pair $((\delta^{f}_{u})^{*}(C u), \overline{\delta}^{f}_{u})$ consisting of a map $(\delta^{f}_{u})^{*}(C u) : (\delta^{f}_{i})^{*}(C i) \to (\delta^{f}_{i'})^{*}(C i')$ of discrete fibrations over B u and a natural isomorphism
  
  over $\delta^{f}_{u}$ . We define $((\delta^{f})^{*} C) u = (\delta^{f}_{u})^{*}(C u)$ ;

• the 2-cell part is defined in a natural way.

Then we can prove the following in the same way as Proposition 3.21.

We consider the case that $\mathfrak{I}$ is the category $\{0 \to 1\}$ . In this case, we write $(\mathbf{DFib}^{\to})_{F}$ for $(\mathbf{DFib}^{\mathfrak{I}})_{F}$ . Let $F : \mathcal{S} \to \mathcal{T}$ be a functor between small categories. By definition, an object of $(\mathbf{DFib}^{\to})_{F}$ is a commutative square

of categories such that $p_{A}$ and $p_{B}$ are small discrete fibrations, and a map $(A, B, G) \to (A', B', G')$ is a square

of categories such that f and g are maps of discrete fibrations over $\mathcal{S}$ and $\mathcal{T}$ , respectively. Such a square is representable if f and g are representable maps of discrete fibrations and the square satisfies the Beck–Chevalley condition. There are representable map functors $\mathbf{dom} : (\mathbf{DFib}^{\to})_{F} \to \mathbf{DFib}_{\mathcal{S}}$ and $\mathbf{cod} : (\mathbf{DFib}^{\to})_{F} \to \mathbf{DFib}_{\mathcal{T}}$ induced by the inclusions $\{0\} \to \{0 \to 1\}$ and $\{1\} \to \{0 \to 1\}$ respectively.

Consider the 2-category $\Theta$ consisting of two 0-cells 0 and 1, two 1-cells $a, b : 0 \to 1$ , and one 2-cell $a \Rightarrow b$ .

For a natural transformation $\sigma : F \Rightarrow G : \mathcal{S} \to \mathcal{T}$ , we have the representable map functors $\mathbf{dom} : (\mathbf{DFib}^{\Theta})_{\sigma} \to (\mathbf{DFib}^{\to})_{F}$ and $\mathbf{cod} : (\mathbf{DFib}^{\Theta})_{\sigma} \to (\mathbf{DFib}^{\to})_{G}$ induced by the inclusions $\{0 \overset{a}{\to} 1\} \to \Theta$ and $\{0 \overset{b}{\to} 1\} \to \Theta$ , respectively.

Proposition 4.22. Let $\mathbb{T}$ be a type theory, $\mathcal{S}$ and $\mathcal{T}$ models of $\mathbb{T}$ , $F, G : \mathcal{S} \to \mathcal{T}$ morphism of models of $\mathbb{T}$ , and $\sigma : F \Rightarrow G$ a natural transformation between the underlying functors. Then $\sigma$ (necessarily uniquely) extends to a 2-morphism $F \Rightarrow G$ of models of $\mathbb{T}$ if and only if there exists a (necessarily unique) representable map functor $\tilde{\sigma} : \mathbb{T} \to (\mathbf{DFib}^{\Theta})_{\sigma}$ such that $\mathbf{dom} \tilde{\sigma} = F_{({-})}$ and $\mathbf{cod} \tilde{\sigma} = G_{({-})}$ .

5.1 Formal definition

We assume that we are given an infinite set of variables $x, y, \dots$ and sufficiently many symbols $\alpha, \beta, \dots$ .

Definition 5.1. Preterms are defined by the following grammar:

The expression $x.B$ means that the variable x is considered to be bound. We always identify $\alpha$ -equivalent preterms. We use the following notations:

• $((x : A) \to B) :\equiv \Pi(A, x.B)$

• $(\lambda(x : A).b) :\equiv \mathsf{abs}(A, x.b)$

• $(a =_{A} b) :\equiv \mathsf{Eq}(A, a, b)$

We write b a for $\mathsf{app}(A, x.B, b, a)$ when the terms A and $x.B$ are clear from the context. The type annotations in $\lambda(x : A).b$ and $a =_{A} b$ are often omitted, and we simply write $\lambda x.b$ and $a = b$ respectively. For preterms a, b and a variable x, the substitution $a[b/x]$ is defined in the ordinary way.

Definition 5.2. A pre-context is a finite sequence of the form

$$ (x_{1} : A_{1}, \dots, x_{n} : A_{n}) $$

with pre-terms $A_{1}, \dots, A_{n}$ and distinct variables $x_{1}, \dots, x_{n}$ . We denote precontexts by $\Gamma, \Delta, \dots$ . We write $x \in \Gamma$ when $\Gamma = (x_{1} : A_{1}, \dots, x_{n} : A_{n})$ and $x = x_{i}$ for some i.

Definition 5.3. A presignature $\Sigma$ consists of a well-ordered set $|\Sigma|$ of symbols and a function $\Sigma$ that assigns to each symbol in $|\Sigma|$ a pair consisting of a precontext and a preterm. We write

$$ \Sigma = (\alpha_{0} : \Gamma_{0} \Rightarrow A_{0}, \alpha_{1} : \Gamma_{1} \Rightarrow A_{1}, \alpha_{2} : \Gamma_{2} \Rightarrow A_{2}, \dots) $$

to mean that $|\Sigma| = \{\alpha_{0} < \alpha_{1} < \alpha_{2} < \dots\}$ and the value $\Sigma(\alpha_{i})$ is $(\Gamma_{i}, A_{i})$ for each $\alpha_{i} \in |\Sigma|$ . We write $(\alpha : \Gamma \Rightarrow A) \in \Sigma$ when $\alpha \in |\Sigma|$ and $\Sigma(\alpha) = (\Gamma, A)$ . For a symbol $\alpha \in |\Sigma|$ , we write $\Sigma|_{\alpha}$ for the restriction of $\Sigma$ to $\{\beta \in |\Sigma| \mid \beta < \alpha\}$ .

Definition 5.4. A judgment is one of the following forms.

For precontexts $\Gamma$ and $\Delta = (y_{1} : B_{1}, \dots, y_{m} : B_{m})$ and a finite sequence of preterms $f = (f_{1}, \dots, f_{m})$ , we write $\Sigma \mid f : \Gamma \to \Delta$ for the finite sequence of judgments

$$ ((\Sigma \mid \Gamma \vdash f_{1} : B_{1}), \dots, (\Sigma \mid \Gamma \vdash f_{m} : B_{m}[f_{1}/y_{1}, \dots, f_{m-1}/y_{m-1}])). $$

For such $\Sigma \mid f : \Gamma \to \Delta$ , we write [f] for the substitution operator $[f_{1}/y_{1}, \dots, f_{m}/y_{m}]$ . We define the set of legal judgments to be the smallest set of judgments closed under the rules listed in Fig. 1. Here we omit the obvious rules for $\equiv$ to be a congruence relation.

Fig. 1. Legal judgments.

Our logical framework has the usual weakening and substitution properties. The following weakening on signatures might be nonstandard since signatures can be infinite.

5.2 Coding type theories

We give several examples of encodings of type theories in our logical framework.

Example 5.7. We define $\Sigma_{DTT}$ to be the following signature.

( $\mathsf{el}$ stands for element.) We call $\Sigma_{DTT}$ the signature for basic dependent type theory. We will see that $\Sigma_{DTT}$ represents the basic dependent type theory $\mathbb{G}$ (Example 5.18).

We consider extending $\Sigma_{DTT}$ by adjoining type constructors.

Example 5.8. $\Pi$ -types in $\mathsf{Type}$ are encoded as follows.

Example 5.9 Identity types in $\mathsf{Type}$ are encoded as follows.

Equality types are encoded as identity types with the following equations called equality reflection.

Example 5.10. A universe (à la Tarski) is encoded by the following symbols.

For nested universes $\mathsf{U}_{0} : \mathsf{U}_{1}$ , we add two pairs of such symbols $(\mathsf{U}_{0}, \mathsf{el}_{\mathsf{U}_{0}})$ and $(\mathsf{U}_{1}, \mathsf{el}_{\mathsf{U}_{1}})$ and a “name” of $\mathsf{U}_{0}$ in $\mathsf{U}_{1}$ :

Since our logical framework can have an infinitely long signature, one can encode infinitely many universes

$$ \mathsf{U}_{0} : \mathsf{U}_{1} : \mathsf{U}_{2} : \dots. $$

One can add dependent products on $\mathsf{U}$ in two ways. In both ways, we add a type constructor

$$ \Pi_{\mathsf{U}} : (A : \mathsf{el}(\mathsf{U}), B : \mathsf{el}(\mathsf{el_{U}}(A)) \to \mathsf{el}(\mathsf{U})) \Rightarrow \mathsf{el}(\mathsf{U}). $$

One way is to add an equation

$$ \_ : (A : \mathsf{el}(\mathsf{U}), B : \mathsf{el}(\mathsf{el_{U}}(A)) \to \mathsf{el}(\mathsf{U})) \Rightarrow \mathsf{el_{U}}(\Pi_{\mathsf{U}}(A, B)) = \Pi(\mathsf{el_{U}}(A), \lambda x.\mathsf{el_{U}}(B x)) $$

assuming $\mathsf{Type}$ has dependent products. The other way is to add symbols and equations in a similar manner to dependent products in $\mathsf{Type}$ . In the latter way, the equation

$$ A : \mathsf{el}(\mathsf{U}), B : \mathsf{el}(\mathsf{el_{U}}(A)) \to \mathsf{el}(\mathsf{U}) \vdash \mathsf{el_{U}}(\Pi_{\mathsf{U}}(A, B)) \equiv \Pi(\mathsf{el_{U}}(A), \lambda x.\mathsf{el_{U}}(B x)) : \mathsf{Type} $$

need not hold, but one can show that $\mathsf{el_{U}}(\Pi_{\mathsf{U}}(A, B))$ and $\Pi(\mathsf{el_{U}}(A), \lambda x.\mathsf{el_{U}}(B x))$ are isomorphic in an appropriate sense.

Example 5.11 Various two-level type theories (Altenkirch et al. Reference Altenkirch, Capriotti, Kraus, Talbot and Regnier2016; Annenkov et al. Reference Annenkov, Capriotti, Kraus and Sattler2023; Voevodsky Reference Voevodsky2013) have a sort of fibrant types as well as a sort of types. We extend $\Sigma_{DTT}$ by the symbols

and several type constructors. For readability, we think of $\mathsf{Fib}$ as a subtype of $\mathsf{Type}$ and often omit $\iota$ . A major difference between $\mathsf{Type}$ and $\mathsf{Fib}$ is that identity types $\mathsf{Id_{Type}}(A, a, b)$ in $\mathsf{Type}$ satisfy axiom K or equality reflection but identity types $\mathsf{Id_{Fib}}(A, a, b)$ in $\mathsf{Fib}$ do not. The induction principle for identity types in $\mathsf{Fib}$ only works for families of $\mathsf{Fib}$ :

Example 5.12. To encode propositional logic, we begin with the following signature.

The equation $\mathsf{mono}$ implies that $\mathsf{true}(P)$ has at most one element. One may extend this signature by adding logical connectives like $\top$ , $\bot$ , $\land$ , $\lor$ and $\supset$ . For example, $\land$ and $\lor$ are encoded as follows.

Example 5.13. To encode predicate logic, we extend the union of the signatures for basic dependent type theory and propositional logic by adding equality and quantifiers. For example, equality and $\forall$ are encoded as follows.

In this encoding, a term can depend on the validity of a proposition, allowing us to write a partial function. For example, a term of type $(x : \mathsf{el}(A)) \to \mathsf{true}(P x) \to \mathsf{el}(B)$ is a partial function from A to B defined on those $a : \mathsf{el}(A)$ satisfying P, where $A : \mathsf{Type}$ , $B : \mathsf{Type}$ and $P : \mathsf{el}(A) \to \mathsf{Prop}$ . One may add the following symbols so that $\bot$ becomes the initial object and $P \lor Q$ the pushout of P and Q under $P \land Q$ .

Example 5.14 Cubical type theory (Cohen et al. Reference Cohen, Coquand, Huber, Mörtberg and Uustalu2018) is an extension of dependent type theory with a formal interval and cofibrant predicates. So we extend the signature for basic dependent type theory with

$\mathbb{I}$ carries a de Morgan algebra structure $(0, 1, \sqcap, \sqcup, (-)')$ , and $\mathsf{Cof}$ has logical connectives $\top, \land, \bot, \lor$ and equalities and quantifiers of the form

Moreover, $\mathsf{eq}_{0}$ and $\mathsf{eq}_{1}$ satisfy equality reflection, elimination operators $\mathsf{elim}_{\bot}^{\mathsf{Type}}$ and $\mathsf{elim}_{\lor}^{\mathsf{Type}}$ are added as in Example 5.13, and $\mathsf{Cof}$ satisfies propositional extensionality:

The composition operation is encoded as follows.

Note that the type of $\mathsf{comp}$ is essentially the same as the type of composition structures in the axiomatic approach to the semantics of cubical type theory given by Orton and Pitts (Reference Orton and Pitts2018). The gluing operation is encoded as follows, assuming that $\mathsf{Type}$ has enough type constructors to define the type $\mathsf{Equiv}(A, B) : \mathsf{Type}$ of equivalences between types $A : \mathsf{Type}$ and $B : \mathsf{Type}$ .

5.3 Syntactic representable map categories

A generating representable morphism in $\mathbb{R}(\Sigma)$ is a morphism isomorphic to the projection

$$ (\Gamma, x : A) \to \Gamma $$

with $\Sigma \mid \Gamma \vdash A : {*}$ . A representable morphism in $\mathbb{R}(\Sigma)$ is a composite of generating representable morphisms.

It is well-known that the syntactic category of a type theory with dependent product types and equality types is locally cartesian closed (Seely Reference Seely1984). The same argument shows that $\mathbb{R}(\Sigma)$ is a representable map category.

We call $\mathbb{R}(\Sigma)$ the syntactic representable map category of $\Sigma$ . It is the representable map category “freely generated by $\Sigma$ ” formulated by the following universal property.

Hence, $\mathbb{R}(\Sigma_{DTT})$ has the same universal property as $\mathbb{G}$ (Corollary 4.13), and thus we have an equivalence $\mathbb{R}(\Sigma_{DTT}) \simeq \mathbb{G}$ in the 2-category $\mathfrak{Rep}$ .

The idea of the proof of Theorem 5.17 is the same as in the interpretation of extensional Marin-Löf type theory in locally cartesian closed categories (Seely Reference Seely1984). The main difficulty is the coherence problem: categorical structures often satisfy equations up to isomorphism, but equations in type theory are strict. A solution to the coherence problem is the splitting technique of Hofmann (Reference Hofmann, Pacholski and Tiuryn1995) which replaces a categorical structure by another structure satisfying equations strictly. We adapt the splitting technique for representable map categories.

Sketch of the proof of Theorem 5.17. 1. If $\Sigma = ()$ , then $\mathbb{R}(\Sigma)$ contains only the empty context, which is the terminal object, and thus $\mathfrak{Rep}(\mathbb{R}(\Sigma), \mathcal{C})$ is contractible. If $\Sigma$ is nonempty and unbounded, then the set of objects of $\mathbb{R}(\Sigma)$ is the union of the sets of objects of $\mathbb{R}(\Sigma|_{\alpha})$ indexed over $\alpha \in |\Sigma|$ and, for objects $\Gamma, \Delta \in \mathbb{R}(\Sigma|_{\alpha})$ , the hom-set $\mathbb{R}(\Sigma)(\Gamma, \Delta)$ is the filtered colimit $\textrm{colim}_{\substack{\beta \in |\Sigma| \\ \alpha < \beta}}\mathbb{R}(\Sigma|_{\beta})(\Gamma, \Delta)$ . From this description one can see that the functor $|\mathfrak{Rep}(\mathbb{R}(\Sigma), \mathcal{C})| \to \lim_{\alpha \in |\Sigma|}|\mathfrak{Rep}(\mathbb{R}(\Sigma|_{\alpha}), \mathcal{C})|$ is in fact an isomorphism.

The others are proved using standard techniques of the semantics of dependent type theory. Consider Item 2 which claims that, given a representable map functor $F : \mathbb{R}(\Sigma') \to \mathcal{C}$ and an object $A \in \mathcal{C}/F \Gamma$ , one can extend F to a representable map functor $G : \mathbb{R}(\Sigma) \to \mathcal{C}$ such that $G \alpha \cong A$ and such an extension is unique up to unique isomorphism. To avoid coherence problems, we use Hofmann’s splitting technique (Hofmann Reference Hofmann, Pacholski and Tiuryn1995). Since $\mathcal{C}$ has pullbacks, we have the pseudo-functor $\mathcal{C}/{-} : \mathcal{C}^{\mathrm{op}} \to \mathfrak{Cat}$ . Hofmann’s splitting gives a 2-functor $U : \mathcal{C}^{\mathrm{op}} \to \mathfrak{Cat}$ and a pseudo-natural equivalence $ U \to (\mathcal{C}/{-})$ . We define a 2-functor $R : \mathcal{C}^{\mathrm{op}} \to \mathfrak{Cat}$ by the pullback

We interpret a context $\Sigma \mid \Gamma \vdash \mathsf{ctx}$ as an object $G \Gamma \in \mathcal{C}$ , a type $\Sigma \mid \Gamma \vdash B : {\Box}$ as an object $G B \in U(G \Gamma)$ , a representable type $\Sigma \mid \Gamma \vdash B : {*}$ as an object $G B \in R(G \Gamma)$ and a term $\Sigma \mid \Gamma \vdash b : B$ as a section of $G B \to G \Gamma$ . Hofmann (Reference Hofmann, Pacholski and Tiuryn1995) constructs sufficient structures on U to interpret equality types and dependent product types, when $\mathcal{C}$ is locally cartesian closed. The same construction works for a representable map category $\mathcal{C}$ to interpret dependent product types over representable types in U and R. To interpret a symbol from $\Sigma'$ , apply F and choose a corresponding element of U via the pseudo-natural equivalence $U \simeq (\mathcal{C}/{-})$ . To interpret $\alpha$ , choose an element of $U(F \Gamma)$ corresponding to $A \in \mathcal{C} / F \Gamma$ . This completes the interpretation of the logical framework in U. Applying the pseudo-natural equivalence $U \simeq (\mathcal{C}/{-})$ , we have a representable map functor $G : \mathbb{R}(\Sigma) \to \mathcal{C}$ . All the choices made in this construction are unique up to unique isomorphism, so G is unique up to unique isomorphism.

5.4 Models of syntactic representable map categories

Given a signature $\Sigma$ , we concretely describe the 2-category $\mathfrak{Mod}_{\mathbb{R}(\Sigma)}$ (Theorem 5.19).

Suppose $\Sigma$ is a signature of the form $\Sigma = (\Sigma', \alpha : \Gamma \Rightarrow {\Box})$ . We define a 2-category $(\mathbf{DFib} \downarrow \Gamma^{(-)})$ as follows:

• the objects are the pairs $(\mathcal{S}, \alpha^{\mathcal{S}})$ consisting of $\mathcal{S} \in \mathfrak{Mod}_{\mathbb{R}(\Sigma')}$ and $\alpha^{\mathcal{S}} \in \mathbf{DFib}_{\mathcal{S}}/\Gamma^{\mathcal{S}}$ ;

• the morphisms $(\mathcal{S}, \alpha^{\mathcal{S}}) \to (\mathcal{T}, \alpha^{\mathcal{T}})$ are the pairs $(F, F_{\alpha})$ consisting of a morphism $F : \mathcal{S} \to \mathcal{T}$ of models of $\mathbb{R}(\Sigma')$ and a map $F_{\alpha} : \alpha^{\mathcal{S}} \to \alpha^{\mathcal{T}}$ of discrete fibrations over $F : \mathcal{S} \to \mathcal{T}$ such that the diagram

  (2)

  
  commutes;

• the 2-morphisms $(F, F_{\alpha}) \Rightarrow (G, G_{\alpha}) : (\mathcal{S}, \alpha^{\mathcal{S}}) \to (\mathcal{T}, \alpha^{\mathcal{T}})$ are the 2-morphisms $\sigma : F \Rightarrow G$ in $\mathfrak{Mod}_{\mathbb{R}(\Sigma')}$ such that there exists a (necessarily unique) natural transformation $\sigma_{\alpha} : F_{\alpha} \Rightarrow G_{\alpha}$ over $\sigma$ .

There is the obvious 2-functor $\mathfrak{Mod}_{\mathbb{R}(\Sigma)} \to (\mathbf{DFib}\downarrow \Gamma^{(-)})$ .

When $\Sigma = (\Sigma', \alpha : \Gamma \Rightarrow {*})$ , the 2-functor $\mathfrak{Mod}_{\mathbb{R}(\Sigma)} \to (\mathbf{DFib} \downarrow \Gamma^{(-)})$ factors through the locally full sub-2-category $(\mathbf{DFib} \downarrow \Gamma^{(-)})_{\mathrm{r}} \subset (\mathbf{DFib} \downarrow\Gamma^{(-)})$ consisting of those objects $(\mathcal{S}, \alpha^{\mathcal{S}})$ such that $\alpha^{\mathcal{S}} \to \Gamma^{\mathcal{S}}$ is a representable map of discrete fibrations over $\mathcal{S}$ and those morphisms $(F, F_{\alpha}) : (\mathcal{S}, \alpha^{\mathcal{S}}) \to (\mathcal{T},\alpha^{\mathcal{T}})$ such that Diagram 2 satisfies the Beck–Chevalley condition.

Suppose $\Sigma = (\Sigma', \alpha : \Gamma \Rightarrow A)$ with $\Sigma'\mid \Gamma \vdash A : {\Box}$ . We define a 2-category $\mathrm{Sect}(A^{(-)})$ as follows:

• the objects are the pairs $(\mathcal{S}, \alpha^{\mathcal{S}})$ consisting of a model $\mathcal{S}$ of $\mathbb{R}(\Sigma')$ and a section $\alpha^{\mathcal{S}}$ of the map $A^{\mathcal{S}} \to \Gamma^{\mathcal{S}}$ of discrete fibrations over $\mathcal{S}$ ;

• the morphisms $(\mathcal{S}, \alpha^{\mathcal{S}}) \to (\mathcal{T}, \alpha^{\mathcal{T}})$ are the morphisms $F : \mathcal{S} \to \mathcal{T}$ of models of $\mathbb{R}(\Sigma')$ such that the diagram

  
  commutes;

• the 2-morphisms $F \Rightarrow G$ are the 2-morphisms of models of $\mathbb{R}(\Sigma')$ .

There is the obvious 2-functor $\mathfrak{Mod}_{\mathbb{R}(\Sigma')} \to\mathrm{Sect}(A^{(-)})$ .

Sketch of the proof. We use the fact that (2-)morphisms of models of a type theory are also representable map functors (Section 4.3). Then we can apply Theorem 5.17 for building not only models of $\mathbb{R}(\Sigma)$ but also (2-)morphisms of models of $\mathbb{R}(\Sigma)$ .

Note that the 2-functors in the statement are locally faithful. It remains to show that those 2-functors are bi-essentially surjective on objects, locally essentially surjective on objects and locally full. We only demonstrate the statement 3. The others can be proved using the same idea.

To show that the 2-functor $\mathfrak{Mod}_{\mathbb{R}(\Sigma)} \to (\mathbf{DFib} \downarrow \Gamma^{(-)})$ is bi-essentially surjective on objects, let $\mathcal{S}$ be a model of $\mathbb{R}(\Sigma')$ and $p : A \to \Gamma^{\mathcal{S}}$ a map of small discrete fibrations over $\mathcal{S}$ . By Theorem 5.17, the representable map functor $(-)^{\mathcal{S}} : \mathbb{R}(\Sigma') \to \mathbf{DFib}_{\mathcal{S}}$ extends to a representable map functor $(-)^{\widetilde{\mathcal{S}}} : \mathbb{R}(\Sigma) \to \mathbf{DFib}_{\mathcal{S}}$ such that $\alpha^{\widetilde{\mathcal{S}}} \cong A$ . This defines a model $\widetilde{\mathcal{S}}$ of $\mathbb{R}(\Sigma)$ such that the restriction of $\widetilde{\mathcal{S}}$ to $\mathbb{R}(\Sigma')$ is isomorphic to $\mathcal{S}$ and $\alpha^{\widetilde{\mathcal{S}}} \cong A$ .

To show that the 2-functor $\mathfrak{Mod}_{\mathbb{R}(\Sigma)} \to (\mathbf{DFib} \downarrow \Gamma^{(-)})$ is locally essentially surjective on objects, let $\mathcal{S}, \mathcal{T}$ be models of $\mathbb{R}(\Sigma)$ , $F : \mathcal{S} \to \mathcal{T}$ a morphism of models of $\mathbb{R}(\Sigma')$ , and $G : \alpha^{\mathcal{S}} \to \alpha^{\mathcal{T}}$ a map of discrete fibrations such that the diagram

commutes. Then $F_{(-)}$ is a representable map functor $\mathbb{R}(\Sigma') \to (\mathbf{DFib}^{\to})_{F}$ by Proposition 4.20 and G is an object of $(\mathbf{DFib}^{\to})_{F}/F_{\Gamma}$ . By Theorem5.17, the representable map functor $F_{(-)} : \mathbb{R}(\Sigma') \to (\mathbf{DFib}^{\to})_{F}$ extends to a representable map functor $\widetilde{F}_{(-)} : \mathbb{R}(\Sigma) \to (\mathbf{DFib}^{\to})_{F}$ such that $\widetilde{F}_{\alpha} \cong G$ . From $\mathbf{dom} \widetilde{F}_{\alpha} \cong \alpha^{\mathcal{S}}$ and $\mathbf{cod} \widetilde{F}_{\alpha} \cong \alpha^{\mathcal{T}}$ , we have $\mathbf{dom} \widetilde{F}_{({-})} \cong ({-})^{\mathcal{S}}$ and $\mathbf{cod} \widetilde{F}_{({-})} \cong ({-})^{\mathcal{T}}$ again by Theorem 5.17. By Proposition 4.20 and Remark 4.21, $\widetilde{F}_{(-)}$ defines a morphism $\mathcal{S} \to \mathcal{T}$ of models of $\mathbb{R}(\Sigma)$ such that the restriction of $\widetilde{F}_{(-)}$ to $\mathbb{R}(\Sigma')$ is F and $\widetilde{F}_{\alpha} \cong G$ .

The local fullness is similarly proved using the representable map category $(\mathbf{DFib}^{\Theta})_{\sigma}$ instead of $(\mathbf{DFib}^{\to})_{F}$ and Proposition 4.22.

Our choice of 2-morphisms of categories with families is quite natural, but there is another choice: the indexed natural transformations between the associated indexed categories (Clairambault and Dybjer Reference Clairambault and Dybjer2014). A difference is that a collection of types is regarded as a set in our definition while it is regarded as a category in the other definition.

Example 5.21. We describe core components of a model $\mathcal{S}$ of cubical type theory (Example 5.14). In addition to the natural model structure $(\mathcal{S}, \mathsf{Type}^{\mathcal{S}}, \mathsf{el}^{\mathcal{S}})$ , it is equipped with a discrete fibration $\mathbb{I}^{\mathcal{S}}$ over $\mathcal{S}$ such that the map $\mathbb{I}^{\mathcal{S}} \to \mathcal{S}$ is representable and a representable map $\mathsf{true}^{\mathcal{S}} \to \mathsf{Cof}^{\mathcal{S}}$ of discrete fibrations over $\mathcal{S}$ that is also a monomorphism.

By Remark 3.10, $\mathbb{I}^{\mathcal{S}}$ is regarded as an object of the category $\mathcal{S}$ with which $\mathcal{S}$ has products. The cubical set model of cubical type theory (Cohen et al. Reference Cohen, Coquand, Huber, Mörtberg and Uustalu2018, Section 8) is turned into this structure. $\mathcal{S}$ is the category of cubical sets: $\mathsf{Type}^{\mathcal{S}}$ is the discrete fibration of fibrant families of cubical sets; $\mathsf{el}^{\mathcal{S}}$ is the discrete fibration of sections of fibrant families of cubical sets; $\mathbb{I}^{\mathcal{S}}$ is the formal interval; in this example, $\mathsf{Cof}^{\mathcal{S}}$ is also representable and constructed from what is called the face lattice; there is a map $\mathsf{Cof}^{\mathcal{S}} \to \Omega$ to the subobject classifier $\Omega$ in $\mathcal{S}$ , and the map $\mathsf{true}^{\mathcal{S}} \to \mathsf{Cof}^{\mathcal{S}}$ is the monomorphism classified by it.

6.1 Democratic models

Usually a bi-initial model of a type theory is a syntactic one and has a special property: every object is represented by a finite sequence of types. We introduce a class of models of a type theory satisfying this property, generalizing the notion of a democratic category with families (Clairambault and Dybjer Reference Clairambault and Dybjer2014).

Note that the terminal object 1 and the context extension $\{b\}^{f}$ are determined only up to unique isomorphism. We include all the terminal objects and all the context extensions in the class of contextual objects so that the class of contextual objects is closed under isomorphisms. We say $\mathcal{S}$ is democratic if every object of $\mathcal{S}$ is contextual and denote by $\mathfrak{Mod}_{\mathbb{T}}^{\mathrm{dem}}$ the full sub-2-category of $\mathfrak{Mod}_{\mathbb{T}}$ consisting of the democratic models.

Democratic models have the following interesting property.

Consequently, $\mathfrak{Mod}_{\mathbb{T}}(\mathcal{S}, \mathcal{T})$ is equivalent to a discrete category.

An easy way to construct a democratic model is to throw away non-contextual objects from an arbitrary model.

$\mathcal{S}^{\heartsuit}$ is indeed a model of $\mathbb{T}$ because $\mathcal{S}^{\heartsuit}$ is closed under context extensions. We call $\mathcal{S}^{\heartsuit}$ the heart of $\mathcal{S}$ .

Let $\mathcal{S}$ be a model of a type theory $\mathbb{T}$ . By definition $\mathcal{S}^{\heartsuit}$ is a democratic model of $\mathbb{T}$ and the obvious inclusion $\mathcal{S}^{\heartsuit} \to \mathcal{S}$ is a morphism of models of $\mathbb{T}$ . The heart $\mathcal{S}^{\heartsuit}$ is the largest democratic model contained in $\mathcal{S}$ in the following sense.

Proposition 6.5. Let $\mathcal{S}$ be a model of a type theory $\mathbb{T}$ . The inclusion $\mathcal{S}^{\heartsuit} \to \mathcal{S}$ induces an isomorphism of categories

$$ \mathfrak{Mod}_{\mathbb{T}}(\mathcal{T}, \mathcal{S}^{\heartsuit}) \cong \mathfrak{Mod}_{\mathbb{T}}(\mathcal{T}, \mathcal{S}) $$

for any democratic model $\mathcal{T}$ of $\mathbb{T}$ .

6.2 The bi-initial model of a type theory

The bi-initial model of a type theory $\mathbb{T}$ is obtained from the Yoneda embedding $\mathbb{T} \to \mathbf{DFib}_{\mathbb{T}}$ .

Proof. The first two claims are obvious. To prove the third, let $u : I \to J$ and $v : J \to K$ be arrows in $\mathcal{S}$ and suppose that the pushforward $v_{*}I \in \mathcal{S}/K$ exists. By definition, for any arrow $w : L \to K$ in $\mathcal{S}$ , we have a bijection $\mathcal{S}/K(L, v_{*}I) \cong \mathcal{S}/J(v^{*}L, I)$ . This means that we have a pullback

and thus $\mathcal{S}/v_{*}I$ is the pushforward of $u : \mathcal{S}/I \to \mathcal{S}/J$ along $v : \mathcal{S}/J \to \mathcal{S}/K$ by Proposition 3.21.

We describe the model $\mathcal{I}(\mathbb{T})$ in more detail. For an object $I \in \mathbb{T}$ , a representable arrow $f : A \to B$ in $\mathbb{T}$ and an object $(b : I \to B) \in \mathbb{T}/B$ , the context extension $\{b\}^{f}$ in the model $(\mathbb{T}, \mathbb{T}/({-}))$ is the pullback in $\mathbb{T}$

Thus, the base category $\mathcal{I}(\mathbb{T})$ is the full subcategory of $\mathbb{T}$ consisting of those objects $A \in \mathbb{T}$ such that the unique arrow $A \to 1$ is representable. For an object $A \in \mathbb{T}$ , the discrete fibration $A^{\mathcal{I}(\mathbb{T})}$ is the comma category $\mathcal{I}(\mathbb{T})/A$ defined by the pullback

When we think of U as the object of types and E as the object of elements, an object in $\mathcal{I}(\mathbb{T})$ is a finite sequence of types, that is, a context. For an object $I \in \mathcal{I}(\mathbb{T})$ , elements of $U^{\mathcal{I}(\mathbb{T})}(I)$ and $E^{\mathcal{I}(\mathbb{T})}(I)$ are types and elements, respectively, indexed over the context I. Hence, the bi-initial model $\mathcal{I}(\mathbb{T})$ generalizes the usual syntactic models of dependent type theories.

We first show the existence of a morphism $\mathcal{I}(\mathbb{T}) \to \mathcal{S}$ . For every object $I \in \mathcal{I}(\mathbb{T})$ , the unique map $I^{\mathcal{S}} \to \mathcal{S}$ of discrete fibrations over $\mathcal{S}$ is representable. In particular, the discrete fibration $I^{\mathcal{S}} \in \mathbf{DFib}_{\mathcal{S}}$ is representable because the category $\mathcal{S}$ has a terminal object. Hence, the restriction of $({-})^{\mathcal{S}} : \mathbb{T} \to \mathbf{DFib}_{\mathcal{S}}$ to $\mathcal{I}(\mathbb{T})$ factors, up to natural isomorphism, as a functor $F : \mathcal{I}(\mathbb{T}) \to \mathcal{S}$ followed by the Yoneda embedding $\mathcal{S} \to \mathbf{DFib}_{\mathcal{S}}$ . For objects $A \in \mathbb{T}$ and $I \in \mathcal{I}(\mathbb{T})$ and an arrow $a : I \to A$ , we define $F_{A}(a) \in A^{\mathcal{S}}(FI)$ to be $\mathcal{S}/FI \cong I^{\mathcal{S}} \overset{a^{\mathcal{S}}}{\longrightarrow} A^{\mathcal{S}}$ , yielding a map $F_{A} : \mathcal{I}(\mathbb{T})/A \to A^{\mathcal{S}}$ of discrete fibrations over $F : \mathcal{I}(\mathbb{T}) \to \mathcal{S}$ .

Clearly, $F : \mathcal{I}(\mathbb{T}) \to \mathcal{S}$ preserves terminal objects and $A \mapsto F_{A}$ is natural. To see the Beck–Chevalley condition, let $f : A \to B$ be a representable arrow in $\mathbb{T}$ . We have to show that the diagram

satisfies the Beck–Chevalley condition. It suffices to show that the composite of squares

(3)

satisfies the Beck–Chevalley condition for all objects $(b : I \to B) \in \mathcal{I}(\mathbb{T})/B$ . By the definition of $F_{({-})}$ , Diagram 3 is equal to the following composite of squares.

(4)

To show that Diagram 4 satisfies the Beck–Chevalley condition for all $(b : I \to B) \in \mathcal{I}(\mathbb{T})/B$ , it suffices to check that the canonical natural transformation $(f^{*}b)^{\mathcal{S}}F(b^{*}f)^{*} \Rightarrow \delta^{f}b^{\mathcal{S}}F$ induced by Diagram 4 is invertible at $\mathrm{id}_{I} \in \mathcal{I}(\mathbb{T})/I$ for all $(b : I \to B) \in \mathcal{I}(\mathbb{T})/B$ . This is straightforward because the right-most square of Diagram 4 is a pullback in $\mathbf{DFib}_{\mathcal{S}}$ and thus satisfies the Beck–Chevalley condition by Corollary 3.15.

To show the uniqueness of morphisms $\mathcal{I}(\mathbb{T}) \to \mathcal{S}$ , let $G : \mathcal{I}(\mathbb{T}) \to \mathcal{S}$ be another morphism of models of $\mathbb{T}$ . By Proposition 6.3, it suffices to show that there exists a 2-morphism $F \Rightarrow G$ . Let $I \in \mathcal{I}(\mathbb{T})$ be an object and let $u : I \to 1$ denote the unique arrow to the terminal object. By the Beck–Chevalley condition for the square

we have the natural isomorphism $G_{I} u^{*} \cong \delta^{u} G : \mathcal{I}(\mathbb{T}) \to I^{\mathcal{S}}$ . Since $\delta^{u} G$ preserves terminal objects, $G_{I}(\mathrm{id}_{I}) \cong G_{I}(u^{*} 1)$ is the terminal object. Thus, by Proposition 3.5, we have $\mathcal{S}/GI \cong I^{\mathcal{S}}$ . For an arrow $a : I \to A$ in $\mathbb{T}$ with $I \in \mathcal{I}(\mathbb{T})$ , the diagram

commutes. This means that $G_{A}(a) \in A^{\mathcal{S}}(GI)$ is given by the composite $\mathcal{S}/GI \cong I^{\mathcal{S}} \overset{a^{\mathcal{S}}}{\longrightarrow} A^{\mathcal{S}}$ . Hence, G has the same definition as F, and thus $F \cong G$ .