2.1 Review

We mainly refer to Annenkov et al. (Reference Annenkov, Capriotti, Kraus and Sattler2023) about 2LTT. Basically, we have two different objects in the theory: types in the HoTT level and exo-types in the meta-level. We have similar types of former constructions at both levels. We always make the distinction between types and exo-types using the superscript $-^e$ . According to this, the main types we care about are $\prod$ , $\sum$ , $+$ , $\times$ , $\mathbf{1}$ , $\mathbf{0}$ , $\mathbb{N}$ , and $\mathbb{N}_{\lt n}$ (finite type of order $n$ ), $\mathcal{U}$ (universe), and the identity type $=$ . Their “exo” versions are denoted by $\prod ^e$ , $\sum ^e$ , $+^e$ , $\times ^e$ , ${\mathbf{1}}^e$ , ${\mathbf{0}}^e$ , $\mathbb{N}^e$ , and $\mathbb{N}^e_{\lt n}$ (exo-finite exo-type of order $n$ ), $\mathcal{U}^e$ (exo-universe), and the exo-equality $\mathsf{=}^{\textit {e}}$ . One can also see Uskuplu (Reference Uskuplu2023) for the detailed definitions. Also, we omit most of the proofs in this section; one can read them in Uskuplu (Reference Uskuplu2023).

Note that these type/exo-type pairs may not coincide in the cases of $+^e$ , $\mathbb{N}^e$ , ${\mathbf{0}}^e$ , and $\mathsf{=}^{\textit {e}}$ . For example, even if $A,B:\mathcal{U}$ , we may not have $A +^e B : \mathcal{U}$ ; namely, this is always an exo-type but not generally a type. The difference in these cases is that the elimination/induction rules of fibrant types cannot be used unless the target is fibrant. For example, we can define functions into any exo-type by recursion on $\mathbb{N}^e$ , but if we want to define a function $f :\mathbb{N} \rightarrow A$ by recursion, we must have $A : \mathcal{U}$ .

Considering these twin definitions, it’s natural to ask whether there is a correspondence between them. We obtain such a correspondence according to the relation between types and exo-types. Annenkov et al. (Reference Annenkov, Capriotti, Kraus and Sattler2023), it is assumed that there is a coercion map $c$ from types to exo-types; for any type $A : \mathcal{U}$ , we have $c(A) : \mathcal{U}^e$ . Another approach, as in Ahrens et al. (Reference Ahrens, North, Shulman and Tsementzis2025), is taking $c$ as an inclusion; in other words, assuming every type is an exo-type. In this work, the second approach is assumed. Therefore, we can apply exo-type formers to types. For example, both $\mathbb{N} + \mathbb{N}$ and $\mathbb{N} +^e \mathbb{N}$ make sense, but both are still exo-types. We will later prove some isomorphisms related to such correspondences.

We define the exo-type of exo-isomorphisms between $A,B : \mathcal{U}^e$ (or briefly isomorphisms) as

\begin{equation*}A \cong B \;:=\; {\sum }^e_{f:A\rightarrow B}{\sum }^e_{g:B\rightarrow A}(f {{\circ }^{\textit {e}}} g {\mathsf{=}^{\textit {e}}} \mathsf{id}_B) \times ^e(g {{\circ }^{\textit {e}}} f {\mathsf{=}^{\textit {e}}} \mathsf{id}_A).\end{equation*}

We define the type of equivalences between $A,B : \mathcal{U}$ as

\begin{equation*}A \simeq B \;:=\; \sum _{f:A\rightarrow B} \left ( \prod _{b:B} \texttt {is-Contr}\left (\sum _{a:A} f(a)=b\right )\right ).\end{equation*}

Also, $f$ function $f : A \rightarrow B$ between types is called quasi-invertible if there is a function $g : B \rightarrow A$ such that $g \circ f = \mathsf{id}_A$ and $f \circ g = \mathsf{id}_B$ where $\mathsf{id}_A : A \rightarrow A$ is the identity map.

Assuming that each type is an exo-type and considering all definitions so far, the correspondence between exo-type formers and type formers can be characterizedFootnote ² as follows.

Remark 2.5. It is worth emphasizing that the inverses of the maps iv, v, and vi can be assumed to exist. There are some models where these hold (for the details, see the discussion below Lemma 2.11 (Annenkov et al., Reference Annenkov, Capriotti, Kraus and Sattler2023)). However, a possible inverse for the map vii would yield a contradiction because the univalence axiom is inconsistent with the uniqueness of identity proofs. This conversion from the exo-equality ( $\mathsf{=}^{\textit {e}}$ ) to the identity ( $=$ ) still has an importance in many proofs later. Thus we denote it by

\begin{equation*}{\mathsf{eqtoid}} : {\prod _{a,b:A}}^e(a{\mathsf{=}^{\textit {e}}} b) \rightarrow (a = b) \,.\end{equation*}

One of its valuable corollaries is the following lemma.

In terms of the relationship between types and exo-types, there are three classes of exo-types. The first one pertains to the notion of fibrant exo-types.

Definition 2.7. An exo-type $A : \mathcal{U}^e$ is called a fibrant exo-type if there is a type $RA : \mathcal{U}$ such that $A$ and $RA$ are exo-isomorphic. In other words, $A$ is fibrant when the following exo-type is inhabited

\begin{equation*}{\mathsf{isFibrant}}(A)\;:=\;{\sum _{RA : \mathcal{U}}}^e (A \cong RA)\,.\end{equation*}

Definition 2.9. Let $A,B: \mathcal{U}^e$ be two fibrant exo-types,and $f:A\rightarrow B$ . Let $RA,RB:\mathcal{U}$ be such that $A\cong RA$ and $B\cong RB$ . We have the following diagram.

We call $f$ a fibrant-equivalence if

\begin{equation*}r_B{{\circ }^{\textit {e}}} f {{\circ }^{\textit {e}}} s_A : RA \rightarrow RB\end{equation*}

is an equivalence.

Proposition 2.10. Let $A,B,C: \mathcal{U}^e$ be fibrant exo-types and $f:A\rightarrow B$ and $g:B \rightarrow C$ . Consider the corresponding diagram.

The following are true:

1. i. If $f$ is an exo-isomorphism, then it is a fibrant-equivalence.

2. ii. If $f,g$ are fibrant-equivalences, then so is $g{{\circ }^{\textit {e}}} f$ .

3. iii. If $f,f'$ are homotopic with respect to $\mathsf{=}^{\textit {e}}$ , and $f$ is a fibrant-equivalence, then $f'$ is a fibrant-equivalence, too.

The second class of exo-types is the cofibrant exo-types.

The following gives a logically equivalent definition of cofibrancy, which is a novel result. Attention should be paid to the use of $=$ and $\mathsf{=}^{\textit {e}}$ in order to indicate whether the terms belong to the type or the exo-type.

Proof. (i $\Rightarrow$ ii) Let $f,g : \prod ^e_{a:A} Y(a)$ be such that $t_a : f(a)=g(a)$ for each $a:A$ . Consider another type family $Y' : A \rightarrow \mathcal{U}$ defined as

\begin{equation*}Y'(a)\;:=\;\sum _{b:B(a)} b=f(a).\end{equation*}

Then both $f'\;:=\;\lambda a.(f(a),{\mathsf{refl}})$ and $g'\;:=\;\lambda a. (g(a),t_a^{-1})$ are terms in $\prod ^e_{a:A}Y'(a)$ . By our assumptions, there is a $FM':\mathcal{U}$ such that

Since the type of paths at a point is contractible, we have each $Y'(a)$ is contractible. By the assumption (i), we get $FM'$ is contractible, and hence

(1) \begin{equation} r'(f)=r'(g). \end{equation}

Using this, we have the following chain of identities:

\begin{equation*} r(f)=r(\pi _1(s'(r'(f'))))=r(\pi _1(s'(r'(g'))))=r(g). \end{equation*}

The first (and the third, by symmetry) identity is obtained as follows: Because $r'$ and $s'$ are exo-inverses of each other, we have

\begin{equation*}(a:A)\rightarrow \pi _1(f'(a)){\mathsf{=}^{\textit {e}}} \pi _1 ((s'(r'(f'))) (a)).\end{equation*}

Thus, by ${\mathsf{funext}}^e$ , we get $\pi _1(f'){\mathsf{=}^{\textit {e}}} \pi _1 ((s'(r'(f'))))$ . Then we applyFootnote ³ $r$ to the equality and make it an identity via $\mathsf{eqtoid}$ because the terms are in $FM'$ which is a type.

The second identity is obtained by applying the function

\begin{equation*}\lambda x. r (\lambda a. \pi _1((s' x) (a))) : FM' \rightarrow FM\end{equation*}

to the identity 1, so we are done.

Note that even if $r$ has an exo-type domain and $s'$ has an exo-type codomain, we can compose these in a way that the resulting map is from a type to a type. Thus, we can apply it to an identity.

(ii $\Rightarrow$ i) Suppose $Y(a)$ is contractible for each $a:A$ . We want to show that $FM$ is contractible where

Let $b_a :Y(a)$ be the center of contraction for each $a:A$ . Then this gives a function $f\;:=\;\lambda a. b_a :\prod ^e_{a:A} Y(a)$ . For any $x:FM$ , since $f(a)=b_a=s(x)(a)$ for each $a:A$ by contractibility, using the assumption (ii), we get $r(f)=r(s(x))$ . Applying $\mathsf{eqtoid}$ to the exo-equality $r(s(x)){\mathsf{=}^{\textit {e}}} x$ , we get $r(f)=x$ . Therefore, $FM$ is contractible with the center of contraction $r(f)$ .

Cofibrant exo-types have the following properties.

Another class of exo-types is the class of sharp ones. This was given in Ahrens et al. (Reference Ahrens, North, Shulman and Tsementzis2025) for the first time.

Definition 2.14 (Definition 2.2 in Ahrens et al. (Reference Ahrens, North, Shulman and Tsementzis2025)). An exo-type $A$ is sharp if it is cofibrant and it has a “fibrant replacement”, meaning that there is a fibrant type $RA$ and a map $r : A \rightarrow RA$ such that for any family of types $Y : RA \rightarrow \mathcal{U}$ , the precomposition map below is a fibrant-equivalence (recall Definition 2.9).

(2) \begin{equation} (\!- {{\circ }^{\textit {e}}} r):\prod _{c:RA} Y(c) \rightarrow {\prod _{a:A}}^e Y(r(a)) \end{equation}

The following lemma gives another definition for sharp exo-types. First, we need an auxiliary definition.

Definition 2.15. Let $A,B: \mathcal{U}^e$ be two fibrant exo-types and $f:A\rightarrow B$ be a map. Let $RA,RB:\mathcal{U}$ be such that $A\cong RA$ and $B\cong RB$ . Take $s_A:RA \rightarrow A$ and $r_B:B \rightarrow RB$ as these isomorphisms. Then $f$ has a fibrant-section if

\begin{equation*}r_B{{\circ }^{\textit {e}}} f {{\circ }^{\textit {e}}} s_A : RA \rightarrow RB\end{equation*}

has a section.

As in the cofibrant exo-types, the notion of sharpness has its own preservation rules. The following proposition gives these rules.

2.2 Lifting cofibrancy from exo-nat to other types

In this section, we give a new result about other inductive types. After this paper was written, the referee helpfully observed that the two examples treated below (List and BinaryTree) can be expressed as $F(A)$ where $A$ is (sharp) cofibrant and $F$ is a container, equivalently, a polynomial functor whose shape exo-type is countable and each position fiber is finite (Abbott et al., Reference Abbott, Altenkirch and Ghani2005; Gambino and Kock, Reference Gambino and Kock2013). Although our original proofs establish (sharpness) cofibrancy directly and were found independently of the container formalism, this perspective is indeed worth recording: the closureFootnote ⁴ of (sharp) cofibrant exo-types under countable coproducts and finite products implies that any such container application preserves cofibrancy. The results below are therefore special cases of this more general closure principle.

2.2.1 List (exo)types

Definition 2.18. For an exo-type $A:\mathcal{U}^e$ , we define the exo-type ${\mathsf{List}}^e (A) : \mathcal{U}^e$ of finite exo-lists of terms of $A$ , which has constructors

• • ${\mathsf{[]}}^e : {\mathsf{List}}^e (A)$

• • $\mathsf{::}^e : A \rightarrow {\mathsf{List}}^e (A) \rightarrow {\mathsf{List}}^e (A)$

Similarly, if $A : \mathcal{U}$ is a type, the type ${\mathsf{List}}(A)$ of finite lists of $A$ has constructors $\mathsf{[]}$ and $\mathsf{::}\,$ .

As in Theorem2.4, we have an obvious map $f : {\mathsf{List}}^e (A) \rightarrow {\mathsf{List}} (A)$ for a type $A$ defined as $f({\mathsf{[]}}^e)\;:=\;{\mathsf{[]}}$ and $f(a\,\mathsf{::}^e\, l)\;:=\;a\, \mathsf{::}\, f(l)$ . We will give some conditions for cofibrancy and sharpness of ${\mathsf{List}}^e (A)$ . Indeed, if we assume $\mathbb{N}^e$ and $A$ are cofibrant, then we can show ${\mathsf{List}}^e (A)$ is cofibrant. The proof is obtained by an isomorphism between a cofibrant exo-type and ${\mathsf{List}}^e(A)$ . The same isomorphism shows that ${\mathsf{List}}^e (A)$ is sharp if $A$ is also sharp. Moreover, we can show the sharpness of ${\mathsf{List}}^e(A)$ in a way analogous to Proposition 2.17(vi).

Lemma 2.19. Let $A : \mathcal{U}^e$ be an exo-type. Then we have

\begin{equation*}\left ( {\sum _{n:\mathbb{N}^e}}^e A^n \right ) \cong {\mathsf{List}}^e(A)\end{equation*}

where $A^{{\mathsf{0}}^e}\;:=\;{\mathbf{1}}^e$ and $A^{({\mathsf{succ}}^e n)} \;:=\; A \times ^e A^n$ .

Proof. Define $\phi : \left (\sum ^e_{n:\mathbb{N}^e} A^n\right ) \rightarrow {\mathsf{List}}^e(A)$ as follows

\begin{eqnarray*} \phi ({\mathsf{0}}^e, \star ^e)&\;:=\;& {\mathsf{[]}}^e\\ \phi ({\mathsf{succ}}^e(n), (a,p))&\;:=\;& a\, \mathsf{::}^e\, \phi (n,p). \end{eqnarray*}

Define $\theta : {\mathsf{List}}^e(A) \rightarrow \left (\sum ^e_{n:\mathbb{N}^e} A^n\right )$ as follows,

\begin{eqnarray*} \theta ({\mathsf{[]}}^e)&\;:=\;& ({\mathsf{0}}^e, \star ^e)\\ \theta (a \,\mathsf{::}^e\, l)&\;:=\;&({\mathsf{succ}}^e(\pi ^e_1\,\theta (l))\,,(a,\pi ^e_2\,\theta (l))). \end{eqnarray*}

Then, it is easy to show by the induction on the constructors that

\begin{equation*}\phi {{\circ }^{\textit {e}}} \theta {\mathsf{=}^{\textit {e}}} \mathsf{id}_{{\mathsf{List}}^e(A)}\quad \text{and}\quad \theta {{\circ }^{\textit {e}}} \phi {\mathsf{=}^{\textit {e}}} \mathsf{id}_{\sum ^e_{n:\mathbb{N}^e} A^n}.\end{equation*}

Proposition 2.20. If $A:\mathcal{U}^e$ is cofibrant and $\mathbb{N}^e$ is cofibrant, then so is ${\mathsf{List}}^e(A)$ .

Proof. By Lemma 2.19 and Proposition 2.13(ii), it is enough to show that $\sum ^e_{n:\mathbb{N}^e} A^n$ is cofibrant. By the assumption and 2.13(iii), we have $A^n$ is a cofibrant exo-type for all $n:\mathbb{N}^e$ . Since we also assume $\mathbb{N}^e$ is cofibrant, we are done by 2.13(iv).

By similar reasoning, we obtain that ${\mathsf{List}}^e(A)$ is sharp if $A$ is sharp and $\mathbb{N}^e$ is cofibrant. However, we can also show the sharpness of ${\mathsf{List}}^e(A)$ , like in Proposition 2.17(vi).

Proposition 2.21. Let $A:\mathcal{U}^e$ be a sharp exo-type. Suppose also that $\mathbb{N}^e$ is cofibrant. Then ${\mathsf{List}}^e(A)$ is sharp.

Proof. Since $A$ is sharp, it is cofibrant. By Proposition 2.20, we have ${\mathsf{List}}^e(A)$ is cofibrant, so it remains to find the fibrant replacement of it.

Since $A$ is sharp, we have a type $RA:\mathcal{U}$ and a map $r_A:A\rightarrow RA$ such that the map 2 is a fibrant-equivalence for any $Y:RA \rightarrow \mathcal{U}$ . We claim that ${\mathsf{List}} (RA)$ is a fibrant replacement of ${\mathsf{List}}^e(A)$ . Define $r:{\mathsf{List}}^e(A)\rightarrow {\mathsf{List}} (RA)$ as

\begin{eqnarray*} r({\mathsf{[]}}^e)&\;:=\;&{\mathsf{[]}}\\ r(a\, \mathsf{::}^e\, l)&\;:=\;&r_A(a) \,\mathsf{::}\, r(l) \end{eqnarray*}

Consider the following commutative diagram for any $Y: {\mathsf{List}}(RA) \rightarrow \mathcal{U}$

The type $FM_Y$ and the isomorphism $u_Y$ are obtained by the cofibrancy of ${\mathsf{List}}^e(A)$ . We want to show that $\alpha _Y$ is an equivalence. First, we define an auxiliary type

\begin{equation*}S : \prod _{t:{\mathsf{List}}(RA)} \left (\prod _{Y:{\mathsf{List}}(RA)\rightarrow \mathcal{U}}\left ( \prod _{x:FM_Y} Y(t)\right )\right ) \end{equation*}

by $S({\mathsf{[]}},Y,x)\;:=\;v_Y(x)({\mathsf{[]}}^e)$ and $S(c \,\mathsf{::}\, l,Y,x)\;:=\;S(l,Y',x')$ where

\begin{eqnarray*} Y'(l)&\;:=\;&Y(c \,\mathsf{::}\, l),\\ x'&\;:=\;& u_{Y'}(T) \end{eqnarray*}

for a $T:\prod ^e_{s:{\mathsf{List}}^e(A)}Y'(r(s))$ defined as follows. For $s:{\mathsf{List}}^e(A)$ , consider the following diagram

The equivalence $\alpha$ is obtained by the sharpness of $A$ . Now we define

\begin{equation*}T(s)\;:=\;\beta (u(\lambda \, a. v_Y(x)(a \,\mathsf{::}^e\, s)))\, (c)\; : Y(c \,\mathsf{::}\, r(s))=Y'(r(s)).\end{equation*}

We also claim that for any $s:{\mathsf{List}}^e(A)$ , $Y:{\mathsf{List}}(RA) \rightarrow \mathcal{U}$ , and $x:FM_Y$ , we have

(3) \begin{equation} S(r(s),Y,x)=v_Y(x) (s). \end{equation}

It follows by induction on ${\mathsf{List}}^e(A)$ . If $s={\mathsf{[]}}^e$ , the $\mathsf{refl}$ term satisfies the identity 3. If $s=b \,\mathsf{::}^e\, s'$ for $s':{\mathsf{List}}^e(A)$ , then we have the following chain of identities.

\begin{eqnarray*} S(r(b \,\mathsf{::}^e\, s'),Y,x)&=&S(r_A(b)\, \mathsf{::}\, r(s'),Y,x)\\ &=&S(r(s'),Y',x')\\ &=&v_{Y'}(u_{Y'} (T))(s')\\ &=&T(s')\\ &=&\beta (u(\lambda \, a. v_Y(x)(a \,\mathsf{::}^e\, s')))\, (r_A(b))\\ &=&(\lambda \, a. v_Y(x)(a \,\mathsf{::}^e\, s')) \,(b)\\ &=&v_Y(x)(b \,\mathsf{::}^e\, s') \end{eqnarray*}

These are obtained by, respectively, the definition of $r$ , the definition of $S$ , the induction hypothesis, the fact that $v_Y'$ is the inverse of $u_Y'$ , the definition of $T$ , the fact that $\beta$ is the inverse of $\alpha =u{{\circ }^{\textit {e}}}(\!-{{\circ }^{\textit {e}}} r)$ , and the definition of the given function. Note that when we have exo-equalities of terms in types, we can use $\mathsf{eqtoid}$ to make them identities.

Now, define $\beta _Y : FM_Y \rightarrow \prod _{t:{\mathsf{List}}(RA)} Y(s)$ as $\beta _Y(x)(t)\;:=\;S(t,Y,x)$ . Then we obtain

\begin{equation*}\alpha _Y(\beta _Y(x))=u_Y(\beta _Y(x) {{\circ }^{\textit {e}}} r)=u_Y(v_Y(x))=x. \end{equation*}

These are obtained by, respectively, the definition of $\alpha _Y$ , the fact that $\beta _Y(x){{\circ }^{\textit {e}}} r= v_Y(x)$ since we can use ${\mathsf{funext}}^e$ for cofibrant exo-types and Equation (3), and the fact that $v_Y$ is the inverse of $u_Y$ .

This proves that $\alpha _Y$ has a section for any $Y:{\mathsf{List}}(RA) \rightarrow \mathcal{U}$ . By Lemma 2.16, we conclude that ${\mathsf{List}}^e(A)$ is sharp.

2.2.2 (Exo)Type of binary trees

Definition 2.22. For exo-type $N,L:\mathcal{U}^e$ , we define the exo-type ${\mathsf{BinTree}}^e (N,L) : \mathcal{U}^e$ of binary exo-trees with node values of exo-type $N$ and leaf values of exo-type $L$ , which has constructors

• • ${\mathsf{leaf}}^e : L \rightarrow {\mathsf{BinTree}}^e (N,L)$

• • ${\mathsf{node}}^e : {\mathsf{BinTree}}^e (N,L) \rightarrow N \rightarrow {\mathsf{BinTree}}^e (N,L) \rightarrow {\mathsf{BinTree}}^e (N,L)$

Similarly, if $N,L : \mathcal{U}$ is a type, the type ${\mathsf{BinTree}} (N,L)$ of binary trees with node values of type $N$ and leaf values of type $L$ constructors $\mathsf{leaf}$ and $\mathsf{node}$ .

We also have a definition for unlabeled binary (exo)trees.

Definition 2.23. The exo-type ${\mathsf{UnLBinTree}}^e : \mathcal{U}^e$ of unlabeled binary exo-trees is constructed by

• • ${\mathsf{u-leaf}}^e : {\mathsf{UnLBinTree}}^e$

• • ${\mathsf{u-node}}^e : {\mathsf{UnLBinTree}}^e \rightarrow {\mathsf{UnLBinTree}}^e \rightarrow {\mathsf{UnLBinTree}}^e$

Similarly, the type ${\mathsf{UnLBinTree}} : \mathcal{U}$ of unlabeled binary trees is constructed by $\mathsf{u-leaf}$ and $\mathsf{u-node}$ .

It is easy to see that if we take $N=L={\mathbf{1}}^e$ , then ${\mathsf{BinTree}}^e (N,L)$ is isomorphic to ${\mathsf{UnLBinTree}}^e$ . However, we have a more general relation between them. For any $N,L:\mathcal{U}^e$ , we can show

(4) \begin{equation} {\mathsf{BinTree}}^e (N,L) \cong {\sum }^e_{t : {\mathsf{UnLBinTree}}^e} \left (N^{\text{\# of nodes of }t} \times ^e L^{\text{\# of leaves of }t}\right ). \end{equation}

Thanks to this isomorphism, we can determine the cofibrancy or the sharpness of ${\mathsf{BinTree}}^e (N,L)$ using the cofibrancy or the sharpness of ${\mathsf{UnLBinTree}}^e$ . Indeed, we will show that if $\mathbb{N}^e$ is cofibrant, then ${\mathsf{UnLBinTree}}^e$ is not only cofibrant but also sharp. Since any finite product of cofibrant (sharp) exo-types is cofibrant (sharp), we can use isomorphism 4 to show ${\mathsf{BinTree}}^e (N,L)$ is cofibrant (sharp) under some conditions. Thus, the main goal is to get cofibrant (sharp) ${\mathsf{UnLBinTree}}^e$ .

We will construct another type that is easily shown to be cofibrant (sharp) to achieve this goal. Let ${\mathsf{Parens}} : \mathcal{U}^e$ be the exo-type of parentheses constructed by ${\mathsf{popen}} : {\mathsf{Parens}}$ and ${\mathsf{pclose}} : {\mathsf{Parens}}$ . In other words, it is an exo-type with two terms. Define an exo-type family

\begin{equation*}{\mathsf{isbalanced}} : {\mathsf{List}}^e ({\mathsf{Parens}}) \rightarrow \mathbb{N}^e \rightarrow \mathcal{U}^e \end{equation*}

where ${\mathsf{isbalanced}}(l,n)\;:=\;{\mathbf{1}}^e$ if the list of parentheses $l$ needs $n$ many opening parentheses to be a balanced parenthesization, and ${\mathsf{isbalanced}}(l,n)\;:=\;{\mathbf{0}}^e$ otherwise. For example, we have

\begin{eqnarray*} &&{\mathsf{isbalanced}}({\mathsf{popen}} \,\mathsf{::}^e\, {\mathsf{pclose}} \,\mathsf{::}^e\, {\mathsf{[]}}^e,{\mathsf{0}}^e)={\mathbf{1}}^e\\ &&{\mathsf{isbalanced}}({\mathsf{popen}} \,\mathsf{::}^e\, {\mathsf{pclose}} \,\mathsf{::}^e\, {\mathsf{pclose}} \,\mathsf{::}^e\, {\mathsf{[]}}^e,{\mathsf{succ}}^e({\mathsf{0}}^e))={\mathbf{1}}^e\\ &&{\mathsf{isbalanced}}({\mathsf{popen}} \,\mathsf{::}^e\, {\mathsf{pclose}} \,\mathsf{::}^e\, {\mathsf{popen}} \,\mathsf{::}^e\, {\mathsf{[]}}^e,{\mathsf{succ}}^e({\mathsf{0}}^e))={\mathbf{0}}^e. \end{eqnarray*}

In other words, the first says that “()” is a balanced parenthesization, the second says that “())” needs one opening parenthesis, and the third says that “()(” is not balanced if we add one more opening parenthesis.

Since ${\mathbf{0}}^e$ and ${\mathbf{1}}^e$ are cofibrant (also sharp), we get ${\mathsf{isbalanced}}(l,n)$ is cofibrant (sharp) for any $l:{\mathsf{List}}^e({\mathsf{Parens}})$ and $n:\mathbb{N}^e$ . Finally, for any $n:\mathbb{N}^e$ define

\begin{equation*}{\mathsf{Balanced}}(n)\;:=\;{\sum }^e_{l:{\mathsf{List}}^e({\mathsf{Parens}})} {\mathsf{isbalanced}}(l,n).\end{equation*}

Lemma 2.24. If $\mathbb{N}^e$ is cofibrant, then for any $n:\mathbb{N}^e$ , the exo-type ${\mathsf{Balanced}}(n)$ is both cofibrant and sharp.

Proof. By definition, $\mathsf{Parens}$ is a finite exo-type. Therefore, it is both cofibrant (Proposition 2.13(iii)) and sharp (Proposition 2.17(v)). Since we assumed $\mathbb{N}^e$ is cofibrant, Proposition 2.20 shows that ${\mathsf{List}}^e({\mathsf{Parens}})$ is cofibrant, and Proposition 2.21 shows that ${\mathsf{List}}^e({\mathsf{Parens}})$ is sharp.

Since ${\mathsf{isbalanced}}(l,n)$ is both cofibrant and sharp for any $l:{\mathsf{List}}^e({\mathsf{Parens}})$ and $n:\mathbb{N}^e$ , the exo-type ${\mathsf{Balanced}}(n)$ is cofibrant by Proposition 2.13(iv) and sharp by Proposition 2.17(iv).

The exo-type that we use to show ${\mathsf{UnLBinTree}}^e$ is both cofibrant and sharp, is ${\mathsf{Balanced}} ({\mathsf{0}}^e)$ . The following result will be analogous to the combinatorial result that there is a one-to-one correspondence between full binary trees and balanced parenthesizations (Even, Reference Even2011).

Proposition 2.25. There is an isomorphism ${\mathsf{UnLBinTree}}^e\cong {\mathsf{Balanced}} ({\mathsf{0}}^e)$ .

Proof. We will define the map and explain the construction. For the proof that it is an isomorphism, we refer to its formalization in our library.

Define first

\begin{equation*}\phi : {\mathsf{UnLBinTree}}^e\rightarrow (n : \mathbb{N}^e) \rightarrow {\mathsf{Balanced}}(n) \rightarrow {\mathsf{Balanced}}(n)\end{equation*}

as follows

\begin{eqnarray*} \phi ({\mathsf{u-leaf}}^e ,\,n,\,b)&\;:=\;&b\\ \phi ({\mathsf{u-node}}^e(t_1,t_2),\,n,\,b)&\;:=\;&\phi (t_2,\,n,\,b') \end{eqnarray*}

where

\begin{eqnarray*} b'\;:=\;({\mathsf{popen}} \,\mathsf{::}^e\, (\pi _1(\phi \,(t_1,\,{\mathsf{succ}}^e(n),\,({\mathsf{pclose}} \,\mathsf{::}^e\, \pi _1(b),\pi _2(b)))))\;,\\ \pi _2(\phi \,(t_1,\,{\mathsf{succ}}^e(n),\,({\mathsf{pclose}} \,\mathsf{::}^e\, \pi _1(b),\pi _2(b))))). \end{eqnarray*}

Using this, the main map $\Phi : {\mathsf{UnLBinTree}}^e \rightarrow {\mathsf{Balanced}}({\mathsf{0}}^e)$ is defined as

\begin{equation*}\Phi (t)\;:=\;\phi (t,{\mathsf{0}}^e,({\mathsf{[]}}^e,\star ^e)).\end{equation*}

The construction basically maps each tree to a balanced parenthesization in the following way. The empty list of parentheses represents a leaf. If trees $t_1$ and $t_2$ have representations $l_1$ and $l_2$ , then the tree ${\mathsf{u-node}}^e(t_1,t_2)$ is represented by the list $l_2(l_1)$ . Figure 1 provides some examples of this conversion.

Figure 1. Examples of the conversion between binary trees and parenthesizations.

The inverse of $\Phi$ is defined precisely to reverse this process, but it needs some auxiliary definitions. One can see the formalization for the details.

This isomorphism provides the results we wanted.

Corollary 2.26. If $\mathbb{N}^e$ is cofibrant, then ${\mathsf{UnLBinTree}}^e$ is both cofibrant and sharp.

Proof. It follows from Lemma 2.24 and Proposition 2.25.

Corollary 2.27. If $\mathbb{N}^e$ is cofibrant, and $N,L:\mathcal{U}^e$ are cofibrant (sharp) exo-types, then ${\mathsf{BinTree}}^e(N,L)$ is cofibrant (sharp).

Proof. It follows from the isomorphism 4 and Corollary 2.26.

2.3 Agda library

This section serves as an introduction to our formalization library (Uskuplu, Reference Uskuplu2025), implemented in Agda.Footnote ⁵ We have opted for Agda over other proof assistants because of its recent feature enhancements that enable us to effectively engage with 2LTT.

Several proof assistants and formalization projects focus on 2LTT. Annenkov et al. (Reference Annenkov, Capriotti and Kraus2019), for instance, the authors of Annenkov et al. (Reference Annenkov, Capriotti, Kraus and Sattler2023) have formalized some of their work in the Lean proof assistant (de Moura et al., Reference de Moura, Kong, Avigad, van Doorn and von Raumer2015). Since Lean lacks direct support for 2LTT, they utilized type classes to distinguish between types and exo-types. Their work operates within a type theory that features universes of types with UIP, where exo-types correspond to the standard types of the proof assistant, while HoTT-level types are represented as types “tagged” with the additional structure of being fibrant. In the case of the Coq proof assistant (Bertot and Castéran, Reference Bertot and Castéran2004), Boulier and Tabareau (Boulier and Tabareau, Reference Boulier and Tabareau2017) adopted a similar strategy within their formalization library (Boulier and Tabareau, Reference Boulier and Tabareau2020). However, Agda offers more direct support for a version of 2LTT.

Agda is a dependently typed functional programming language that is developed in the programming logic group at Chalmers University of Technology, with implementation described in Ulf Norell’s PhD thesis (Norell, Reference Norell2007). Thanks to its typing system, Agda can be used as a proof assistant, allowing users to prove mathematical theorems, generally in a constructive setting. The main features of Agda are the following (Bove et al., Reference Bove, Dybjer and Norell2009):

• • It is based on MLTT and supports a rich family of strictly positive inductive and inductive-recursive data types and families.

• • Agda offers an interactive proof development environment. Users can incrementally construct formal proofs, interact with the system to provide hints, and receive immediate feedback on the validity of their proofs.

• • Agda supports very flexible pattern matching.

As previously mentioned, we have chosen to employ Agda for formalizing our theory due to its favorable environment for working with 2LTT, facilitated by a combination of both longstanding and recently introduced flags. In Agda, a flag typically refers to a command-line option or compiler directive that you can use to modify the behavior of the Agda compiler and development environment when working with Agda code. Flags are used to control various aspects of type-checking, optimization, error reporting, and more. They allow you to customize your Agda development environment to suit your specific needs and requirements.

In our library, we mainly use four flags: –two-level, –cumulativity, –without-K, and –exact-split.

The flag-two level enables a new sort called SSet. This provides two distinct universes for us. Note that while it is common to assume typical ambiguityFootnote ⁶ in papers, the formalization works with polymorphic universes.

Agda Code 1. Universes defined in the library.

The flag –cumulativity enables the subtyping rule Set i $\leq$ SSet i for any level i. Thanks to the flag, we can take types as arguments for the operations/formations that are originally defined for exo-types, as we discussed in Section 2.1.

Historical note. Earlier drafts of this manuscript featured a counter-example that demonstrated a soundness bug which appeared when the compiler flags – two-level and –cumulativity were enabled at the same time: after coercing a fibrant type from Set i into SSet i, Agda incorrectly allowed pattern matching on its constructors, bypassing the intended “no elimination from fibrant to non-fibrant” guard. The bug was first reported as Issue #5761 and later isolated in the more general Issue #7503. It has since been fixed upstream (see pull request #7504, merged 19 September 2024 and released with Agda 2.8.0). The type-checker now inspects the principal sort of an index rather than its declared sort, so the offending code is rightly rejected. Because current Agda versions compile our library without workarounds, the illustrative bug figure and its accompanying discussion are no longer needed and have been removed from this paper. Readers interested in the examples can still find the code snippets in the GitHub issues.

Agda Code 2. The coercion map used in $\texttt {2LTT\_C}$ .

The flag – without K – enables a key property. It is used to disable the uniqueness of identity proofs (called Axiom K) in Agda for the identity type of the terms of types in Set. Disabling the Axiom K with the flag has a significant impact on the behavior of Agda. When this flag is used, Agda will treat equality between types (in Set) as non-trivial, meaning that even if you have a proof of equality between two types, you cannot directly substitute one for the other in all contexts. Since this does not affect the types in SSet, the exo-equality is not affected by this change; thus, we indeed obtain the identity type and the exo-equality separately as we desired.

The flag – exact-split – enables that Agda definitions by pattern matching are definitional equalities. The flag is used to enforce stricter and more precise pattern matching in dependent function definitions, helping to catch potential issues related to coverage and termination checking.

2.3.1 Overview of library structure

Table 1 gives a bird’s-eye view of the repository (Uskuplu, Reference Uskuplu2025): the left-hand column lists every top-level file or folder in the 2LTT-Agda tree, while the right-hand column highlights its mathematical role or the main results formalized there. Use this map as a quick index when navigating the code: for example, all fibrancy lemmas live under 2LTT/Coercion/, cofibration material under 2LTT/Cofibration/, and the no-cumulativity variant under 2LTT_C/.

Table 1. The general structure of the library

After introducing the two universes as in Agda Code 1, we declare each standard type-former twice – once in the fibrant universe $\mathcal{U}$ and once in the exo-universe $\mathcal{U}^{e}$ . The definitions are syntactically identical; only the ambient universe changes. Agda Code 3 illustrates this pattern for the coproduct: the ordinary operator $+$ inhabits $\mathcal{U}$ , whereas its exo counterpart $+^{e}$ lives in $\mathcal{U}^{e}$ . Throughout the library we mark every exo object (types, constructors, and fields) with the superscript “ $e$ ” to keep the distinction explicit.

Recall that Theorem2.4 shows that whenever an exo–type former is applied to types, the resulting exo-type canonically coerces to its fibrant counterpart. Agda Code 4 shows the instance for coproducts: for types $A,B : \mathcal{U}$ , a value of $A +^{e} B$ is sent to the ordinary sum $A + B$ by straightforward pattern matching on ${\mathsf{inl}}^e$ and ${\mathsf{inr}}^e$ .

A tempting next step is to define an inverse $A + B \to A +^{e} B$ , yet Agda’s elimination discipline prevents this. As Code 5 illustrates, the type-checker refuses to eliminate the fibrant term $A + B$ when the target lies in the exo-universe, and the hole remains unfillable. The restriction is precisely what keeps the system sound, so no inverse map can be supplied without extra assumptions.

Agda Code 3. Definition of coproduct type and exo-type in the library.

Agda Code 4. Theorem2.4.iv.

Agda Code 5. Agda does not allow to define $\texttt {inv-map-coprod}^e$ .

The first of our three exo-type classes is that of fibrant exo-types, defined in Agda Code 6. Given an exo-type $B : \mathcal{U}^{e}$ , the record $\texttt {isFibrant}\,B$ packages a fibrant witness $C : \mathcal{U}$ and an exo-isomorphism $B \,\cong \, C$ . The choice differs slightly from that in Ahrens et al. (Reference Ahrens, North, Shulman and Tsementzis2025), where fibrancy is defined as $\texttt {isFibrant}(A) \;\;:=\;\; \sum ^{e}_{C:\mathcal{U}}\,(A {\mathsf{=}^{\textit {e}}} C)$ and the fibrant match is by definition the same carrier $A$ . Under Axiom T3Footnote ⁷ the two formulations are logically equivalent, but the record style used here is more convenient in Agda. When we merely have an equality $A {\mathsf{=}^{\textit {e}}} B$ (with $A : \mathcal{U}^{e}$ and $B : \mathcal{U}$ ), we may still transport terms to obtain an isomorphism, whereas an explicit $\cong$ -witness gives the isomorphism immediately. The definition conforms cleanly with our notion of fibrant-equivalence (Definition 2.9), introduced for proof engineering purposes. We also postulated T3 in Agda and provided the equivalent definition for $\texttt {isFibrant}$ , but never used it anywhere else in the library.

Agda Code 6. The definition being fibrant exo-type and T3-axiom.

The second class consists of cofibrant exo-types. In Agda Code 7, mirroring Definition 2.11, we first formulate cofibrancy relative to a single family $Y : B \to \mathcal{U}$ via the predicate $\texttt {isCofibrant}$ - $\texttt {at}$ . The wrapper $\texttt {isCofibrant}(B)$ then declares $B$ cofibrant precisely when this condition holds for every such family $Y$ . This formulation supports a dedicated version of function extensionality for cofibrant exo-types. Agda Code 8 shows the essential fragment of the $\texttt {FUNEXT}$ module: the map ${\beta }$ coerces a dependent exo-type of functions to its fibrant match, while $\texttt {FEP}$ states that two (coerced via $\beta$ ) exo-functions are equal as soon as they agree pointwise. We proved the equivalence as in Proposition 2.12, but the remaining auxiliary proofs are omitted here for brevity.

This specialized version of function extensionality is not just a curiosity; it is crucial in our proof that the dependent exo-sum preserves sharpness, named ${\Sigma }^{e}$ - $\texttt {preserve}$ - $\texttt {Sharp}$ in the library, which is Proposition 2.17(vi). That argument proceeds through a lengthy chain of equivalences, and at one stage we must handle dependent exo-types of functions whose fibrancy follows from the cofibrancy of their domain. To push equivalences through such spaces, we require a functoriality lemma defined in Agda Code 9 whose proof hinges on the function extensionality principle for cofibrant exo-types introduced above. This is a telling example of how mechanization tightens the theory: the informal derivation in Ahrens et al. (Reference Ahrens, North, Shulman and Tsementzis2025), although compelling on paper, silently skipped the functoriality bookkeeping that only became apparent during formalization.

Agda Code 7. The definition being a cofibrant exo-type.

Agda Code 8. The function extensionality for cofibrant exo-types.

The last structural class is that of sharp exo-types. Agda Code 10 gives the Agda record that realizes Definition 2.14. Because Agda keeps the dependent exo-type of function $\Pi ^{e}$ and its fibrant match strictly separate, the formal version must shuttle explicitly between the two when expressing that the pre-composition map is a fibrant-equivalence. These transports – routine but verbose – are handled inside the record fields and therefore suppressed in the paper-level formulation of the definition.

Agda Code 9. The functoriality rule is about the dependent exo-type of functions where the domain is cofibrant. This says if $A$ is a cofibrant exo-type and $P,Q:A\to \mathcal{U}$ are two dependent types such that there is a function $F:\prod ^e_{a:A} P(a) \to Q(a)$ where each $F(a)$ is an equivalence, then fibrant matches of $\prod ^e_{a:A} P(a)$ and $\prod ^e_{a:A} Q(a)$ are also equivalent.

Agda Code 10. The definition being a sharp exo-type.

Agda Code 11. An example of Agda’s different behaviors about level checking.

As noted in Remark 2.28, we keep a second copy of the library, 2LTT_C, which is compiled without the –cumulativity flag. Except for a few low-level details, the two code bases are line-for-line identical, so all results in this paper carry over verbatim. The flag does, however, affect universe inference. When –cumulativity is off, the checker can almost always infer the required universe levels automatically. With the flag on, the same term may yield an UnsolvedConstraints error, forcing the user to spell out the levels by hand. Agda Code 11 illustrates the difference: the first definition type-checks in 2LTT_C; in 2LTT we must supply the explicit level parameters for each isEquiv witness and for the equivalence type $\_\simeq \_$ .

The 2LTT and 2LTT_C repositories together provide a complete, flag-parametrized formalization of every concept used in this paper: two-level universes, main type-formers, and the three structural classes (fibrant, cofibrant, and sharp exo-types). Both variants compile cleanly with Agda 2.8.0 and the standard library 2.2, and all theorems cited here are verified verbatim. Upcoming work will enrich the code base with the semantics of diagram signatures and further results of 2LTT, but the current release already stands as a stable reference implementation for the theory developed in this article.

3.1 Review

Common models of type theories arise in the category with families. A category with families (CwF) consists of the following:

• • A category $\mathcal{C}$ with a terminal object $ 1_{\mathcal{C}} : \mathcal{C}$ . Its objects are called contexts, and $1_{\mathcal{C}}$ is called the empty context.

• • A presheaf $\texttt {Ty} : \mathcal{C}^{\textrm {{op}}} \rightarrow {\mathsf{Set}}$ . If $ A : \texttt {Ty}(\Gamma )$ , then we say $A$ is a type over $\Gamma$ .

• • A presheaf $\texttt {Tm} : (\!\int \texttt {Ty})^{\textrm {{op}}} \rightarrow {\mathsf{Set}}$ . If $a : \texttt {Tm}(\Gamma , A)$ , then we say $a$ is a term of $A$ .

• • For any $\Gamma : \mathcal{C}$ and $A : \texttt {Ty}(\Gamma )$ , there is an object $\Gamma .A :\mathcal{C}$ , a morphism $p_A : \Gamma .A \rightarrow \Gamma$ , and a term $q_A : \texttt {Tm}(\Gamma .A,A[p_A])$ with the universal property: for any object $\Delta : \mathcal{C}$ , a morphism $\sigma : \Delta \rightarrow \Gamma$ , and a term $a:\texttt {Tm}(\Delta ,A[\sigma ])$ , there is a unique morphism $\theta : \Delta \rightarrow \Gamma .A$ such that $p_A \circ \theta = \sigma$ and $q_A[\theta ]=a$ . This operation is called the context extension.

Note that for all contexts $\Gamma : \mathcal{C}$ and types $A : \texttt {Ty}(\Gamma )$ , there is a natural isomorphism

\begin{equation*}\texttt {Tm}(\Gamma ,A)\cong \mathcal{C}/{\Gamma }((\Gamma ,\mathsf{id}_{\Gamma }),(\Gamma .A,p_A)).\end{equation*}

Indeed, this follows from the universal property of the context extension by taking $\Delta \;:=\;\Gamma$ and $\sigma \;:=\;\mathsf{id}_{\Gamma }$ . This observation says that the terms of $A$ over $\Gamma$ can be regarded as the sections of $p_A:\Gamma .A \rightarrow \Gamma$ .

The proposition below is a useful fact for the rest of the section.

Proposition 3.1. Let $\sigma :\Delta \rightarrow \Gamma$ be a context morphism and $A:\texttt {Ty}(\Gamma )$ . There exists a morphism $\sigma ^{+}:\Delta .A[\sigma ] \rightarrow \Gamma .A$ that makes the following diagram into a pullback square:

Rather than presenting a specific instance of a CwF, we will offer a more extensive range of CwF examples. The category of presheaves is an archetypal example of a CwF (Hofmann, Reference Hofmann1997). Let $\mathcal{C}$ be a (small) category, and let $\widehat {\mathcal{C}}$ be its category of presheaves. The CwF structure on $\widehat {\mathcal{C}}$ , denoted by $(\widehat {\texttt {Ty}}, \widehat {\texttt {Tm}})$ is defined in the following manner:

• • Contexts are presheaves $\mathcal{C}^{\textrm {{op}}} \rightarrow {\mathsf{Set}}$ .

• • The constant presheaf that takes the value $\star$ in the category of sets can be characterized as the terminal object $1_{\widehat {\mathcal{C}}}$ .

• • Recall that $\widehat {\texttt {Ty}}$ is a presheaf on $\widehat {\mathcal{C}}$ .

  If $P:\widehat {\mathcal{C}}$ , then $\widehat {\texttt {Ty}}(P)$ is the underlying set of the category of presheaves $\widehat {\int P}$ over the category of elements $\int P$ . In other words, a type $A:\widehat {\texttt {Ty}}(P)$ is a functor $\left (\int P \right )^{\textrm {{op}}} \rightarrow {\mathsf{Set}}$ .

  If $\phi :Q \rightarrow P$ is a morphism in $\widehat {\mathcal{C}}$ and $A:\widehat {\texttt {Ty}}(P)$ , we define the type substitution $A[\phi ]:\widehat {\texttt {Ty}}(Q)$ as

  \begin{equation*}A[\phi ](\Gamma ,x)\;:=\; A (\Gamma ,\phi _{\Gamma }(x))\end{equation*}
  where $x:Q_{\Gamma }$ .

• • Recall that $\widehat {\texttt {Tm}}$ is a presheaf on $\int \widehat {\texttt {Ty}}$ .

  For $P:\widehat {\mathcal{C}}$ and $A:\widehat {\texttt {Ty}}(P)$ , we define

  \begin{equation*}\widehat {\texttt {Tm}}(P,A) \;:=\; \left \lbrace a : \prod _{\Gamma : \mathcal{C},\, x: P_{\Gamma }} A(\Gamma ,x) \mid \text{ if } \sigma : \Delta \rightarrow \Gamma , \, x : P_{\Gamma }\text{, then } a(\Gamma ,x)[\sigma ]=a(\Delta ,x[\sigma ])\right \rbrace .\end{equation*}

  If $\phi :Q \rightarrow P$ is a morphism in $\widehat {\mathcal{C}}$ , $A:\widehat {\texttt {Ty}}(P)$ , and $a:\widehat {\texttt {Tm}}(P,A)$ , we define the term substitution $a[\phi ]:\widehat {\texttt {Tm}}(Q,A[\phi ])$ as

  \begin{equation*}a[\phi ](\Gamma ,x)\;:=\; a (\Gamma ,\phi _{\Gamma }(x))\end{equation*}
  where $x:Q_{\Gamma }$ .

• • For $P:\widehat {\mathcal{C}}$ and $A:\widehat {\texttt {Ty}}(P)$ , the context $P.A$ is again a presheaf over $\mathcal{C}$ defined by

  \begin{equation*}P.A(\Gamma )\;:=\;\prod_{x : P(\Gamma )} A(\Gamma , x).\end{equation*}

  If $\sigma : \Delta \rightarrow \Gamma$ is a morphism in $\mathcal{C}$ and $(x,a):P.A (\Gamma )$ , then we define

  \begin{equation*}(x,a)[\sigma ]=P.A(\sigma ) (x,a)\;:=\;(x[\sigma ],a[\sigma ]).\end{equation*}

  The morphism $p_A:P.A \rightarrow P$ is defined by the first projection. In other words, for $\Gamma : \mathcal{C}$ and $(x ,a): P.A(\Gamma )$ , we have $(p_A)_{\Gamma }(x,a)=x$ . The term $q_A: \widehat {\texttt {Tm}}(P.A,A[p_A])$ is given by the second projection. In other words, for $\Gamma :\mathcal{C}$ and $(x,a): P.A(\Gamma )$ , we have $q_A(\Gamma , (x,a))=a$ . Note that $A[p_A](\Gamma ,(x,a))=A(\Gamma ,p_A(x,a))=A(\Gamma ,x)$ .

  It remains to verify the universal property for the context extension.

  Let $Q:\widehat {\mathcal{C}}$ , $\tau :Q\rightarrow P$ , and $b:\widehat {\texttt {Tm}}(Q,A[\tau ])$ . Define $\theta :Q \rightarrow P.A$ as follows: for $\Gamma : \mathcal{C}$ and $x:Q_{\Gamma }$ , we have

  \begin{equation*}\theta _{\Gamma }(x)\;:=\; (\tau _{\Gamma }(x),b(\Gamma ,x)).\end{equation*}
  It is straightforward to verify the defining rules: $p_A\circ \theta =\tau$ because
  \begin{equation*}(p_A\circ \theta )_{\Gamma }(x)=p_A(\tau _{\Gamma }(x),b(\Gamma ,x))=\tau _{\Gamma }(x),\end{equation*}
  and $q_A[\theta ]=b$ because
  \begin{equation*}q_A[\theta ](\Gamma ,x)=q_A(\Gamma , \theta _{\Gamma }(x))=q_A(\Gamma , (\tau _{\Gamma }(x),b(\Gamma ,x)))=b(\Gamma ,x).\end{equation*}
  Since the defining properties of $p_A$ and $q_A$ determine the map $\theta$ , it is uniquely determined.

Another well-known example is the category $\mathsf{SSet}$ . Let $\mathcal{C}=\bigtriangleup$ be the simplex category. The presheaf category $\widehat {\bigtriangleup }$ is called the category of simplicial sets, denoted by $\mathsf{SSet}$ . Like any other presheaf category, $\mathsf{SSet}$ has a CwF structure $(\widehat {\texttt {Ty}},\widehat {\texttt {Tm}})$ , but it has another CwF structure. We only define a new type presheaf $\texttt {Ty} : {\mathsf{SSet}}^{\textrm {{op}}} \rightarrow {\mathsf{Set}}$ as follows: $\texttt {Ty}(P)$ is a subset of $\widehat {\texttt {Ty}}(P)$ such that $A : \texttt {Ty}(P)$ if the display map $P.A \rightarrow P$ is a Kan fibration. With the induced term presheaf $\texttt {Tm}$ obtained by $\widehat {\texttt {Tm}}$ , we obtain a CwF $({\mathsf{SSet}}, \texttt {Ty}, \texttt {Tm})$ . It is easy to prove this structure satisfies all axioms of being a CwF (Capriotti, Reference Capriotti2016). We always refer to this new CwF structure on $\mathsf{SSet}$ unless stated otherwise.

$\mathsf{SSet}$ is not a unique example of a presheaf category having two CwF structures. There are many of them, and they will be useful to talk about two-level structures in the later sections.

Now, we need to establish the meanings of particular type formers within a CwF and to examine the requirements that must be fulfilled for the CwF to possess these type formers. Although this analysis could be applied to various standard type formers, our focus will be solely on those indispensable for the rest of the study. One can see the definitions of type formers in a CwF from (Hofmann, Reference Hofmann1997) and (Uskuplu, Reference Uskuplu2023). We only provide what they mean in a presheaf CwF.

We start with the $\Pi$ types in a presheaf Cwf.

Let $A:\widehat {\texttt {Ty}}(P)$ and $B:\widehat {\texttt {Ty}}(P.A)$ . First, we need to define $\Pi (A,B):\widehat {\texttt {Ty}}(P)$ . Recall that $\Pi (A,B)$ should be a presheaf over $\int P$ . For each $\Gamma :\mathcal{C}$ and $x:P_{\Gamma }$ , the type $\Pi (A,B)(\Gamma ,x)$ consists of the elements $f$ in the categorical product

\begin{equation*} f: \mathbf{\prod }_{\Delta :\mathcal{C},\, \sigma :\Delta \rightarrow \Gamma ,\, a: A (\Delta , x[\sigma ])} B(\Delta , (x[\sigma ],a)) \end{equation*}

such that if $\Theta : \mathcal{C}$ and $\tau : \Theta \rightarrow \Delta$ , then

(5) \begin{equation} f(\Delta ,\sigma ,a)[\tau ]=f(\Theta ,\sigma \circ \tau ,a[\tau ]). \end{equation}

If $\tau :(\Upsilon ,y)\rightarrow (\Gamma ,x)$ is a morphism in $\int P$ , namely, $\tau : \Upsilon \rightarrow \Gamma$ is a morphism in $\mathcal{C}$ such that $x[\tau ]=y$ , for each $f : \Pi (A,B)(\Gamma ,x)$ , $\Delta : \mathcal{C}$ , $\sigma : \Delta \rightarrow \Upsilon$ , and $a : A (\Delta , y[\sigma ])$ , we define $f[\tau ](\Delta ,\sigma ,a)\;:=\;f(\Delta , \tau \circ \sigma , a)$ . Using the compatibility condition 5, we indeed obtain a presheaf $\Pi (A,B)$ on $\int P$ .

Second, for each $b:\widehat {\texttt {Tm}}(P.A,B)$ , we need to define a term $\lambda (b):\widehat {\texttt {Tm}}(P,\Pi (A,B))$ . Recall that $\lambda (b)$ should be an element in

\begin{equation*}\mathbf{\prod }_{\Gamma : \mathcal{C}, x : P_{\Gamma }} \Pi (A,B)(\Gamma ,x).\end{equation*}

Now, for $\Gamma :\mathcal{C}$ , $x : P_{\Gamma }$ , $\Delta :\mathcal{C}$ , $\sigma :\Delta \rightarrow \Gamma$ , and $a : A(\Delta , x[\sigma ])$ we define

\begin{equation*}\lambda (b)(\Gamma ,x) (\Delta ,\sigma ,a)\;:=\;b(\Delta ,x[\sigma ],a).\end{equation*}

This definition makes sense because the term $b$ is in $\mathbf{\prod }_{\Gamma : \mathcal{C}, z : P.A_{\Gamma }} B(\Gamma ,z)$ .

Third, for each $f:\widehat {\texttt {Tm}}(P,\Pi (A,B))$ and $a:\widehat {\texttt {Tm}}(P,A)$ , we need to define a term ${\mathsf{app}}(f,a):\widehat {\texttt {Tm}}(P,B[a])$ . Recall that it should be in

\begin{equation*}\mathbf{\prod }_{\Gamma : \mathcal{C}, x : P_{\Gamma }} B[a](\Gamma ,x).\end{equation*}

Now, for $\Gamma :\mathcal{C}$ , $x : P_{\Gamma }$ , we define

\begin{equation*}{\mathsf{app}}(f,a)(\Gamma ,x)\;:=\; f(\Gamma ,x)(\Gamma ,\mathsf{id},a).\end{equation*}

Next, we provide $\Sigma$ types in terms of the presheaf CwFs.

Let $A:\widehat {\texttt {Ty}}(P)$ and $B:\widehat {\texttt {Ty}}(P.A)$ . First, we need to define $\Sigma (A,B):\widehat {\texttt {Ty}}(P)$ . Recall that $\Sigma (A,B)$ should be a presheaf over $\int P$ . For each $\Gamma :\mathcal{C}$ and $x:P_{\Gamma }$ , we define

\begin{equation*}\Sigma (A,B)(\Gamma ,x)\;:=\;\lbrace (a,b) |\, a : A(\Gamma ,x),\, b: B(\Gamma ,(x,a)) \rbrace . \end{equation*}

For a morphism $\sigma : (\Delta ,y)\rightarrow (\Gamma ,x)$ in $\int P$ and $(a,b):\Sigma (A,B)(\Gamma ,x)$ , we define

\begin{equation*}(a,b)[\sigma ]\;:=\;(a[\sigma ],b[\sigma ]).\end{equation*}

One can define the operations $\langle \_,\_ \rangle$ , $\pi _1$ , and $\pi _2$ in an obvious way, and it is easy to prove the coherence rules.

Thirdly, the presheaf CwFs also satisfy the extensional identities.

Let $A:\widehat {\texttt {Ty}}(P)$ . Recall that ${\mathsf{Eq}}_A$ should be a presheaf over $\int \Gamma .A.A[p_A]$ . For each $\Gamma :\mathcal{C}$ , $x:P_{\Gamma }$ , and $a,b: A(\Gamma ,x)$ , we define

\begin{equation*}{\mathsf{Eq}}_A(\Gamma ,x,a,b)\;:=\; \begin{cases} \{\star \} & \text{if } a = b\\ \emptyset & \text{if } a \neq b \end{cases}.\end{equation*}

The morphism ${\mathsf{refl}}^e_A : P.A \rightarrow P.A.A[p_A].{\mathsf{Eq}}_A$ is defined for each $\Gamma : \mathcal{C}$ as $(x,a) \mapsto (x,a,a,\star )$ . Since $p_{{\mathsf{Eq}}}: P.A.A[p_A].{\mathsf{Eq}}_A \rightarrow P.A.A[p_A]$ is the first projection, we indeed have $p_{{\mathsf{Eq}}}\circ {\mathsf{refl}}^e_A$ is the diagonal map.

If $B : \widehat {\texttt {Ty}}(P.A.A[p_A].{\mathsf{Eq}}_A)$ , the function $J^e:\widehat {\texttt {Tm}}(P.A, B[{\mathsf{refl}}^e_A]) \rightarrow \widehat {\texttt {Tm}}(P.A.A[p_A].{\mathsf{Eq}}_A, B)$ is defined as follows: If $\alpha : \widehat {\texttt {Tm}}(P.A, B[{\mathsf{refl}}^e_A])$ , this means we have

\begin{equation*}\alpha : \prod _{\substack {\Gamma : \mathcal{C} \\ (x,a): P.A_{\Gamma }}} B(\Gamma ,(x,a,a,\star ))\quad \text{ and } \quad J^e(\alpha ) : \prod _{\substack {\Gamma : \mathcal{C}\\ (x,a,b,p): P.A.A[p_A].{\mathsf{Eq}}_{\Gamma }}} B(\Gamma ,(x,a,b,p)).\end{equation*}

Thus, we define $J^e(\alpha )(\Gamma ,(x,a,b,p))\;:=\;\alpha (\Gamma ,(x,a))$ , which makes sense because $p : {\mathsf{Eq}}(a,b)$ means $a=b$ and $p=\star$ . It is easy to prove the coherence rules.

Since the intensional identity type former, $\mathsf{Id}$ , is a particular case of $\mathsf{Eq}$ , we can say that every presheaf CwF supports intensional identity types. If we also assume univalence for intensional identities, it is not true in general that every presheaf CwF supports such an identity. For example, the presheaf CwF on $\mathsf{Set}$ supports the identity type and the uniqueness of identity proof (UIP), but UIP contradicts with univalence (HoTTBook, 2013). Nonetheless, it has been established that the simplicial set CwF, denoted as $\mathsf{SSet}$ , does provide support for univalent identity types (Kapulkin and Lumsdaine, Reference Kapulkin and Lumsdaine2021). Indeed, this category stands as one of the widely recognized models not only for Martin-Löf Type Theory but also for Homotopy Type Theory.

In the case of intensional identity types, we can also establish a definition for what constitutes a “contractible” type. This notion serves as a crucial component in the overall definition of cofibrancy.

Definition 3.2. Let $A:\texttt {Ty}(\Gamma )$ be a type. We call $A$ a contractible type if there is a term in the following type over $\Gamma$ :

\begin{equation*}{\mathsf{isContr}}(A)\;:=\;\Sigma (A,\Pi (A[p_A],\mathsf{Id}_A)).\end{equation*}

Having such a term means that there is a term $c : \texttt {Tm}(\Gamma ,A)$ called center of contraction, such that for any term $a: \texttt {Tm}(\Gamma .A,A[p_A])$ there is a term $p: \texttt {Tm}(\Gamma ,\mathsf{Id}_A[c^+,a^+])$ . In other words, we have a section map $c:\Gamma \rightarrow \Gamma .A$ to $p_A$ such that the following diagram, where $h$ is the contracting homotopy commutes:

Finally, we will show that for any (small) category $\mathcal{C}$ , the presheaf Cwf $(\widehat {\mathcal{C}},\widehat {\texttt {Ty}},\widehat {\texttt {Tm}})$ supports a natural number type.

Since $\mathbb{N}$ should be a presheaf over $\int 1_{\widehat {C}}$ , for each $\Gamma :\mathcal{C}$ and $x:(1_{\widehat {C}})_{\Gamma }$ we define $\mathbb{N}(\Gamma ,x)\;:=\; \mathbf{N}$ , the external set of natural numbers. The term ${\mathsf{0}} : \widehat {\texttt {Tm}}(1_{\widehat {C}}, \mathbb{N})$ is obtained by the morphism $1_{\widehat {C}} \rightarrow 1_{\widehat {C}}.\mathbb{N}$ which we define ${\mathsf{0}}_{\Gamma }(x)\;:=\;(x,0)$ for any $\Gamma :\mathcal{C}$ and $x:(1_{\widehat {C}})_{\Gamma }$ . The morphism ${\mathsf{succ}}:1_{\widehat {C}}.\mathbb{N} \rightarrow 1_{\widehat {C}}.\mathbb{N}$ is defined as ${\mathsf{succ}}_{\Gamma }(x,k)\;:=\;(x, k+1)$ for any $\Gamma :\mathcal{C}$ and $(x,k):(1_{\widehat {C}}.\mathbb{N})_{\Gamma }$ .

For any $P :\widehat {\mathcal{C}}$ and $\sigma : P \rightarrow 1_{\widehat {\mathcal{C}}}$ , if $B : \widehat {\texttt {Ty}}(P.\mathbb{N}[\sigma ])$ with $b_0$ and $b_s$ , the function $J^{\mathbb{N}}_B:P.\mathbb{N}[\sigma ] \rightarrow P.\mathbb{N}[\sigma ].B$ is defined as, for each $\Gamma :\mathcal{C}$ , and $(x,k):(P.\mathbb{N}[\sigma ])_{\Gamma }$

\begin{equation*}(J^{\mathbb{N}}_B)_{\Gamma }(x,k)\;:=\; \begin{cases} {b_0(x)} & \text{if } k = 0\\ {b_s\big ((x,k'),\,(J^{\mathbb{N}}_B)_{\Gamma }(x,k')\big )} & \text{if } k = k' + 1 \end{cases}.\end{equation*}

Clearly, $J^{\mathbb{N}}_B \circ {\mathsf{0}}[\sigma ]=b_0$ holds. For the other equality, we have

\begin{equation*}(J^{\mathbb{N}}_B \circ {\mathsf{succ}}[\sigma ])_{\Gamma }(x,k)=(J^{\mathbb{N}}_B)_{\Gamma }(x,k+1)=(b_s)_{\Gamma }\big ((x,k),\,(J^{\mathbb{N}}_B)_{\Gamma }(x,k)\big ).\end{equation*}

Also, it is easy to prove the remaining coherence rules.

Based on the example of presheaf CwF we discussed earlier, it becomes clear that CwFs can be used to model dependent type theory. A CwF provides a structure that covers contexts, types, terms, and substitutions. Moreover, if a CwF has enough type formers, it allows us to work with dependent products, dependent sums, and other inductive types. If you’re interested in delving into the details and exploring similar constructions related to CwFs, you can refer to Capriotti (Reference Capriotti2016). Our primary focus here is to establish the necessary background to discuss models where exo-nat is cofibrant.

We already know that a presheaf CwF is a model of Martin-Löf type theory with usual type formers and extensional identity types. The simplicial set CwF $\mathsf{SSet}$ is a model of Martin-Löf type theory with usual type formers and intensional identity types.

Let us recall that this study on semantics aims to gain insight into the models of 2LTT with a cofibrant exo-nat. Once we have acquired models of various type theories, a natural question arises: can we merge these models to obtain a comprehensive model of 2LTT? The answer to this question is affirmative, as it is indeed possible to combine two CwF structures in the same category in a manner that ensures their compatibility and coherence.

Example 3.8. Recall $\widehat {\texttt {Ty}}$ denotes the presheaf CwF structure. The simplicial set presheaf $\mathsf{SSet}$ originally had a presheaf CwF structure. Recall that

\begin{equation*}\texttt {Ty}(P)\;:=\;\{A : \widehat {\texttt {Ty}}(P) \mid p_A : P.A \rightarrow P \text{ is a Kan fibration}\}\end{equation*}

gives another type functor. Taking $\texttt {Ty}^f=\texttt {Ty}$ and $\mathcal{c} : \texttt {Ty}^f \rightarrow \widehat {\texttt {Ty}}$ as the inclusion, we obtain $\mathsf{SSet}$ as a two-level Cwf.

In a similar vein to how we can build a presheaf CwF from any arbitrary (small) category when the category itself is a CwF, we can proceed to construct a two-level CwF. This particular construction, which we refer to as the presheaf two-level CwF, will serve as our primary focus and model of interest.

Similar to how a type theory can be interpreted within the framework of a category with families, two-level type theories can be interpreted using a two-level category with families. Below, we provide the precise definition for such an interpretation.

Our only assumption concerning the coercion morphism $\texttt {Ty}^f \rightarrow \texttt {Ty}^e$ is that it is a natural transformation. However, in the context of a two-level model with enough type formers, we have the semantic counterpart of Theorem2.4. This theorem furnishes nice inversion rules from types to exo-types, and its proof heavily relies on the preservation of context extension and the elimination rules associated with the type formers (Annenkov et al., Reference Annenkov, Capriotti, Kraus and Sattler2023).

3.2 Models with cofibrant exo-nat

With the foundational knowledge established thus far, we are now equipped to delve into the discussion of potential models of 2LTT that satisfy the condition of having a cofibrant exo-nat. However, before proceeding to where this discussion takes place, let us first provide the semantic definition of “cofibrancy” for an exo-type.

The naturality conditions give the following: In Figure 2, all vertical arrows are context substitutions. When the commutative sides are appropriately composed, the top and bottom isomorphisms are equal, which can be expressed as the cube “commuting.” In Figure 3, the vertical arrows are context substitutions, and the square is commutative.

Figure 2. Naturality condition for $\Theta ^{\texttt {Ty}}$ , where $\sigma :\Delta \rightarrow \Gamma$ and $\tau : \Upsilon \rightarrow \Delta$ in $\mathcal{C}$ .

Figure 3. Naturality condition for $\Theta ^{\texttt {Tm}}$ , where $\sigma :\Delta \rightarrow \Gamma$ and $\tau : \Upsilon \rightarrow \Delta$ in $\mathcal{C}$ .

In the following definition, recall that all products exist in the category of sets.

In simpler terms, the first requirement stated in Definition 3.14 ensures that $\prod ^e_{a:\mathbb{N}^e} Y(a)$ has a fibrant match, while the second requirement ensures that the $\mathsf{funext}$ condition for cofibrant exo-types holds. Therefore, we have the semantic counterpart of Proposition 2.12:

Lemma 3.15. Let $(\mathcal{C},\texttt {Ty}^e,\texttt {Tm}^e,\texttt {Ty}^f,\texttt {Tm}^f)$ be a two-level CwF model of 2LTT equipped with the coercion $\mathcal{c}:\texttt {Ty}^f\!\to \!\texttt {Ty}^e$ . Fix an exo-type $A:\texttt {Ty}^e(\Gamma )$ and a context morphism $\sigma :\Delta \to \Gamma$ . Assume that condition (1) of Definition 3.12 holds, namely we have a natural transformation

\begin{equation*} \Theta ^{\texttt {Ty}}_{\Delta }\;:\; \texttt {Ty}^f(\Delta .A[\sigma ])\;\longrightarrow \;\texttt {Ty}^f(\Delta ) \quad \text{ with } \quad \mathcal{c}_{\Delta }\!\bigl (\Theta ^{\texttt {Ty}}_{\Delta }(Y)\bigr ) \;\cong \; {\prod \nolimits }^{e}\!(A,\mathcal{c}_{\Delta .A[\sigma ]}(Y)). \end{equation*}

Then following statements are equivalent: For every $Y:\texttt {Ty}^f(\Delta .A[\sigma ])$ ,

1. i. there is a map, natural in $\Delta$ ,

   \begin{equation*}\Theta ^{\texttt {Tm}}_{\Delta }:\texttt {Tm}^f\left (\Delta .A[\sigma ],{\mathsf{isContr}}(Y)\right ) \rightarrow \texttt {Tm}^f\left (\Delta ,{\mathsf{isContr}}(\Theta _{\Delta }^{\texttt {Ty}}(Y))\right ).\end{equation*}

2. ii. if $d,c:\prod _{a:A}\texttt {Tm}^f(\Delta ,Y[a])$ are such that the terms $\texttt {Tm}^f\bigl (\Delta ,\mathsf{Id}_{Y[a]}(d_a,c_a)\bigr )$ exist for all $a:A$ then the terms

   \begin{equation*} \texttt {Tm}^f\Bigl (\Delta ,\; \mathsf{Id}_{\Theta ^{\texttt {Ty}}_{\Delta }(Y)} \bigl (\psi (d),\psi (c)\bigr ) \Bigr ) \end{equation*}
   also exists, where $\psi :\prod _{a:A}\texttt {Tm}^f(\Delta ,Y[a])\to \texttt {Tm}^f(\Delta ,\Theta ^{\texttt {Ty}}_{\Delta }(Y))$ is the inverse in the isomorphism of condition (1).

We now give a canonical instance of a CWF with exo-nat products.

Example 3.16. Let $\mathcal{C}$ be a simplicial model category (Hovey, Reference Hovey1999). Write ${\mathcal{C}}/{\Gamma }$ for the slice over a context $\Gamma \in \mathcal{C}$ . Define $\texttt {Ty}(\Gamma )$ as the set of fibrations over $\Gamma$ (with suitable coherence conditions (Lumsdaine and Warren, Reference Lumsdaine and Warren2015)). Define for $A: \texttt {Ty}(\Gamma )$ the set $\texttt {Tm}(\Gamma ,P)$ as the hom-set ${\mathcal{C}}/{\Gamma }[\Gamma , \Gamma .A ]$ .

Fibrations are closed under pullback in any model category, and products are pullbacks of the projections. Hence the product of a countable family of fibrations is again a fibration. Given a family $Y\colon \mathbf N\to \texttt {Ty}(\Gamma )$ , set

\begin{equation*} \Omega _\Gamma (Y)\;\;:=\;\;\prod _{n:\mathbf N} Y_n \;\in \;\texttt {Ty}(\Gamma ). \end{equation*}

Because sections are morphisms in the slice, the universal property of products induces a canonical natural isomorphism

\begin{equation*} \texttt {Tm} \bigl (\Gamma ,\Omega _\Gamma (Y)\bigr ) \;\cong \; \prod _{n:\mathbf N}\texttt {Tm} \bigl (\Gamma ,Y_n\bigr ), \end{equation*}

verifying clause (1) of Definition 3.14 .

Since $\mathcal{C}$ is simplicial, each slice $\mathcal{C}/\Gamma$ is tensored and cotensored with simplicial sets, giving an adjunction

(6) \begin{equation} (\!-\!)\!\otimes \!\Delta [1]\;\dashv \;(\!-\!)^{\Delta [1]} \;:\;\mathcal{C}/\Gamma \;\longrightarrow \;\mathcal{C}/\Gamma . \end{equation}

For a fibration $p:A\rightarrow \Gamma$ define the identification object

\begin{equation*} \mathsf{Id}_A\;\;:=\;\;A^{\Delta [1]} . \end{equation*}

Because $(\!-\!)^{\!\Delta [1]}$ is the right adjoint in (6), it preserves limits, yielding the natural isomorphism

(7) \begin{equation} \Bigl (\prod _{n:\mathbf N} Y_n\Bigr )^{\!\Delta [1]} \;\cong \; \prod _{n:\mathbf N} Y_n^{\Delta [1]} . \end{equation}

Let

\begin{equation*} d,c\;:\;\prod _{n:\mathbf N}\texttt {Tm}(\Gamma ,Y_n), \qquad h_n\;:\;\texttt {Tm}\bigl (\Gamma ,\mathsf{Id}_{Y_n}(d_n,c_n)\bigr ). \end{equation*}

Transport the tuple $(h_n)_{n:\mathbf N}$ across (7); via the slice/section identification, this yields a single

\begin{equation*} h\;:\;\texttt {Tm} \bigl (\Gamma ,\mathsf{Id}_{\Omega _\Gamma (Y)}(d,c)\bigr ). \end{equation*}

Restricting $h$ along the projections recovers each $h_n$ , so the correspondence is bijective. Because (7) is natural in $\Gamma$ (it is the mate of (6)), clause (2) of Definition 3.14 is satisfied naturally.

In Theorem3.17, we provide a class of two-level CwFs that satisfy the axiom that $\mathbb{N}^e$ is a cofibrant exo-type.

Proof. Recall that $\mathbb{N}^e : \widehat {\texttt {Ty}}(1_{\widehat {\mathcal{C}}})$ . For any $P :\widehat {\mathcal{C}}$ , the context morphism $\sigma : P \rightarrow 1_{\widehat {\mathcal{C}}}$ is unique, so we omit substitutions over such morphisms and write $P.\mathbb{N}^e$ instead of $P.\mathbb{N}^e[\sigma ]$ .

First, we define the map $\Theta ^{\texttt {Ty}}_{P}:\texttt {Ty}^f(P.\mathbb{N}^e)\rightarrow \texttt {Ty}^f(P)$ . For any $Y : \texttt {Ty}^f(P.\mathbb{N}^e) = \widehat {\mathcal{C}}(P.\mathbb{N}^e, \texttt {Ty})$ , we need $\Theta ^{\texttt {Ty}}_{P}(Y): \texttt {Ty}^f(P)=\widehat {\mathcal{C}}(P, \texttt {Ty})$ . We denote $\Theta ^{\texttt {Ty}}_{P}(Y)$ by $\tilde {Y}$ for easier reading. Now, for $\Gamma : \mathcal{C}$ and $x : P_{\Gamma }$ , we define

\begin{equation*}\tilde {Y}_{\Gamma }(x)\;:=\; \Omega _{\Gamma }\left ( (Y_{\Gamma }(x,n))_n\right )\end{equation*}

where the map $\Omega _{\Gamma } : \prod _{\mathbf{N}} \texttt {Ty}(\Gamma ) \rightarrow \texttt {Ty}(\Gamma )$ is obtained by the assumption of having exo-nat products, Definition 3.14. The definition makes sense since we have $Y_{\Gamma }(x,n) : \texttt {Ty}(\Gamma )$ . We want to show

\begin{equation*} \mathcal{c}_P(\tilde {Y}) \cong {\prod }^e(\mathbb{N}^e, \mathcal{c}_{P.\mathbb{N}^e}(Y)).\end{equation*}

Both are elements in $\widehat {\texttt {Ty}}(P)$ , namely, presheaves over $\int P$ . Thus, it is enough to define a natural transformation $G: \mathcal{c}_P(\tilde {Y}) \rightarrow \prod ^e(\mathbb{N}^e, \mathcal{c}_{P.\mathbb{N}^e}(Y))$ such that for any $\Gamma : \mathcal{C}$ and $x : P_{\Gamma }$ , the map

\begin{equation*}G_{\Gamma ,x} : \mathcal{c}_P(\tilde {Y})(\Gamma ,x) \rightarrow {\prod }^e(\mathbb{N}^e, \mathcal{c}_{P.\mathbb{N}^e}(Y)) (\Gamma ,x)\end{equation*}

is an isomorphism of sets, namely, a bijection. If we elaborate on these a little further, we obtain the following. By definition, $\mathcal{c}_P(\tilde {Y})(\Gamma ,x)=\texttt {Tm}(\Gamma , \tilde {Y}_{\Gamma }(x))$ . Also, ${\prod }^e(\mathbb{N}^e, \mathcal{c}_{P.\mathbb{N}^e}(Y)) (\Gamma ,x)$ consists of the elements

\begin{equation*}f : \prod _{\substack {\Delta : \mathcal{C} \\ \sigma : \Delta \rightarrow \Gamma \\ n : \mathbb{N}^e (\Delta , x[\sigma ])}} \mathcal{c}_{P.\mathbb{N}^e}(Y)(\Delta , x[\sigma ], n)(=\texttt {Tm}(\Delta , Y_{\Delta }(x[\sigma ],n)))\end{equation*}

such that if $\Upsilon : \mathcal{C}$ and $\tau : \Upsilon \rightarrow \Delta$ , then $f(\Delta , \sigma , n)[\tau ]=f(\Upsilon , \sigma \circ \tau , n[\tau ])$ . By the definition of $\mathbb{N}^e$ , we have $\mathbb{N}^e(\Delta ,x[\sigma ])=\mathbf{N}$ , external natural number set, and having exo-nat products provides us

\begin{equation*}\prod _{n :\mathbf{N}} \texttt {Tm}(\Delta , Y_{\Delta }(x[\sigma ],n)) \cong \texttt {Tm}(\Delta , \tilde {Y}(x[\sigma ])).\end{equation*}

Thus, the range of $G_{\Gamma ,x}$ can be written as

\begin{equation*}f : \prod _{\substack {\Delta : \mathcal{C} \\ \sigma : \Delta \rightarrow \Gamma }} \texttt {Tm}(\Delta , \tilde {Y}_{\Delta }(x[\sigma ]))\end{equation*}

such that if $\Upsilon : \mathcal{C}$ and $\tau : \Upsilon \rightarrow \Delta$ , then $f(\Delta , \sigma )[\tau ]=f(\Upsilon , \sigma \circ \tau )$ . This elaboration allows us to easily perceive that this function is a bijection because it is a standard application of the Yoneda Lemma. The naturality condition of this operation is also easily satisfied because the substitution is functorial. Thus, we have confirmed the first stage of our claim.

By Lemma 3.15 the contractibility requirement is already satisfied; therefore, $\mathbb{N}^e$ is a cofibrant exo-type in the presheaf two-level CwF.