2.1 Internal languages

Formally we will work with models of type theories, but we will construct types and terms of those models in a syntactic way using their internal languages. Let $\mathcal{E} = (\mathbb{C}^{\mathcal{E}}, \mathbb{T}^{\mathcal{E}}, \mathbb{E}^{\mathcal{E}},\dots)$ be a model of a type theory. For a context $\Gamma \in \mathbb{C}^{\mathcal{E}}$ and a type $\Gamma \vdash X$ , we introduce a variable x and write $(\Gamma, x : X)$ for the context $\Gamma.X$ . For another type $\Gamma \vdash Y$ , the weakening $\Gamma, x : X \vdash Y$ is interpreted as the reindexing $\Gamma.X \vdash Y \cdot \chi(X)$ . For an element $\Gamma \vdash a : X$ and a type $\Gamma, x : X \vdash Y(x)$ , the substitution $\Gamma \vdash Y(a)$ is interpreted as the reindexing $\Gamma \vdash Y \cdot \bar{a}$ , where $\bar{a} : \Gamma \to \Gamma.X$ is the section of $\Gamma.X \to \Gamma$ corresponding to the element $\Gamma \vdash a : X$ . All type and term constructors of the type theory are soundly interpreted in $\mathcal{E}$ in a natural way. Note that types and terms built in the internal language are stable under reindexing.

2.2 W-types with reductions

We will later use W-types with reductions to construct higher inductive types. So that we can use them internally in type theory we give below a new, split formulation. This is based on the non-dependent special case of the version in Swan (Reference Swan2018).

Let $\mathcal{E} = (\mathbb{C}^{\mathcal{E}}, \mathbb{T}^{\mathcal{E}}, \mathbb{E}^{\mathcal{E}},\dots)$ be a model of a type theory with dependent product types, dependent sum types and extensional identity types. A polynomial with reductions over a context $\Gamma \in \mathbb{C}^{\mathcal{E}}$ consists of the following data:

• a type $\Gamma \vdash Y$ of constructors;

• a type $\Gamma, y : Y \vdash X(y)$ of arities;

• a type $\Gamma, y : Y \vdash R(y)$ together with an element $\Gamma, y : Y, r : R(y) \vdash k(y, r) : X(y)$ which we refer to as the reductions.

An algebra for a polynomial with reductions (Y, X, R, k) over $\Gamma \in \mathbb{C}^{\mathcal{E}}$ is a type $\Gamma \vdash W$ together with an element $\Gamma, y : Y, \alpha : X(y) \to W \vdash s(y, \alpha) : W$ such that $\Gamma, y : Y, \alpha : X(y) \to W, r : R(y) \vdash s(y, \alpha) =\alpha(k(y, r))$ . Algebras for (Y, X, R, k) form a category in the obvious way and we refer to the initial algebra (if it exists) as the W-type with reductions for the polynomial with reductions.

3.1 Higher inductive types in Orton–Pitts models

We are still working with a model $\mathcal{E}$ of type theory that satisfies Assumption 4. We will show how to construct higher inductive types in $\widetilde{\mathcal{E}}$ . Our techniques are fairly general, although we will focus on the higher inductive types (HITs) that we will need for the main theorem. The techniques developed by Coquand, Huber and MÖrtberg in Coquand et al. (Reference Coquand, Huber and MÖrtberg2018) are already very close to working in arbitrary Orton–Pitts models. The main exception is that the underlying objects for the HITs are given by certain initial algebras, which are constructed directly for cubical sets. This definition does not quite work for cubical assemblies for two reasons. Firstly we are using a different cube category, and secondly we are working internally in assemblies. Rather than proving the same results again for cubical assemblies, we will use a more general approach based on W-types with reductions that covers both cases. The first author already showed in Swan (Reference Swan2018), Section 4 that (non-split) W-types with locally decidable reductions can be constructed in any category of presheaf assemblies and we will see later how to ensure that we get in fact split W-types with reductions in presheaf assemblies.

Finally, we will also make some minor adjustments related to the fact that we do not assume the interval object has reversals.

When we construct higher inductive types, we will use formulations based on $\mathsf{Path}$ types, following Coquand, Huber and MÖrtberg. Technically these formulations can only be stated in cubical type theory, and not in intensional type theory in general. However, it is straightforward to derive versions based on $\mathsf{Id}$ types using the equivalence of $\mathsf{Path}$ and $\mathsf{Id}$ types, which are then valid in $\widetilde{\mathcal{E}}$ . We note that although computation rules hold definitionally for both point and path constructors for the $\mathsf{Path}$ type versions, after translating to $\mathsf{Id}$ types, the definitional equality only holds for point constructors. However, neither definitional equality will be needed for our end result.

We will review a key idea in the techniques developed by Coquand et al. (Reference Coquand, Huber and MÖrtberg2018) for constructing higher inductive types. We first recall the Orton–Pitts definition of Kan composition.

We now recall the Coquand–Huber–MÖrtberg definitions of homogeneous Kan composition and transport. In order to apply their techniques to arbitrary Orton–Pitts models, we will make the following minor adjustment. They work in a setting where the interval type has an involution operation, allowing them to use a ‘one directional’ definition of Kan composition. That is, instead of the definition above, which takes a direction argument $\varepsilon \in \{0, 1\}$ , they only consider the simpler definition with $\varepsilon$ fixed as 0, where the other direction can be derived using the involution operator, if necessary. In the more general setting of an arbitrary Orton–Pitts model, we need to follow the Orton–Pitts definition of Kan composition, and similarly define homogenous composition and transport so that they take a direction as input.

We can now see the key lemma used in the construction of higher inductive types.

Proof. This appears as Coquand et al. (Reference Coquand, Huber and MÖrtberg2018, Lemma 2.5). We simply observe that given a transport operator for both directions $\varepsilon \in \{0, 1\}$ we can produce also a Kan composition operator for both directions, by applying the Coquand–Huber–MÖrtberg argument twice, once for each direction.

We can also think of types with homogeneous Kan composition as ‘locally fibrant.’ We show below how to use W-types with reductions to construct local fibrant replacements, i.e. how to freely add homogeneous Kan composition to arbitrary types.

• When $a : A$ , we add an element $\mathsf{inc}(a)$ to $\mathsf{LFR}(A)$ .

• When $\varphi : \mathbb{F}$ , $\varepsilon \in \{0, 1\}$ and $u : \sum_{i : \mathbb{I}} ((i = \varepsilon) \vee \varphi) \;\to\; \mathsf{LFR}(A)$ , we add an element $\mathsf{hcomp}(\varphi, \varepsilon, u)$ to $\mathsf{LFR}(A)$ .

• If $p : \varphi$ and $\varepsilon$ and u are as above then $\mathsf{hcomp}(\varphi, \varepsilon, u)$ reduces to $u(1 - \varepsilon, p)$ .

Formally, we define the constructors Y to be the coproduct $A + (\mathbb{F} \times 2)$ . We take the arity $X(\mathsf{inl}(a))$ to be the empty type for $a : A$ and $X(\mathsf{inr}(\varphi, \varepsilon))$ to be $\sum_{i : \mathbb{I}} \varphi \vee (i = \varepsilon)$ for $(\varphi, \varepsilon) : \mathbb{F} \times 2$ . We take the reductions $R(\mathsf{inl}(a))$ to be $\bot$ for $a : A$ and $R(\mathsf{inr}(\varphi, \varepsilon))$ to be $\varphi$ together with the map $p : \varphi \vdash (1 - \varepsilon, p) : \sum_{i : \mathbb{I}} \varphi \vee (i = \varepsilon)$ .

Theorem. The model $\widetilde{\mathcal{E}}$ supports suspensions.

Proof. Suppose we are given a type $\Gamma \vdash_{\widetilde{\mathcal{E}}} X$ . We first construct the naÏve suspension, $\mathsf{Susp}_0(X)$ as the pushout below.

We next take the local fibrant replacement, to get $\mathsf{LFR}(\mathsf{Susp}_0(X))$ . This is then an initial $\mathsf{Susp}(X)$ algebra, as defined by Coquand et al. (Reference Coquand, Huber and MÖrtberg2018), Section 2.2 and so we can then proceed with the same proof as they do there, observing that their argument produces by symmetry a transport operator in both directions.

Theorem The model $\widetilde{\mathcal{E}}$ supports propositional truncation.

Proof. Suppose we are given a type $\Gamma \vdash_{\widetilde{\mathcal{E}}}A$ . We first define the underlying object of $\| A \|$ to be the W-type with reductions defined as follows.

• When $a : A$ , we add an element $\mathsf{inc}(a)$ to $\| A \|$ .

• When $\varphi : \mathbb{F}$ , $\varepsilon \in \{0, 1\}$ and $u : \sum_{i : \mathbb{I}} ((i = \varepsilon) \vee \varphi) \;\to\; \|A \| $ , we add an element $\mathsf{hcomp}(\varphi, \varepsilon, u)$ to $\| A \|$ .

• If $p : \varphi$ and $\varepsilon$ and u are as above, then $\mathsf{hcomp}(\varphi, \varepsilon, u)$ reduces to $u(1-\varepsilon, p)$ .

• If $x, y : \| A \|$ and $i : \mathbb{I}$ , then $\| A \|$ contains an element of the form $\mathsf{sq}(x, y, i)$ .

• If x, y, i are as above and $i = 0$ , then $\mathsf{sq}(x, y, i)$ reduces to x.

• If x, y, i are as above and $i = 1$ , then $\mathsf{sq}(x, y, i)$ reduces to y.

Formally, we define this by taking the coproduct of two polynomials with reductions. The first is the one we used before for $\mathsf{LFR}$ . The second has constructors $Y := \mathbb{I}$ , with the arity defined by $X(i) := 2$ , and reductions $R(i) := (i = 0) \lor (i = 1)$ together with the map $p : (i = 0) \lor (i = 1) \vdash k(p) : 2$ defined by $k(p) = 0$ if $p : i = 0$ and $k(p) = 1$ if $p : i = 1$ .

The remainder of the proof is the same as the syntactic description of propositional truncation by Coquand et al. (Reference Coquand, Huber and MÖrtberg2018), Section 3.3.4.

We now construct a new higher inductive type, which is a simplified version of the higher inductive type $\mathcal{J}_F$ defined by Rijke et al. (Reference Rijke, Shulman and Spitters2020, Section 2.2). Given families of types $\Gamma \vdash_{\widetilde{\mathcal{E}}} A$ and $\Gamma, a : A \vdash_{\widetilde{\mathcal{E}}} B(a)$ we will construct a higher inductive type $\mathcal{K}^\Gamma_B$ defined as follows.

• When $a : A$ and $f : B(a) \to \mathcal{K}_B$ , we add an element $\mathsf{ext}(a, f)$ to $\mathcal{K}_B$ .

• When $a : A$ , $f : B(a) \to \mathcal{K}_B$ and $b : B(a)$ we add an element $\mathsf{isext}(a, f, b)$ to $\mathsf{Path}(\mathsf{ext}(a, f), f(b))$ .

We require that $\mathcal{K}_B$ satisfies the following elimination rule. Suppose we are given a family of types $\Gamma, x : \mathcal{K}_B \vdash_{\widetilde{\mathcal{E}}} P(x)$ together with the terms below.

\begin{align*} R &: \prod_{a : A} \prod_{f : B(a) \to \mathcal{K}_B} \left(\prod_{b : B(a)} P(f(b))\right) \to P(\mathsf{ext}(a, f)) \\[4pt] S &: \prod_{a : A} \prod_{f : B(a) \to \mathcal{K}_B} \,\prod_{f' : \prod_{b : B(a)} P(f(b))} \,\prod_{b : B(a)} \prod_{i : \mathbb{I}} P(\mathsf{isext}(a, f, b)(i))\end{align*}

Suppose further that S satisfies the equalities

\begin{align*} S(a, f, f', b, 0) &= R(a, f, f') \\[4pt] S(a, f, f', b, 1) &= f'(b)\end{align*}

Then we have a choice of term $\Gamma, x : \mathcal{K}_B \vdash s(x) : P(x)$ satisfying the following computation rules for $a : A$ , $f : B(a) \to \mathcal{K}_B$ and $b :B(a)$ .

\begin{align*} s(\mathsf{ext}(a, f)) &= R(a, f, s \circ f) \\[4pt] s(\mathsf{isext}(a, f, b)(i)) &= S(a, f, s \circ f, b, i)\end{align*}

Moreover the choice of term is strictly preserved by reindexing.

We use the techniques developed by Coquand, Huber and MÖrtberg together with W-types with reductions for constructing the actual objects. In order to give $\mathcal{K}^\Gamma_B$ the structure of a fibration, we need to define a composition operator. We will do this by freely adding an $\mathsf{hcomp}$ operator, and then combining it with a transport operator, which we will explicitly define.

Definition 10. Let $\Gamma \vdash_{\mathcal{E}} X$ be a type. We define the naÏve cone, $\mathsf{Cone}(X)$ to be the following pushout ^{Footnote 3} .

We can now define $\mathcal{K}^\Gamma_B$ to be the following W-type with reductions.

• When $a : A$ , $c : \mathsf{Cone}(B(a))$ and $f : B(a) \to \mathcal{K}_B$ , we add an element $\mathsf{pastecone}(a, c, f)$ to $\mathcal{K}_B$ .

• If a, c, f are as above and c is of the form $\mathsf{inr}(b, 1)$ for $b : B(a)$ , then $\mathsf{pastecone}(a, c, f)$ reduces to f(b).

• When $\varphi : \mathbb{F}$ , $\varepsilon : \{ 0, 1 \}$ and $u : \sum_{i : \mathbb{I}} ((i = 0) \vee \varphi) \;\to\; \mathcal{K}_B$ , we add an element $\mathsf{hcomp}(\varphi, \varepsilon, u)$ to $\mathcal{K}_B$ .

• If $p : \varphi$ , and $\varepsilon$ and u are as above then $\mathsf{hcomp}(\varphi, \varepsilon, u)$ reduces to $u(1 - \varepsilon, p)$ .

To check that this really is a W-type with reductions, we need to define the polynomial with reductions. We take it to be the coproduct of the following two polynomials with reductions.

We define the first component of the coproduct as follows. We take the constructors Y to be $\sum_{a : A} \mathsf{Cone}(B(a))$ and the arities X(a, c) to be B(a). We take the reductions $R(a, \mathsf{inl}(\ast))$ to be $\bot$ and $R(a, \mathsf{inr}(b, i))$ to be $(i = 1)$ together with the map $R(a, \mathsf{inr}(b, i)) \vdash b : B(a)$ . Note that $R : \left(\sum_{a : A}\mathsf{Cone}(B(a))\right) \to \mathbb{F}$ is well-defined because we have $(0 = 1) = \bot$ by propositional extensionality.

The second component in the coproduct is the polynomial with reductions that we used for local fibrant replacement.

Proof. We give a proof for the direction $\varepsilon =0$ . The construction for the other direction is exactly the same by symmetry.

Suppose we are given $\varphi : \mathbb{F}$ and a path $\gamma$ in $\Gamma$ which is constant on $\varphi$ . We need to define a transport operator, which is a map $t : \mathcal{K}_{B(\gamma(0))} \to \mathcal{K}_{B(\gamma(1))}$ such that t is the identity when $\varphi$ is true. Formally this map can be defined by giving an appropriate algebra structure on $\mathcal{K}_{B(\gamma(1))}$ and then using the initiality of $\mathcal{K}_{B(\gamma(0))}$ . However, for clarity we will present the proof as an argument by higher recursion on the definition of $\mathcal{K}_{B(\gamma(0))}$ .

We need to show how to define $t(\mathsf{pastecone}(a, c, f))$ and $t(\mathsf{hcomp}(\psi, \varepsilon, u))$ , and then check that the definition respects the reduction equations. For the latter, we define the transport operator so that it preserves the $\mathsf{hcomp}$ structure, which determines it uniquely, following Coquand et al. (Reference Coquand, Huber and MÖrtberg2018). For the former, we recall that $\mathsf{Cone}(B(a))$ was defined as a pushout, and so we can split into a further two cases. Either c is of the form $\mathsf{inl}(\ast)$ , or it is of the form $\mathsf{inr}(b, i)$ where $b : B(a)$ and $i : \mathbb{I}$ . Now in addition to the reduction equation, we have to also satisfy $t(\mathsf{inl}(\ast)) = t(\mathsf{inr}(b, 0))$ in order to eliminate out of the pushout.

Write $t_A$ for the transport $A(\gamma(0)) \to A(\gamma(1))$ and $t_B$ for the transport $\prod_{a : A(\Gamma(0))} B(a) \to B(t_A(a)) $ ensuring that $t_A(a) = a$ and $t_B(b) = b$ when $\varphi = \top$ , for all $a : A(\gamma(0))$ and $b : B(a)$ . Write $t_B^{-1}$ for the homotopy inverse $\prod_{a : A(\Gamma(0))} B(t_A(a)) \to B(a)$ , again ensuring that $t_B^{-1}(b) = b$ when $\varphi = \top$ . Since we are only guaranteed the existence of a homotopy inverse, not a strict inverse, we do not necessarily have $t_B^{-1} \circ t_B = 1_{B(a)}$ . We can however construct paths $p : \prod_{a : A(\Gamma(0))} \prod_{b : B(a)} \mathbb{I} \to B(a)$ satisfying for all $a : A(\Gamma(0))$ and $b : B(a)$ that $p(a, b, 0) = t_B^{-1}(t_B(b)) $ and $p(a, b, 1) = b$ . Furthermore, we may assume that for any a, b and i, if $\varphi = \top$ then $p(a, b, i) = b$ .

We define $t(\mathsf{pastecone}(a, \mathsf{inl}(\ast), f))$ to be of the form $\mathsf{pastecone}(t_A(a), \mathsf{inl}(\ast), f')$ , where we still need to define a function $f' : B(t_A(a)) \to \mathcal{K}_{B(\gamma(1))}$ . Note that we may assume by recursion that for each $b : B(a)$ , t(f(b)) has already been defined and belongs to $\mathcal{K}_{B(\gamma(1))}$ . Hence, we can simply define f’ to be $t \circ f \circ t_B^{-1}$ .

The obvious first attempt at defining $t(\mathsf{pastecone}(a, \mathsf{inr}(b, i), f))$ would be $\mathsf{pastecone}(t_A(a), \mathsf{inr}(t_B(b), i), t \circ f \circ t_B^{-1})$ . Note however that this does not satisfy the reduction equations. This is because when $i = 1$ , $\mathsf{pastecone}(a, \mathsf{inr}(b, i), f)$ reduces to f(b) and $\mathsf{pastecone}(t_A(a), \mathsf{inr}(t_B(b), i), t \circ f \circ t_B^{-1})$ reduces to $t(f(t_B^{-1}(t_B(b))))$ which is not necessarily strictly equal to t(f(b)). We fix this using the $\mathsf{hcomp}$ constructor, following the construction of homotopy pushouts in Coquand et al. (Reference Coquand, Huber and MÖrtberg2018), Section 2.3. We define $\psi : \mathbb{F}$ to be $\varphi \vee (i = 0) \vee (i = 1)$ . We then define $u : \sum_{j : \mathbb{I}} (\psi \vee (j = 0)) \to \mathcal{K}_{B(\gamma(a))}$ as follows.

\begin{equation*} u(j, \ast) :\equiv \begin{cases} \mathsf{pastecone}(t_A(a), \mathsf{inr}(t_B(b), i), t \circ f \circ t_B^{-1}) & j = 0 \\ \mathsf{pastecone}(a, \mathsf{inr}(b, i), t \circ f) & \varphi = \top \\ \mathsf{pastecone}(t_A(a), \mathsf{inl}(\ast), t \circ f \circ t_B^{-1}) & i = 0 \\ t(f(p(a, b, j))) & i = 1 \end{cases} \end{equation*}

We then define $t(\mathsf{pastecone}(a, \mathsf{inr}(b, i), f))$ to be $\mathsf{hcomp}(\psi, 0, u)$ . The reduction equation for $\mathsf{hcomp}$ then ensures that we do satisfy the reduction equation for $\mathsf{pastecone}$ and also retain the necessary equations for the pushout and furthermore ensures that the resulting map $t : \mathcal{K}_{B(\gamma(0))} \to \mathcal{K}_{B(\gamma(1))}$ is (one direction of) a transport operator.

Theorem We construct a fibration structure for each $\mathcal{K}_B$ , which is strictly preserved by reindexing.

Proof. By Lemmas 11 and 8.

Proof.

\begin{align*} \mathsf{ext}(a, f) &:\equiv \mathsf{pastecone}(a, \mathsf{inl}(\ast), f) \\ \mathsf{isext}(a, f, b)(i) &:\equiv \mathsf{pastecone}(a, \mathsf{inr}(b, i), f) \end{align*}

Proof. Suppose we are given a family of types $\Gamma, x : \mathcal{K}_B \vdash_{\widetilde{\mathcal{E}}} P(x)$ together with the terms below.

\begin{align*} R &: \prod_{a : A} \prod_{f : B(a) \to \mathcal{K}_B} \left(\prod_{b : B(a)} P(f(b))\right) \to P(\mathsf{ext}(a, f)) \\ S &: \prod_{a : A} \prod_{f : B(a) \to \mathcal{K}_B} \, \prod_{f' : \prod_{b : B(a)} P(f(b))} \, \prod_{b : B(a)} \prod_{i : \mathbb{I}} P(\mathsf{isext}(a, f, b)(i)) \end{align*}

We need to define a term $\Gamma, x : \mathcal{K}_B \vdash s(x) : P(x)$ satisfying the appropriate equalities. We define s by higher recursion on the construction of $\mathcal{K}_B$ . We first deal with the case $s(\mathsf{pastecone}(a, c, f))$ . Recalling that $\mathsf{Cone}(B(a))$ is defined as a pushout, we can split into the two cases $c = \mathsf{inl}(\ast)$ and $c = \mathsf{inr}(b, i)$ for some $b : B(a)$ and $i : \mathbb{I}$ .

We define

\begin{align*} s(\mathsf{pastecone}(a, \mathsf{inl}(\ast), f)) &:\equiv R(a, f, s \circ f) \\ s(\mathsf{pastecone}(a, \mathsf{inr}(b, i), f)) &:\equiv S(a, f, s \circ f, b, i) \end{align*}

It is straightforward to check that this does preserve the reduction and pushout equations and so does give a well defined map. One can show it is a section again by higher recursion and the computation rules are satisfied by definition.

Finally, to define $s(\mathsf{hcomp}(\varphi, \varepsilon, u))$ , we use the fibration structure on $\Gamma, x : \mathcal{K}_B \vdash_{\mathcal{E}} P(x)$ .

3.2 Internal cubical models

Let $\mathcal{S}$ be a model of dependent type theory with dependent product types, dependent sum types, extensional identity types, unit type, finite colimits, W-types and a countable chain of universes. We also assume that every context of $\mathcal{S}$ is isomorphic to $1.X$ for some type $1 \vdash_{\mathcal{S}} X$ . In particular, the category $\mathbb{C}^{\mathcal{S}}$ is finitely complete so that internal categories in $\mathbb{C}^{\mathcal{S}}$ make sense. Let $\Box$ denote the internal category in $\mathbb{C}^{\mathcal{S}}$ in which the objects are the natural numbers and the morphisms from n to m are the order-preserving functions $\mathbf{2}^{n} \to \mathbf{2}^{m}$ . Note that $\mathcal{S}$ has a natural number object since it has W-types. Spitters (Reference Spitters2016) has observed that $\Box$ is equivalent to the opposite of the category of free finitely generated distributive lattices. We will refer to internal presheaves over $\Box$ as internal cubical objects.

Theorem Under those assumptions, the category of internal cubical objects in $\mathcal{S}$ is part of a model of type theory that satisfies Assumption 4.

It is shown in Orton and Pitts (Reference Orton and Pitts2018) that, when $\mathcal{S} = \mathbf{Set}$ , the category of presheaves over $\Box$ satisfies all the axioms of Orton and Pitts if we take $\mathbb{F}$ to be the presheaf of locally decidable propositions. The proof works for an arbitrary $\mathcal{S}$ and one can show that the category of internal cubical objects in $\mathcal{S}$ is part of a model of type theory satisfying Assumption 3 except the existence of cofibrant W-types with reductions (see also Uemura Reference Uemura2019). To construct cofibrant W-types with reductions, we recall the following from Swan (Reference Swan2018).

Theorem Let $\mathcal{E}$ be a locally cartesian closed category with finite colimits and disjoint coproducts and W-types, and let $\mathbf{C}$ be an internal category in $\mathcal{E}$ . Then the category $\mathcal{P}(\mathbf{C})$ of internal presheaves over $\mathbf{C}$ has all locally decidable W-types with reductions.

We furthermore observe that one can show that this construction is stable under pullback up to isomorphism using a technique similar to the one used by Gambino and Hyland for ordinary W-types (Gambino and Hyland Reference Gambino and Hyland2004). The reason is that pointed polynomial endofunctors are stable under pullback because they are constructed from $\Sigma$ types, $\Pi$ types and pushouts, all of which are preserved by pullback, and in locally cartesian closed categories, the initial algebras of such pointed endofunctors are also stable under pullback. However, to ensure that the construction is strictly preserved requires a little more work.

We show how to use the non-split version above to construct split W-types with reductions. The essential idea is to carry out the construction given above “pointwise,” expanding out the method suggested by Coquand et al. (Reference Coquand, Huber and MÖrtberg2018, Section 2.2). Since we define cubical sets here as a category of presheaves in the usual, contravariant sense, we work with contravariant presheaves here, although the original proof in Swan (Reference Swan2018) is phrased in terms of covariant presheaves. We also make minor adjustments to fit with the ‘split’ version appearing in Section 2.2.

Suppose that we are given a context $\Gamma \in \mathcal{P}(\mathbf{C})$ together with a type $Y \in \mathcal{P}(\int_\mathbf{C} \Gamma)$ , a type $X \in \mathcal{P}(\int_C \{Y\})$ , a locally decidable monomorphism $R \rightarrowtail Y$ and a map $k : \prod_{y : R} X(y)$ over $\int_\mathbf{C} \Gamma$ .

We need to show how to define a strict version of the W-type with reductions W(Y, X, R). We will refer to the new strict version as W’(Y, X, R). This should be an element of $\mathcal{P}(\int_\mathbf{C} \Gamma)$ , so in particular we need to define a family of types $W'(Y, X, R)(c, \gamma)$ indexed by objects c of $\mathbf{C}$ and elements $\gamma : \Gamma(c)$ .

We fix such a c and $\gamma$ . We first note that we have a locally decidable polynomial with reductions $Y_\gamma, X_\gamma, R_\gamma$ in the internal presheaf category $\mathcal{P}(\int_\mathbf{C} \mathbf{C}(-,c))$ given by reindexing along the map $\mathbf{C}(-,c) \to \Gamma$ given by Yoneda. We then carry out the ‘non-strict’ construction to get a presheaf $W(Y_\gamma, X_\gamma, R_\gamma)$ on $\int_\mathbf{C} \mathbf{C}(-,c)$ and finally we define $W'(Y, X, R)(c, \gamma)$ to be $W(Y_\gamma, X_\gamma, R_\gamma) (c, 1_c)$ .

For completeness, we unfold the definitions to obtain the following explicit description of $W'(Y, X, R)(c, \gamma)$ . We first define the dependent W-type $N_0$ of normal forms indexed by the objects (d, f) of $\int_\mathbf{C} \mathbf{C}(-,c)$ .

If (d, f) is an object of $\int_\mathbf{C} \mathbf{C}(-,c)$ , we add an element to $N_0(d, f)$ of the form $\sup\!(y, \alpha)$ whenever y is an element of $Y(d, \Gamma(f)(\gamma))$ that does not belong to the subobject $R(d, \Gamma(f)(\gamma))$ and $\alpha$ is an element of the following type.

\begin{equation*} \prod_{g : e \to d} N_0(e, f \circ g)^{X(f \circ g, \Gamma(f\circ g)(\gamma), Y(g)(y))}\end{equation*}

The next step is to define maps $N_0(d, f) \to N_0(e, f \circ g)$ whenever $g \colon e \to d$ and $f \colon d \to c$ in $\mathbf{C}$ . Say that we are given an element of $N_0(d, f)$ of the form $\sup\!(y, \alpha)$ . We recall that $N_0(g)(\!\sup\!(y, \alpha))$ is defined by splitting into cases depending on whether or not y belongs to the subobject $R(d, \Gamma(f)(\gamma))$ . If it does, we define $N_0(g)(\!\sup\!(y, \alpha))$ to be $\alpha(g,k(y))$ . Otherwise, we define $N_0(g)(\!\sup\!(y, \alpha))$ to be $\sup\!(Y(g)(y), \alpha')$ where $\alpha'(h, x)$ is defined to be $\alpha(g \circ h, x)$ .

We then define N(d, f) for each $f : d \to c$ to be the subobject of $N_0(d, f)$ consisting of hereditarily natural elements and verify that this does indeed define a presheaf on $\int_\mathbf{C} \mathbf{C}(-,c)$ . But this is identical to Swan (Reference Swan2018), Section 4 so we omit the details.

If we then define $W'(Y, X, R)(c, \gamma)$ to be $N(c, 1_c)$ , then this is strictly stable under reindexing by definition.

One can construct by recursion an isomorphism between N(d, f) and $W(Y, X, R)(d, \Gamma(f)(\gamma))$ for each $f : d \to c$ . In particular, this gives us an isomorphism between $N(c, 1_c)$ and $W(Y, X, R)(c, \gamma)$ , and so we have a canonical isomorphism between $W'(Y, X, R)(c, \gamma)$ and $W(Y, X, R)(c, \gamma)$ . It follows that we can assign an initial algebra structure to $W'(Y, X, R)(c, \gamma)$ by transferring the algebra structure on $W(Y, X, R)(c, \gamma)$ via the isomorphism.

3.3 Discrete types

We introduce a class of types in an Orton–Pitts model for future use. Let $\mathcal{E}$ be a model of type theory satisfying Assumption 4.

The proofs of the following propositions are found in Uemura (Reference Uemura2019).

4.1 Failure of Church’s thesis in internal cubical models

Let $\mathcal{S}$ be a model of type theory as in Section 3.2. We have seen that the category $\mathcal{P}(\Box)$ of internal cubical objects in $\mathcal{S}$ is part of a model of type theory satisfying Assumption 3. In this section, we show the following theorem.

Theorem The negation of Church’s Thesis holds in the model of univalent type theory $\widetilde{\mathcal{P}(\Box)}$ .

To prove Theorem 4.1, we recall from Uemura (Reference Uemura2019) the notion of a codiscrete presheaf and some useful propositions. The constant presheaf functor $\Delta : \mathcal{S} \to \mathcal{P}(\Box)$ extends to a morphism of cwf’s and preserves (at least up to isomorphism) several type constructors. Here, we only need the following.

A constant presheaf $\Delta X$ is regarded as a type in $\widetilde{\mathcal{P}(\Box)}$ by the following proposition and Proposition 16.

For types $1 \vdash_{\mathcal{S}} X$ and $x : X \vdash_{\mathcal{S}} Y(x)$ , one can define a type $x : \Delta X \vdash_{\mathcal{P}(\Box)} \nabla_{X}Y(x)$ called the codiscrete presheaf which has the following properties.

Proof of Theorem 4.1. We define types $f : \mathbb{N} \to \mathbb{N} \vdash C'(f) :\equiv \sum_{e : \mathbb{N}}\prod_{x : \mathbb{N}}\sum_{z : \mathbb{N}}T(e, x, z) \times U(x) = f(x)$ and $f : \mathbb{N} \to \mathbb{N} \vdash C(f) :\equiv \left\|{C'(f)}\right\|$ . Let N denote the natural number object in $\mathcal{S}$ . Then Church’s Thesis is interpreted in $\widetilde{\mathcal{P}(\Box)}$ as $\prod_{f : \Delta(N \to N)}[\![{C}]\!]^{\widetilde{\mathcal{P}(\Box)}}(f)$ by Proposition 19. We will construct two functions in $\widetilde{\mathcal{P}(\Box)}$ :

• $\prod_{f : \Delta(N \to N)}[\![{C}]\!]^{\widetilde{\mathcal{P}(\Box)}}(f) \to \nabla_{N \to N}[\![{C'}]\!]^{\mathcal{S}}(f)$ ;

• $\left(\prod_{f : \Delta(N \to N)}\nabla_{N \to N}[\![{C'}]\!]^{\mathcal{S}}(f)\right) \to \mathbf{0}$ .

Then we readily get a function $\left(\prod_{f : \Delta(N \to N)}[\![{C}]\!]^{\widetilde{\mathcal{P}(\Box)}}(f)\right) \to \mathbf{0}$ .

For the former one, it suffices to give a function $[\![{C'}]\!]^{\widetilde{\mathcal{P}(\Box)}}(f) \to \nabla_{N \to N}[\![{C'}]\!]^{\mathcal{S}}(f)$ for all $f : \Delta(N \to N)$ by the recursion principle of the propositional truncation because the codomain is a proposition by Proposition 22. By the adjunction $(-)_{0} \dashv \nabla_{N \to N}$ , it suffices to give a function $[\![{C'}]\!]^{\widetilde{\mathcal{P}(\Box)}}_{0} \to [\![{C'}]\!]^{\mathcal{S}}$ but we have an isomorphism $[\![{C'}]\!]^{\widetilde{\mathcal{P}(\Box)}} \cong (\Delta[\![{C'}]\!]^{\mathcal{S}})_{0} = [\![{C'}]\!]^{\mathcal{S}}$ by Proposition 19.

For the latter function, observe that $\prod_{f : \Delta(N \to N)}\nabla_{N \to N}[\![{C'}]\!]^{\mathcal{S}}(f) \cong \nabla_{1}\left(\prod_{f : N \to N}[\![{C'}]\!]^{\mathcal{S}}(f)\right)$ and that $\nabla_{1}\mathbf{0} \cong \mathbf{0}$ . Then we apply $\nabla_{1}$ to the function $\left(\prod_{f : N \to N}[\![{C'}]\!]^{\mathcal{S}}(f)\right) \to \mathbf{0}$ in $\mathcal{S}$ obtained from the inconsistency of Church’s Thesis with the axiom of choice and function extensionality.

5.1 Null types in Orton–Pitts models

Let $\widetilde{\mathcal{E}}$ be an Orton–Pitts model.

Proof. We show that, for any $a : A$ , the function $k_{a} :\equiv \lambda x.\lambda b.x : X \to (B(a) \to X)$ is an isomorphism in the internal language of $\mathcal{E}$ . Since B is well-supported, $k_{a}$ is injective. To prove surjectivity, we assume that $f : B(a) \to X$ is given. By the well-supportedness of B, there exists some element $b : B(a)$ . We show that $f = k_{a}(f(b))$ . Assume $b' : B(a)$ is given. Since B is a proposition in $\widetilde{E}$ , we have a path $p : \mathbb{I} \to B(a)$ such that $p0 = b'$ and $p1 = b$ . By the discreteness of X the path $f \circ p : \mathbb{I} \to X$ is constant, which implies that $f(b') = f(b)$ . Hence, we have $f = k_{a}(f(b))$ by function extensionality.

We easily deduce the following corollaries.

Proposition 33. Let $a : A \vdash_{\widetilde{\mathcal{E}}} B(a)$ be a proposition and $\Gamma \vdash_{\widetilde{\mathcal{E}}} X$ and $\Gamma \vdash_{\widetilde{\mathcal{E}}} Y$ types. Then there exists a term of type

\begin{equation*} \Gamma \vdash \mathsf{isNull}_{B}(X) \to \mathsf{isNull}_{B}(Y) \to \mathsf{isNull}_{B}(X + Y). \end{equation*}

Proof. We proceed in the internal language of $\mathcal{E}$ . Suppose that X and Y has a B-null structure. Assume that $a : A$ is given. Since the function $(X + Y) \to (B(a) \to (X + Y))$ factors as

it suffices to show that the function $\Phi : ((B(a) \to X) + (B(a) \to Y)) \to (B(a) \to (X + Y))$ is an isomorphism. The injectivity of $\Phi$ follows from the well-supportedness of B. To prove the surjectivity, we assume that $f : B(a) \to (X + Y)$ is given. We show that $(\forall_{b : B(a)}fb \in X) \lor (\forall_{b : B(a)}fb \in Y)$ . Since B is well-supported, there exists some element $b_{0} : B(a)$ . We know that $fb_{0} \in X \lor fb_{0} \in Y$ . Suppose that $fb_{0} \in X$ . Assume $b : B(a)$ is given. Since B is a proposition in $\widetilde{\mathcal{E}}$ , we have a path $p : \mathbb{I} \to B(a)$ such that $p0 = b_{0}$ and $p1 = b$ . Since the exponential functor $(-)^{\mathbb{I}}$ preserves colimits because it has a right adjoint, we have $(\forall_{i : \mathbb{I}}f(pi) \in X) \lor (\forall_{i : \mathbb{I}}f(pi) \in Y)$ . Now $f(p0) \in X$ and thus we have $\forall_{i : \mathbb{I}}f(pi) \in X$ . In particular, $fb \in X$ . In a similar manner, we have $\forall_{b : B(a)}fb \in Y$ assuming $fb_{0} \in Y$ . Hence, we get $(\forall_{b : B(a)}fb \in X) \lor (\forall_{b : B(a)}fb \in Y)$ .

By Corollary 18, and Propositions 24, 30 and 33, for any type X defined in dependent type theory only using dependent product types, dependent sum types, identity types, unit type, disjoint finite coproducts and natural numbers, the underlying type of $[\![{X}]\!]^{\widetilde{\mathcal{E}}_{B}}$ is equal to $[\![{X}]\!]^{\widetilde{\mathcal{E}}}$ for any well-supported proposition B in $\widetilde{\mathcal{E}}$ . In particular, in this case if $[\![{X}]\!]^{\widetilde{\mathcal{E}}}$ is inhabited, then so is $[\![{X}]\!]^{\widetilde{\mathcal{E}}_{B}}$ .

Example 34. Markov’s Principle is the following axiom.

\begin{equation*} \forall_{\alpha : \mathbb{N} \to \mathbf{2}}\neg\neg(\exists_{n : \mathbb{N}}\alpha(n)) \to \exists_{n : \mathbb{N}}\alpha(n) \end{equation*}

It is equivalent to the type

\begin{equation*} \prod_{\alpha : \mathbb{N} \to \mathbf{2}}\prod_{p : (\prod_{n : \mathbb{N}}\alpha(n) \to \mathbf{0}) \to \mathbf{0}}\left\|{\sum_{n : \mathbb{N}}\alpha(n)}\right\|. \end{equation*}

For a decidable predicate $\alpha : \mathbb{N} \to \mathbf{2}$ , the proposition $\left\|{\sum_{n : \mathbb{N}}\alpha(n)}\right\|$ is equivalent to the type $\sum_{n : \mathbb{N}}\alpha(n) \times \prod_{k : \mathbb{N}}\alpha(k) \to n \leq k$ which is defined without propositional truncation. Hence, if the model $\mathcal{E}$ of extensional dependent type theory satisfies Markov’s Principle, then so does the model $\widetilde{\mathcal{E}}_{B}$ of univalent type theory for any well-supported proposition B in $\widetilde{\mathcal{E}}$ .

We now show how to define nullification operators in Orton–Pitts models. Following Rijke et al. (Reference Rijke, Shulman and Spitters2020), Section 2.2, we will first define an operator $\mathcal{J}_B$ , although we will only consider the case of nullification, since that is all we need here.

• When $x : X$ , then $\mathcal{J}_B(X)$ contains an element $\alpha^B_X(x)$ .

• When $a : A$ and $f : B(a) \to \mathcal{K}_B$ , then $\mathcal{J}_B(X)$ contains an element $\mathsf{ext}(a, f)$ .

• When $a : A$ , $f : B(a) \to \mathcal{K}_B$ and $b : B(a)$ then $\mathsf{Id}(\mathsf{ext}(a, f), f(b))$ contains an element $\mathsf{isext}(a, f, b)$ .

Proof. $\mathcal{J}_B(X)$ differs from $\mathcal{K}_B$ by having an extra point constructor $\alpha^B_X : X \to \mathcal{J}_B(X)$ .

We define A’ to be the type $A + X$ and define the family of types $a : A' \vdash B'(a)$ as follows.

\begin{align*} B'(\mathsf{inl}(a)) &:\equiv B(a) \\ B'(\mathsf{inr}(x)) &:\equiv 0 \end{align*}

We can then take $\mathcal{J}_B(X)$ to be $\mathcal{K}_{B'}$ , as defined in Section 3.1. We take $\alpha^B_X(x)$ to be $\mathsf{ext}(\mathsf{inr}(x), \bot_{\mathcal{K}_B})$ where $\bot_{\mathcal{K}_B}$ is the unique map from 0 to $\mathcal{K}_B$ .

Theorem. $\widetilde{\mathcal{E}}$ has a nullification operator $\mathcal{L}_{B}$ for every type $a : A \vdash_{\widetilde{\mathcal{E}}} B(a)$ .

Proof. This follows from Rijke et al. (Reference Rijke, Shulman and Spitters2020, Theorem 2.18), observing that for the case of nullification the pushout appearing there is just a suspension, which we have already shown how to implement in Theorem 3.1, and we showed in Lemma 35 how to implement their $\mathcal{J}$ operator.