2. Definition of $\mathsf{V}^0$ and Its Basic Properties

The ideas behind $\mathsf{V}^0$ trace back to The type theoretic interpretation of constructive set theory by Aczel (Reference Aczel, Macintyre, Pacholski and Paris1978). In op. cit., Aczel constructed a model of set theory in dependent type theory relying upon a non-trivial defined equality relation on the underlying type of the model in order to (hereditarily) force set-extensionality. This underlying type of the model is what we in modern parlance would call a $\mathsf{W}$ -type. To construct $\mathsf{V}^0$ , we opt to carve out a subtype of a $\mathsf{W}$ -type rather than take such a quotient. Instead of defining an equivalence relation which identifies the elements of the $\mathsf{W}$ -type which represent the same set, we shall identify a subtype of the $\mathsf{W}$ -type which contains only the canonical representations of each (iterative) set. Thus, we get a model of set theory in type theory where the equality is interpreted as the identity type and no further non-trivial identifications are required.

In this section, we will review the definition of $\mathsf{V}^0$ and prove some properties about it. In particular, we will show that $\mathsf{V}^0$ is an h-set. In order to define $\mathsf{V}^0$ , we start by recalling the $\mathsf{W}$ -type Aczel used: “the unrestricted iterative hierarchy”. It is the type of well-founded trees with branching types chosen freely from a fixed universe ${\mathcal{U}}_\ell$ .

Definition 1 (). Given a universe ${\mathcal{U}}_\ell$ , we define the type $\mathsf{V}^\infty_\ell$ as

\begin{equation*} {\mathsf {V}}^\infty_\ell := \mathsf {W}_{A : \mathcal {U}_\ell }\,A \end{equation*}

We will usually omit the universe level $\ell$ for ${\mathcal{U}}_\ell$ and ${\mathsf{V}}^\infty_\ell$ , and write simply $\mathcal{U}$ and $\mathsf{V}^\infty$ . When seeing $\mathsf{V}^\infty$ as a type of sets, an element $\mathsf{sup}\,A\,f :{\mathsf{V}^\infty }$ represents a set whose elements are indexed by the type $A : \mathcal{U}$ . The function $f : A\to {\mathsf{V}^\infty }$ picks out the element at each index. Since the function $f$ need not be injective, the same element can be picked out several times. Indeed, the role of Aczel’s equivalence relation on this type was to erase this multiplicity. If we instead omit this further identification, $\mathsf{V}^\infty$ can be seen as a type of multisets (Gylterud Reference Gylterud2020).

Notation 2. Given $x :{\mathsf{V}^\infty }$ , we follow Aczel (Reference Aczel, Macintyre, Pacholski and Paris1978) and define a pair of operations $\overline{x} : \mathcal{U}$ and $\widetilde x : \overline x\to {\mathsf{V}^\infty }$ , as follows:

\begin{equation*} \overline{\mathsf{sup}\,A\,f} \mathrel{:=} A \qquad \widetilde{\mathsf{sup}\,A\,f} \mathrel{:=} f \end{equation*}

We present two characterizations of equality in $\mathsf{V}^\infty$ as both are useful in different contexts. We note that both characterizations rely on univalence.

The first is an instance of a more general characterization of equality in $\mathsf{W}$ -types (Gylterud Reference Gylterud2020, Lemma 1). It states that two elements are equal if they have equivalent underlying indexing types and this equivalence is coherent with respect to the functions picking out the elements.

Theorem 3 (Gylterud Reference Gylterud2020, Theorem 1, ). For two elements $x,y :{\mathsf{V}^\infty }$ the canonical map

\begin{equation*} \left (x = y\right ) \to \left (\sum _{e :\,\overline {x} \simeq \overline {y}} \widetilde {x} \sim \widetilde {y} \circ e\right ) \end{equation*}

which sends $\mathsf{refl}$ to $(\mathsf{id\mbox{-}equiv}, \mathsf{refl\mbox{-}htpy})$ , is an equivalence.

The second characterization of equality in $\mathsf{V}^\infty$ states that two elements in $\mathsf{V}^\infty$ are equal when the functions picking out the elements are fiberwise equivalent. Intuitively, this means that they pick out the same elements the same number of times. One can think of this characterization of equality as a higher level generalization of the axiom of extensionality.

Theorem 4 (Gylterud Reference Gylterud2020, Theorem 2, ). For two elements $x,y :{\mathsf{V}^\infty }$ the canonical map

\begin{equation*} \left (x = y\right ) \to \prod _{z : {\mathsf {V}^\infty }} \mathsf {fib}\,\widetilde {x}\,z \simeq \mathsf {fib}\,\widetilde {y}\,z \end{equation*}

which sends $\mathsf{refl}$ to ${\lambda }z.\mathsf{id\mbox{-}equiv}$ , is an equivalence.

Proof. We reproduce the proof here for convenience. We have the following chain of equivalences:

\begin{align*} \left (x = y\right ) \simeq \left (\sum _{e :\,\overline{x}\, \simeq \,\overline{y}} \widetilde{x} \sim \widetilde{y} \circ e\right ) \simeq \left (\prod _{z :{\mathsf{V}^\infty }} \mathsf{fib}\,\widetilde{x}\,z \simeq \mathsf{fib}\,\widetilde{y}\,z\right ) \end{align*}

The first equivalence is the one constructed in Theorem3. The second equivalence is proven by Gylterud (Reference Gylterud2020, Lemma 5). One directly checks that the constructed equivalence computes as desired for $\mathsf{refl}$ .

We will not dwell much on our structures being models of material set theory, but rather focus on their structural properties in this paper. However, we will define the elementhood relation on $\mathsf{V}^\infty$ following Gylterud (Reference Gylterud2020). This elementhood relation, and its well-foundedness, will be used in later constructions.

Definition 5 (Elementhood, ). We define $ \in \,:\,{\mathsf{V}^\infty }\to {\mathsf{V}^\infty }\to \mathcal{U}$ by

\begin{equation*} x \in y := \mathsf {fib}\,\widetilde {y}\,x \end{equation*}

In particular, for canonical elements we get

\begin{equation*} \left (x \in \mathsf {sup}\,A\,f \right ) \equiv \left (\sum _{a:A}f\,a=x\right ) \end{equation*}

The exensionality property of Theorem4 can now be reformulated as an equivalence

\begin{equation*} \left (x = y\right ) \simeq \left (\prod _{z:{\mathsf {V}^\infty }} z \in x \simeq z \in y\right ) \end{equation*}

By virtue of univalence, we obtain this extensionality result without taking quotients by set extensionality or bisimulation like Aczel does. In particular, we are able to avoid working with quotients or setoids while still achieving the equivalence above.

Note that $x \in y$ need not be an h-proposition, that is, $y$ could contain several instances of $x$ . This is because, as discussed above, there is no restriction on the function $\widetilde{y}$ and its fibers. We will soon focus our attention to a subtype of $\mathsf{V}^\infty$ where these fibers are h-propositions, that is, where they have at most one inhabitant. But first, we will look at how some familiar sets can be represented in $\mathsf{V}^\infty$ .

We define the empty set as follows:

\begin{equation*} \emptyset := \mathsf {sup}\,\mathsf {empty}\,\mathsf {empty}\mbox {-}\mathsf {elim} \end{equation*}

This represents the empty set since for any $x :{\mathsf{V}^\infty }$ , the type $x \in \emptyset$ is empty.

Given $x :{\mathsf{V}^\infty }$ , we can construct the singleton containing $x$ as follows:

\begin{equation*} \{ x \} := \mathsf {sup}\,\mathsf {unit}\,({\lambda }\_.x) \end{equation*}

The type $x \in \{ x \}$ is inhabited by $(\mathsf{tt},\mathsf{refl})$ . Indeed, for any $y :{\mathsf{V}^\infty }$ , there is an equivalence $\left (y \in \{ x \} \right ) \simeq \left (y = x\right )$ .

We can also construct the unordered pair of two elements $x, y :{\mathsf{V}^\infty }$ :

\begin{equation*} \{ x, y \} := \mathsf {sup}\,\mathsf {bool}\,({\lambda }b.\mathsf {if}\,b\,\mathsf {then}\,x\,\mathsf {else}\,y) \end{equation*}

For any $z :{\mathsf{V}^\infty }$ , the type $z \in \{x,y\}$ is equivalent to $(z = x)\,+\,(z = y)$ . Note in particular that the type $x \in \{x,x\}$ is equivalent to $(x = x)\,+\,(x = x)$ , which contains at least two distinct elements. Thus, $\{x,x\}$ is a multiset that contains two copies of $x$ . Using images one can whittle this down to an iterative set, see the forthcoming paper (Gylterud and Stenholm Reference Gylterud and Stenholm2023) for details on the various types of pairing in higher h-levels.

In order to construct a universe of sets, we need to ensure that the $\in$ -relation is h-proposition valued, that is, that any element occurs at most once in a set. As the type $x \in y$ is the type of homotopy fibers of $\widetilde{y}$ over $x$ , this type would be an h-proposition if $\widetilde{y}$ was an embedding:

We write $\mathsf{is\mbox{-}emb}\,f$ for the type of proofs that $f$ is an embedding and $f : A \hookrightarrow B$ for ${\sum }_{f : A\to B}\,\mathsf{is\mbox{-}emb}\,f$ . A key observation about embeddings is

This motivates Gylterud’s definition of iterative sets in HoTT/UF (Gylterud Reference Gylterud2018):

Definition 8 (Iterative sets, ). We define $\mathsf{is\mbox{-}iterative\mbox{-}set} :{\mathsf{V}^\infty }\to \mathcal{U}$ as

\begin{equation*} \mathsf {is\mbox {-}iterative\mbox {-}set}\,(\mathsf {sup}\,A\,f) := (\mathsf {is\mbox {-}emb}\,f) \times \left ( \prod _{a : A} \mathsf {is\mbox {-}iterative\mbox {-}set}\,(f\,a) \right ) \end{equation*}

The idea is to pick out those elements $x :{\mathsf{V}^\infty }$ for which the function that selects elements is an embedding and such that the elements of $x$ satisfy the same criterion, recursively. This means that any $y :{\mathsf{V}^\infty }$ element is a member of $x$ at most once and, consequently, $x$ encode a set rather than a multiset. For these sets, the $\in$ -relation becomes h-proposition valued by Lemma 7, as desired.

Not all the elements in $\mathsf{V}^\infty$ are iterative sets. For example, the unordered pair $\{ \emptyset, \emptyset \}$ from above is not an iterative set as the function in the definition is not an embedding.Footnote ² On the other hand, the empty set, $\emptyset$ , is an iterative set, since $\mathsf{empty}\mbox{-}\mathsf{elim}$ is always an embedding, regardless of the codomain. Moreover, for any iterative set $x : \mathsf{V}^0$ , the singleton $\{x\}$ is an iterative set since any map from an h-proposition into an h-set is an embedding (we will see that $\mathsf{V}^0$ is an h-set in Theorem12). Furthermore, if $x$ and $y$ are distinct iterative sets then $\{ x, y \}$ is also an iterative set. To see this, it suffices to verify that the below map ${\varphi } : \mathsf{bool}\to \mathsf{V}^0$ is an embedding if it is injective:

\begin{equation*} {\varphi } \, b := \mathsf {if}\,b\,\mathsf {then}\,x\,\mathsf {else}\,y \end{equation*}

Given $b_1, b_2 : \mathsf{bool}$ , either $b_1 = b_2$ , in which case we are done, or $b_1 \neq b_2$ , in which case we get a path between $x$ and $y$ , from which the result follows by assumption.

Definition 9 (Type of iterative sets, ). We define the type of iterative sets as follows:

\begin{equation*} \mathsf {V}^0 := {\sum }_{x : {\mathsf {V}^\infty }}\,\mathsf {is\mbox {-}iterative\mbox {-}set}\,x \end{equation*}

We will extend the previously introduced notation to apply to iterative sets:

Moreover, the elementhood relation $\in$ defined on $\mathsf{V}^\infty$ gives rise to an elementhood relation for $\mathsf{V}^0$ given by projecting out the underlying elements in $\mathsf{V}^\infty$ and applying $\in$ : for any $x,y : \mathsf{V}^0$ we let $x \in y := \mathsf{pr}_1 \, x \in \mathsf{pr}_1 \, y$ . We use the same notation for both relations, as it will be clear from context which one is meant.

Having an embedding $\mathsf{V}^0 \hookrightarrow {\mathsf{V}^\infty }$ means that equality in $\mathsf{V}^0$ is exactly equality of the corresponding elements in $\mathsf{V}^\infty$ . Since we have already characterized equality in $\mathsf{V}^\infty$ , we can use this characterization to show that $\mathsf{V}^0$ is an h-set.

Proof. For $(x,p), (y,q) : \mathsf{V}^0$ , we have a chain of equivalences:

\begin{equation*} \left ((x,p) =_{\mathsf {V}^0} (y,q)\right ) \simeq \left (x =_{{\mathsf {V}^\infty }} y\right ) \simeq \left (\prod _{z : {\mathsf {V}^\infty }} z \in x \simeq z \in y\right ) \end{equation*}

The first equivalence is the characterization of equality in subtypes. The second is Theorem4. Note that $z \in x \equiv \mathsf{fib}\,\widetilde{x}\,z$ , and $\widetilde{x}$ is an embedding by $p$ . Thus $z \in x$ is an h-proposition. The same holds for $z \in y$ . Thus, the rightmost type in the chain of equivalences above is a family of equivalences between h-propositions and is thus an h-proposition. It then follows that the type $(x,p) =_{\mathsf{V}^0} (y,q)$ is an h-proposition.

Given a type $A : \mathcal{U}$ and an embedding $f : A \hookrightarrow \mathsf{V}^0$ , we can construct an element of $\mathsf{V}^0$ . This function is the counterpart to $\mathsf{sup}$ for $\mathsf{V}^\infty$ , and while it is not formally a constructor it behaves like one in that the recursion and elimination principles, with fitting computation rules, are provable for it (Gylterud and Stenholm Reference Gylterud and Stenholm2023).

Definition 14 (). We define the following function:

\begin{align*} &\mathsf{sup}^0 : \left (\sum _{A : \mathcal{U}} A \hookrightarrow \mathsf{V}^0\right )\to \mathsf{V}^0 \\ &\mathsf{sup}^0\,(A,f) := \left (\mathsf{sup}\,A\,(\pi _0 \circ f), \_\right ) \end{align*}

Similarly, given an element of $\mathsf{V}^0$ , we can extract the underlying type and embedding.

Definition 15 (). We define the following function:

\begin{align*} &\mathsf{desup}^0 : \mathsf{V}^0\to \left (\sum _{A : \mathcal{U}} A \hookrightarrow \mathsf{V}^0\right ) \\ &\mathsf{desup}^0\,(\mathsf{sup}\,A\,f, \_) := (A, (f, \_)) \end{align*}

By virtue of being a $\mathsf{W}$ -type, $\mathsf{V}^\infty$ is the initial algebra to the polynomial functor

\begin{equation*} X \mapsto \left (\sum _{A : \mathcal {U}} A \to X\right ) \end{equation*}

Similarly, $\mathsf{V}^0$ is the initial algebra for the functor $X \mapsto \left (\sum _{A : \mathcal{U}} A \hookrightarrow X\right )$ , even though this functor is not polynomial. The initiality induces an equivalence $\mathsf{V}^0 \simeq \left (\sum _{A : \mathcal{U}} A \hookrightarrow \mathsf{V}^0\right )$ , realized by the maps $\mathsf{sup}^0$ and $\mathsf{desup}^0$ above. These results are due to Gylterud and Stenholm (Reference Gylterud and Stenholm2023), who extend this construction to a whole hierarchy of functors $X \mapsto \left (\sum _{A : \mathcal{U}} A \hookrightarrow _n X\right )$ , for $n : \mathbb{N}_{-1}$ . Each of these has an initial algebra, given by a higher level generalization of $\mathsf{V}^0$ .

3. $\mathsf{V}^0$ as a Universe à la Tarski

The type $\mathsf{V}^0$ can be thought of as a type of material sets, in the sense that $\mathsf{V}^0$ together with the binary relation $\in$ is a model of constructive set theory (Gylterud Reference Gylterud2018). This section demonstrates that, more type-theoretically, $\mathsf{V}^0$ can be organized into a universe à la Tarski. In this way, $\mathsf{V}^0$ becomes a universe of h-sets which is itself an h-set. Furthermore, $\mathsf{V}^0$ is a strict universe in the sense that the decoding from codes to types is definitional. For instance, the decoding of the code for the natural numbers is definitionally equal to the type of natural numbers, and the decoding of a $\Pi$ - or $\Sigma$ -type of a family is the actual $\Pi$ - or $\Sigma$ -type of the decoding of the family.

We begin by defining the decoding family for our universe, $\mathsf{V}^0$ , as the underlying index type for each of its elements.

Definition 16 (Decoding, ). We define the decoding function $\mathsf{El}^0\,: \mathsf{V}^0 \to \mathcal{U}$ by

\begin{equation*} \mathsf {El}^0\,x :=\overline x \end{equation*}

It is easy to prove that the decoding of each code in $\mathsf{V}^0$ is also an h-set:

Note that for any $A : \mathcal{U}$ and embedding $f : A \hookrightarrow \mathsf{V}^0$ we have the definitional equality $\mathsf{El}^0\,(\mathsf{sup}^0\,\left (A, \right )f) \equiv A$ . That is, if we construct a code for a type in $\mathcal{U}$ using $\mathsf{sup}^0$ (which is what we usually do), then the decoding of this code is definitionally equal to the type we started with. This is very convenient when working with the universe $\mathsf{V}^0$ , especially for formalization.

As a universe, $\mathsf{V}^0$ contains codes of all the traditional type formers as long as they are present in the underlying universe, $\mathcal{U}$ . Using $\mathsf{sup}^0$ , one can construct a code for a given type $A : \mathcal{U}$ in $\mathsf{V}^0$ if there is an embedding $A \hookrightarrow \mathsf{V}^0$ . In fact, there is a code for $A$ in $\mathsf{V}^0$ precisely when it can be embedded into $\mathsf{V}^0$ .

Proposition 18 (). For any $A : \mathcal{U}$ there is an equivalence

\begin{equation*} \left (A \hookrightarrow \mathsf {V}^0\right ) \simeq \left (\sum _{a : \mathsf {V}^0} \mathsf {El}^0\,a = A\right ) \end{equation*}

Proof. The maps back and forth are

\begin{align*} &{\alpha } : \left (A \hookrightarrow \mathsf{V}^0\right )\to \sum _{a : \mathsf{V}^0} \mathsf{El}^0\,a = A \\ &{\alpha }\,f := (\mathsf{sup}^0\,\left (A,f\right ), \mathsf{refl}) \\ \\ &{\beta } : \left (\sum _{a : \mathsf{V}^0} \mathsf{El}^0\,a = A\right )\to (A \hookrightarrow \mathsf{V}^0) \\ &{\beta }\,(a, \mathsf{refl}) := \widetilde{a} \end{align*}

We compute as follows:

\begin{gather*}{\alpha }({\beta }(a,\mathsf{refl})) \equiv {\alpha } (\widetilde a) \equiv (\mathsf{sup}^0\,\left (\overline a,\widetilde a\right ),\mathsf{refl}) = (a,\mathsf{refl}) \\ \mathsf{pr}_1 \left ({\beta }({\alpha }\,f)\right ) \equiv \mathsf{pr}_1 \left ({\beta }(\mathsf{sup}^0\,\left (A,f\right ), \mathsf{refl})\right ) \equiv \mathsf{pr}_1 \left (\widetilde{\mathsf{sup}^0\,\left (A,f\right )}\right ) \equiv \mathsf{pr}_1\,f \end{gather*}

Thus, $\alpha$ is a quasi-equivalence and therefore an equivalence.

We emphasize that the definitional equation $\mathsf{El}^0\,(\mathsf{sup}^0\, (A, f)) \equiv A$ simplifies the definition of $\alpha$ as we may then use $\mathsf{refl}$ for the second argument. Moreover, ${\beta } \circ{\alpha }$ definitionally preserves the function underlying the embedding. The same is not true of the witness that this function is an embedding, but such witnesses belong to a contractible type and can safely be ignored.

3.1 Basic types

We now construct codes for some basic types in $\mathsf{V}^0$ .

Proposition 19 (). $\mathsf{V}^0$ contains the empty type, unit type and booleans.

Proof. We define the elements $\mathsf{empty}^0, \mathsf{unit}^0, \mathsf{bool}^0 : \mathsf{V}^0$ as follows:

\begin{align*} \mathsf{empty}^0 &:= \emptyset \\ \mathsf{unit}^0 &:= \{\emptyset \} \\ \mathsf{bool}^0 &:= \{ \emptyset, \{\emptyset \} \} \end{align*}

There were all verified to be iterative sets in Section 2.

We note that the expected equations hold up to definitional equality:

\begin{equation*} \mathsf {El}^0\,\mathsf {empty}^0 \equiv \mathsf {empty}, \qquad \mathsf {El}^0\,\mathsf {unit}^0 \equiv \mathsf {unit} \qquad \text {and} \qquad \mathsf {El}^0\,\mathsf {bool}^0 \equiv \mathsf {bool}. \end{equation*}

Proposition 20 (). $\mathsf{V}^0$ contains the natural numbers.

Proof. By Proposition 18 it is enough to construct an embedding $\mathbb{N} \hookrightarrow \mathsf{V}^0$ . Here there is a choice of encoding of the naturals in $\mathsf{V}^0$ and several encodings are possible. We will use the von Neumann encoding and show that this is an embedding.

First, we define the successor function in $\mathsf{V}^0$ :

\begin{align*} &\mathsf{suc}^0 : \mathsf{V}^0\to \mathsf{V}^0 \\ &\mathsf{suc}^0\,x := \mathsf{sup}^0\,(\overline{x}\,+\,\mathsf{unit}, \varphi _{x}) \end{align*}

In the above, $\varphi _{x} : \overline{x}\,+\,\mathsf{unit}\to \mathsf{V}^0$ is defined as follows:

\begin{align*} &\varphi _{x}\,(\mathsf{inl}\,a) := \widetilde{x}\, a \\ &\varphi _{x}\,(\mathsf{inr}\,b) := x \end{align*}

To see that the map $\varphi _{x}$ is an embedding, note that for any $z : \mathsf{V}^0$ the fiber $\mathsf{fib}\,\varphi _x\,z$ is equivalent to $(z \in x)\,+\,(x = z)$ . Both summands are h-propositions and they are disjoint: if they were both inhabited we could derive $x \in x$ which contradicts the well-foundedness of $\in$ (Gylterud and Stenholm Reference Gylterud and Stenholm2023).

The von Neumann encoding of the natural numbers is then the function:

\begin{align*} &f : \mathbb{N}\to \mathsf{V}^0 \\ &f\,\mathsf{0} := \emptyset \\ &f\,(\mathsf{s}\,n) := \mathsf{suc}^0\,(f\,n) \end{align*}

It remains to show that $f$ is an embedding. As $\mathbb{N}$ and $\mathsf{V}^0$ are both h-sets it suffices that $f$ is injective. Observe that $\overline{f\,x} \simeq \mathsf{Fin}\,x$ , so if $f\,n = f\,m$ then $\mathsf{Fin}\,n \simeq \mathsf{Fin}\,m$ from which $n = m$ follows.

Having shown that $f : \mathbb{N}\to \mathsf{V}^0$ is an embedding, we define the (code for the) natural numbers in $\mathsf{V}^0$ as follows:

\begin{equation*} \mathbb {N}^0 := \mathsf {sup}^0\,(\mathbb {N}, f) \end{equation*}

Note, again, that the decoding holds up to definitional equality:

\begin{equation*} \mathsf {El}^0\,\mathbb {N}^0 \equiv \mathbb {N} \end{equation*}

3.2 Type formers

We now turn to closing $\mathsf{V}^0$ under the standard type formers. For these constructions we will need ordered pairs.

Lemma 21 (). There is an ordered pairing operation $\langle \_,\_\rangle : \mathsf{V}^0 \times \mathsf{V}^0 \hookrightarrow \mathsf{V}^0$ .

Proof. Ordered pairs are constructed using the Norbert Wiener encoding. The details of this construction can be found in the proof of Gylterud and Stenholm (Reference Gylterud and Stenholm2023, Theorem 7).

Proposition 22 (). $\mathsf{V}^0$ is closed under $\prod$ -types.

Proof. Let $x : \mathsf{V}^0$ and $y : \mathsf{El}^0\,x\to \mathsf{V}^0$ . By Gylterud and Stenholm (Reference Gylterud and Stenholm2023, Lemma 12), there is an embedding:

\begin{equation*} \mathsf {graph}_{x,y} : \left (\prod _{a : \mathsf {El}^0\,x} \mathsf {El}^0\,(y\,a)\right ) \hookrightarrow \mathsf {V}^0 \end{equation*}

This map sends $\varphi : \prod _{a : \mathsf{El}^0\,x} \mathsf{El}^0\,(y\,a)$ to the element $\mathsf{sup}^0\,\left (\mathsf{El}^0\,x,{\lambda }a.\langle \widetilde{x}\,a,\widetilde{(y\,a)}\,(\varphi \,a)\rangle \right )$ . The $\prod$ -type is then defined as follows:

\begin{equation*} \prod ^0\,x\,y := \mathsf {sup}^0\,\left (\prod _{a : \mathsf {El}^0\,x} \mathsf {El}^0\,(y\,a), \mathsf {graph}_{x,y}\right ) \end{equation*}

The decoding holds up to definitional equality:

\begin{equation*} \mathsf {El}^0\,\left (\prod ^0\,x\,y\right ) \equiv \mathop\prod\limits_{a : \mathsf {El}^0\,x} \mathsf {El}^0\,(y\,a) \end{equation*}

Corollary 23 (). $\mathsf{V}^0$ is closed under (nondependent) function types. Let $x \to ^0 y$ denote the code for the type $\mathsf{El}^0\,x\to \mathsf{El}^0\,y$ .

Proposition 24 (). $\mathsf{V}^0$ is closed under $\Sigma$ -types.

Proof. Let $x : \mathsf{V}^0$ and $y : \mathsf{El}^0\,x\to \mathsf{V}^0$ . Define a putative embedding as follows:

\begin{align*} &f : \left (\sum _{a : \mathsf{El}^0\,x} \mathsf{El}^0\,(y\,a)\right )\to \mathsf{V}^0 \\ &f\,(a,b) := \langle \widetilde{x}\,a,\widetilde{(y\,a)}\,b\rangle \end{align*}

This is the composition of two embeddings: $\langle \_,\_\rangle$ and ${\lambda }(a,b).(\widetilde{x}\,a,\widetilde{(y\,a)}\,b)$ and therefore an embedding. The last function is an embedding because $\widetilde{x}$ is an embedding and as is $\widetilde{(y\,a)}$ for every $a : \mathsf{El}^0\,x$ . We may now define the code for $\Sigma$ -types:

\begin{equation*} \Sigma ^0\,x\,y := \mathsf {sup}^0\,\left (\sum _{a : \mathsf {El}^0\,x} \mathsf {El}^0\,(y\,a)\,f\right ) \end{equation*}

The decoding holds up to definitional equality:

\begin{equation*} \mathsf {El}^0\,\left (\Sigma ^0\,x\,y\right ) \equiv \sum _{a : \mathsf {El}^0\,x} \mathsf {El}^0\,(y\,a) \end{equation*}

Corollary 25 (). $\mathsf{V}^0$ is closed under cartesian products. Let $x \times ^0 y$ be the code for $\mathsf{El}^0\,x \times \mathsf{El}^0\,y$ .

In order to construct coproducts in $\mathsf{V}^0$ we need two lemmas about embeddings.

Lemma 26 (). Given types $Y$ , $Z$ and h-set $X$ with a point $x_0 : X$ , any embedding $f : X \times Y \hookrightarrow Z$ gives rise to an embedding by fixing the first coordinate: $f\,(x_0,\_) : Y \hookrightarrow Z$ .

Proof. We need to show that for any $z : Z$ , the fiber of $f\,(x_0,\_)$ over $z$ is an h-proposition. But the following chain of equivalences holds:

\begin{equation*} \left (\sum _{y : Y} f\,(x_0,y) = z\right ) \simeq \left (\sum _{y : Y} \sum _{\sum _{x : X} (x = x_0)} f\,(x,y) = z\right ) \simeq \left (\sum _{((x,y),p) : \mathsf {fib}\,f\,z} x = x_0\right ) \end{equation*}

The last type is an h-proposition since $\mathsf{fib}\,f\,z$ is an h-proposition by Lemma 7 and for each $((x,y),p) : \mathsf{fib}\,f\,z$ , the type $x = x_0$ is an h-proposition.

Lemma 27 (). Given types $X$ , $Y$ , and $Z$ together with embeddings $f : X \hookrightarrow Z$ and $g : Y \hookrightarrow Z$ . If $f\,x\,\neq \,g\,y$ for all $x : X$ and $y : Y$ then the following map is an embedding:

\begin{align*} &h : X\,+\,Y\to Z \\ &h\,(\mathsf{inl}\,x) := f\,x \\ &h\,(\mathsf{inr}\,y) := g\,y \end{align*}

Proof. Let $s,t : X\,+\,Y$ . We need to show that $\mathsf{ap}\,h : s = t\to h\,s = h\,t$ is an equivalence. Using induction on coproducts, there are two kinds of cases to consider: when $s$ and $t$ lie in different summands, and when they lie in the same one.

First, suppose without loss of generality that $s \equiv \mathsf{inl}\,x$ and $t \equiv \mathsf{inr}\,y$ . In this case, we need to show that $\mathsf{ap}\,h : \mathsf{inl}\,x = \mathsf{inr}\,y\to f\,x = g\,y$ is an equivalence. But both types are empty, so any map between them is an equivalence.

Now, suppose without loss of generality that $s \equiv \mathsf{inl}\,x$ and $t \equiv \mathsf{inl}\,x'$ . We need to show that $\mathsf{ap}\,h : \mathsf{inl}\,x = \mathsf{inl}\,x'\to f\,x = f\,x'$ is an equivalence. But note that the following diagram commutes:

Since both $\mathsf{ap}\,f$ and $\mathsf{ap}\,\mathsf{inl}$ are equivalences it follows that $\mathsf{ap}\,h$ is an equivalence.

Proposition 28 (). $\mathsf{V}^0$ is closed under coproducts.

Proof. Let $x,y : \mathsf{V}^0$ . Define the map

\begin{align*} &f : \mathsf{El}^0\,x\,+\,\mathsf{El}^0\,y\to \mathsf{V}^0 \\ &f\,(\mathsf{inl}\,a) := \langle \mathsf{empty}^0, \widetilde{x}\,a\rangle \\ &f\,(\mathsf{inr}\,b) := \langle \mathsf{unit}^0, \widetilde{y}\,b\rangle \end{align*}

By Lemma 26 both ${\lambda }a.\langle \mathsf{empty}^0, \widetilde{x}\,a\rangle$ and ${\lambda }b.\langle \mathsf{unit}^0, \widetilde{y}\,b\rangle$ are embeddings. Moreover, suppose $\langle \mathsf{empty}^0, \widetilde{x}\,a\rangle = \langle \mathsf{unit}^0, \widetilde{y}\,b\rangle$ for some $a : \mathsf{El}^0\,x$ and $b : \mathsf{El}^0\,y$ . It then follows that $\mathsf{empty}^0 = \mathsf{unit}^0$ , which is a contradiction. Therefore, by Lemma 27 we conclude that $f$ is an embedding.

We now define the coproduct:

\begin{equation*} x\,+^0\,y := \mathsf {sup}^0\,(\mathsf {El}^0\,x\,+\,\mathsf {El}^0\,y, f) \end{equation*}

Note that the decoding holds up to definitional equality:

\begin{equation*} \mathsf {El}^0\,(x\,+^0\,y) \equiv \mathsf {El}^0\,x\,+\,\mathsf {El}^0\,y \end{equation*}

Proposition 29 (). $\mathsf{V}^0$ is closed under identity types.

Proof. Let $x : \mathsf{V}^0$ and $a,a' : \mathsf{El}^0\,x$ . Define the following map:

\begin{align*} &f : a = a'\to \mathsf{V}^0 \\ &f\,p := \emptyset \end{align*}

This is an embedding as it is a map from an h-proposition into an h-set. The identity type in $\mathsf{V}^0$ is then defined as follows:

\begin{equation*} \mathsf {Id}^0\,x\,a\,a' := \mathsf {sup}^0\,(a = a', f) \end{equation*}

The decoding holds up to definitional equality:

\begin{equation*} \mathsf {El}^0\,(\mathsf {Id}^0\,x\,a\,a') \equiv \left (a = a'\right ) \end{equation*}

We emphasize that $\mathsf{El}^0\,x$ is an h-set for any for any $x : \mathsf{V}^0$ . Accordingly, $\mathsf{Id}^0\,x\,a\,a'$ is necessarily a proposition for any $a,a' : \mathsf{El}^0\,x$ . In particular, $\mathsf{Id}^0\,x\,a\,a'$ satisfies UIP. As this identity type represents internalizes the ambient identity type, other expected properties of the identity type (such as function extensionality) also hold.

3.3 Set quotients

In order to define set quotients in $\mathsf{V}^0$ , we must assume that these quotients exist in our starting universe $\mathcal{U}$ . More specifically, we first assume that there is a function of the following type

\begin{equation*} {-}/{-} : \prod _{A : \mathcal {U}} (A \to A \to \mathcal {U}) \to \mathsf {hSet}_{\mathcal {U}} \end{equation*}

We then ensure that $A / R$ realizes the quotient of $A$ by the relation $R$ by requiring a map $[{-}]_R : A\to A / R$ such that $R\,a\,b\to [a]_R = [b]_R$ for all $a,b : A$ . We also assume a suitable elimination principle: given a family of h-sets $P : A / R\to \mathsf{hSet}_{\mathcal{U}}$ , we can construct a function $\prod _{x : A / R} P\,x$ from a function $q : \prod _{x : A / R} P\,[a]_R$ , which coheres with the map $R\,a\,b\to [a]_R = [b]_R$ . The require that function we get satisfies a coherence condition, and if we precompose it with the quotient map $[{-}]_R$ we get back $q$ . (For the exact assumptions, see the formalization .)

While we do not need to assume that $R : A\to A\to \mathcal{U}$ is an equivalence relation (h-propositional, symmetric, reflexive, and transitive), the constructions below will use the fact any $R$ induces an equivalence relation $|R| : A\to A\to \mathsf{hProp}_{\mathcal{U}}$ defined by $|R|\,a\,b := \left ([a]_R = [b]_R\right )$ .

To streamline the process, we will use an interesting formulation of equivalence relations:

Lemma 30. Given a relation $R : A\to A\to \mathsf{hProp}_{\mathcal{U}}$ , the following are equivalent:

• • $R$ is an equivalence relation

• • $R\,a\,b \simeq \prod _{c:A} \left (R\,a\,c \simeq R\,b\,c\right )$ for all $a,b:A$

• • $R\,a\,b \simeq \left (R\,a =_{A\to \mathcal{U}} R\,b\right )$ for all $a,b:A$

Proof. Since $R$ is an (h-propositional) binary relation, the above statements are all h-propositions. The last two are equivalent by function extensionality and univalence. It thus remains to show that being an equivalence relation is equivalent to one of the last two – we choose the middle one.

Assume that $R$ is an equivalence relation. Everything in sight is an h-proposition, so the equivalences are logical equivalences. Thus, assume that $R\,a\,b$ . Then we get a map $\prod _{c:A} R\,a\,c \leftrightarrow R\,b\,c$ by transitivity and symmetry. In the other direction, if $\prod _{c:A} R\,a\,c \leftrightarrow R\,b\,c$ , choose $c = a$ in order to obtain $R\,a\,a \leftrightarrow R\,a\,b$ . Since $R$ is reflexive, we get $R\,a\,b$ .

Conversely, assume $R\,a\,b \simeq \prod _{c:A} \left (R\,a\,c \simeq R\,b\,c\right )$ for all $a,b:A$ . To show reflexivity, let $b=a$ and notice that $\prod _{c:A} \left (R\,a\,c \simeq R\,a\,c\right )$ has a canonical element, from which we obtain $R\,a\,a$ . Symmetry is obtained by observing that $ \prod _{c:A} \left (R\,a\,c \simeq R\,b\,c\right ) \simeq \prod _{c:A} \left (R\,b\,c \simeq R\,a\,c\right )$ and hence $R\,a\,b \simeq R\,b\,a$ . For transitivity, remember that $R\,a\,b$ gives $\prod _{c:A} \left (R\,a\,c \simeq R\,b\,c\right )$ , thus if we have $R\,b\,c$ we get $R\,a\,c$ by following the backwards direction of the equivalence.

The property $R\,a\,b \simeq \prod _{c:A} \left (R\,a\,c \simeq R\,b\,c\right )$ for all $a,b:A$ essentially states that the equivalence classes of $R$ behave well. Tangentially, we note that this requirement make sense even when $R$ is a general binary family, not only of h-propositions. Thus, this property might make for interesting future study.

Proposition 31 (). $\mathsf{V}^0$ is closed under set quotients. That is, given $a:\mathsf{V}^0$ and $R : \mathsf{El}^0\,a\to \mathsf{El}^0\,a\to \mathcal{U}$ there is $a /^0 R : \mathsf{V}^0$ such that $\mathsf{El}^0\,(a /^0 R ) \equiv (\mathsf{El}^0\,a) / R$ .

Proof. By Proposition 18, it suffices to construct an embedding $\mathsf{El}^0\,a / R \hookrightarrow \mathsf{V}^0$ . We define $f :\,\mathsf{El}^0\,a\to \mathsf{V}^0$ and prove that for any $x,x':\mathsf{El}^0\,a$ we have $\left ([x]_R = [x']_R\right ) \simeq \left (f\,x = f\,x'\right )$ . By the elimination principle for set quotients, this will induce an embedding $\mathsf{El}^0\,a / R \hookrightarrow \mathsf{V}^0$ .

Thus, let $f\,x = \mathsf{sup}^0\,\left (\sum _{y:\mathsf{El}^0\,a}|R|\,x\,y,\widetilde a \circ \mathsf{pr}_1\right )$ , and observe the chain of equivalences:

\begin{align*} \left (f\,x = f\,x'\right ) & \equiv \left (\mathsf{sup}^0\,\left (\sum _{y:\mathsf{El}^0\,a}|R|\,x\,y,\widetilde a \circ \mathsf{pr}_1\right ) = \mathsf{sup}^0\,\left (\sum _{y:\mathsf{El}^0\,a}|R|\,x'\,y,\widetilde a \circ \mathsf{pr}_1\right )\right ) \\ & \simeq \sum _{{\alpha } : \left (\sum _{y:\mathsf{El}^0\,a}|R|\,x\,y\right ) \simeq \left (\sum _{y:\mathsf{El}^0\,a}|R|\,x'\,y\right )} \widetilde a \circ \mathsf{pr}_1 = \widetilde a \circ{\alpha }\circ \mathsf{pr}_1 \\ & \simeq \sum _{{\alpha } : \left (\sum _{y:\mathsf{El}^0\,a}|R|\,x\,y\right ) \simeq \left (\sum _{y:\mathsf{El}^0\,a}|R|\,x'\,y\right )} \mathsf{pr}_1 ={\alpha }\circ \mathsf{pr}_1 \\ & \simeq \prod _{y : \mathsf{El}^0\,a} |R|\,x\,y \simeq |R|\,x'\,y \\ & \simeq |R|\,x\,x' \\ & \equiv \left ([x]_R = [x']_R\right ) \end{align*}

Note that we have used the characterization of equivalence relations given by Lemma 30 in the next to last step.

Remark 32. We note that our choice of embedding to define $a /^0 R : \mathsf{V}^0$ is similar to the construction of set quotients using type-theoretic replacement due to Rijke (Reference Rijke2017).

3.4 Using the type formers

Using the types and type formers in $\mathsf{V}^0$ we can construct new types. The decoding of these composite types will then hold up to definitional equality. For example, given elements $x,y : \mathsf{V}^0$ , a map $f : \mathsf{El}^0\,x\to \mathsf{El}^0\,y$ and $b : \mathsf{El}^0\,y$ , we can define the code for the fiber of $f$ over $b$ as

\begin{equation*} \mathsf {fib}^0\,f\,b := \Sigma ^0\,x\,({\lambda }a.\mathsf {Id}^0\,y\,(f\,a)\,b) \end{equation*}

After applying the decoding function, we get obtain the following definitional equality:

\begin{equation*} \mathsf {El}^0\,\left (\mathsf {fib}^0\,f\,b\right ) \equiv \mathsf {fib}\,f\,b \end{equation*}

3.5 Universes

The flexible handling of hierarchies of universes is a key feature of dependent type theory. It makes it easy to formalize higher order concepts and mathematical structures. Our universe construction retains this ability, and in this subsection we demonstrate that by observing that the types $\mathsf{V}^0_0, \mathsf{V}^0_1, \ldots, \mathsf{V}^0_{\ell }, \ldots$ form a hierarchy of universes, where each universe occurs as a type with a code in the next.

Proposition 33 (). For any universe level $\ell$ there is a code $\mathsf{V}^0_{\ell }\mbox{-}\mathsf{code} : \mathsf{V}^0_{\ell + 1}$ for $\mathsf{V}^0_{\ell }$ with the definitional equality $\mathsf{El}^0\,\mathsf{V}^0_{\ell }\mbox{-}\mathsf{code} \equiv \mathsf{V}^0_{\ell }$

Proof. Given a universe level $\ell$ , we need to construct an embedding $\mathsf{V}^0_{\ell } \hookrightarrow \mathsf{V}^0_{\ell ^+}$ . For this, we start by constructing an embedding ${\mathsf{V}^\infty }_{\ell } \hookrightarrow {\mathsf{V}^\infty }_{\ell ^+}$ . Thus define the map

\begin{align*} &\varphi :{\mathsf{V}^\infty }_{\ell }\to {\mathsf{V}^\infty }_{\ell ^+} \\ &\varphi \,(\mathsf{sup}\,A\,f) := \mathsf{sup}\,A\,(\varphi \circ f) \end{align*}

(Note that we are using cumulative universes in the ambient type theory, so $A : \mathcal{U}_{\ell ^+}$ whenever $A : \mathcal{U}_{\ell }$ .) To show that $\varphi$ is an embedding, let $\mathsf{sup}\,A\,f, \mathsf{sup}\,B\,g :{\mathsf{V}^\infty }_{\ell }$ be arbitrary elements. We need to show that

\begin{equation*} \mathsf {ap}\,\varphi : \mathsf {sup}\,A\,f = \mathsf {sup}\,B\,g \to \mathsf {sup}\,A\,(\varphi \circ f) = \mathsf {sup}\,B\,(\varphi \circ g) \end{equation*}

is an equivalence. First, note that the following diagram commutes:

The map $\mathsf{sup}$ is an equivalence, so $\varphi$ is an embedding if and only if the top map is an embedding. Thus we need to show the following to be an equivalence:

\begin{equation*} \mathsf {ap}\,({\lambda }(X,h).(X,\varphi \circ h)) : (A,f) = (B,g) \to (A,\varphi \circ f) = (B,\varphi \circ g) \end{equation*}

While we might hope to argue that this is a fiberwise embedding and therefore the total map is an embedding as well. Unfortunately, our induction hypothesis does not state that $\varphi$ is an embedding, i.e., that $\mathsf{ap}$ is an equivalence for all elements. It only ensures that it is an equivalence for some elements. We must then take a different path and instead note that the diagram below commutes.

The vertical maps are provided by Theorem3. Using 3-for-2, the top map is an equivalence if and only if the bottom one is an equivalence. We now note that it suffices to check this property on fibers, so we need to show that given $e : A \simeq B$ , the following map is an equivalence:

\begin{equation*} ({\lambda }H.\mathsf {ap}\,\varphi \circ H) : f \sim g \circ e \to \varphi \circ f \sim \varphi \circ g \circ e \end{equation*}

We now recall that postcomposition by a family of maps is an equivalence if it is a family of equivalences. Finally, we must argue that $\mathsf{ap}\,\varphi : f\,a = g\,(e\,a)\to \varphi \,(f\,a) = \varphi \,(g\,(e\,a))$ is an equivalence for all $a : A$ . This follows from the induction hypothesis. Thus, we conclude that $\varphi :{\mathsf{V}^\infty }_{\ell }\to {\mathsf{V}^\infty }_{\ell ^+}$ is an embedding.

To argue that this equivalence restricts to $\mathsf{V}^0$ , we must show this equivalence sends iterative sets to iterative sets. Thus, let $\mathsf{sup}\,A\,f :{\mathsf{V}^\infty }_{\ell }$ be such that $f$ is an embedding and $f\,a$ is an iterative set for all $a : A$ . We must argue that $\mathsf{sup}\,A\,(\varphi \circ f)$ is an iterative set. But the map $\varphi \circ f$ is an embedding as the composition of two embeddings. Moreover, by the induction hypothesis, for any $a : A$ , $\varphi \,(f\,a)$ is an iterative set since $f\,a$ is an iterative set.

Therefore, $\varphi$ is an embedding from $\mathsf{V}^0_{\ell }$ into $\mathsf{V}^0_{\ell ^+}$ . The code for $\mathsf{V}^0_{\ell }$ in $\mathsf{V}^0_{\ell ^+}$ is thus defined as

\begin{equation*} \mathsf {V}^0_{\ell }\mbox {-}\mathsf {code} := \mathsf {sup}^0\,(\mathsf {V}^0_{\ell }, \varphi ) \end{equation*}

Note that the decoding holds up to definitional equality:

\begin{equation*} \mathsf {El}^0\,\mathsf {V}^0_{\ell }\mbox {-}\mathsf {code} \equiv \mathsf {V}^0_{\ell } \end{equation*}

Proposition 34. $\mathsf{V}^0$ is not a univalent universe.

Proof. For $x, y : \mathsf{V}^0$ , $x = y$ is an h-proposition as $\mathsf{V}^0$ is an h-set, but $\mathsf{El}^0\,\,x \simeq \mathsf{El}^0\,\,y$ is in general a proper h-set.

4. $\mathsf{V}^0$ as a Category

The universe structure on $\mathsf{V}^0$ induces a category with a full and faithful functor into $\mathcal{hSet}_{\mathcal{U}}$ . In this section, we define this category and show that it is closed under many essential constructions (finite limits and colimits, exponentials, and more). This category provides another concrete way for us to measure the adequacy of $\mathsf{V}^0$ as a replacement for $\mathsf{hSet}_{\mathcal{U}}$ ; the former induces a closely related category to the latter, sharing many similar properties.

Note that we will take category to denote what the HoTT Book calls “precategory,” univalent category to denote what the book calls “category,” and strict category to denote a category where the objects form an h-set. Hence, $\mathcal{V}$ is a strict category as $\mathsf{V}^0$ is an h-set.

The following holds more-or-less by construction:

Clearly, $\mathcal{V}$ is not a univalent category since it possesses objects with non-trivial automorphisms, but the type of objects in $\mathcal{V}$ is an h-set. Still, one might ask if $\mathsf{El}^0$ is an equivalence of categories. This does not appear to be true in general, but can be implied by further axioms. For instance, the axiom of choice implies that $\mathsf{El}^0$ is an equivalence. The core of this is whether every type in $\mathcal{U}$ can be equipped with an iterative set-structure – a property known as well-founded materialization. We discuss this further in Section 6.4.

Fortunately, even without additional axioms we are able to show that $\mathcal{V}$ retains much of the essential structure of $\mathcal{hSet}_{\mathcal{U}}$ and that $\mathsf{El}^0$ preserves many important categorical constructions. In order to show that $\mathcal{V}$ is closed under some categorical structure, it therefore suffices to break the process into two stages:

1. 1. Show that $\mathcal{hSet}_{\mathcal{U}}$ is closed under for example, finite limits, exponentials, etc.

2. 2. Show that the objects involved land in the image of $\mathsf{El}^0$ .

Better still, $\mathcal{hSet}_{\mathcal{U}}$ is well-studied and known to be closed under all the categorical structures we consider (Rijke and Spitters Reference Rijke and Spitters2015). Our task is therefore reduced only to showing that various objects of $\mathcal{hSet}_{\mathcal{U}}$ land in the image of $\mathsf{El}^0$ . For this, we repeatedly capitalize on the fact that the decoding $\mathsf{El}^0$ holds up to definitional equality; it ensures that the final step can be rephrased as follows: show that there exists an iterative structure on the objects involved. This pattern is used repeatedly to prove the following result:

As $\mathcal{V}$ has pullbacks/pushouts and terminal/initial object we directly get:

We defer further categorical considerations of $\mathcal{V}$ to future work and instead turn our attention to slice categories of $\mathcal{V}$ , which play an important role in the study of it as a model of type theory.

4.1 Slice categories of $\mathcal{V}$

Similar methods to the ones above also apply when showing that the slice categories $\mathcal{V}/a$ are well-behaved. In particular, $\mathsf{El}^0\,$ induces a full and faithful functor $\mathcal{V}/a \to \mathcal{hSet}_{\mathcal{U}}/\mathsf{El}^0\,a$ . We can use this fact to deduce, for example, that $\mathcal{V}/a$ is cartesian closed.

Proposition 39 (). For any $a : \mathsf{V}^0$ , $\mathcal{V} / a$ has finite limits.

Proof. This can be proved using the standard argument: products in a slice category are realized by pullbacks in the underlying category and connected limits are realized by limits of the underlying diagram. We could also argue by noting that $\mathcal{hSet}_{\mathcal{U}}/\mathsf{El}^0\,a$ is finitely complete and that limits of diagrams in the image of $\mathsf{El}^0\,$ remain in the image of $\mathsf{El}^0\,$ .

We give a bit more details in the following proof as it showcases the usefulness of being able to encode things directly in $\mathsf{V}^0$ , combined with the fact that $\mathsf{El}^0\,$ strictly decodes to the expected thing in $\mathcal{U}$ .

Proposition 40 (). For any $a : \mathsf{V}^0$ , $\mathcal{V} / a$ has exponentials.

Proof. Given $(x,f), (y,g) : \mathsf{Ob}\,(\mathcal{V} / a)$ , define their exponential as the element

\begin{equation*} \mathsf {exp}\,(x,f)\,(y,g) := \Sigma ^0\,a\,(\lambda \,i.\,\mathsf {fib}^0\,g\,i \to ^0 \mathsf {fib}^0\,f\,i) \end{equation*}

Note that we have the following definitional equality:

\begin{equation*} \mathsf {El}^0\,(\mathsf {exp}\,(x,f)\,(y,g)) \equiv \sum _{i : \mathsf {El}^0\,a} \mathsf {fib}\,g\,i \to \mathsf {fib}\,f\,i \end{equation*}

This is the exponential in $\mathcal{hSet}_{\mathcal{U}}/\mathsf{El}^0\,a$ , so $\mathsf{exp}\,(x,f)\,(y,g)$ is an exponential object in $\mathcal{V}/a$ .

Corollary 41. $\mathcal{V}$ is locally cartesian closed and $\mathsf{El}^0$ is a locally cartesian closed functor.

Finally, the following proposition foreshadows the next section where we build a model of type theory on $\mathcal{V}$ . In that section, we wish to interpret types in context $a$ as elements of $\mathcal{V}/a$ and to realize substitution as pullback. It is well-known, however, that the result is merely pseudofunctorial and therefore insufficient to form a (strict) model of type theory (Curien et al. Reference Curien, Garner and Hofmann2014; Seely Reference Seely1984). In the specific case of $\mathcal{hSet}_{\mathcal{U}}$ , there is a well-known pseudo-natural equivalence between the slice category $\mathcal{hSet}_{\mathcal{U}}/a$ and the functor category $[a,\mathcal{hSet}_{\mathcal{U}}]$ , which remedies the coherence issues. This equivalence restricts to the full subcategory determined by $\mathcal{V}$ :

Proposition 42 (). Given $a : \mathsf{V}^0$ and writing $a$ for the corresponding discrete category associated with $\mathsf{El}^0\,a$ , there is a canonical equivalence $\mathcal{V} / a \simeq [a,\mathcal{V}]$ .

Proof. The equivalence is constructed in the standard way. The functor from $[a,\mathcal{V}]$ to $\mathcal{V} / a$ sends $F : \mathsf{Ob}\,[a,\mathcal{V}]$ to the element $\Sigma ^0\,a\,F$ together with the first projection. In the other direction, an element $(x,f) : \mathsf{Ob}\,(\mathcal{V} / a)$ is mapped to the functor $\lambda i. \mathsf{fib}^0\,f\,i$ .

5. $\mathsf{V}^0$ as a Category with Families

Having established that $\mathsf{V}^0$ organizes into a well-behaved category, we now take this a step further by showing that $\mathsf{V}^0$ supports a model of extensional type theory. Since our goal is to do this very formally, our task is threefold: first, we must specify what we mean by a model of type theory. To this end, we have formalized a particular presentation of a CwF (Dybjer Reference Dybjer, Berardi and Coppo1996). This extends a category with the additional structure required to interpret dependent type theory. Next, we show that $\mathcal{V}$ can be equipped with this additional structure. Finally, since our definition of a CwF does not prescribe closure under any connectives, we detail how to extend a CwF with various connectives and show that the CwF structure on $\mathcal{V}$ supports these extensions.

5.2 Equipping $\mathcal{V}$ with a CwF structure

We now turn to equipping $\mathcal{V}$ with a CwF structure. We begin by defining $\mathsf{Ty}_{\mathcal{V}}$ as follows:

\begin{equation*} \mathsf {Ty}_{\mathcal {V}}\,X := \mathsf {El}^0\,X \to \mathsf {V}^0 \end{equation*}

Intuitively, a type in context $X$ is precisely an $X$ -indexed family of sets. There is, however, a major subtlety in this definition that should be emphasized: the version of the definition where $\mathsf{V}^0$ is replaced by $\mathsf{hSet}_{\mathcal{U}}$ would be incorrect. We have required that $\mathsf{Ty}_{\mathcal{V}}\,X$ always be an h-set as it is assumed to be an $\mathsf{hSet}_{\mathcal{U}}$ valued presheaf. Therefore, it is only after finding an adequate “h-set of h-sets” that we can define the set model of type theory in this manner.

The definition of the presheaf of terms is also reasonably direct:

\begin{equation*} \mathsf {Tm}_{\mathcal {V}}\,(X, A) = \prod _{x : \mathsf {El}^0\,X} \mathsf {El}^0\,(A\,x) \end{equation*}

We now show that, along with $\mathcal{V}$ , these two definitions assemble into a CwF.

Proposition 45 (). $\mathcal{V}$ can be equipped with a CwF structure.

Proof. We have given the putative definitions of $\mathsf{Ty}_{\mathcal{V}}$ and $\mathsf{Tm}_{\mathcal{V}}$ . We note that it is straightforward to ensure that both are suitably functorial. The functorial action is given by precomposition and all the required equations hold on-the-nose.

It remains to show that these three pieces of data satisfy the required properties of a CwF. We have already shown that $\mathcal{V}$ has a terminal object, so it remains to discuss the interpretation of context extension. Fix $X : \mathsf{V}^0$ and $A : \mathsf{Ty}_{\mathcal{V}}\,X$ . We define $X.\,A : \mathsf{V}^0$ as $\Sigma ^0\,X\,A$ . The natural equivalence then follows from the $\eta$ principle of dependent sums.

By virtue of Proposition 42, we further note that types $A$ in context $X$ in this model are realized up to equivalence by families $\mathsf{El}^0\,A \to \mathsf{El}^0\,X$ and terms are likewise determined by sections. By presenting $\mathsf{Ty}_{\mathcal{V}}$ and $\mathsf{Tm}_{\mathcal{V}}$ in terms of (dependent) products rather than families and sections, we are able to equip both with strictly functorial actions.

We emphasize that the accomplishment here is not in the definition itself; it mirrors the naïve definition of the set model of type theory as presented by for example, Hofmann (Reference Hofmann1997). What is crucial is that $\mathsf{V}^0$ retains enough of the good behavior of $\mathsf{hSet}_{\mathcal{U}}$ to support such a straightforward definition of the CwF structure while still managing to be an h-set itself.

Remark 46. We note that there are many closely related presentations of models of type theory (categories with attributes (Cartmell Reference Cartmell1986), contextual categories (Cartmell Reference Cartmell1986; Streicher Reference Streicher1991), comprehension categories (Jacobs Reference Jacobs1993, Reference Jacobs1999), natural models (Awodey Reference Awodey2018; Fiore Reference Fiore2012), and so on). We have opted for CwFs because the CwF structure on $\mathsf{V}^0$ is particularly simple and enjoys an exceptional number of definitional equalities. In particular, as opposed to other models which recover terms indirectly as sections to display maps, CwFs require the presheaf of terms as part of their data. This allows us to choose a particular definitional representative for the type of terms in our model and we are then able to explicitly select dependent functions. We shall see that this makes closing $(\mathcal{V}, \mathsf{Ty}_{\mathcal{V}}, \mathsf{Tm}_{\mathcal{V}})$ under various constructions particularly straightforward, as most naturality conditions hold definitionally.

5.3 Further structure on $\mathsf{V}^0$ as a cwF

While we have constructed a CwF structure on $\mathcal{V}$ , we have thus far only shown that the model interprets the basic structural rules of type theory, but not that it is closed under any connectives. The process of extending the model with new connectives is essentially modular: for each connective, we specify the relevant structure on top of a CwF necessary to interpret it and then show that the CwF $(\mathcal{V},\mathsf{Ty}_{\mathcal{V}},\mathsf{Tm}_{\mathcal{V}})$ supports this additional structure.

We illustrate the process with $\Pi$ -types. First, we must define a $\Pi$ -structure on a CwF.

Definition 47 ( $\prod$ -structure, ). A $\prod$ -structure on a CwF $({\mathcal{C}},\mathsf{Ty}_{\mathcal{C}},\mathsf{Tm}_{\mathcal{C}})$ is defined by the following:

• • An operation $\mathsf{pi}_{\mathcal{C}} : \prod _{\Gamma : \mathsf{Ob}\,\mathcal{C}} \mathsf{Ty}_{\mathcal{C}}\,\Gamma\to \mathsf{Ty}_{\mathcal{C}}\,(\Gamma .\,A)\to \mathsf{Ty}_{\mathcal{C}}\,\Gamma$ , natural in $\Gamma$ .

• • For any $\Gamma : \mathsf{Ob}\,{\mathcal{C}}$ , $A : \mathsf{Ty}_{\mathcal{C}}\,\Gamma$ , and $B : \mathsf{Ty}_{\mathcal{C}}\,(\Gamma .\,A)$ an isomorphism $\alpha _{\mathsf{pi}_{\mathcal{C}}}$ between $\mathsf{Tm}_{\mathcal{C}}\,(\Gamma, \mathsf{pi}_{\mathcal{C}}\,\Gamma \,A\,B)$ and $\mathsf{Tm}_{\mathcal{C}}\,(\Gamma .A, B)$ , natural in $\Gamma$ .

Remark 48. One may unpack the content of $\alpha _{\mathsf{pi}_{\mathcal{C}}}$ to see that it includes the introduction, elimination, $\beta$ -, and $\eta$ -rules for $\prod$ -types. The additional requirement of naturality enforces the stability of the introduction and elimination rules under substitution.

Lemma 49 (). The CwF $(\mathcal{V},\mathsf{Ty}_{\mathcal{V}},\mathsf{Tm}_{\mathcal{V}})$ supports a $\prod$ -structure.

Proof. We begin by defining $\mathsf{pi}_{\mathcal{V}}$ as follows:

\begin{equation*} \mathsf {pi}_{\mathcal {V}}\,\Gamma \,A\,B := \lambda (\gamma : \mathsf {El}^0\,(\Gamma )).\, \prod ^0\,(A\,\gamma )\,(\lambda a.B\,(\gamma, a)) \end{equation*}

Naturality in $\Gamma$ is a straightforward computation. The definition of $\alpha _{\mathsf{pi}_{\mathcal{V}}}$ , after unfolding, reduces to the manifestly natural equivalence induced by currying:

\begin{equation*} \prod _{\gamma : \Gamma }\prod _{y : Y(\gamma )} Z(\gamma, y) \simeq \prod _{p : \sum _{\gamma : \Gamma } Y(\gamma )} Z(p) \end{equation*}

We have formalized both the definition of $\Pi$ -structures and the particular $\Pi$ -structure on $(\mathcal{V},\mathsf{Ty}_{\mathcal{V}},\mathsf{Tm}_{\mathcal{V}})$ in $\texttt{Agda}$ . However, already some small inconveniences emerge. For instance, in the definition of naturality for $\alpha{\mathsf{pi}_{\mathcal{V}}}$ , we must specify an equality dependent on the proof witnessing naturality of $\mathsf{pi}_{\mathcal{V}}$ . The dependence is straightforward in this case, but becomes more complex for the later structures. Accordingly, we present only paper proofs for them.

Furthermore, $(\mathcal{V},\mathsf{Ty}_{\mathcal{V}},\mathsf{Tm}_{\mathcal{C}})$ also supports dependent sums.

Definition 50 ( $\Sigma$ -structure). A $\Sigma$ -structure on a CwF $({\mathcal{C}},\mathsf{Ty}_{\mathcal{C}},\mathsf{Tm}_{\mathcal{C}})$ consists of the following two pieces of data:

• • An operation $\mathsf{sig}_{\mathcal{C}} : \prod _{\Gamma : \mathsf{Ob}\,\mathcal{C}} \mathsf{Ty}_{\mathcal{C}}\,\Gamma\to \mathsf{Ty}_{\mathcal{C}}\,(\Gamma .\,A)\to \mathsf{Ty}_{\mathcal{C}}\,\Gamma$ , natural in $\Gamma$ .

• • For any $\Gamma : \mathsf{Ob}\,{\mathcal{C}}$ , $A : \mathsf{Ty}_{\mathcal{C}}\,\Gamma$ , and $B : \mathsf{Ty}_{\mathcal{C}}\,(\Gamma .\,A)$ a natural isomorphism $\alpha _{\mathsf{sig}_{\mathcal{C}}}$ between $\mathsf{Tm}_{\mathcal{C}}\,(\Gamma, \mathsf{sig}_{\mathcal{C}}\,\Gamma \,A\,B)$ and pairs $\sum _{a : \mathsf{Tm}_{\mathcal{C}}(\Gamma, A)} \mathsf{Tm}_{\mathcal{C}}\,(\Gamma, B \cdot ({\mathsf{id}}.a))$ .

Lemma 51. The CwF $(\mathcal{V},\mathsf{Ty}_{\mathcal{V}},\mathsf{Tm}_{\mathcal{V}})$ supports a $\Sigma$ -structure.

Proof. We define $\mathsf{sig}_{\mathcal{V}}$ as follows:

\begin{equation*} \mathsf {sig}_{\mathcal {V}}\,\Gamma \,A\,B := \lambda (x : \mathsf {El}^0\,\Gamma ).\, \Sigma ^0\,(A\,x)\,(\lambda a.B\,(x,a)) \end{equation*}

The remaining structure follows directly. In particular, even though the naturality requires complex path algebra to state properly in the specific CwF on $\mathcal{V}$ all these paths are given by reflexivity.

We next consider a representative inductive type: booleans.

Definition 52 ( $\mathsf{bool}$ -structure). A $\mathsf{bool}$ -structure on a CwF $({\mathcal{C}},\mathsf{Ty}_{\mathcal{C}},\mathsf{Tm}_{\mathcal{C}})$ consists of the following two pieces of data:

• • An operation $\mathsf{bool}_{\mathcal{C}} : \prod _{\Gamma : \mathsf{Ob}\,\mathcal{C}} \mathsf{Ty}_{\mathcal{C}}\,\Gamma$ natural in $\Gamma$ .

• • A pair of operations $\mathsf{true}_{\mathcal{C}},\mathsf{false}_{\mathcal{C}} : \prod _{\Gamma : \mathsf{Ob}\,\mathcal{C}} \mathsf{Tm}_{\mathcal{C}}\,(\Gamma, \mathsf{bool}_{\mathcal{C}})$ also natural in $\Gamma$ .

Furthermore, the following canonical map induced by substitution must be an isomorphism for each $\Gamma : \mathsf{Ob}\,{\mathcal{C}}$ and $A : \mathsf{Ty}_{\mathcal{C}}\,(\Gamma .\,\mathsf{bool}_{\mathcal{C}})$ :

\begin{equation*} \mathsf {Tm}_{\mathcal {C}}\,(\Gamma .\,\mathsf {bool}_{\mathcal {C}}, A) \to \mathsf {Tm}_{\mathcal {C}}\,(\Gamma, \lambda \gamma .\,A(\gamma, \mathsf {true}_{\mathcal {C}})) \times \mathsf {Tm}_{\mathcal {C}}\,(\Gamma, \lambda \gamma .\,A(\gamma, \mathsf {false}_{\mathcal {C}})) \end{equation*}

Lemma 53. The CwF $(\mathcal{V},\mathsf{Ty}_{\mathcal{V}},\mathsf{Tm}_{\mathcal{V}})$ supports a $\mathsf{bool}$ -structure.

Proof. We define $\mathsf{bool}_{\mathcal{V}}$ as follows:

\begin{equation*} \mathsf {bool}_{\mathcal {V}}\,\Gamma = {\lambda } x.\mathsf {bool}^0 \end{equation*}

By construction, $\mathsf{Tm}_{\mathcal{V}}\,(\Gamma, \mathsf{bool}_{\mathcal{V}})$ is definitionally equal to $\mathsf{El}^0\,\Gamma \to \mathsf{bool}$ and so we define $\mathsf{true}_{\mathcal{V}}$ as $\lambda \gamma .\,\mathsf{true}$ and $\mathsf{false}_{\mathcal{V}}$ as $\lambda \gamma .\,\mathsf{false}$ . The calculations that these definitions are natural and that the required map is an isomorphism are routine.

By similar considerations, we may define and close $(\mathcal{V},\mathsf{Ty}_{\mathcal{V}},\mathsf{Tm}_{\mathcal{V}})$ under many other connectives already constructed in Section 3: extensional identity types, natural numbers, and universes among others. Putting all of this together, we conclude the following:

Theorem 54. $\mathcal{V}$ supports a model of extensional type theory with the standard connectives.

We note that this result, combined with Proposition 42 and the series of results about $\mathsf{El}^0$ preserving various categorical connectives can be summarized by the informal slogan: $\mathcal{V}$ supports a model of type theory which internalizes the set-level fragment of the ambient type theory.

6. Relationship to Set Models of Type Theory and Other Set Universes in HoTT/UF

The idea of set theoretic semantics of type theory is of course an old and natural one. An early reference where this is written down more formally is the master thesis of Salvesen (Reference Salvesen1984, Chapter 5). As discussed in the introduction, the work presented in this paper goes back to the model of CZF in type theory of Aczel (Reference Aczel, Macintyre, Pacholski and Paris1978). Aczel also interpreted extensional type theory with universes in an extension of CZF with a hierarchy of inaccessible sets (Aczel, Reference Aczel, Altenkirch, Reus and Naraschewski1999). In fact, Aczel’s V occurs already in the PhD thesis of Leversha (Reference Leversha1976), where it was used to represent ordinals constructively. Various earlier works have also relied on Aczel’s V to model type theory. For instance, Werner (Reference Werner1997) modeled the core system of $\texttt{Coq}$ in ZFC and vice versa, using Aczel’s encoding of sets. A refinement by Barras (Reference Barras2010, Reference Barras2012) models the core system of $\texttt{Coq}$ system in intuitionistic ZF and formalizes the model in $\texttt{Coq}$ (The Coq Development Team 2021). More recently, Palmgren (Reference Palmgren2019) presented an interpretation of extensional MLTT (Martin-Löf Reference Martin-Löf, Cohen, Łoś, Pfeiffer and Podewski1982) into intensional MLTT via setoids, also relying on Aczel’s V. Palmgren’s work was also formalized in $\texttt{Agda}$ .

Aczel’s V was revisited in HoTT/UF by Gylterud (Reference Gylterud2018, Reference Gylterud2020) who observed that this gives a universe of multisets, but that one can restrict it, as in Definition 9, to get a universe of h-sets. These universes of (multi)sets has recently also been further studied by Escardó and de Jong who has their own $\texttt{Agda}$ formalization as part of the $\texttt{TypeTopology}$ project (Escardó and de Jong Reference Escardó and de Jong2023). Among many other things, they have two more proofs of Theorem12 formalized. Various HITs for representing finite multisets have also been considered in HoTT/UF (Angiuli et al. Reference Angiuli, Cavallo, Mörtberg and Zeuner2021; Basold et al. Reference Basold, Geuvers and van der Weide2017; Choudhury and Fiore Reference Choudhury and Fiore2019; Frumin et al. Reference Frumin, Geuvers, Gondelman and van der Weide2018; Joram and Veltri Reference Joram, Veltri, Naumowicz and Thiemann2023; Veltri Reference Veltri and Kobayashi2021); however, these are of course not sufficient to model full type theory.

We will now discuss other approaches to constructing strict categories of sets in HoTT/UF that could also serve as internal models of type theory. These often require various extensions of the quite minimal univalent type theory that we have relied on in this paper.