2. Simplicial homotopy type theory

Recall that HoTT is Martin–Löf dependent type theory plus Voevodsky’s Univalence Axiom. We use all the standard notation from HoTT as in The Univalent Foundations Program (2013). In particular, we use $a=_Ab$ for the identity type between two terms $a,b$ of type $A$ , or $a=b$ when $A$ is clear from the context. A function $f:A \to B$ is an equivalence if it has contractible fibers. The fact that $f$ is an equivalence is a proposition, which we denote as $\textsf {isEquiv}(f)$ . Given two types $A$ and $B$ , we write $A\simeq B$ for the type of equivalences between $A$ and $B$ . The Univalence Axiom establishes that the map (defined by path induction)

\begin{equation*} (A=B) \to (A\simeq B) \end{equation*}

is an equivalence. For two functions $f,g:A \to B$ , we write

\begin{equation*}f \sim g :\equiv \prod _{a:A}f(a)=_B g(a).\end{equation*}

An element of this type is called a homotopy. The non-dependent version of function extensionality states that the natural map

\begin{equation*} (f=_{A \to B}g) \to (f\sim g) \end{equation*}

is an equivalence. Voevodsky proved that univalence implies function extensionality. Throughout this work, we make liberal use of function extensionality. In particular, whenever we say a function $f$ is “unique,” we mean that for any other function $g$ (satisfying the same property as $f$ ) we have a path $p:f=g$ or a homotopy $q:f\sim g$ .

In order to obtain sHoTT, for technical reasons, the theory is layered, with a separate layer containing the interval $\unicode{x1D7DA}$ . This facilitates the construction of extension types, which give families of hom types with fixed endpoints. The type $\unicode{x1D7DA}$ is equipped with a total order $\leq$ , a bottom element $0$ , and a distinct top element $1$ . In a type $X$ , an arrow in $X$ can be defined as a map $f: \unicode{x1D7DA} \to X$ , with the source and target of $f$ being the image of the endpoints $0, 1 : \unicode{x1D7DA}$ . Although we can do this ordinary type theory, we need to add extra equalities $f(0)=x$ and $f(1)=y$ . The new type former, extension type, allows to introduce definitional equalities instead. We refer to Riehl and Shulman (Reference Riehl and Shulman2017) for all the details, or to Bardomiano Martínez (Reference Bardomiano Martínez2025) for a light introduction.

In this type theory, we can define some familiar simplicial shapes:

$ \Delta ^0:\equiv \{t:{\textbf {1}}|\top \},$

$ \Delta ^1:\equiv \{t:\unicode{x1D7DA}|\top \},$

$ \Delta ^2:\equiv \{\langle t_1,t_2\rangle :\unicode{x1D7DA}\times \unicode{x1D7DA}|t_2\leq t_1 \},$

$ \partial \Delta ^1:\equiv \{t:\unicode{x1D7DA}|(t\equiv 0)\vee (t\equiv 1)\},$

$ \partial \Delta ^2:\equiv \{\langle t_1,t_2\rangle :\Delta ^2| (0\equiv t_2\leq t_1)\vee (t_1\equiv t_2)\vee ( t_2\leq t_1 \equiv 1)\},$

$ \Lambda _1^2:\equiv \{\langle t_1,t_2\rangle :\Delta ^2| (t_1\equiv 1)\vee (t_2\equiv 0)\}.$

Note that for example, we have a map $i: \Lambda _1^2 \to \Delta ^2$ , we will use this map often. In general, type elimination rules for shapes (Riehl and Shulman Reference Riehl and Shulman2017, Figure 3) allow us to construct functions from a shape $\phi$ to a type $A$ , that is, the type of functions $\phi \to A$ . Given $x,y:A$ , we can define the type

\begin{equation*} {\textsf {hom}}_A(x,y):\equiv \left \langle \Delta ^1\rightarrow A\middle |_{[x,y]}^{\partial \Delta ^1}\right \rangle . \end{equation*}

A term in this type is called an arrow in $A$ . Essentially, an arrow is a function $\Delta ^1 \to A$ with $f(0)\equiv x$ and $f(1)\equiv y$ .

Definition 2.1 (Riehl and Shulman Reference Riehl and Shulman2017, Theorem 5.5). A type $A$ is Segal if and only if the map induced by $i:\Lambda _1^2 \to \Delta ^2$

\begin{equation*} A^{\Delta ^2} \to A^{\Lambda _1^2} \end{equation*}

is an equivalence.

An element of the proposition ${\textsf {isSegal}}(A)$ is a proof that the type $A$ is Segal.

Intuitively, a type $A$ is Segal whenever the composite of two arrows is uniquely determined. This composition is unital and associative (Riehl and Shulman Reference Riehl and Shulman2017, Propositions 5.8 and 5.9). The identity for the composition is defined as ${\textsf {id}}_x:{\textsf {hom}}_A(x,x)$ as ${\textsf {id}}_x(t)\equiv x$ for all $t:\Delta ^1$ . An arrow $f$ in a Segal type $A$ is an isomorphism if it has a left and a right inverse, that is, if the type

\begin{equation*}\textsf {isiso}(f):\equiv \left ( \sum \limits _{g:{\textsf {hom}}_A(y,x)}g\circ f={\textsf {id}}_x \right ) \times \left ( \sum \limits _{h:{\textsf {hom}}_A(y,x)}f\circ h={\textsf {id}}_y \right )\end{equation*}

is inhabited. The proposition of isomorphisms between terms $x,y:A$ is denoted $x\cong y$ . By path induction, there is a map

\begin{equation*} {\textsf {idtoiso}}: (x=y) \to (x\cong y). \end{equation*}

An element of the proposition ${\textsf {isRezk}}(A)$ is a proof that the Segal type $A$ is Rezk.

Riehl and Shulman (Reference Riehl and Shulman2017, Appendix A) prove that in the bisimplicial sets semantics of sHoTT, Segal types and Rezk types correspond to Segal spaces and complete Segal spaces, respectively. We often think of Segal spaces as pre- $\infty$ -categories and complete Segal spaces as $\infty$ -categories. The same intuition passes to Segal and Rezk types. Therefore, Segal types are the synthetic pre- $\infty$ -categories and Rezk types are the synthetic $\infty$ -categories. Moreover, the Rezk types in which all arrows are isomorphisms are what we think of as $\infty$ -groupoids. These types are also characterized by the fact that map defined by path induction

\begin{equation*} a=_Ab \to {\textsf {hom}}_A(x,y) \end{equation*}

is an equivalence (Riehl and Shulman Reference Riehl and Shulman2017, Proposition 10.10). These types are called discrete.

Recall from Buchholtz and Weinberger (Reference Buchholtz and Weinberger2023) that a type family $ Q:B\to \mathcal{U}$ is an inner family if the following proposition is true

\begin{equation*} {\textsf {isInner}}(Q):\equiv \prod _{\alpha :\Delta ^2 \to B }\prod _{\delta :\prod _{t:\Lambda _1^2}Q(\alpha (i(t)))} {\textsf {isContr}}\left (\left \langle \prod _{t:\Delta ^2}Q(\alpha (t))\middle |_{\delta }^{\Lambda _1^2} \right \rangle \right ). \end{equation*}

The significance of inner families can be understood via Proposition 2.3 (Buchholtz and Weinberger Reference Buchholtz and Weinberger2023, Propositions 4.1.5 and 4.1.6). We explain briefly the motivation behind this result. First, given a type $A$ , we can think of it as a type family over $\textbf {1}$ sending the element $*:{\textbf {1}}$ to $A$ . This gives us the function $\pi :A \to {\textbf {1}}$ . Then $A$ is a Segal type if and only if the diagram

has a diagonal filler that is unique up to homotopy (see Definition 2.1). Now, Proposition 2.3 establishes a relative version of this characterization of Segal types.

Since $B$ is a Segal type, if $Q:B \to \mathcal{U}$ is a type family over the Segal type $B$ and $\pi :\sum _{b\;:\;B}Q(b) \to B$ is the canonical projection from its total type, then Proposition 2.3 can be rephrased by saying that the diagram

has a diagonal filler that is unique up to homotopy, see Buchholtz and Weinberger (Reference Buchholtz and Weinberger2023, Observation 2.4.1).

Finally, the last piece of background we need are isoinner families (Buchholtz and Weinberger Reference Buchholtz and Weinberger2023). A type family $ P:B\to \mathcal{U}$ over a Segal type $ B$ is called isoinner family if the following proposition is true:

\begin{equation*} {\textsf {isIsoinner}}(P):\equiv {\textsf {isInner}}(P)\times \prod _{b\;:\;B} {\textsf {isRezk}}(P(b)). \end{equation*}

Although isoinner families can be defined over arbitrary types, whenever we work with isoinner families we will always assume the base type involved is a Rezk type, see Remark 3.8. The reason is because we are only interested in exponentials over $\infty$ -categories. We will also use the following fact from Riehl and Shulman (Reference Riehl and Shulman2017, Proposition 10.9): if $ B$ is a Rezk type and $ X$ is any type or shape, then $ B^X$ is also a Rezk type.

3. The Segal and Rezk type completion

The notion we study in this section is essential to correctly formulate Condition 5 in Theorem 4.1. In Section 3.1, we define the Segal completion of a type and establish some basic properties. In an analogous way, we spell out Rezk completion of a type in Section 3.2.

3.1 Segal type completion

Definition 3.1. A Segal type completion for a type $ A$ consists of a Segal type $S$ and a map $ \iota :A \to S$ such that for any Segal type $ X$ the map

(1) \begin{equation} \iota ^* :\equiv \_\circ \iota :(S\to X)\to (A\to X) \end{equation}

is an equivalence, that is, if

\begin{equation*} {\textsf {isCompletion}}^A(S,\iota ):\equiv {\textsf {isSegal}}(S)\times \left (\prod _{X\;:\;\mathcal{U}}{\textsf {isSegal}}(X)\to \textsf {isEquiv}(\iota ^*)\right ). \end{equation*}

Remark 3.2. The types ${\textsf {isSegal}}(S)$ , $\textsf {isEquiv}(i^*)$ are propositions. Given $X:U$ the type ${\textsf {isSegal}}(X) \to \textsf {isEquiv}(i^*)$ is also a proposition, and therefore $\prod _{X\;:\;\mathcal{U}}{\textsf {isSegal}}(X) \to \textsf {isEquiv}(i^*)$ is a proposition. From this, we see that ${\textsf {isCompletion}}^A(S,i)$ is a proposition.

Remark 3.3. For this work, we do not need a full implementation of Definition 3.1 (or its relative version Definition 3.6). However, it should be possible to obtain this construction via modalities Rijke et al., (Reference Rijke, Shulman and Spitters2020). Explicitly, we can get modal extensions of sHoTT using multimodal type theory (Daniel Gratzer et al. Reference Daniel Gratzer, Nuyts and Birkedal2021). For example, this has been done in Gratzer et al. (Reference Gratzer, Weinberger and Buchholtz2024, Reference Gratzer, Weinberger and Buchholtz2025) to build the $\infty$ -category of $\infty$ -groupoids and to define the Yoneda embedding, respectively. We leave the construction of this extension of the type theory for future work.

Definition 3.4. Let $ A$ be a type. We define

\begin{equation*} \textsf {Completion}(A):\equiv \sum _{S:\mathcal{U}}\sum _{\iota :A\to S}{\textsf {isCompletion}}^A(S,\iota ). \end{equation*}

Remark 3.5. Note that whenever a completion $ (S,\iota )$ exists, it is unique up to equivalence, and this follows by the standard argument using the universal property. By the Univalence Axiom, this can be reformulated by saying that $\textsf {Completion}(A)$ is a proposition.

If we have $(S,\iota ,p):\textsf {Completion}(A)$ , then we call the underlying Segal type $S$ the Segal type completion of the type $ A$ . In the Segal space model structure on $ \mathbf{ssSet}$ , this corresponds to the fibrant replacement of a Reedy fibrant bisimplicial set by a Segal space, see Proposition 5.5.

The equivalence (1) tells us that the fibers are contractible; for any $\psi :A\to X$ we have

\begin{equation*}{\textsf {isContr}}\left (\sum _{\varphi :S\to X}\varphi \circ \iota =\psi \right ).\end{equation*}

Unfolding the above means that for any $\psi :A\to X$ , where $X$ is a Segal type, there exists a unique $\varphi :S\to X$ such that $\varphi \circ \iota =\psi$ . We can put this pictorially by saying that any $\psi :A\to X$ uniquely factors through $\iota$ as in the following diagram:

We will often refer only to the Segal space $S$ and assume that the map $\iota :A \to S$ is given and available for use. By uniqueness in the above, we simply mean that the type $\sum _{\varphi :S \to X} \psi \sim \varphi \circ \iota$ is contractible. The homotopy in the center of contraction is omitted most of the time, and we only make reference to the function. We carry this convention forward below.

We can also consider a relative version of this universal property. For the time being, fix a Segal type $B$ . We use the following notation:

\begin{equation*} \mathcal{U} /B :\equiv \sum _{S:\mathcal{U}}S\to B. \end{equation*}

Furthermore, we will refer to an element $(S, \phi )$ of this type by leaving implicit the type $S$ and mentioning that we have a map of type $S \to B$ .

Recall from Buchholtz and Weinberger (Reference Buchholtz and Weinberger2023) that the relative function type between functions $\pi :A\to B$ and $\xi :E\to B$ is given by the pullback diagram:

Note that if we assume further that $E$ is a Segal type and $A$ is a type or shape, then by Riehl and Shulman (Reference Riehl and Shulman2017, Corollary 5.6] the types $E^A$ and $B^A$ are Segal types. From Buchholtz and Weinberger (Reference Buchholtz and Weinberger2023, Proposition 4.1.3) we have that inner maps are pullback stable. Therefore, ${\textsf {Fun}}_{/B}(A,E)$ is a Segal type. An element $\iota : {\textsf {Fun}}_{/B}(A,E)$ is a function making the following diagram commute:

Thus, we call the elements of ${\textsf {Fun}}_{/B}(A,E)$ functions over $B$ .

Definition 3.6. Let $ A\to B$ a type over $ B$ . A relative Segal type completion for $ A \to B$ consists of a Segal type over $B$ , $S \to B$ , and a map $ \iota :{\textsf {Fun}}_{/B}(A,S)$ such that for any Segal type $X$ over $B$ the map

(2) \begin{equation} \iota _{/B}^*:{\textsf {Fun}}_{/B}(S,X)\to {\textsf {Fun}}_{/B}(A,X) \end{equation}

is an equivalence, that is, if

\begin{equation*} {\textsf {isCompletion}}_{/B}^A(S,\iota ):\equiv {\textsf {isSegal}}(S)\times \left ( \prod _{X\;:\;\mathcal{U}/B}{\textsf {isSegal}}(X)\to \textsf {isEquiv}(\iota _{/B}^*)\right ). \end{equation*}

Note that whenever $ S\to B$ exists, then it is unique up to equivalence.

Definition 3.7. Let $ A \to B$ be a type over $ B$ . We define

\begin{equation*} \textsf {Completion}_{/B}(A):\equiv \sum _{S:\mathcal{U}/B}\sum _{ \iota :{\textsf {Fun}}_{/B}(A,S) }{\textsf {isCompletion}}_{/B}^A(S,\iota ). \end{equation*}

Remark 3.8. Note that in the definitions above, we do not need to assume that the type $B$ is Segal. However, we will add this assumption in our result Theorem 4.1. Furthermore, we will note that the Segal completion of a type $A$ over a Segal type $B$ is equivalent to the completion over the point (Proposition 3.9). This will force us to consider inner families over a Segal type $B$ since they correspond exactly to maps from Segal types into $B$ (Proposition 2.3). For similar reasons, we will add the assumption that $B$ is Rezk in Corollary 4.5 and use isoinner families.

We will say often that $S$ is a Segal type completion relative to the type $B$ leaving implicit the map $\xi :S\to B$ , and that $\iota :{\textsf {Fun}}_{/B}(A,S)$ exists. The equivalence (2) tells us that the fibers are contractible: for any $\psi :{\textsf {Fun}}_{/B}(A,X)$ we have

\begin{equation*}{\textsf {isContr}}\left (\sum _{\varphi :{\textsf {Fun}}_{/B}(S,X)}\varphi \circ \iota =\psi \right ).\end{equation*}

Just as we did before, unfolding the above means that for any $\psi :{\textsf {Fun}}_{/B}(A,X)$ , where $\delta :X\to B$ is a map between Segal types, there exists a unique $\varphi :{\textsf {Fun}}_{/B}(S,X)$ such that $\varphi \circ \iota =\psi$ . The picture for this situation is the commutative diagram below:

For the next section, it will be useful to know that for a type $A$ its associated Segal type completion $S$ is also universal relative to any Segal type $B$ , and vice versa. Informally, the categorical interpretation we give to this is that having the Segal type completion over the single-point Segal space is equivalent to having it over any slice (by a Segal space).

Proposition 3.9. Let $A$ be any type and $ B$ a Segal type. Assume further we have a commutative diagram

where $ S$ is a Segal type. Then:

\begin{equation*}{\textsf {isCompletion}}^A(S,\iota ) \simeq {\textsf {isCompletion}}_{/B}^A(S,\iota ).\end{equation*}

Proof. Since $ S$ is a Segal type is enough to show that

\begin{equation*} \left (\prod _{X\;:\;\mathcal{U}}{\textsf {isSegal}}(X)\to \textsf {isEquiv}(\iota ^*)\right ) \simeq \left (\prod _{X\;:\;\mathcal{U}/B}{\textsf {isSegal}}(X)\to \textsf {isEquiv}(\iota _{/B}^*)\right ).\end{equation*}

We can construct a function

\begin{equation*}\varphi :\left (\prod _{X\;:\;\mathcal{U}}{\textsf {isSegal}}(X)\to \textsf {isEquiv}(\iota ^*)\right ) \to \left (\prod _{X\;:\;\mathcal{U}/B}{\textsf {isSegal}}(X)\to \textsf {isEquiv}(\iota _{/B}^*)\right )\end{equation*}

as follows.

Assume that $\prod _{X\;:\;\mathcal{U}}{\textsf {isSegal}}(X)\to \textsf {isEquiv}(\iota ^*)$ and consider a function $\delta :X\to B$ , where $ X$ a Segal type. Our main goal is to prove that $\iota _{/B}^*$ is an equivalence by showing that it has contractible fibers. Let $j:{\textsf {Fun}}_{/B}(A,X)$ a function. The assumption produces the equivalence $\iota ^* : Y^S \to Y^A$ for every Segal type $Y$ , by instantiating on $X$ this gives us unique functions $h:S\to B$ and $g:S\to X$ which make the following diagrams commutative:

Since $j:{\textsf {Fun}}_{/B}(A,X)$ then by uniqueness of $h:S\to B$ , we must have $h=\delta \circ g$ , and using the equivalence $g$ must be unique. Thus, we have the following diagram:

The above shows that $ \iota _{/B}^*$ is an equivalence, and this concludes the construction of $ \varphi$ .

Similarly, we can construct a function

\begin{equation*}\psi :\left (\prod _{X\;:\;\mathcal{U}/B}{\textsf {isSegal}}(X)\to \textsf {isEquiv}(\iota _{/B}^*)\right ) \to \left (\prod _{X\;:\;\mathcal{U}}{\textsf {isSegal}}(X)\to \textsf {isEquiv}(\iota ^*)\right ).\end{equation*}

We assume that $\prod _{X\;:\;\mathcal{U}/B}{\textsf {isSegal}}(X)\to \textsf {isEquiv}(\iota _{/B}^*)$ and consider $j:A\to X$ a function where $X$ is a Segal type. Now we want to observe that the function $\iota ^*$ has contractible fibers. We get the function $j\times \textrm{Id}_B:A\times B\to X\times B$ and construct the commutative diagram:

This implies that $(j,\pi ):{\textsf {Fun}}_{/B}(A,X\times B)$ . Since $S$ is the Segal type completion of $ A$ relative to $B$ , the assumption says that the function $\iota _{/B}^*$ is an equivalence. So there exists a unique function $f:S\to X\times B$ making the following diagram commutative:

From this we obtain:

To show uniqueness, if we had a map $g:S\to X$ fitting in the triangle above, then we certainly get:

By uniqueness we have that $f=(g,\xi )$ , from which we finally conclude $p_1\circ f= g$ . After obtaining $ \varphi$ and $ \psi$ , we can then show that $ \varphi \circ \psi$ and $ \psi \circ \varphi$ are homotopic to the corresponding identity functions. This follows using the universality of each completion.

3.2 The Rezk type completion

In the same way as in the previous section, we define the Rezk type completion of a type. The results that we obtained for Segal type completions are also true for Rezk type completions. We formally state the relative version of Rezk completions in Definition 3.10.

Analogous to Definition 3.6, we state

Definition 3.10. Let $ A\to B$ be a type over $ B$ . A relative Rezk type completion for $ A \to B$ consists of a Rezk type over $B$ , $R \to B$ and a map $ \iota :{\textsf {Fun}}_{/B}(A,R)$ such that for any Rezk type $X$ over $B$ the map

(3) \begin{equation} \iota _{/B}^*:{\textsf {Fun}}_{/B}(R,X)\to {\textsf {Fun}}_{/B}(A,X) \end{equation}

is an equivalence, that is, if

\begin{equation*} \textsf {isRCompletion}_{/B}^A(S,\iota ):\equiv {\textsf {isRezk}}(R)\times \left ( \prod _{X\;:\;\mathcal{U}/B}{\textsf {isRezk}}(X)\to \textsf {isEquiv}(\iota _{/B}^*)\right ). \end{equation*}

Definition 3.11. Let $ A \to B$ a type over $ B$ . We define

\begin{equation*} \textsf {RCompletion}_{/B}(A):\equiv \sum _{R:\mathcal{U}/B}\sum _{ \iota :{\textsf {Fun}}_{/B}(A,R) }\textsf {isRCompletion}_{/B}^A(R,\iota ). \end{equation*}

Definition 3.12. Let $A$ be any type. We define the Rezk type completion of $A$ as the relative Rezk completion of $A$ over the Rezk type $\textbf {1}$ . In the notation above, we omit $\textbf {1}$ and simply write $\textsf {isRCompletion}^A(S,\iota )$ .

The proof of the following proposition is the same Proposition 3.9, with only minor adjustments in the assumptions.

Proposition 3.13. Let $A$ be any type and $ B$ a Rezk type. Assume further we have a commutative diagram

where $ R$ is a Rezk type. Then:

\begin{equation*}\textsf {isRCompletion}^A(R,\iota ) \simeq \textsf {isRCompletion}_{/B}^A(R,\iota ).\end{equation*}

4. Conduché’s theorem

The next theorem characterizes and, at the same time, allows defining exponentiable functors between Segal types. As we mentioned in the introduction, this result is analogous to the one given by Ayala, Francis, and Rozenblyum in Ayala et al. (Reference Ayala, Francis and Rozenblyum2018, Lemma 5.16] for quasi-categories. Its proof is the main focus of this section.

We proceed to prove the main result Theorem 4.1. Our main contribution is the equivalence of conditions (1) and (5). The equivalence between (1) and (2) follows from the work of Buchholtz and Weinberger (Reference Buchholtz and Weinberger2023). Also, the equivalence between (2), (3), and (4) follows from Niefield (Reference Niefield1982); see Remark 4.2.

Remark 4.2. Before proceeding with the proof, we give some motivation for the conditions of the theorem. If a map $ f:E \to B$ in a category $ \mathcal{C}$ is exponentiable, then we have a triple adjunction

In this situation, the internal hom in the slice $ \mathcal{C}/B$ category is given by

\begin{equation*} [f:E \to B,-]_{/B}:\equiv \prod _f \circ f^* .\end{equation*}

In general, this formula is the semantic interpretation of the type family in (2) of Theorem 4.1, $ \lambda b. E(b)\to P(b)$ . In our framework of sHoTT, the extra condition of being an inner family is a natural one since maps between Segal types are equivalent to inner families (Proposition 2.3). Conditions (2)–(4) simply express the fact that exponentiation happens over the point, over $ B$ , or over a type dependent on $ B$ . A detailed explanation behind (5) can be found in Section 5.2.1.

Proof. (1) $\Leftrightarrow$ (2). This is immediate from Proposition 2.3 where it is proved that the total space of a type family $P:B\to \mathcal{U}$ over a Segal type $B$ is a Segal type if and only if $P$ is an inner family.

$(4)\Leftrightarrow (3) \Leftrightarrow (2)$ is a classical result that can be found, for example, in Niefield (Reference Niefield1982). This is because the conditions express the fact we have internal hom objects in the corresponding slices (see Remark 4.2).

(1) $\Rightarrow$ (5). Our global assumption is that for any inner family $P:B \to \mathcal{U}$ , the type $\sum _{b\;:\;B}\left (E(b)\to P(b)\right )$ is a Segal type. The additional data from (5) gives us the commutative diagram

where $ p_1$ and $ p_1$ are the projections. The function $q$ denotes the map $ \lambda (t, e).(\alpha (t),e)$ , and $ f$ is the projection map. In particular, we have that $ \alpha \circ p_1 = f\circ q$ and $ \alpha \circ i \circ p_2 = f\circ q \circ \iota$ .

Our main goal is to show that $ f\circ q: F_1 \to B$ is the Segal type completion relative to $ B$ of $ f\circ q \circ \iota : F_2 \to B$ . Note that $F_1$ is a Segal type: since inner maps are defined by a right-lifting property, then they are stable under pullbacks, and by assumption $f$ is inner, therefore $p_1$ is inner, which implies in particular that $F_1$ is Segal. Then we just have to show that it is universal, that is, that for any Segal type $X$ over $B$ , this is map $k:X \to B$ , and $\psi :F_2 \to X$ such that $k \circ \psi = \alpha \circ i \circ p_2$ , the induced map $(f\circ q)_{/B}^*: {\textsf {Fun}}_{/B}(F_1,X) \to {\textsf {Fun}}_{/B}(F_2,X)$ is an equivalence. We simply have to show that $\psi : F_2 \to X$ factors through a $\iota$ via a unique map $\varphi : F_1 \to X$ . So we must construct this map exists and justify that it is unique.

By Straightening/Unstraightening, we may assume that $X \equiv \sum _{b\;:\;B} P(b)$ and that $k$ is the first projection.

The map $\varphi$ . To build the map $\varphi$ , we apply the global hypothesis to a lifting problem involving $\sum _{b\;:\;B}(E(b) \to P(b))$ . So first, we define the adequate lifting problem. For this, we can assume that $\psi (t,e)\equiv (b_t,e_t):\sum _{b\;:\;B}P(b)$ . For each $(t,e):F_2$ , there is a path $ p_t: b_t=\alpha (i(t))$ given by $ \alpha \circ i \circ p_2 =k \circ \psi$ . Then we can consider the transport map

\begin{equation*}p_t^*:P(\alpha (i(t)))\to P(b_t)\end{equation*}

and its inverse

\begin{equation*}(p_t^{-1})^*: P(b_t) \to P(\alpha (i(t))).\end{equation*}

Using this, we can define

\begin{equation*}\nu :\equiv \lambda t.\lambda e. (p_t^{-1})^*(e_t):\prod _{t:\Lambda _1^2}\left (E(\alpha (i(t)))\to P(\alpha (i(t)))\right ).\end{equation*}

This gives us the lifting problem:

Now we can use the global assumption, and the $\sum$ -type in the right corner is Segal. Then $\pi '$ is inner, since it is a map between Segal types (Proposition 2.3). Therefore, there exists a unique

\begin{equation*}\xi :\left \langle \prod _{t:\Delta ^2}(E(\alpha (t))\to P(\alpha (t)))\middle |_{\nu }^{\Lambda _1^2} \right \rangle \end{equation*}

a diagonal filler of the square. The construction of

\begin{equation*}\varphi :\left (\sum _{t:\Delta ^2}E(\alpha (t))\right )\to \left (\sum _{b\;:\;B}P(b) \right )\end{equation*}

is simply given by the formula: $\lambda (t,e).(\alpha (t),\xi _t(e))$ .

Factorization through $\iota$ . It remains to show we have a homotopy $\varphi \circ \iota \sim \psi$ . Consider $(t,e):\sum _{t:\Delta ^2}E(\alpha (t))$ . Then by definition, we have

\begin{equation*}\varphi \circ \iota (t,e)\equiv (\alpha (i(t)),\xi _{i(t)}(e))\text{ and }\psi (t,e)\equiv (b_t,e_t).\end{equation*}

Using the characterization of paths in the total space, we obtain $p_t:\alpha (i(t))=b_t$ , and also there is an equality

\begin{equation*}p_t^*(\xi _{i(t)}(e))\equiv p_t^*(\nu _t(e))\equiv p_t^*((p_t^{-1})^*(e_t))=e_t.\end{equation*}

Therefore, $\varphi \circ \iota (t,e)=\psi (t,e),$ it gives us the required homotopy.

Uniqueness of $\varphi$ . To prove uniqueness, the idea is to use the uniqueness of $\xi$ as solution for the lifting problem we defined above. Then we can compare the resulting maps, $\varphi '$ and $\varphi$ , via a direct computation.

Let

\begin{equation*}\varphi ':\left (\sum _{t:\Delta ^2}E(\alpha (t))\right )\to \left (\sum _{b\;:\;B}P(b)\right )\end{equation*}

be a map over $B$ and a homotopy $\bar r : \varphi ' \circ \iota \sim \psi$ . We can assume that $\varphi '(t,e)\equiv (b_t',e_t')$ . There is a homotopy $q:\alpha \circ p_1 \sim g\circ \varphi '$ . For any $(t,e):\sum _{t:\Delta ^2}E(\alpha (t))$ , we get a path $q_t:\alpha (t)=b_t'$ , which gives rise to the transport map $q_t^*:P(\alpha (t))\to P(b_t')$ .

Similarly, if $(t,e):\sum _{t:\Lambda _1^2}E(\alpha (i(t)))$ then

\begin{equation*}\bar {r}_t:\varphi '(\iota (t,e))\equiv (b_{i(t)}',e_t')=(b_t,e_t) \equiv \psi (t,e).\end{equation*}

This is a path in a total space, so it is given by $r_t:b_{i(t)}'=b_t$ and $d_t:r_t^*(e_t')=e_t$ , where $r_t^*:P(b_{i(t)}')\to P(b_t)$ is the transport map.

We first make the following observation:

(4) \begin{equation} \prod _{(t,e):F_2}\prod _{p_t:\alpha (i(t))=b_t}\prod _{q_{i(t)}:\alpha (i(t))=b_{i(t)}'}\prod _{r_t:b_{i(t)}'=b_t} (q_{i(t)}^{-1})^*((r_t^{-1})^*(e_t))=(p_t^{-1})^*(e_t). \end{equation}

By induction on the paths, we assume that $p_t\equiv {\textsf {refl}} :\alpha (i(t))=\alpha (i(t))$ , $q_{i(t)}\equiv {\textsf {refl}} :\alpha (i(t))=\alpha (i(t))$ , and $r_t\equiv {\textsf {refl}}:\alpha (i(t))=\alpha (i(t))$ . In this case, the transport maps are identities, so we can use ${\textsf {refl}}:e_t=e_t$ . Using the map $\varphi '$ , we construct

\begin{equation*} \xi ':\left \langle \prod _{t:\Delta ^2}(E(\alpha (t))\to P(\alpha (t)))\middle |_{\nu }^{\Lambda _1^2} \right \rangle \end{equation*}

by $\xi '\equiv \lambda t.\lambda e. (q_t^{-1})^*(e_t')$ . To see this indeed gives the correct type to $\xi '$ , we evaluate on $t:\Lambda _1^2$ . Our aim is to construct for each $e:E(\alpha (i(t)))$ a path $\xi _{i(t)}'(e)=\nu _t(e)$ , unfolding the definitions this is a path

\begin{equation*}(q_{i(t)}^{-1})^*(e_{i(t)}')=(p_t^{-1})^*(e_t).\end{equation*}

Note that this follows from (4) in combination with the path $d_t:r_t^*(e_t')=e_t$ . The uniqueness of $\xi$ gives us the equality $\xi '=\xi$ .

Finally, we now show that for all $(t,e):\sum _{t:\Delta ^2}E(\alpha (t))$ , $\varphi (t,e)=\varphi '(t,e)$ , so we need $\varphi (t,e)\equiv (\alpha (t),\xi _t(e))=(b_t',e_t')\equiv \varphi '(t,e)$ . First, there is a path $q_t:\alpha (t)=b_t'$ , and using that $\xi =\xi '$ we have

\begin{equation*} q_t^*(\xi _t(e))=q_t^*(\xi _t'(e))=q_t^*((q_t^{-1})^*(e_t'))=e_t'. \end{equation*}

To sum up, we have shown that for any Segal type $X$ and map $\psi : F_1 \to X$ , the map $\psi$ factors uniquely through $\iota$ . The above proves that $F_1$ is a Segal type completion for $F_2$ .

(5) $\Rightarrow$ (1). We assume that $F_1$ is the relative Segal type completion of $F_2$ . Let $P: B \to \mathcal{U}$ an inner type family. We can reduce the problem of showing (1) using the equivalence (1) $\Leftrightarrow$ (2). So it is enough to show that the type family

\begin{equation*}P:\equiv \lambda b.E(b)\to P(b):B\to \mathcal{U}\end{equation*}

is an inner family. This amounts to showing that the projection map

\begin{equation*}\pi :\left (\sum _{b\;:\;B}(E(b)\to P(b))\right )\to B\end{equation*}

is right orthogonal to the horn inclusion $i:\Lambda _1^2 \to \Delta ^2$ .

The lifting problem. Consider a lifting problem

(5)

This means we have a partial section $\delta :\prod _{t:\Lambda _1^2}\big (E(\alpha (i(t)))\to P(\alpha (i(t)))\big )$ . We define the function

\begin{equation*}\psi :\left (\sum _{t:\Lambda _1^2}E(\alpha (i(t)))\right )\to \left (\sum _{b\;:\;B}P(b)\right )\end{equation*}

as $\psi :\equiv \lambda t.\lambda e.(\alpha (i(t)),\delta _t(e))$ . We illustrate this in a commutative diagram:

Since $P$ is an inner family, its total space is Segal. Using that $F_1$ is the Segal type completion, then we can complete the diagram above with a unique map $\varphi : F_1 \to \sum _{b\;:\;B} P(b)$ over $B$ such that $\varphi \circ \iota = \psi$ .

Solution to the lifting problem. We can use the map $\varphi$ to construct a map filling the square (5). In what follows, we can assume that $\varphi (t,e)\equiv (b_t,e_t)$ . We have paths

\begin{equation*}p_t:\alpha (t)=b_t \text{ and } \bar {q}_t:\psi (t,e)\equiv (\alpha (i(t)),\delta _t(e))=(b_{i(t)},e_{i(t)})\equiv \varphi (\iota (t,e)).\end{equation*}

The second path amounts to,

\begin{equation*} q_t:\alpha (i(t))=b_{i(t)} \text{ and } d_t:q_t^*(\delta _t(e))=e_{i(t)} \end{equation*}

where again $q_t^*:P(\alpha (i(t)))\to P(b_{i(t)})$ is the transport map. The element

\begin{equation*}\xi :\left \langle \prod _{t:\Delta ^2}(E(\alpha (t))\to P(\alpha (t)))\middle |_{\delta }^{\Lambda _1^2} \right \rangle \end{equation*}

is given by the formula $\xi _t(e):\equiv (p_t^{-1})^*(e_t)$ .

We verify that $\xi$ is indeed a solution to the lifting problem (5). To do this, first observe that we have an element of the type

(6) \begin{equation} \prod _{(t,e):F_2}\prod _{p_{i(t)}:\alpha (i(t))=b_{i(t)}}\prod _{q_t:\alpha (i(t))=b_{i(t)}}p_{i(t)}^*=q_t^*. \end{equation}

Indeed, by path induction we can assume that $p_{i(t)}\equiv {\textsf {refl}} :\alpha (i(t))=\alpha (i(t))$ and $q_t\equiv {\textsf {refl}} :\alpha (i(t))=\alpha (i(t))$ . Moreover, in this case the transport maps are identities; therefore, the claimed equality holds. From (6), we get that for all $(t,e):F_2$ ,

\begin{equation*} \xi _{i(t)}(e)\equiv (p_{i(t)}^{-1})^*(e_{i(t)})=(q_{t}^{-1})^*(e_{i(t)})=\delta _t(e). \end{equation*}

This shows that $\xi$ is indeed a solution.

Uniqueness of the solution. We must show that $\xi$ is unique up to homotopy. Assume an element

\begin{equation*}\xi ':\left \langle \prod _{t:\Delta ^2}(E(\alpha (t))\to P(\alpha (t)))\middle |_{\delta }^{\Lambda _1^2} \right \rangle .\end{equation*}

Then we can define the function

\begin{equation*} \varphi ':\left (\sum _{t:\Delta ^2}E(\alpha (t))\right )\to \left (\sum _{b\;:\;B}P(b)\right ) \end{equation*}

is defined as $\lambda (t,e).(\alpha (t),\xi _t'(e)).$ We observe that if $(t,e):\sum _{t:\Lambda _1^2}E(\alpha (i(t)))$ , then

\begin{equation*} \varphi '(\iota (t,e))\equiv \varphi '(i(t),e)\equiv (\alpha (i(t)),\xi _{i(t)}'(e))\equiv (\alpha (i(t)),\delta _t(e))\equiv \psi (t,e). \end{equation*}

where the middle definitional equality holds since $\xi _{i(t)}'(e)\equiv \delta _t(e)$ . By uniqueness, we have $\varphi =\varphi '$ . This means that

\begin{equation*}\varphi (t,e)\equiv (b_t,e_t)=(\alpha (t),\xi _t'(e))\equiv \varphi '(t,e)\end{equation*}

This equality implies that for $p_t^{-1}:b_t=\alpha (t)$ . We also get an equality $(p_t^{-1})^*(e_t)=\xi _t'(e)$ . Therefore, $\xi _t(e)=\xi _t'(e)$ , proving the uniqueness of the extension $\xi$ . This shows that the type $\sum _{b\;:\;B}(E(b)\to P(b))$ is a Segal type, which is now equivalent to (1).

The terminology “functor” in the definition above is justified by the functorial behavior of functions between Segal types (Riehl and Shulman Reference Riehl and Shulman2017, Proposition 6.1). We have reserved the name exponentiable functor till this point in view of Remark 4.4. From Theorem 4.1 and Corollary 4.5, it would seem that we have two notions of exponential functors, one for Segal types and the other for Rezk types. However, both coincide when we restrict to Rezk types.

5. The bisimplicial sets semantics of sHoTT

In this section, we check that our synthetic definitions of Segal type completion and exponentiable functors are semantically correct. In Section 5.1, we verify that the Segal type completion is consistent with the usual semantics. We conclude with the semantics of exponentiable functors in Section 5.2. We use the fact sHoTT has a semantics in the category of bisimplicial sets $ \mathbf{ssSet}.$ The details of the semantics are found in Riehl and Shulman (Reference Riehl and Shulman2017). We also recommend (Bardomiano Martínez Reference Bardomiano Martínez2025, Section 6) for a discussion.

Throughout this section, we will use the following notation and terminology:

• • $\Delta$ will denote the category whose objects are the non-empty linearly ordered finite sets, $[n] :\equiv \{0\leq 1 \leq \cdots \leq n \}$ with $n\in \mathbb{N}$ , and whose morphisms are the order-preserving maps between linearly ordered sets.

• • The representable presheaf $\Delta ^{op} \to \mathbf{Set}$ given by $[n] \in \Delta$ is noted by $\Delta [n]$ .

• • For each $ n\in \mathbb{N}$ , $ F(n): \Delta ^{op} \times \Delta ^{op} \to \mathbf{Set}$ is defined as

  \begin{equation*} F(n)_{k,l} :\equiv \Delta ([k],[n]). \end{equation*}
  This just means we are looking at the constant bisimplicial set $\Delta ([k],[n])$ . Two particular examples are $F(1)$ and $F(0)$ , the last of which is the terminal object.

• • The category of simplicial sets is cartesian closed: for $X,Y \in \mathbf{sSet}$ , the mapping simplicial set is defined as

  \begin{equation*}Map(X,Y)_n :\equiv \mathbf{sSet}(X\times \Delta [n],Y).\end{equation*}

• • The category of bisimplicial sets is cartesian closed: for $X,Y \in \mathbf{ssSet}$ , the mapping bisimplicial set is defined as

  \begin{equation*}(Y^X)_{kl} :\equiv \mathbf{ssSet}(F(k)\times \Delta [l] \times X,Y).\end{equation*}

5.1 Segal type completion and Segal space completion

In this section, we check the soundness of a Segal type completion by comparing it with the Segal space completion defined in bisimplicial sets. Furthermore, since in general we want to consider dependent types, we need a relative version. Given a Segal space $ B$ , we consider the induced model structure on the slice $ \mathbf{ssSet}/B$ . The existence of such a Segal completion is given by the fibrant replacement in the Segal model structure on $ \mathbf{ssSet}$ , and in the relative case, the fibrant replacement in the slice $ \mathbf{ssSet}/B$ (see Proposition 5.5.)

Recall that for objects $ \pi :A\to B$ and $ \xi :S \to B$ in $ \mathbf{ssSet}/B$ , the relative mapping space is denoted as $ Map_{/B}(A,S)$ . This space is obtained by the pullback square:

On the other hand, the Segal space of functions between $ \pi$ and $ \xi$ is given by the following pullback square:

Observe that $ Fun_{/B}(A,S)_0=Map_{/B}(A,S)$ .

Definition 5.1. Let $ \pi :A\to B$ and $\xi : S \to B$ in $ \mathbf{ssSet}/B$ . Assume further that $ S$ is a Segal space and there is a map $ \iota : A\to S\in \mathbf{ssSet}/B.$ We say that $ S$ is a Segal space completion relative to $ B$ for $ A$ if for any Segal space over $ B$ , $ \delta :X \to B$ , the induced map

\begin{equation*} Map_{/B}(S,X)\to Map_{/B}(A,X) \end{equation*}

is an equivalence of spaces.

This definition is the generalization of the completion of a Segal space into a complete one as defined by Rezk (Reference Rezk2001). A related notion of completion of a precategory into a category in the context of homotopy type theory due to Ahrens, Kapulkin, and Shulman appears in Ahrens et al. (Reference Ahrens, Kapulkin and Shulman2015), where the authors use the suggestive name “Rezk completion.”

Observation 5.2. Note that the interpretation of the map (2) from Definition 3.6 into bisimplicial sets gives us an equivalence between Segal spaces,

\begin{equation*}Fun_{/B}(S,X)\to Fun_{/B}(A,X), \end{equation*}

which is just to say that we have a level-wise equivalence of spaces. Thus, for any function $ \pi :A\to B$ , with $ B$ a Segal type, the relative Segal type completion $\xi :S\to B$ for the type $ A$ gives us a Segal space completion relative to $ B$ for the Reedy fibrant bisimplicial set $ A$ .

We will show that the semantic interpretation of our notion of Segal type completion from Definition 3.6 is equivalent to Definition 5.1. For this, we need the result from Riehl and Verity (Reference Riehl and Verity2022, 1.2.22. Proposition]; the slice category $ \mathbf{ssSet}/B$ is cotensored over simplicial sets, we have an equivalence of spaces $ Map_{/B}(F(n)\times S, X) \simeq Map_{/B}(S, F(n)\pitchfork _B X)$ (here we think of $F(n)$ as a space). Note that since $ X$ is a Segal space over $ B$ , so is $ F(n)\pitchfork _B X$ , as this cotensor is constructed via the pullback

Lemma 5.3. Let $ \pi :A\to B$ and $\xi : S \to B$ in $ \mathbf{ssSet}/B$ , where $ B$ is a Segal space. Assume that there is a map $ \iota : A\to S\in \mathbf{ssSet}/B$ exhibiting $ S$ as the Segal space completion relative to $ B$ for $ A$ . Then for any Segal space $ X$ together with $ \delta :X\to B$ , the induced map

\begin{equation*}Fun_{/B}(S,X)\to Fun_{/B}(A,X) \end{equation*}

is an equivalence of Segal spaces.

Proof. First, for any $ n\geq 0$ we have the following:

\begin{align*} Fun_{/B}(S,X)_n & = Map(F(n),Fun_{/B}(S,X)) \\ & = Map(F(n),X^S\times _{B^S} F(0)) \end{align*}

\begin{align*} & = Map(F(n),X^S)\times _{Map(F(n),B^S)} F(0) \\ & = Map(F(n)\times S,X)\times _{Map(F(n)\times S,B)} F(0) \\ & = Map_{/B}(F(n)\times S, X) \\ & = Map_{/B}(S, F(n)\pitchfork _B X), \end{align*}

We are relying on the fact that $ F(n)\pitchfork _B X$ is a Segal space (see the previous paragraph). Similarly, we get that $ Fun_{/B}(A,X)_n = Map_{/B}(A, F(n)\pitchfork _B X)$ for all $ n\geq 0$ . By assumption, $ S\to B$ is the Segal space completion relative to $ B$ for $ A$ . Hence, for $ F(n)\pitchfork _B X$ , we have an equivalence of spaces

\begin{equation*} Map_{/B}(S, F(n)\pitchfork _B X) \to Map_{/B}(A, F(n)\pitchfork _B X) \end{equation*}

for all $ n\geq 0$ . This gives us the equivalence between Segal spaces:

\begin{equation*}Fun_{/B}(S,X)\to Fun_{/B}(A,X). \end{equation*}

Corollary 5.4. Given a Segal type $ B$ and any type $ A$ over $ B$ . The semantics of a relative Segal type completion for the type $ A$ coincide with Definition 5.1.

Proof. This now follows from Observation 5.2 and Lemma 5.3.

Of course, we also have the non-relative version of this soundness result, and for this it is enough to take $ B$ to be the terminal object. To finalize this section, we observe that the fibrant replacement in $ \mathbf{ssSet}/B$ coincides with Segal completion relative to $ B$ . Since the model structure on $ \mathbf{ssSet}/B$ is induced by the one from $ \mathbf{ssSet}$ , it will be enough to verify this fact for $ B=1$ .

Proposition 5.5. Let $ A$ be a Reedy fibrant bisimplicial set. If the Segal space completion of $ A$ exists, then it defines a fibrant replacement of $A$ in the Segal space model structure on $\mathbf{ssSet}$ .

Proof. Recall that a map $ i:A \to S$ is a weak equivalence in $ \mathbf{ssSet}$ if is a local map, that is, a map such that

\begin{equation*} i^*:Map(S,X) \to Map(A,X) \end{equation*}

is an equivalence of spaces for any Segal space $ X$ . Then, it is clear that if $ S$ is a fibrant replacement, it must be a Segal space completion.

Conversely, if $ S$ is the Segal space completion, then it induces equivalences like the above. Therefore, $ i$ is indeed a weak equivalence. $ S$ is a Segal space by assumption, so it must be a fibrant replacement in $ \mathbf{ssSet}$ .

5.2 Exponentiable functors

Here, we verify that our notion of exponentiable functor is semantically correct. Ayala, Francis, and Rozenblyum prove in Ayala et al. (Reference Ayala, Francis and Rozenblyum2018) the following result, which characterizes exponentiable functors between $ \infty$ -categories (quasi-categories). In order to be consistent with the literature and avoid introducing unnecessary notation clashing with that of the previous section, we make the following clarifications:

• • By $\infty$ -category, we mean quasi-category.

• • The ordinal $[0]$ seen as a subobject of $[n]$ is denoted $\{i\}$ where $i$ is the image of the unique object via the map $[0] \to [n]$ .

• • $\{i\lt j\}$ is the subobject of $[n]$ given by the map $[1] \to [n]$ such that $0 \mapsto i$ and $1 \mapsto j$ .

• • $\mathbf{ssSet}_{Segal}$ denotes the subcategory, consisting of Segal spaces, of the category of bisimplicial sets $\mathbf{ssSet}$ .

• • Sometimes, we think of the ordinal $[n]$ as an $\infty$ -category. Whenever this happens, it is usually clear from the context.

Theorem 5.6. The following conditions on a functor $ \pi :\mathcal{E} \to \mathcal{B}$ between $ \infty$ -categories are equivalent.

1. (1) The functor $ \pi$ is an exponentiable fibration.

2. (2) For each functor $ [2] \to \mathcal{B}$ , the diagram of pullbacks

   

   is a pushout of $ \infty$ -categories.

For a full explanation of the theorem, we recommend the original reference Ayala et al. (Reference Ayala, Francis and Rozenblyum2018). We will focus on Condition 2 to see this is exactly Condition 5 of Corollary 4.5. This last condition involves objects that are defined by the following pullback square

We remark that the arrow on the far left is not a fibration because $F(1) \sqcup _{F(0)} F(1)$ is not a Segal space. Nevertheless, the diagrams express the fact that $ E_{F(1) \sqcup _{F(0)} F(1)}$ and $ E_{F(2)}$ are the fibers of $ f$ over $ F(1) \sqcup _{F(0)} F(1)$ and $ F(2)$ , respectively.

The map $ E_{F(1) \sqcup _{F(0)} F(1)} \to E_{F(2)}$ shows $ E_{F(2)}$ as the Segal type completion of $ E_{F(1) \sqcup _{F(0)} F(1)}$ . Therefore, when we interpret the square in Condition 5 from Corollary 4.5 into $ \mathbf{ssSet}_{Segal}$ . This shows that $ E_{F(2)}$ is the fibrant replacement of $ E_{F(1)}\coprod _{F(0)}E_{F(1)}$ in $ \mathbf{ssSet}_{Segal}$ . This is just to say that the diagram

is a pushout square of Segal spaces. When $ E$ and $ B$ are Rezk spaces, this is exactly Condition 2 of Theorem 5.6. On the other hand, the category of simplicial sets can be embedded into bisimplicial sets via $ p_1^*:\mathbf{sSet} \to \mathbf{ssSet}$ as defined in Joyal and Tierney (Reference Joyal and Tierney2007). Furthermore, this is shown to provide a Quillen adjunction between the Joyal model structure on $ \mathbf{sSet}$ and the complete Segal space model on $ \mathbf{ssSet}$ . This inclusion preserves exponentials.

5.2.1 On profunctors and correspondences

Since we work in the original formulation sHoTT, we cannot yet incorporate all conditions of Theorem 5.6 into our Theorem 4.1. Some of the obstructions include the construction of the Segal type of discrete types and the core $\infty$ -groupoid construction. The composition of profunctors appears naturally in Conduché’s theorem. Condition 5 in Corollary 4.5 carries similar information in the synthetic framework. Given its relevance, in this last section, we explain why this is not yet a theorem. However, it might be possible to prove this in the new modal extensions of sHoTT Gratzer et al. (Reference Gratzer, Weinberger and Buchholtz2024, Reference Gratzer, Weinberger and Buchholtz2025), or in new extensions there in. We leave this task for future work, which can also include the construction of a universe of correspondences.

The result in Theorem 5.6 is expressed and proved using correspondences between $ \infty$ -categories. If we have categories $ \mathcal{C}$ and $ \mathcal{D}$ , a correspondence from $ \mathcal{C}$ to $\mathcal{D}$ is category $ \mathcal{M}$ which contains $\mathcal{C}$ and $ \mathcal{D}$ as full subcategories, it is equipped with a functor $ \pi :\mathcal{M}\to \{0\lt 1\}$ such that $ \mathcal{C}= \pi ^{-1}(0)$ and $ \mathcal{D}= \pi ^{-1}(1)$ . In contrast, a profunctor from $\mathcal{C}$ to $\mathcal{D}$ is a functor $ P:\mathcal{C}\times \mathcal{D}^{op}\to \mathbf{Set}$ . There is a bicategorical equivalence between profunctors from $ \mathcal{C}$ to $ \mathcal{D}$ and correspondences from $ \mathcal{C}$ to $ \mathcal{D}$ . Switching to the realm of $ \infty$ -categories, we recall the following definition:

Definition 5.7. A correspondence between $ \infty$ -categories $ \mathcal{C}$ and $ \mathcal{D}$ is a pair of pullbacks:

This is simply a functor between $ \infty$ -categories $ \mathcal{M}\to [1]$ with fibers $ \mathcal{C}$ over $ 0$ and $ \mathcal{D}$ over $ 1$ .

It is a well-known fact that a profunctor $ P:\mathcal{C}\times \mathcal{D}^{op}\to \mathbf{Set}$ can also be defined as a two-sided discrete fibration over $ \mathcal{C} \times \mathcal{D}$ by work of Street (Reference Street2006). This is a functor $ \mathcal{E}\to \mathcal{C} \times \mathcal{D}$ , that is, a discrete Grothendieck fibration over $ \mathcal{D}$ and a discrete Grothendieck opfibration over $ \mathcal{C}$ .

Taking into account the limitations of sHoTT, for us it would make sense to momentarily think of profunctors as two-sided discrete fibrations. Let $ P:A \to B \to \mathcal{U}$ be a two-variable type family over Segal types $ A$ and $ B$ . From Riehl and Shulman (Reference Riehl and Shulman2017), we say that $ P$ is a two-sided discrete fibration if for all $ a:A$ and $ b:B$ , the type families

\begin{equation*} \lambda x.P(x,b):A \to \mathcal{U} \text{ and } \lambda y.P(a,y):B \to \mathcal{U} \end{equation*}

are contravariant and covariant, respectively. The most famous two-sided discrete fibration over a Segal type $ B$ is the $ ``{\textsf {hom}}''$ type family

\begin{equation*} \lambda x.\lambda y.{\textsf {hom}}_B(x,y): B \to B \to \mathcal{U}. \end{equation*}

More generally, let $ f:E \to B$ be a function between Segal types, $ a, \, b:B$ , and $ u:{\textsf {hom}}_B(a,b)$ , then

\begin{equation*} \lambda x.\lambda y.{\textsf {hom}}_E^u(x,y): E_a \to E_b \to \mathcal{U} \end{equation*}

is a two-sided discrete fibration. The type $ {\textsf {hom}}_E^u(x,y)$ denotes the type of arrows in $ E$ that start at $ x:E_a$ and end at $ y:E_b$ .

Weinberger provides in Weinberger (Reference Weinberger2024) the following characterization of two-sided discrete families:

Proposition 5.8. Given $ P:A \to B \to \mathcal{U}$ a two-side type family over Rezk types, the following are equivalent:

1. (1) The family $ P$ is a two-sided discrete fibration.

2. (2) The family $ P$ is cartesian over $ A$ and cocartesian over $ B$ , and for all $ a:A, \, b:B$ the bifibers $ P(a,b)$ are discrete types.

We have not introduced cartesian and cocartesian type families in sHoTT, and this is the main topic of Buchholtz and Weinberger (Reference Buchholtz and Weinberger2023). However, the meaning of such concepts is in practice the same one as for $ \infty$ -categories. Therefore, the first two conditions of the second point in Proposition 5.8 simply mean that the type families

\begin{equation*} \lambda x.P(x,b):A \to \mathcal{U} \text{ and } \lambda y.P(a,y):B \to \mathcal{U} \end{equation*}

are cartesian and cocartesian, respectively, for all $ a:A$ and $ b:B$ together with some compatibility condition. This is what Weinberger (Reference Weinberger2024) defines as two-sided cartesian family. Given $ P:A \to B \to \mathcal{U}$ and $ Q: B \to C \to \mathcal{U}$ two two-sided type families, there is a natural composition to obtain another two-sided type family:

\begin{equation*} Q \odot P \equiv \lambda a.\lambda c.\sum _{b\;:\;B} P(a,b)\times Q(b,c):A \to C \to \mathcal{U}. \end{equation*}

The result in Weinberger (Reference Weinberger2024, Proposition 5.5] shows that if the families $ P$ and $ Q$ are two-sided cartesian, then $ Q\odot P$ is again a two-sided cartesian family. Unfortunately, even if both $ P$ and $ Q$ are two-sided discrete fibrations, it does not follow that $ Q\odot P$ is a two-sided discrete fibration. Instead, to make sense of the composition in this case, we consider the discrete type completion of $ Q \odot P$ .

If we have a function $ f:E \to B$ between Segal types, a condition we would like to add to Theorem 4.1 is the following: For any $ a,\, b,\, c: B, \, u:{\textsf {hom}}_B(a,b), \, v:{\textsf {hom}}_B(b,c)$ and $ x:E_a, \, z:E_c$ , the canonical map induced by the composition

\begin{equation*} \left (\sum _{y:E_b} {\textsf {hom}}_E^u(x,y) \times {\textsf {hom}}_E^v(y,z)\right ) \to {\textsf {hom}}_E^{v \circ u} (x,z) \end{equation*}

exhibits $ {\textsf {hom}}_E^{v \circ u} (x,z)$ as the discrete type completion of

\begin{equation*} \sum _{y:E_b} {\textsf {hom}}_E^u(x,y) \times {\textsf {hom}}_E^v(y,z).\end{equation*}

The problem arises because Condition 5 of Corollary 4.5 encodes the composition of correspondences in sHoTT. These are Segal (Rezk) types over $ F(1)$ . In Stevenson (Reference Stevenson2018), it is shown that correspondences from $ \mathcal{C}$ to $ \mathcal{D}$ are the same as $ \mathcal{C}\times \mathcal{D}^{op} \to \mathcal{S}$ , where $ \mathcal{S}$ denotes the $ \infty$ -category of spaces, and furthermore, that theses correspondences are the same as bifibration. This is done by endowing the category of correspondences $ Corr(\mathcal{C},\mathcal{D})$ and the category $ \mathbf{sSet}/(\mathcal{C}\times \mathcal{D})$ , with model structures, respectively, such that they are Quillen equivalent and where the fibrant objects of $ \mathbf{sSet}/(\mathcal{C}\times \mathcal{D})$ are the bifibrations. Both of these models are Quillen equivalent to $ \mathbf{sSet}/(\mathcal{C} \times \mathcal{D}^{op})$ endowed with the covariant model structure, that is, this encodes profunctors.

We venture to say that until sHoTT is further enhanced, to be more expressive, the analogous result from Stevenson (Reference Stevenson2018) is out of reach. By this, we just mean we cannot yet formulate a full and precise relation between correspondences and two-sided discrete fibrations and profunctors.