2.1.1 Combinatory algebras

A CA $\mathscr{A}$ consists of a set $\mathscr{A}$ and a binary “application” operation:

\begin{align*} (-)\cdot (?): \mathscr{A}\times \mathscr{A}\rightarrow \mathscr{A} \end{align*}

such that there exist elements (“combinators”) $\textsf{k},\textsf{s}\in \mathscr{A}$ satisfying $\textsf{k}\cdot a \cdot b = a$ and $\textsf{s} \cdot a \cdot b \cdot c = a \cdot c \cdot (b \cdot c)$ . From now on, we adopt the convention of left associativity and may omit the application symbol.

CAs enjoy a property known as “combinatory completeness,” which, roughly speaking, allows them to mimic the $\lambda$ -calculus (Schönfinkel, Reference Schönfinkel1924; Curry, Reference Curry1930). A polynomial over $\mathscr{A}$ is a formal expression built from the grammar:

\begin{align*} t ::= x\in \mathscr{V} \;|\; a\in \mathscr{A} \;|\; t \cdot t \end{align*}

where $\mathscr{V}$ is a countably infinite set of variables. Clearly, we can talk about free variables of a polynomial and substitution $t[a/x]$ of the element $a\in \mathscr{A}$ for the free variable $x\in \mathscr{V}$ in the polynomial $t$ . Moreover, closed polynomials have an obvious interpretation in $\mathscr{A}$ . Combinatory completeness says that for every polynomial $t$ over $\mathscr{A}$ and every variable $x$ , there exists a polynomial $\lambda x.\, t$ such that $\textsf{FV}(\lambda x.\, t) \subseteq \textsf{FV}(t) - \{x\}$ and for all $a\in \mathscr{A}$ :

\begin{align*} (\lambda x.\, t) a = t[a/x] \end{align*}

Indeed, an example of a CA is the set of $\lambda$ -terms up to $\beta$ -equivalence, where application is $\lambda$ -application, $\textsf{k}:= \lambda x y.\, x$ and $\textsf{s} = \lambda f g x.\, fx(gx)$ . (Kleene’s first algebra $\mathscr{K}_1$ , mentioned in the previous section, is not a combinatory algebra but a partial combinatory algebra: the application of one element to another may not be defined (Turing machines may not halt).)

The objects of the category $\mathbf{Asm}(\mathscr{A})$ of assemblies over the CA $\mathscr{A}$ are pairs $(X,\Vdash _X)$ consisting of a set $X$ and a “realizability” relation $\Vdash _X \subseteq \mathscr{A}\times X$ (written infix) such that $\forall x\in X. \, \exists a\in \mathscr{A}. \, a \Vdash _X x$ . Assemblies are sometimes thought of as datatypes, whose “values” are implemented (by their realizers) in the programming language given by the CA. A morphism $f:(X,\Vdash _X) \rightarrow (Y,\Vdash _Y)$ is a set-function $f:X\rightarrow Y$ such that $\exists e\in \mathscr{A}.\, \forall x\in X. \, \forall a\in \mathscr{A}. \, a\Vdash _X x \rightarrow ea \Vdash _Y f(x)$ . We say that the function $f$ is realized by $e$ and write $e\Vdash f$ . Identities in $\mathbf{Asm}(\mathscr{A})$ are identity functions, which are realized by $\lambda x.\, x$ . Composition is also inherited from $\mathbf{Set}$ : if $e\Vdash f: X\rightarrow Y$ and $e^{\prime}\Vdash g:Y \rightarrow Z$ then $\lambda x.\, e^{\prime}(ex) \Vdash gf:X \rightarrow Z$ .

An assembly $X$ is modest when elements are uniquely determined by any of their realizers: $\forall x,x^{\prime}\in X. \, \forall a\in \mathscr{A}. \, a \Vdash _X x \land a \Vdash _X x^{\prime} \rightarrow x=x^{\prime}$ . An assembly $X$ is partitioned when the relation $\Vdash _X$ is actually a function:

\begin{align*} \left \Vert - \right \Vert _X : X \rightarrow \mathscr{A} \end{align*}

(so each element of $X$ has exactly one realizer). Of course, an assembly $X$ may be both modest and partitioned, in which case the function $\left \Vert - \right \Vert _X$ is an injection. Thus, every modest partitioned assembly can be identified with a subset of $\mathscr{A}$ ; this is used in the construction of an impredicative universe of modest partitioned assemblies in $\mathbf{PAsm}(\mathscr{A})$ .

2.1.2 Typed combinatory algebras

Typed (partial) combinatory algebras were introduced by Longley (Reference Longley1999). A typed combinatory algebra (TCA) is built over a “type system”: a non-empty set $\mathscr{T}$ of “types” that is closed under the operation $\rightarrow$ (which associates with the right by convention). (Normally one would also consider products, but for present purposes we need not.) A TCA $\mathscr{A}$ over the type system $\mathscr{T}$ is a family

\begin{align*} (\mathscr{A}_A)_{A\in \mathscr{T}} \end{align*}

of non-empty sets indexed by types, together with a family of typed application operations:

\begin{align*} (-) \cdot _{A,B} (?): \mathscr{A}_{A\rightarrow B} \times \mathscr{A}_A \rightarrow \mathscr{A}_B \end{align*}

(again we tend to omit the symbol) such that the following typed combinators are required to exist and satisfy the given equations:

\begin{align*} &\textsf{k}_{A,B} \in \mathscr{A}_{A\rightarrow B \rightarrow A} &\textsf{k}_{A,B} a b = a \\ &\textsf{s}_{A,B,C} \in \mathscr{A}_{(A \rightarrow B \rightarrow C) \rightarrow (A \rightarrow B) \rightarrow A \rightarrow C} &\textsf{s}_{A,B,C} f g a = f a (g a) \end{align*}

TCAs are combinatorially complete in a typed sense (see Bauer (Reference Bauer2022) for details).

If $\mathscr{A}$ is a TCA, then an object of $\mathbf{Asm}(\mathscr{A})$ is a triple $(X,A,\Vdash _X)$ , where $X\in \mathbf{Set}$ , $A\in \mathscr{T}$ is the “realizer type” and $\Vdash _X \subseteq \mathscr{A}_A \times X$ (so realizers of elements of $X$ all have the same type) is such that $\forall x\in X. \, \exists a\in \mathscr{A}_A. \, a \Vdash _X x$ . A morphism $(X,A,\Vdash _X)\rightarrow (Y,B,\Vdash _Y)$ is a function $f:X\rightarrow Y$ such that $\exists e\in \mathscr{A}_{A\rightarrow B}.\, \forall x\in X. \, \forall a\in \mathscr{A}. \, a\Vdash _X x \rightarrow ea \Vdash _Y f(x)$ .

Any CA can be regarded as a TCA with a single type $U=U\rightarrow U$ (in such a way that the ensuing categories of assemblies are isomorphic). Furthermore, any TCA can be equipped with a “unit” type such that the ensuing category of assemblies remains unchanged up to equivalence. That is, given any type system $\mathscr{T}$ , we consider the augmented type system $\mathscr{T}^1 := \mathscr{T} \cup \{ 1\}$ , where $1\rightarrow A := A$ and $A \rightarrow 1 := 1$ . Then we obtain the augmented TCA $\mathscr{A}^1$ over $\mathscr{T}_1$ by setting $\mathscr{A}^1_1 := 1$ (the terminal set). The augmented application operation is given by:

\begin{align*} &a \cdot _{1\rightarrow A} * := a &* \cdot _{A\rightarrow 1} a := * \end{align*}

For the combinators, it suffices to specify the following (where $A,A^{\prime}\in \mathscr{T}^1$ ):

\begin{align*} &\textsf{k}_{1,A} := * &\textsf{k}_{A,1} := \lambda x.\, x \\ &\textsf{s}_{1,A,A^{\prime}} := \lambda f a.\, fa =: \textsf{s}_{A,1,A^{\prime}} &\textsf{s}_{A,A^{\prime},1} := * \end{align*}

Here, we use typed combinatorial completeness of $\mathscr{A}$ . Checking the relevant equations is straightforward.

Proof. $\mathbf{Asm}(\mathscr{A})$ is a full subcategory of $\mathbf{Asm}(\mathscr{A}^1)$ . For any $X=(X,1,\Vdash _X)\in \mathbf{Asm}(\mathscr{A}^1)$ , we define an isomorphism:

\begin{align*} f: X \rightarrow X^{\prime} := \left (X,A,\left \Vert -\right \Vert _{X^{\prime}}\right ) \end{align*}

where for all $x\in X$ , we set

\begin{align*} \left \Vert x \right \Vert _{X^{\prime}} := a_0 \end{align*}

for an arbitrarily chosen $a_0 \in \mathscr{A}_A$ . The function $f:= \textsf{id}_X$ is realized by $a_0 \in \mathscr{A}_A = \mathscr{A}_{1\rightarrow A}$ . The inverse $f^{-1} := \textsf{id}_X$ is realized by $*\in \mathscr{A}_1 = \mathscr{A}_{A\rightarrow 1}$ .

2.1.3 Categories

A category, like a TCA, gives rise to an a priori typed notion of realizability: the types are the objects of the category. Suppose $\mathbf{R}$ is a category with a terminal object $1$ . Then we have the functor:

\begin{align*} \Pi := \mathbf{R}(1,-): \mathbf{R} \rightarrow \mathbf{Set} \end{align*}

The objects of the category $\mathbf{Asm}(\mathbf{R})$ of assemblies over the “realizer category” $\mathbf{R}$ are triples $(X,A,\Vdash _X)$ , where $X\in \mathbf{Set}$ , $A\in \mathbf{R}$ (a realizer type is now an object from the realizer category) and $\Vdash _X \subseteq \Pi A \times X$ is such that $\forall x\in X. \, \exists a\in \Pi A. \, a \Vdash _X x$ . A morphism $(X,A,\Vdash _X)\rightarrow (Y,B,\Vdash _Y)$ is a function $f:X\rightarrow Y$ such that $\exists e:A\rightarrow B \in \mathbf{R}.\, \forall x\in X. \, \forall a\in \mathscr{A}. \, a\Vdash _X x \rightarrow \Pi (e)(a) = e\circ a \Vdash _Y f(x)$ .

Given a TCA $\mathscr{A}$ with a unit type $1$ ( $\mathscr{A}_1 = 1$ ), we may build a category $\mathbf{R}(\mathscr{A})$ with a terminal object. The objects of $\mathbf{R}(\mathscr{A})$ are the types of $\mathscr{A}$ . A morphism $A \rightarrow B$ is a “computable” function $k: \mathscr{A}_A \rightarrow \mathscr{A}_B$ . The function $k$ is computable iff there exists $e\in \mathscr{A}_{A\rightarrow B}$ representing it, that is, for all $a\in \mathscr{A}_A$ we have $k(a)=ea$ . Note that computable functions $\mathscr{A}_1 \rightarrow \mathscr{A}_A$ are in bijective correspondence with elements of $\mathscr{A}_A$ : a function $k:\mathscr{A}_1 \rightarrow \mathscr{A}_A$ is represented by $\lambda x.\, k(*)$ .

2.2.1 Intervals

Let $\mathbf{R}$ be a category with a terminal object. An interval (cogroupoid) $\mathbb{I}\in \mathbf{R}$ is a diagram of the form

We require that $\mathbb{I}_0=1$ is terminal. $\mathbb{I}_0$ and $\mathbb{I}_1$ are, respectively, known as the “object of coobjects” and the “object of coarrows.” The diagram

(1)

is required to be a pushout. Maps $\mathbb{I}_1 \rightarrow A$ are thought of as paths in $A$ , and so, the pushout allows us to concatenate two paths $\alpha, \beta :\mathbb{I}_1 \rightarrow A$ that match nose to tail: $\beta 0 = \alpha 1$ . The result is $[\beta, \alpha ]:\mathbb{I}_2 \rightarrow A$ (a path with twice the length of $\alpha$ and $\beta$ ). Likewise, the following is a pushout.

To round off the definition, the cogroupoid axioms are required to hold. The first set makes sure that the end point (or, respectively, cosource and cotarget) maps $0,1$ play nicely with cocomposition $2$ and coidentity $*$ .

The second set makes sure that the inverse operation $\sigma$ behaves as expected.

The next two axioms are coidentity and coassociativity, respectively.

Lastly, we have coinverse laws.

Example 4. $(\mathbf{Gpd},\mathbf{I})$ is a realizer category. The object of coarrows $\mathbf{I}_1$ is the “walking isomorphism”:

The maps $0$ and $1$ pick out the corresponding end points of $\mathbf{I}_1$ . The map $\sigma$ sends $i\mapsto i^{-1}$ .

$\mathbf{I}_2$ has three objects and, again, one arrow in each hom set.

The maps $i_0$ and $i_1$ send $i\in \mathbf{I}_1$ to the synonymous (eponymous, even) morphisms in $\mathbb{I}_2$ . The map $2$ picks out the composite $i_1 i_0$ .

Continuing the trend, $\mathbf{I}_3$ has four objects and one arrow in each hom set.

The map $j_0$ sends $i_0 \mapsto i_0$ and $i_1 \mapsto i_1$ ; the map $j_1$ sends $i_0 \mapsto i_1$ and $i_1 \mapsto i_2$ .

2.2.2 Homotopies

The interval $\mathbb{I}\in \mathbf{R}$ endows the ambient category with the structure of a (2,1)-category (in fact, a strict $\omega$ -category, see Warren (Reference Warren2012), Theorem 1.12). The higher cells are given by homotopies with respect to $\mathbb{I}$ .

A homotopy $H:f\Rightarrow g:A\rightarrow B$ with respect to the interval $\mathbb{I}$ is a map $H:A\times \mathbb{I}_1 \rightarrow B$ making the following diagram in $\mathbf{R}$ commute.

Given a homotopy $H:A\times \mathbb{I}_1 \rightarrow B$ , we can find its domain and codomain respectively by:

\begin{align*} &\textsf{dom}(H) := H \circ \left \langle A, 0* \right \rangle &\textsf{cod}(H) := H \circ \left \langle A, 1* \right \rangle \end{align*}

As the functor $(-)\times A$ possesses a right adjoint, the following square is a pushout.

If $H,H^{\prime}: A \times \mathbb{I}_1 \rightarrow B$ such that $H^{\prime}(A \times 0) = H (A \times 1)$ , then the morphism $[H^{\prime},H]:A\times \mathbb{I}_2 \rightarrow B$ is given by:

\begin{align*} \mu \left [ \lambda (H^{\prime} \circ \textsf{swap}), \lambda (H \circ \textsf{swap}) \right ] \circ \textsf{swap} \end{align*}

If $\textsf{dom}(H^{\prime}) = \textsf{cod}(H)$ , then their vertical composition is defined using this universal morphism:

\begin{equation*} H^{\prime} \circ H := [H^{\prime},H] \circ \left (A\times 2\right ) \end{equation*}

(note the overloading of the composition symbol $\circ$ ).

The horizontal composition $H^{\prime} \ast H$ of $H:A\times \mathbb{I}_1\rightarrow B$ and $H^{\prime}:B\times \mathbb{I}_1\rightarrow C$ is given by the following composite in $\mathbf{R}$ .

The identity homotopy at $f:A\rightarrow B$ is given by $f\pi _1: A\times \mathbb{I}_1 \rightarrow B$ , and the inverse of a homotopy $H:A\times \mathbb{I}_1 \rightarrow B$ is $H\circ (A\times \sigma )$ .

2.2.3 Fundamental groupoids

By considering maps out of $\mathbb{I}$ into a fixed object $A\in \mathbf{R}$ , we obtain the fundamental groupoid of $A$ . That is, we have a 2-functor:

\begin{equation*} \Pi = (-)^{\mathbb {I}} : \mathbf {R} \rightarrow \mathbf {Gpd} \end{equation*}

A quick way to see this is because the contravariant hom functor takes colimits (used in the definition of an interval) to limits (used in the definition of a category).

The fundamental groupoid of $A$ has as objects points in $A$ , that is, maps $\mathbb{I}_0\rightarrow A$ . A morphism $\alpha :a\rightarrow b$ is a path $\alpha$ in $A$ making the following diagram commute.

The composition of $\alpha :a\rightarrow b$ with $\beta :b\rightarrow c$ is defined by:

\begin{equation*} \beta \circ \alpha := [\beta, \alpha ] \circ 2 \end{equation*}

( $2$ reparameterizes the double-length path). The identity at $a$ is $a*$ and the inverse of $\alpha$ is $\alpha \sigma$ .

If $f:A\rightarrow B \in \mathbf{R}$ , the functor $\Pi (f):\Pi A \rightarrow \Pi B$ is given by post-composition (in $\mathbf{R}$ ) with $f$ . The composition law for functors holds because $f[\beta, \alpha ] = [f\beta, f\alpha ]$ by the universal property of the pushout (1).

A natural isomorphism $\phi : F\Rightarrow G: \mathbf{C}\rightarrow \mathbf{D}$ between functors (between categories or groupoids) is equivalently given by a functor:

\begin{align*} \phi :\mathbf{C} \times \mathbf{I}_1 \rightarrow \mathbf{D} \end{align*}

such that $\phi (-,0)=F$ and $\phi (-,1)=G$ . In this way, a homotopy $H:f\Rightarrow g:A\rightarrow B\in \mathbf{R}$ gives rise to a natural transformation:

\begin{equation*} \Pi (H) : \Pi A \times \mathbf {I}_1 \rightarrow \Pi B \end{equation*}

by setting:

\begin{align*} &\Pi (H)(a,0) = \Pi (H)(a) := H \circ \langle a,0 \rangle : \mathbb{I}_0\rightarrow B \\ &\Pi (H)(a,1) = \Pi (H)(a) := H \circ \langle a,1 \rangle : \mathbb{I}_0\rightarrow B \\ &\Pi (H)(\alpha, i) = \Pi (H)(\alpha ) := H \circ \langle \alpha, \mathbb{I}_1 \rangle : \mathbb{I}_1\rightarrow B \\ &\Pi (H)(\alpha, i^{-1}) = \Pi (H)(\alpha ) := H \circ \langle \alpha, \sigma \rangle : \mathbb{I}_1\rightarrow B \end{align*}

Given a homotopy $H:f\Rightarrow g:A\rightarrow B$ and a path $\alpha :a\rightarrow b\in \Pi A$ , the path $\Pi (H)(\alpha, i) \in \Pi B$ can be thought of as the diagonal path through a naturality square. Indeed, we have the following.

Lemma 6 (Warren, personal communication). With the above data, the following diagram commutes in $\Pi B$ .

Proof. We show that the diagonal is equal to the left-bottom boundary; that these are equal to the top-right boundary is a symmetric argument. Using the definition of composition in $\Pi B$ , we have to show

\begin{align*} H \circ \langle \alpha, \mathbb{I}_1 \rangle = \left [H \circ \langle \alpha 1 *, \mathbb{I}_1 \rangle, H \circ \langle \alpha, 0 *\rangle \right ] \circ 2 \end{align*}

This simplifies to

\begin{align*} H \circ (\alpha \times \mathbb{I}_1) \circ \left [ \left \langle 1 *, \mathbb{I}_1 \rangle, \langle \mathbb{I}_1, 0 * \right \rangle \right ] \circ 2 \end{align*}

But

\begin{align*} \left [\left \langle 1 *, \mathbb{I}_1 \rangle, \langle \mathbb{I}_1, 0 * \right \rangle \right ] \circ 2 \end{align*}

is a decomposition of the diagonal $\Delta _{\mathbb{I}_1}$ (see Warren (Reference Warren2012), p. 212).

2.2.4 Squares

Let $(\mathbf{R},\mathbb{I})$ be a realizer category. Define the (edge-symmetric) double category $\Box \mathbf{R}(A,B)$ to have

• • objects: maps $A\rightarrow B \in \mathbf{R}$ ;

• • both horizontal and vertical morphisms: maps $A\times \mathbb{I}_1 \rightarrow B$ ;

• • 2-cells: commutative squares of maps $A\times \mathbb{I}_1 \rightarrow B$ .

Denote by $\blacksquare \mathbf{R}(A,B)$ the (edge-symmetric) double category with the same objects and morphisms as $\Box \mathbf{R}(A,B)$ , but with 2-cells maps $A\times \mathbb{I}_1 \times \mathbb{I}_1 \rightarrow B$ .

As a consequence of Lemma 6, every 2-cell $\phi \in \blacksquare \mathbf{R}(A,B)$ determines a 2-cell $\partial (\phi )\in \Box \mathbf{R}(A,B)$ :

where, for example, $\phi _{01} := \phi \circ \langle A, 0*, 1* \rangle : A \rightarrow B$ and $\phi _{\mathbb{I}_1 1} := \phi \circ \langle A\pi _1, \mathbb{I}_1\pi _2, 1*\pi _2 \rangle : A\times \mathbb{I}_1 \rightarrow B$ . This determines a double functor:

\begin{align*} \partial _{\mathbf{R}(A,B)}: \blacksquare \mathbf{R}(A,B) \rightarrow \Box \mathbf{R}(A,B) \end{align*}

for each $A,B\in \mathbf{R}$ (we sometimes just write $\partial$ ).

Remark 7. Some of the results to follow rely on the hypothesis that, for all $A,B\in \mathbf{R}$ , $\partial _{\mathbf{R}(A,B)}$ is an isomorphism of double categories. As shorthand for this, we will say that $\partial _{\mathbf{R}}$ is an isomorphism.

This hypothesis holds whenever $(\mathbf{R},\mathbb{I})$ is finitely complete as a (2,1)-category, for then the cotensor $\mathbf{I}_1 \pitchfork B$ is necessarily isomorphic to the internal hom $B^{\mathbb{I}_1}$ (Warren, Reference Warren2008, Lemma 3.25):

\begin{align*} \mathbf{R}\left (A, \mathbf{I}_1 \pitchfork B \right ) \cong \mathbf{Gpd}\left ( \mathbf{I}_1, \mathbf{R}(A,B) \right ) \cong \mathbf{R}\left (A\times \mathbb{I}_1, B \right ) \cong \mathbf{R}\left (A,B^{\mathbb{I}_1}\right ) \end{align*}

2.2.5 Untyped realizer categories

We know from other settings (Birkedal, Reference Birkedal2000a,b; Robinson and Rosolini, Reference Robinson and Rosolini2001; Lietz and Streicher, Reference Lietz and Streicher2002) that impredicativity is intimately related to untypedness at the level of realizers. A “universal object” allows us to turn the a priori typed notion of realizability given by a category into an untyped one. Traditionally, an object in a category is universal iff every object in the category is a retract of it. In the higher setting, we allow for pseudoretracts.

Definition 8. An object $U\in \mathbf{R}$ is universal iff for every object $A\in \mathbf{R}$ , there is a section and retraction

together with a homotopy:

\begin{align*} \rho _A: r_A s_A \Rightarrow \textsf{id}_A \end{align*}

In particular, an untyped realizer category provides an up-to-homotopy model of the untyped $\lambda$ -calculus: $U^U$ is a pseudoretract of $U$ .

3.2.1 Dependent products

The notion of dependent product for path categories studied as (Definition 5.2 van den Berg and Moerdijk (Reference van den Berg and Moerdijk2018)) is as follows. A path category $\mathbf{C}$ is said to have homotopy $\Pi$ -types iff for any two fibrations $f: X \rightarrow J$ and $\alpha : J \rightarrow I$ , there is an object $\Pi _\alpha (F): \Pi _\alpha X \rightarrow I$ in $\mathbf{C}(I)$ together with an “evaluation map” $\textsf{ev}: \alpha ^* \Pi _\alpha X \rightarrow X$ over $J$ with the following universal property: if there is a map $g: Y \rightarrow I$ and a map $m: \alpha ^* Y \rightarrow X$ over $J$ , then there exists a unique map $n: Y \rightarrow \Pi _\alpha X$ such that $\textsf{ev} \circ \alpha ^* n$ and $m$ are fiberwise homotopic over $J$ (type-theoretically, this is the $\beta$ -law). If the uniqueness criterion on $n$ is dropped, we are left with weak homotopy $\Pi$ -types. Type-theoretically, these correspond to dependent functions that do not necessarily satisfy the $\eta$ -law and thus do not necessarily satisfy function extensionality. If the $\beta$ -law holds on the nose, then we have (weak) $\Pi$ -types (see den Besten (Reference den Besten2020)) for a more in-depth study of dependent products in path categories).

Proof. Given fibrations $G: (X, A,{\left \Vert -\right \Vert }_X) \rightarrow (Y, B,{\left \Vert -\right \Vert }_Y)$ and $F: Y \rightarrow (Z, C,{\left \Vert -\right \Vert }_Z)$ , the objects of the underlying groupoid of $\Pi _F X$ are tuples $(z,H,e,\epsilon )$ , where $z\in Z$ , $H: F\downarrow z \rightarrow X$ in the slice over $Y$

(4)

( $F\downarrow z$ is the “homotopy fiber,” constructed as a pseudopullback, see Proposition 13), $e \in \Pi (A^{B\times C^{\mathbb{I}_1}})$ , and $\epsilon$ is a natural isomorphism such that $(\mu (e), \epsilon ) \Vdash H$ .

Observe that any $r: z \rightarrow z^{\prime}$ induces a morphism $F\downarrow r: F\downarrow z \rightarrow F\downarrow z^{\prime}$ of partitioned groupoidal assemblies defined by $(y,u: Fy \rightarrow z) \mapsto (y, ru)$ and identity on morphisms; the morphism is realized by $(e^r, \epsilon ^r)$ , where

\begin{align*} e^r &:= \textsf{id}_{B\times C^{\mathbb{I}_1}} \\ \epsilon ^r_{(y,u)} &:= \left \langle B, \lambda \left (\partial ^{-1}{\left \Vert (r,u)\right \Vert }_Z\right ) \right \rangle \end{align*}

and where ${\left \Vert (r,u) \right \Vert }_Z$ denotes the following commutative square.

Naturality follows double functoriality of $\partial ^{-1}$ .

A morphism $(r, \psi, f, \zeta ): (z,H,e,\epsilon ) \rightarrow (z^{\prime},H^{\prime},e^{\prime},\epsilon ^{\prime})$ in $\Pi _F X$ consists of a morphism $r:z\rightarrow z^{\prime}$ , a natural isomorphism $\psi : H \Rightarrow H^{\prime} \circ (F\downarrow r)$ over $Y$ and a path $f:e\rightarrow e^{\prime} \circ e^r = e^{\prime}$ , and a natural isomorphism $\zeta$ satisfying

\begin{align*} &\zeta (-,0,-) = \epsilon &&\zeta (-,1,-) = \left (\epsilon ^{\prime} \ast (F\downarrow r) \right ) \circ \left (\Pi (e^{\prime}) \ast \epsilon ^r \right ) \end{align*}

as well as

\begin{align*} (\mu (f)\circ \textsf{swap},\zeta )\Vdash \psi \end{align*}

As in the proof of Proposition 12, $\zeta$ is uniquely determined. The realizer type is $C \times A^{B\times C^{\mathbb{I}_1}}$ and the realizability functor is given by:

\begin{align*}{\left \Vert -\right \Vert }_{\Pi _F X} := \left \langle{\left \Vert -\right \Vert }_Z \circ \pi _1, \pi _3 \right \rangle \end{align*}

The fibration $\Pi _F(G): \Pi _F X \rightarrow Z$ is given by the first projection and realized by $(\pi _1, \textsf{id})$ . The chosen lift of $r$ at $(z,H,e,\epsilon )$ is

\begin{align*} (r, \psi, f): (z,H,e,\epsilon ) \rightarrow (z^{\prime},H^{\prime},e,\epsilon ^{\prime}) \end{align*}

where

\begin{align*} H^{\prime} &:= H \circ \left ( F \downarrow r^{-1} \right ) \\ \epsilon ^{\prime} &:= \left (\epsilon \ast F\downarrow r^{-1} \right ) \circ \left ( e \ast \epsilon ^r \right ) \\ \psi &:= \textsf{id}_H \\ f &:= \textsf{id}_e \end{align*}

The third of these definitions makes sense because $(F\downarrow r^{-1}) \circ (F\downarrow r) = \textsf{id}_{F\downarrow z}$ .

We now define the evaluation map $\textsf{ev}: F^* \Pi _F X \rightarrow X$ . First, let us compute $F^*\Pi _F X$ . Objects of the underlying groupoid of $F^*\Pi _F X$ are tuples $(y,z,H,e,\epsilon )$ , where $y\in Y$ , $z=Fy$ , and $H$ , $e$ , $\epsilon$ are as above. A morphism $(q, r, \psi, f): (y,z,H,e,\epsilon ) \rightarrow (y^{\prime},z^{\prime},H^{\prime},e^{\prime},\epsilon ^{\prime})$ consists of morphisms $q: y\rightarrow y^{\prime}$ and $r=F(q): z \rightarrow z^{\prime}$ , as well as $\psi$ and $f$ as described above. The realizer type of $F^*\Pi _F X$ is $B \times C \times A^B$ and the realizability functor is given by:

\begin{align*}{\left \Vert -\right \Vert }_{F^*\Pi _F X} := \left \langle{\left \Vert -\right \Vert }_Y \circ \pi _1,{\left \Vert -\right \Vert }_Z \circ \pi _2, \pi _4 \right \rangle \end{align*}

Define the map $\textsf{ev}$ by:

\begin{align*} \textsf{ev}(y,z,H,e,\epsilon ) &:= H \left ( y, \textsf{id}_z \right ) \\ \textsf{ev}(q,r,\psi, f) &:= H^{\prime}(q) \circ \psi _{\left (y,\textsf{id}_z\right )} \end{align*}

The argument $q$ of $H^{\prime}$ is here considered as a morphism $(y,r) \rightarrow (y^{\prime}, \textsf{id}_{z^{\prime}})$ in $F\downarrow z^{\prime}$ . The map $\textsf{ev}$ lives in the slice over $Y$ thanks to (4). It is realized by $( \textsf{ev} \circ \langle \pi _3, \pi _1 \rangle, \epsilon ^{\prime} )$ , where we overload notation and use $\textsf{ev}$ for the evaluation map from $\mathbf{R}$ , and $\epsilon ^{\prime}_{(y,z,H,e,\epsilon )} := \epsilon _y$ . This is natural as we know that there is a natural isomorphism $\zeta$ making the following diagram commute.

For the universal property, assume we have a map $R: (W, D, \left \Vert -\right \Vert _{W}) \rightarrow Z$ and a map $S: F^*W \rightarrow X$ over $Y$ .

We construct a map $T$ as in the following diagram.

Define

\begin{align*} Tw &:= \left ( Rw, H, e, \epsilon \right ) \\ T(v:w\rightarrow w^{\prime}) &:= \left ( R(v), \psi, f \right ) \end{align*}

where the components $H$ , $e$ , $\epsilon$ , $\psi$ , and $f$ are defined below.

First,

\begin{align*} &H: F\downarrow Rw \rightarrow X \\ &H := S \circ \left [ \sigma _1, \sigma _2 \right ] \end{align*}

Here,

\begin{align*} &\sigma _1: F\downarrow Rw \rightarrow Y \\ &\sigma _1(y,r) := r^* y \\ &\sigma _1(q:(y,r)\rightarrow (y^{\prime},r^{\prime})) := \overline{r^{\prime}}(y^{\prime}) \circ q \circ \overline{r}(y)^{-1} \end{align*}

is realized by $(\pi _1, \delta )$ , where $\delta _y := \left \Vert \overline{r}(y) \right \Vert _Y$ and $\sigma _2: F\downarrow Rw \rightarrow W$ is the constantly $w$ functor, realized by $( \left \Vert w \right \Vert _W *, \delta ^{\prime})$ , where $\delta _w := \textsf{id}_{\left \Vert w \right \Vert _W}$ . We can form $[\sigma _1,\sigma _2]$ because $F(r^*y) = Rw$ and

\begin{align*} F \left ( \overline{r^{\prime}}(y^{\prime}) \circ q \circ \overline{r}(y)^{-1} \right ) = r^{\prime}qr^{-1} = r^{\prime}qq^{-1}{r^{\prime}}^{-1} = \textsf{id}_{Rw} \end{align*}

Next, pick a realizer $(e^S, \epsilon ^S) \Vdash S$ and define $e \in \Pi (A^{B\times C^{\mathbb{I}_1}})$ to be the exponential transpose of

The natural isomorphism $\epsilon$ is defined by the following pasting diagram.

The $\Pi$ -types being constructed are weak because we have had to make a choice of realizer $(e^S,\epsilon ^S)$ for each $S$ .

As for $\psi : H \Rightarrow H^{\prime} \circ (F\downarrow R(v))$ , we must have

\begin{align*} &\psi (y,r,0) = H(y,r) = S\left ( r^* y, w \right ) \\ &\psi (y^{\prime},r^{\prime},1) = H^{\prime}(y^{\prime}, R(v)r^{\prime}) = S\left ( (R(v)r^{\prime})^* y^{\prime}, w^{\prime} \right ) \end{align*}

So we define

\begin{align*} \psi (q,i) := S\left ( \overline{R(v)}\left ( (r^{\prime})^*(y^{\prime}) \right ), v \right ) \circ H(q) = H^{\prime}(q) \circ S\left ( \overline{R(v)}\left ( r^*(y) \right ), v \right ) \end{align*}

The argument $q$ of $H$ is regarded as a morphism $(y,r)\rightarrow (y^{\prime},r^{\prime})$ in $F\downarrow Rw$ , whereas $q$ qua argument of $H^{\prime}$ is regarded as a morphism $(y,R(v)r)\rightarrow (y^{\prime},R(v)r^{\prime})$ in $F\downarrow Rw^{\prime}$ . Finally, we define $f:e\rightarrow e^{\prime}$ (recall that $(e,\epsilon ) \Vdash H$ ) to be the exponential transpose of

To complete the proof, we show that

\begin{align*} S = \textsf{ev} \circ F^*T \end{align*}

On objects:

\begin{align*} \textsf{ev}\left ( F^*T (y,w) \right ) &= \textsf{ev}(y,Tw) \\ &= \textsf{ev}(y,Rw,H,e,\epsilon ) \\ &= H\left (y,\textsf{id}_{Rw} \right ) \\ &= S(y, w) \end{align*}

and on morphisms:

\begin{align*} \textsf{ev}\left ( F^*T (q,v) \right ) &= \textsf{ev}(q,T(v)) \\ &= \textsf{ev}(q,R(v),\psi, f) \\ &= H^{\prime}\left (q:(y,R(v)) \rightarrow \left (y^{\prime},\textsf{id}_{Rw^{\prime}}\right )\right ) \circ \psi _{\left ( y, \textsf{id}_{Rw} \right )} \\ &= S\left ( q \circ \left (\overline{R(v)}(y)\right )^{-1}, w^{\prime} \right ) \circ S\left ( \overline{R(v)}(y), v \right ) \\ &= S(q,v) \end{align*}

The fact that that $S = \textsf{ev} \circ F^*T$ holds on the nose is down to the fact that isofibrations of groupoids are exponentiable.