2.1 Basic constructions on reflexive graphs

The most basic reflexive graph structure is the discrete one provided by its identity type.

On the other extreme, we may consider the codiscrete reflexive graph structure.

Definition 15 (Codiscrete reflexive graph). A type A can be, conversely, equipped with the codiscrete reflexive graph structure ${\triangledown}{A}$ that identifies any two elements:

\begin{align*} \lvert {\triangledown}{A} \rvert & :\equiv A \\ x \approx_{{\triangledown}{A}}y & :\equiv \mathbf{1} \\ \mathsf{rx}_{{\triangledown}{A}}{x} & :\equiv * \end{align*}

Definition 16 (Total reflexive graph). If $\mathcal{A}$ is a reflexive graph and $\mathcal{B}$ is a displayed reflexive graph over $\mathcal{A}$ , then the total reflexive graph $\mathcal{A}.\mathcal{B}$ can be defined with vertices in $\sum_{(x:\lvert \mathcal{A} \rvert)}\lvert \mathcal{B} \rvert(x)$ and edges $(a_0,b_0) \approx_{\mathcal{A}.\mathcal{B}} (a_1,b_1)$ in $\sum_{(p:a_0\approx_{\mathcal{A}} a_1)} b_0 \approx_{\mathcal{B}}^{p} b_1$ , with reflexivity data given by $\mathsf{rx}_{\mathcal{A}.\mathcal{B}}{(a,b)} :\equiv (\mathsf{rx}_{A}{a},\mathsf{rx}_{B}{b})$ . Naturally, we have a homomorphism $\pi_{\mathcal{B}}\colon \mathcal{A}.\mathcal{B} \to \mathcal{A}$ of reflexive graphs:

\begin{align*} \lvert \pi_{\mathcal{B}} \rvert\,(x,y) & :\equiv x \\ \pi_{\mathcal{B}}^{\approx}\,(p,q) & :\equiv p \\ \pi_{\mathcal{B}}^{\mathsf{rx}}\,(x,y) & :\equiv \mathbf{refl} \end{align*}

We can generalise Definition 17 to arbitrary families.

Definition 18 (Product of reflexive graphs). Let A be a type and let $\mathcal{B}(x)$ be a reflexive graph for each $x:A$ . The product $\prod_{(x:A)}\mathcal{B}(x)$ of the family of $\mathcal{B}$ reflexive graphs is defined like so:

Definition 19 (Coproduct of reflexive graphs). Let A be a type and let $\mathcal{B}(x)$ be a reflexive graph for each $x:A$ . The coproduct $\coprod_{(x:A)}\mathcal{B}(x)$ of the family $\mathcal{B}$ of reflexive graphs is defined like so:

Later on in Example 73, we will see how to refactor the construction of the coproduct of reflexive graphs in terms of a lens over the discrete reflexive graph ${\triangle}{A}$ .

Definition 20 (Tensor and cotensor of reflexive graphs). Let A be a type and let $\mathcal{B}$ be a reflexive graph; then, the tensor and cotensor of $\mathcal{B}$ by A are defined, respectively, using coproducts and products as below:

Definition 21 (Constant displayed reflexive graph). Let $\mathcal{A}$ and $\mathcal{B}$ be reflexive graphs. The constant displayed reflexive graph over $\mathcal{A}$ induced by $\mathcal{B}$ is the displayed reflexive graph specified $\mathcal{A}^*\mathcal{B}$ below:

\begin{align*} \lvert \mathcal{A}^*\mathcal{B} \rvert(x) & :\equiv \lvert \mathcal{B} \rvert \\ u \approx_{\mathcal{A}^*\mathcal{B}}^{p} v & :\equiv u \approx_{\mathcal{B}} v \\ \mathsf{rx}_{\mathcal{A}^*\mathcal{B}}^{x}{u} & :\equiv \mathsf{rx}_{\mathcal{B}}{u} \end{align*}

Thus, evidently, we have $(\mathcal{A}^*\mathcal{B})(x)\equiv \mathcal{B}$ for all $x:\lvert \mathcal{A} \rvert$ .

Definition 23 (Subgraph comprehension). Let $\mathcal{A}$ be a reflexive graph, and let P(x) be a proposition for each vertex $x:\mathcal{A}$ . We define the comprehension of P to be the following reflexive graph:

Displayed edges and reflexivity data are defined in terms of those of $\mathcal{C}$ like so:

2.2 Duality involution for reflexive graphs

Definition 26 (Opposite reflexive graph). We define the opposite of a reflexive graph $\mathcal{A}$ as follows:

\begin{align*} \lvert \mathcal{A}^{\mathsf{op}} \rvert &:\equiv \lvert \mathcal{A} \rvert \\ x \approx_{\mathcal{A}^{\mathsf{op}}} y &:\equiv y\approx_{\mathcal{A}} x \\ \mathsf{rx}_{\mathcal{A}^{\mathsf{op}}}{x} &:\equiv \mathsf{rx}_{\mathcal{A}}{x} \end{align*}

Definition 28 (Total opposite of a displayed reflexive graph). Let $\mathcal{A}$ be a reflexive graph, and let $\mathcal{B}$ be a displayed reflexive graph over $\mathcal{A}$ . We define the total opposite $\mathcal{B}^{\widetilde{\mathsf{op}}}$ of $\mathcal{B}$ to be the following displayed reflexive graph over $\mathcal{A}^{\mathsf{op}}$ ,

\begin{align*} \lvert \mathcal{B}^{\widetilde{\mathsf{op}}} \rvert(x) &:\equiv \lvert \mathcal{B} \rvert(x) \\ u \approx_{\mathcal{B}^{\widetilde{\mathsf{op}}}}^{p} v &:\equiv v\approx_{\mathcal{B}}^{p}u \\ \mathsf{rx}_{\mathcal{B}^{\widetilde{\mathsf{op}}}}^{x}{u} &:\equiv \mathsf{rx}_{\mathcal{B}}^{x}{u}\text{,} \end{align*}

so that we have $(\mathcal{A}.\mathcal{B})^{\mathsf{op}} \equiv \mathcal{A}^{\mathsf{op}}.\mathcal{B}^{\widetilde{\mathsf{op}}}$ .

Note that Definition 28 does not define the actual “opposite” of a displayed reflexive graph $\mathcal{B}$ , which would naturally have the same base as $\mathcal{B}$ ; opposites of arbitrary displayed reflexive graphs do not make sense (for the same reason that Bénabou’s definition of opposites applies only to displayed categories that are additionally fibrations).

Observation 27 extends to the following duality involution on displayed reflexive graphs vs Definition 28.

2.3 Path objects and the univalence condition

Conversely, assume that every co-fan is contractible.

When it causes no ambiguity, we will write $\lfloor p\rfloor_{\mathcal{A}}$ for $ \lfloor p\rfloor^{x,y}_{\mathcal{A}}$ .

Then, we define $[\![{-}]\!]^{x,y}_{\mathcal{A}}$ by instantiation.

Next we prove that $[\![{-}]\!]^{x,y}_{\mathcal{A}}$ is a section of $\lfloor{-}\rfloor^{x,y}_{\mathcal{A}}$ .

Finally, we prove that $[\![{-}]\!]^{x,y}_{\mathcal{A}}$ is a retraction of $\lfloor{-}\rfloor^{x,y}_{\mathcal{A}}$ .

1. (1) Each fan $\{x\}_{\mathcal{A}}^{+}$ is a proposition.

2. (2) Each co-fan $\{x\}_{\mathcal{A}}^{-}$ is a proposition.

3. (3) Each fan $\{x\}_{\mathcal{A}}^{+}$ is contractible with centre $(x,\mathsf{rx}_{\mathcal{A}}{x})$ .

4. (4) Each co-fan $\{x\}_{\mathcal{A}}^{-}$ is contractible with centre $(x,\mathsf{rx}_{\mathcal{A}}{x})$ .

5. (5) Each of the functions $\mathsf{idToEdge}_{x,y} : x =_{\lvert \mathcal{A} \rvert} y \to x\approx_{\mathcal{A}}y$ defined in Construction 32 is an equivalence.

We shall refer to a univalent reflexive graph as a path object .

2.4 Path algebra in a path object

Edges in an arbitrary reflexive graph cannot be concatenated or inverted; when the reflexive graph is univalent, i.e. a path object, the type of edges becomes (by Lemma 34) equivalent to the first level of the canonical $\infty$ -groupoid structure on the type of vertices, and as such brooks fully coherent concatenation and inversion operations. We will need these operations only rarely — and indeed, the central thesis of the theory of path objects in univalent foundations is that it is often possible to avoid painful path algebra by very careful choices of path object structure.

Of course, there is no end to the possible combinators, so we have presented just a few representative examples that we will make use of.

Lemma 38 (Pre-concatenation equivalence). For a path object $\mathcal{A}$ and an edge $p:x\approx_{\mathcal{A}}y$ , we have a family of equivalences induced by pre-concatenation with p as below,

\[ z:\lvert \mathcal{A} \rvert \vdash (p{{\,\centerdot\,}}_{\mathcal{A}} -) \colon y \approx_{\mathcal{A}}z \to x\approx_{\mathcal{A}}z \]

with chosen inverse $(p^{-1}_{\mathcal{A}} {{\,\centerdot\,}}_{\mathcal{A}} -) : x\approx_{\mathcal{A}}z \to y\approx_{\mathcal{A}}z$ .

2.5 Univalent families and reflexive graph images

Definition 39 (Reflexive graph image). The reflexive graph image of a type A under a family of types $x:A\vdash B(x):\mathbf{type}$ is defined to be the following reflexive graph $A/B$ :

\begin{align*} \lvert A/B \rvert & :\equiv A \\ x \approx_{A/B} y & :\equiv \mathsf{Equiv}(B(x),B(y)) \\ \mathsf{rx}_{A/B}{x} & :\equiv \mathsf{idnEquiv}_{B(x)} \end{align*}

1. (1) For each $x:A$ , the type $\sum_{(y:A)}\mathsf{Equiv}(B(x),B(y))$ is a proposition.

2. (2) For each $x:A$ , the type $\sum_{(y:A)}\mathsf{Equiv}(B(x),B(y))$ is a contractible.

3. (3) For each $x,y:A$ the canonical map $x =_{A} y\to \mathsf{Equiv}(B(x),B(y))$ sending $\mathbf{refl}$ to the identity equivalence is an equivalence.

4. (4) The reflexive graph image $A/B$ is univalent.

One often refers to a univalent family of types (U,E) as a universe to emphasise that it may be closed under various type constructors, etc.

Definition 41 (Propositional reflexive graph image). Let B(x) be a proposition for each $x:A$ . The propositional reflexive graph image of A under B is defined to be the following reflexive graph $A/_{-1}B$ :

\begin{align*} \lvert A/_{-1}B \rvert & :\equiv A \\ x \approx_{A/_{-1}B} y & :\equiv (B(x)\to B(y))\times(B(y)\to B(x)) \\ \mathsf{rx}_{A/_{-1}B}{x} & :\equiv (\lambda{u}\mathpunct{.}{u},\lambda{u}\mathpunct{.}{u}) \end{align*}

2.6 Basic closure properties of path objects

Conversely, assume that ${\triangledown}{A}$ is univalent. For any two $x,y:A$ , we evidently have $*:x\approx_{{\triangledown}{A}}y$ and thus $[\![{*}]\!]^{x,y}_{{\triangledown}{A}} : x =_{A} y$ by Lemma 33.

3.1 Lenses of reflexive graphs

We introduce two dual kinds of lens of reflexive graphs.

Definition 54 (Oplax covariant lens). Let $\mathcal{A}$ be a reflexive graph. An oplax covariant lens of reflexive graphs over $\mathcal{A}$ is defined to be a family of reflexive graphs $\mathcal{B}(x)$ indexed in $x:\lvert \mathcal{A} \rvert$ equipped with the following data:

\begin{align*} x,y:\lvert \mathcal{A} \rvert;\; p : x \approx_{\mathcal{A}} y, u : \lvert \mathcal{B}(x) \rvert & \vdash \mathsf{push}_{\mathcal{B}}^{p}u : \lvert \mathcal{B}(y) \rvert \\ x:\lvert \mathcal{A} \rvert, u:\lvert \mathcal{B}(x) \rvert & \vdash \mathsf{pushRx}_{\mathcal{B}}^{x}u : \mathsf{push}_{\mathcal{B}}^{\mathsf{rx}_{\mathcal{A}}{x}}{u} \approx_{\mathcal{B}(x)} u \end{align*}

Definition 55 (Lax contravariant lens). Let $\mathcal{A}$ be a reflexive graph. A lax contravariant lens of reflexive graphs over $\mathcal{A}$ is defined to be a family of reflexive graphs $\mathcal{B}(x)$ indexed in $x:\lvert \mathcal{A} \rvert$ equipped with the following data:

\begin{align*} x,y:\lvert \mathcal{A} \rvert;\;p:x\approx_{\mathcal{A}}y, u:\lvert \mathcal{B}(y) \rvert & \vdash \mathsf{pull}_{\mathcal{B}}^{p}u : \lvert \mathcal{B}(x) \rvert \\ x:\lvert \mathcal{A} \rvert,u:\lvert \mathcal{B}(x) \rvert & \vdash \mathsf{pullRx}_{\mathcal{B}}^{x}u: u \approx_{\mathcal{B}(x)} \mathsf{pull}_{\mathcal{B}}^{\mathsf{rx}_{\mathcal{A}}{x}}{u} \end{align*}

In the context of Definition 54 or Definition 55, we will refer to $\mathcal{B}$ as a lens of path objects or a univalent lens when each $\mathcal{B}(x)$ is univalent.

Figure 1.

Summary of univalence lemmas for (displayed) reflexive graphs.

Figure 2.

Summary of univalence lemmas for reflexive graph lenses.

Definition 56 (Display of oplax covariant lenses). Let $\mathcal{B}$ be an oplax covariant lens over a reflexive graph $\mathcal{A}$ . The (covariant) display of $\mathcal{B}$ over $\mathcal{A}$ is defined to be the following displayed reflexive graph $\mathsf{disp}^+_{\mathcal{A}}\mathcal{B}$ over $\mathcal{A}$ :

\begin{align*} x:\lvert \mathcal{A} \rvert & \vdash \lvert \mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}} \rvert(x) :\equiv \lvert \mathcal{B}(x) \rvert \\ p:x\approx_{\mathcal{A}}y, u:\lvert \mathcal{B}(x) \rvert,v:\lvert \mathcal{B}(y) \rvert & \vdash u \approx_{\mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}}}^{p} v :\equiv \mathsf{push}_{\mathcal{B}}^{p}u \approx_{\mathcal{B}(y)} v \\ x:\lvert \mathcal{A} \rvert,u:\lvert \mathcal{B}(x) \rvert & \vdash \mathsf{rx}_{\mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}}}^{x}{u} :\equiv \mathsf{pushRx}_{\mathcal{B}}^{x}{u} \end{align*}

Definition 57 (Display of lax contravariant lenses). Let $\mathcal{B}$ be a lax contravariant lens over a reflexive graph $\mathcal{A}$ . The (contravariant) display of $\mathcal{B}$ over $\mathcal{A}$ is defined to be the following displayed reflexive graph $\mathsf{disp}^-_{\mathcal{A}}{\mathcal{B}}$ over $\mathcal{A}$ :

\begin{align*} x:\lvert \mathcal{A} \rvert & \vdash \lvert \mathsf{disp}^-_{\mathcal{A}}{\mathcal{B}} \rvert(x) :\equiv \lvert \mathcal{B}(x) \rvert \\ p:x\approx_{\mathcal{A}}y, u:\lvert \mathcal{B}(x) \rvert,v:\lvert \mathcal{B}(y) \rvert & \vdash u \approx_{\mathsf{disp}^-_{\mathcal{A}}{\mathcal{B}}}^{p} v :\equiv u\approx_{\mathcal{B}(x)}\mathsf{pull}_{\mathcal{B}}^{p}{v} \\ x:\lvert \mathcal{A} \rvert,u:\lvert \mathcal{B}(x) \rvert & \vdash \mathsf{rx}_{\mathsf{disp}^-_{\mathcal{A}}{\mathcal{B}}}^{x}{u} :\equiv \mathsf{pullRx}_{\mathcal{B}}^{x}{u} \end{align*}

Computation 58 (Components of the display of oplax covariant lenses). Let $\mathcal{B}$ be an oplax covariant lens over a reflexive graph $\mathcal{A}$ . For any $x:\lvert \mathcal{A} \rvert$ , the component $(\mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}})(x)$ is precisely the following:

\begin{align*} \lvert (\mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}})(x) \rvert & \equiv \lvert \mathcal{B}(x) \rvert \\ u\approx_{(\mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}})(x)} v & \equiv \mathsf{push}_{\mathcal{B}}^{\mathsf{rx}_{\mathcal{A}}{x}}{u} \approx_{\mathcal{B}(x)} v \\ \mathsf{rx}_{(\mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}})(x)}{u} & \equiv \mathsf{pushRx}_{\mathcal{B}}^{x}{u} \end{align*}

We likewise have the following description of fans:

\[ \{u\}_{(\mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}})(x)}^{+} \equiv \{\mathsf{push}_{\mathcal{B}}^{\mathsf{rx}_{\mathcal{A}}{x}}{u}\}_{\mathcal{B}(x)}^{+} \]

Computation 59 (Components of the display of lax contravariant lenses). Let $\mathcal{B}$ be a lax contravariant lens over a reflexive graph $\mathcal{A}$ . For any $x:\lvert \mathcal{A} \rvert$ , the component $(\mathsf{disp}^-_{\mathcal{A}}{\mathcal{B}})(x)$ is unfolds as follows:

\begin{align*} \lvert (\mathsf{disp}^-_{\mathcal{A}}{\mathcal{B}})(x) \rvert & \equiv \lvert \mathcal{B}(x) \rvert \\ u\approx_{(\mathsf{disp}^-_{\mathcal{A}}{\mathcal{B}})(x)} v & \equiv u \approx_{\mathcal{B}(x)} \mathsf{pull}_{\mathcal{B}}^{\mathsf{rx}_{\mathcal{A}}{x}}{v} \\ \mathsf{rx}_{(\mathsf{disp}^-_{\mathcal{A}}{\mathcal{B}})(x)}{u} & \equiv \mathsf{pullRx}_{\mathcal{B}}^{x}{u} \end{align*}

We have the following description of co-fans:

\[ \{u\}_{(\mathsf{disp}^-_{\mathcal{A}}{\mathcal{B}})(x)}^{-} \equiv \{\mathsf{pull}_{\mathcal{B}}^{\mathsf{rx}_{\mathcal{A}}{x}}{u}\}_{\mathcal{B}(x)}^{-} \]

3.2 Unbiased dependent lenses

In this section, we describe a slightly strange technical device that generalises both oplax covariant and lax contravariant lenses that we name the unbiased dependent lens. The reason for introducing these is to characterise the identity types of structures involving mixed variance (including, for example, the type of reflexive graphs!).

Definition 62 (Unbiased dependent lens). Let $\mathcal{A}$ be a reflexive graph. A unbiased dependent lens over $\mathcal{A}$ is defined to be a family of reflexive graphs $\mathcal{B}(p)$ indexed in edges $p:x\approx_{\mathcal{A}}y$ equipped with the following data:

\begin{align*} x,y:\lvert \mathcal{A} \rvert;\;p:x\approx_{\mathcal{A}}y,u:\lvert \mathcal{B}(\mathsf{rx}_{\mathcal{A}}{x}) \rvert & \vdash \mathsf{lext}_{\mathcal{B}}^{p}{u} : \lvert \mathcal{B}(p) \rvert \\ x,y:\lvert \mathcal{A} \rvert;\;p:x\approx_{\mathcal{A}}y,u:\lvert \mathcal{B}(\mathsf{rx}_{\mathcal{A}}{y}) \rvert & \vdash \mathsf{rext}_{\mathcal{B}}^{p}{u} : \lvert \mathcal{B}(p) \rvert \\ x:\lvert \mathcal{A} \rvert,u:\lvert \mathcal{B}(\mathsf{rx}_{\mathcal{A}}{x}) \rvert & \vdash \mathsf{extRx}_{\mathcal{B}}^{x} {u} : \mathsf{lext}_{\mathcal{B}}^{\mathsf{rx}_{\mathcal{A}}{x}}{u} \approx_{\mathcal{B}(\mathsf{rx}_{\mathcal{A}}{x})} \mathsf{rext}_{\mathcal{B}}^{\mathsf{rx}_{\mathcal{A}}{x}}{u} \\ x:\lvert \mathcal{A} \rvert,u:\lvert \mathcal{B}(\mathsf{rx}_{\mathcal{A}}{x}) \rvert & \vdash \mathsf{rextRx}_{\mathcal{B}}^{x} {u} : u \approx_{\mathcal{B}(\mathsf{rx}_{\mathcal{A}}{x})} \mathsf{rext}_{\mathcal{B}}^{\mathsf{rx}_{\mathcal{A}}{x}}{u} \end{align*}

Definition 63 (Display of unbiased dependent lenses). Let $\mathcal{B}$ be a unbiased dependent lens over a reflexive graph $\mathcal{A}$ . The display of $\mathcal{B}$ over $\mathcal{A}$ is defined to be the following displayed reflexive graph $\mathsf{disp}^\pm_{\mathcal{A}}$ over $\mathcal{A}$ :

\begin{align*} x:\lvert \mathcal{A} \rvert & \vdash \lvert \mathsf{disp}^\pm_{\mathcal{A}}\mathcal{B} \rvert(x) :\equiv \lvert \mathcal{B}(\mathsf{rx}_{\mathcal{A}}{x}) \rvert \\ p : x\approx_{\mathcal{A}}y, u:\lvert \mathcal{B}(x) \rvert, v : \lvert \mathcal{B}(y) \rvert & \vdash u \approx_{{\mathsf{disp}^\pm_{\mathcal{A}}}^{p}\mathcal{B}} v :\equiv \mathsf{lext}_{\mathcal{B}}^{p}{u} \approx_{\mathcal{B}(p)} \mathsf{rext}_{\mathcal{B}}^{p}{u} \\ x:\lvert \mathcal{A} \rvert, u:\lvert \mathcal{B}(x) \rvert & \vdash \mathsf{rx}_{{\mathsf{disp}^\pm_{\mathcal{A}}}^{x}\mathcal{B}}{u} :\equiv \mathsf{extRx}_{\mathcal{B}}^{x} {u} \end{align*}

The following Construction 66 shows that unbiased dependent lenses generalise both oplax covariant and lax contravariant lenses up to definitional equality.

Computation. We compute the displays as follows.

\begin{align*} & \lvert \mathsf{disp}^\pm_{\mathcal{A}}[\mathcal{B}]_+^\pm \rvert(x) \equiv \lvert [\mathcal{B}]_+^\pm \rvert(\mathsf{rx}_{\mathcal{A}}{x}) \equiv \lvert \mathcal{B}(x) \rvert \equiv \lvert \mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}} \rvert(x) \\ & \lvert \mathsf{disp}^\pm_{\mathcal{A}} [\mathcal{B}]_-^\pm \rvert(x) \equiv \lvert [\mathcal{B}]_-^\pm \rvert(\mathsf{rx}_{\mathcal{A}}{x}) \equiv \lvert \mathcal{B}(x) \rvert \equiv \lvert \mathsf{disp}^-_{\mathcal{A}}{\mathcal{B}} \rvert(x) \\ & u \approx_{\mathsf{disp}^\pm_{\mathcal{A}}[\mathcal{B}]_-^\pm}^{p} v \equiv \mathsf{lext}_{[\mathcal{B}]_+^\pm}^{p}{u} \approx_{[\mathcal{B}]_+^\pm(p)} \mathsf{rext}_{[\mathcal{B}]_+^\pm}^{p}{v} \equiv \mathsf{push}_{\mathcal{B}}^{p}{u} \approx_{\mathcal{B}(y)} u \equiv u \approx_{\mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}}}^{p}v \\ & u \approx_{\mathsf{disp}^\pm_{\mathcal{A}}[\mathcal{B}]_-^\pm}^{p} v \equiv \mathsf{lext}_{[\mathcal{B}]_-^\pm}^{p}{u} \approx_{[\mathcal{B}]_-^\pm(p)} \mathsf{rext}_{[\mathcal{B}]_-^\pm}^{p}{v} \equiv u \approx_{\mathcal{B}(y)} \mathsf{pull}_{\mathcal{B}}^{p}{v} \equiv u \approx_{\mathsf{disp}^-_{\mathcal{A}}{\mathcal{B}}}^{p}v \\ & \mathsf{rx}_{\mathsf{disp}^\pm_{\mathcal{A}}[\mathcal{B}]_-^\pm}^{x}{u} \equiv \mathsf{extRx}_{[\mathcal{B}]_+^\pm}^{x} {u} \equiv \mathsf{push}_{\mathcal{B}}^{p}{u} \equiv \mathsf{rx}_{\mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}}}^{x}{u} \\ & \mathsf{rx}_{\mathsf{disp}^\pm_{\mathcal{A}}[\mathcal{B}]_-^\pm}^{x}{u} \equiv \mathsf{extRx}_{[\mathcal{B}]_-^\pm}^{x} {u} \equiv \mathsf{pull}_{\mathcal{B}}^{p}{u} \equiv \mathsf{rx}_{\mathsf{disp}^-_{\mathcal{A}}{\mathcal{B}}}^{x}{u} & & \end{align*}

3.3 Duality involution for reflexive graph lenses

We have seen in Definition 26 and Observation 27, the duality involution for reflexive graphs that flips the direction of edges. This involution hosts a (definitional) equivalence between the types of oplax covariant and lax contravariant lenses, respectively. By virtue of this equivalence, from a result about all oplax covariant lenses one obtains automatically a dual result about all lax contravariant lenses.

Definition 67 (Total opposite of a lens). Let $\mathcal{A}$ be a reflexive graph, and let $\mathcal{B}$ be an oplax covariant lens of reflexive graphs over $\mathcal{A}$ . We define the total opposite of $\mathcal{B}$ to be the following lax contravariant lens $\mathcal{B}^{\widetilde{\mathsf{op}}}$ over $\mathcal{A}^{\mathsf{op}}$ :

\begin{align*} \mathcal{B}^{\widetilde{\mathsf{op}}}(x) & :\equiv \mathcal{B}(x)^{\mathsf{op}} \\ \mathsf{pull}_{\mathcal{B}^{\widetilde{\mathsf{op}}}}^{p}{u} & :\equiv \mathsf{push}_{\mathcal{B}}^{p}{u} \\ \mathsf{pullRx}_{\mathcal{B}^{\widetilde{\mathsf{op}}}}^{x}{u} & :\equiv \mathsf{pushRx}_{\mathcal{B}}^{x}{u} \end{align*}

Conversely, if $\mathcal{B}$ is a lax contravariant lens, we define its total opposite to be the following lax contravariant lens $\mathcal{B}^{\widetilde{\mathsf{op}}}$ over $\mathcal{A}^{\mathsf{op}}$ :

\begin{align*} \mathcal{B}^{\widetilde{\mathsf{op}}}(x) & :\equiv \mathcal{B}(x)^{\mathsf{op}} \\ \mathsf{push}_{\mathcal{B}^{\widetilde{\mathsf{op}}}}^{p}{u} & :\equiv \mathsf{pull}_{\mathcal{B}}^{p}{u} \\ \mathsf{pushRx}_{\mathcal{B}^{\widetilde{\mathsf{op}}}}^{x}{u} & :\equiv \mathsf{pullRx}_{\mathcal{B}}^{x}{u} \end{align*}

The purpose of total opposites is to exhibit a duality involution identifying oplax covariant and lax contravariant lenses.

Although many other definitions can be obtained directly by duality, we will continue to specify various concepts and results for both oplax covariant and lax contravariant lenses in order to facilitate easy reference.

3.4 Definitional lenses of reflexive graphs

An (oplax covariant, lax contravariant) lens equips a family with a pushforward or pullback operator together with a witness of a unit law holding up to a directed edge. That the unit law holds only up to an edge is the meaning of the “oplax/lax” terminology. An important class of lenses are the ones in which the unit law holds up to definitional equality, so that the oplax or lax witness or the unit law can be given by the reflexivity datum of the component reflexive graph. We study these definitional lenses in the present section, with an eye to developing the theory of polynomials and partial products of lenses in Section 3.4.4.

Definition 71 (Definitional lens). A definitional covariant lens is an oplax covariant lens $\mathcal{B}$ over $\mathcal{A}$ in which the following equations hold definitionally:

\begin{align*} x:\lvert \mathcal{A} \rvert, u:\lvert \mathcal{B}(x) \rvert & \vdash \mathsf{push}_{\mathcal{B}}^{\mathsf{rx}_{\mathcal{A}}{x}}{u} \equiv u \\ x:\lvert \mathcal{A} \rvert,u:\lvert \mathcal{B}(x) \rvert & \vdash \mathsf{pushRx}_{\mathcal{B}}^{x}{u} \equiv \mathsf{rx}_{\mathcal{B}(x)}{u} \end{align*}

By the same token, a definitional contravariant lens is a lax contravariant lens in which the following equations hold definitionally:

\begin{align*} x:\lvert \mathcal{A} \rvert, u:\lvert \mathcal{B}(x) \rvert & \vdash \mathsf{pull}_{\mathcal{B}}^{\mathsf{rx}_{\mathcal{A}}{x}}{u} \equiv u \\ x:\lvert \mathcal{A} \rvert,u:\lvert \mathcal{B}(x) \rvert & \vdash \mathsf{pullRx}_{\mathcal{B}}^{x}{u} \equiv \mathsf{rx}_{\mathcal{B}(x)}{u} \end{align*}

Example 73 (Definitional lenses over discrete reflexive graphs). Let $\mathcal{B}(x)$ be a family of reflexive graphs indexed in $x:A$ . Then we may turn $\mathcal{B}$ canonically into an oplax covariant lens or a lax contravariant lens over the discrete reflexive graph ${\triangle}{A}$ as follows using transport:

\begin{align*} x,y:A;\; p:x =_{A} y, u:\lvert \mathcal{B}(x) \rvert &\vdash \mathsf{push}_{\mathcal{B}}^{p}u :\equiv p_*u\\ x,y:A;\; p:x =_{A} y, u:\lvert \mathcal{B}(y) \rvert &\vdash \mathsf{pull}_{\mathcal{B}}^{p}u :\equiv p^{-1}_*u \end{align*}

The total reflexive graph of the display of $\mathcal{B}$ qua covariant lens is then definitionally equal to the coproduct $\coprod_{(x:A)}\mathcal{B}(x)$ from Definition 19.

Computation 74 (Display of definitional lenses). The display of definitional lenses is, naturally, more simple than that of ordinary lenses. If $\mathcal{B}$ is a definitional covariant lens over $\mathcal{A}$ , its display computes as follows:

\begin{align*} \lvert \mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}} \rvert(x) &\equiv \lvert \mathcal{B}(x) \rvert \\ u \approx_{\mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}}}^{p:x\approx_{\mathcal{A}}y} v &\equiv \mathsf{push}_{\mathcal{B}}^{p}{u} \approx_{\mathcal{B}(y)} v \\ \mathsf{rx}_{\mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}}}^{x}{u} &\equiv \mathsf{rx}_{\mathcal{B}(x)}{u} \end{align*}

Likewise, if $\mathcal{B}$ is a definitional contravariant lens, its display computes as follows:

\begin{align*} \lvert \mathsf{disp}^-_{\mathcal{A}}{\mathcal{B}} \rvert(x) &\equiv \lvert \mathcal{B}(x) \rvert \\ u \approx_{\mathsf{disp}^-_{\mathcal{A}}{\mathcal{B}}}^{p:x\approx_{\mathcal{A}}y} v &\equiv u \approx_{\mathcal{B}(x)} \mathsf{pull}_{\mathcal{B}}^{p}{v} \\ \mathsf{rx}_{\mathsf{disp}^-_{\mathcal{A}}{\mathcal{B}}}^{x}{u} &\equiv \mathsf{rx}_{\mathcal{B}(x)}{u} \end{align*}

3.4.1 Example: definitional lenses from univalent families

Example 75 (Definitional lenses of from univalent families). Any univalent family of path objects has a canonical definitional lens structure. In particular, given a univalent family of path objects $(U,\mathcal{E})$ , we can equip $\mathcal{E}$ with the structure of a covariant and a contravariant lens over the reflexive graph image $U/\mathcal{E}$ :

\begin{align*} A_0,A_1:U;\; f : A_0\approx_{U/\mathcal{E}} A_1;\; a : \lvert \mathcal{E}(A_0) \rvert & \vdash \mathsf{push}_{\mathcal{E}}^{\,f}{a} :\equiv f a \\ A_0,A_1:U;\; f : A_0\approx_{U/\mathcal{E}} A_1;\; a : \lvert \mathcal{E}(A_1) \rvert & \vdash \mathsf{pull}_{\mathcal{E}}^{\,f}{a} :\equiv f^{-1} a \\ A:U;\; a:\lvert \mathcal{E}(A) \rvert & \vdash \mathsf{pushRx}_{\mathcal{E}}^{A} {a} :\equiv \mathsf{rx}_{\mathcal{E}(A)}{a} \\ A:U;\; a:\lvert \mathcal{E}(A) \rvert & \vdash \mathsf{pullRx}_{\mathcal{E}}^{A}{a} :\equiv \mathsf{rx}_{\mathcal{E}(A)}{a} \end{align*}

Example 76 (Definitional lenses from universes). Any univalent family of types (U,E), e.g. a universe, determines a univalent family of path objects $(U,{\triangle}{E})$ . In this case, $U/\lvert {\triangle}{E} \rvert\equiv U/E$ is precisely the canonical path object structure on U determined by equivalences between components of E, and we may consider the corresponding definitional lens structures on ${\triangle}{E}$ by specialising Example 75.

Example 77 (Definitional lenses from subuniverses). Let $(U,\mathcal{E})$ be a univalent family, and let P(A) be a proposition for every $A:U$ , We can restrict $\mathcal{E}$ to a family of path objects $\mathcal{E}_P$ over $\{{A:U/\lvert \mathcal{E}| \rvert}{P(A)}\}$ setting $\mathcal{E}_P(A,h) :\equiv \mathcal{E}(A)$ . Then $\mathcal{E}_P$ can be equipped with the structure of a definitional covariant and contravariant lens over $U/\lvert \mathcal{E} \rvert$ in the following way:

\begin{align*} A, B:\lvert \{{A:U/\lvert \mathcal{E} \rvert}{P(A)}\} \rvert;\; f : A\approx_{U/\lvert \mathcal{E} \rvert}B, x : \lvert \mathcal{E}(\pi_1{A}) \rvert & \vdash \mathsf{push}_{\mathcal{E}_P}^{\,f}{x} :\equiv fx \\ A, B:\lvert \{{A:U/\lvert \mathcal{E} \rvert}{P(A)}\} \rvert;\; f : A\approx_{U/\lvert \mathcal{E} \rvert}B, x : \lvert \mathcal{E}(\pi_1{B}) \rvert & \vdash \mathsf{pull}_{\mathcal{E}_P}^{\,f}{x} :\equiv f^{-1}{x} \\ A:\lvert \{{A:U/\lvert \mathcal{E} \rvert}{P(A)}\} \rvert, x : \lvert \mathcal{E}(\pi_1{A}) \rvert & \vdash \mathsf{pushRx}_{\mathcal{E}_P}^{A}{x} :\equiv \mathsf{rx}_{\mathcal{E}(\pi_1A)}{x} \\ A:\lvert \{{A:U/\lvert \mathcal{E} \rvert}{P(A)}\} \rvert, x : \lvert \mathcal{E} \rvert(\pi_1{A}) & \vdash \mathsf{pullRx}_{\mathcal{E}_P}^{A}{x} :\equiv \mathsf{rx}_{\mathcal{E}(\pi_1A)}{x} \end{align*}

3.4.2 Example: definitional lenses for finite ordinals

For each $n:\mathbb{N}$ , let $\mathcal{F}(n)$ be the following path object with vertices in the standard finite set with n elements:

\[ \mathcal{F}(n) :\equiv \{{i:{\triangle}{\mathbb{N}}}|{i < n}\} \]

The reflexive graph image of $\mathbb{N}$ under $\mathcal{F}$ is not univalent because $\mathbb{N}$ is a set and thus each $m =_{\mathbb{N}} n$ is a proposition, whereas there is a proper set of equivalences from $\mathcal{F}(m)$ to $\mathcal{F}(m)$ for $m \geq 2$ . Therefore, we cannot apply Example 75 nor Example 77 to obtain a good definitional lens structure on $\mathcal{F}$ . Nonetheless, we can impose with some ingenuity a more restricted path object structure on $\mathbb{N}$ over which it is not difficult to exhibit $\mathcal{F}$ as a definitional lens.

Example 78. We may classify the augmented simplices by the following reflexive graph structure $\Delta_{a}$ with vertices in the natural numbers and edges given by monotone equivalences between finite ordinals:

We shall deduce that Example 78 yields a path object, using the fact that the category of augmented simplices is gaunt in the sense that there is at most one monotone isomorphism between any two finite ordinals.

Lemma 79 (A path object classifying augmented simplices). The reflexive graph $\Delta_{a}$ of augmented simplices is univalent.

Proof. Fixing $m:\mathbb{N}$ , we must check that the fan $\{m\}_{\Delta_{a}}^{+}$ is a proposition. Unfolding definitions, we have $ \{m\}_{\Delta_{a}}^{+} \equiv \sum_{(n:\mathbb{N})} \sum_{(f:\mathsf{Equiv}(\lvert \mathcal{F}(m) \rvert,\lvert \mathcal{F}(n) \rvert))} \mathsf{isMonotone}\,f $ . As there can be at most one monotone equivalence between $\lvert \mathcal{F}(m) \rvert$ and some $\lvert \mathcal{F}(n) \rvert$ , it remains only to observe that $\lvert \mathcal{F}(n) \rvert$ being equinumerous with $\lvert \mathcal{F}(m) \rvert$ implies $m=n$ .

Example 80 (Definitional lens structure of finite ordinals). We may exhibit definitional covariant and contravariant lens structures on $\mathcal{F}$ over $\Delta_{a}$ like so:

\begin{align*} m,n:\mathbb{N};\; (\,f,f^\leq) : m \approx_{\Delta_{a}} n, i : \lvert \mathcal{F}(m) \rvert & \vdash \mathsf{push}_{\mathcal{F}}^{(f,f^\leq)}{i} :\equiv f i \\ m,n:\mathbb{N};\; (\,f,f^\leq) : m \approx_{\Delta_{a}} n, i : \lvert \mathcal{F}(n) \rvert & \vdash \mathsf{pull}_{\mathcal{F}}^{(f,f^\leq)}{i} :\equiv f^{-1} i \\ m:\mathbb{N};\; i:\lvert \mathcal{F}(m) \rvert & \vdash \mathsf{pushRx}_{\mathcal{F}}^{m}{i} :\equiv \mathsf{rx}_{\mathcal{F}(m)}{i} \\ m:\mathbb{N};\; i:\lvert \mathcal{F}(m) \rvert & \vdash \mathsf{pullRx}_{\mathcal{F}}^{m}{i} :\equiv \mathsf{rx}_{\mathcal{F}(m)}{i} \end{align*}

3.4.3 Definitional replacement of lenses

We have seen that in some examples of definitional lenses, the reflexive graph structure on the base required some ingenuity. In this section, we describe a general method to reparameterise any lens as a definitional lens over a different base.

Construction 81 (Flattening of lenses). Let $\mathcal{B}$ be an oplax covariant or lax contravariant lens of reflexive graphs over a reflexive graph $\mathcal{A}$ . We may equip $\lvert \mathcal{A} \rvert$ with a new reflexive graph structure $\mathcal{A}_{{\downarrow}\mathcal{B}}^{+}$ or $\mathcal{A}_{{\downarrow}\mathcal{B}}^{-}$ , respectively, called the flattening of $\mathcal{B}$ onto $\mathcal{A}$ .

Lemma 82 (Flattening of lenses of path objects onto path objects). Let $\mathcal{B}$ be an oplax covariant or lax contravariant lens of path objects over a path object $\mathcal{A}$ . Assuming function extensionality, the flattening of $\mathcal{B}$ onto $\mathcal{A}$ is univalent.

Proof. We consider only the covariant case, as the contravariant one is similar. Fixing $x:\lvert \mathcal{A} \rvert$ , we must check that the fan $\{x\}_{\mathcal{A}_{{\downarrow}\mathcal{B}}^{+}}^{+}$ is a proposition. First, we compute the fan up to definitional equality:

As each $\mathcal{B}(y)$ is univalent, we deduce from function extensionality that the cotensor $\lvert \mathcal{B}(x) \rvert\pitchfork\mathcal{B}(y)$ is univalent. Therefore, each fan $\{\mathsf{push}_{\mathcal{B}}^{p}\}_{\lvert \mathcal{B}(x) \rvert\pitchfork\mathcal{B}(y)}^{+}$ is contractible and so we have the following equivalences:

As $\mathcal{A}$ is assumed to be univalent, $\{x\}_{\mathcal{A}}^{+}$ is a proposition and thus so is $\{x\}_{\mathcal{A}_{{\downarrow}\mathcal{B}}^{+}}^{+}$ .

Construction 83 (The definitional replacement of a lens). Let $\mathcal{B}$ be an oplax covariant lens of reflexive graphs over a reflexive graph $\mathcal{A}$ . The covariant definitional replacement of $\mathcal{B}$ is defined to be the following definitional covariant lens $\mathsf{drep}^+_{\mathcal{A}}\mathcal{B}$ over the flattening $\mathcal{A}_{{\downarrow}\mathcal{B}}^{+}$ with $\mathcal{B}$ as its underlying family of reflexive graphs.

\begin{align*} (\mathsf{drep}^+_{\mathcal{A}}\mathcal{B})(x) &:\equiv \mathcal{B}(x) \\ \mathsf{push}_{\mathsf{drep}^+_{\mathcal{A}}\mathcal{B}}^{(p,p')}{u} &:\equiv p'u \end{align*}

On the other hand, if $\mathcal{B}$ is a lax contravariant lens of reflexive graphs over a reflexive graph $\mathcal{A}$ , we define the contravariant definitional replacement of $\mathcal{B}$ to be the following definitional contravariant lens $\mathsf{drep}^-_{\mathcal{A}}\mathcal{B}$ over the flattening $\mathcal{A}_{{\downarrow}\mathcal{B}}^{-}$ :

\begin{align*} (\mathsf{drep}^-_{\mathcal{A}}\mathcal{B})(x) &:\equiv \mathcal{B}(x) \\ \mathsf{pull}_{\mathsf{drep}^-_{\mathcal{A}}\mathcal{B}}^{(p,p')}{u} &:\equiv p'u \end{align*}

3.4.4 Polynomials and partial products of reflexive graphs

Any family of types $x:A\vdash B(x) : \mathbf{type}$ induces a polynomial type operator $\mathbf{P}_{{A}}({C, B}) :\equiv \sum_{(x:A)}\prod_{(u:B(x))}C$ ; the universal property of $\mathbf{P}_{{A}}({C, B})$ is that of a partial product (Dyckhoff and Tholen Reference Dyckhoff and Tholen1987). In this section, we generalise the construction of polynomials to lenses of reflexive graphs; the definitions of the polynomial reflexive graphs will then mirror the description of covariant and contravariant partial products in a 2-category (Johnstone Reference Johnstone1993; Johnstone Reference Johnstone2002). We then show how to specialise the construction of polynomials to obtain suitable path object structures on the partial map classifier of a given dominance as in synthetic domain theory (Rosolini Reference Rosolini1986).

In this section, we assume function extensionality.

Construction 84. Let $\mathcal{B}$ be a definitional covariant lens of reflexive graphs over a reflexive graph $\mathcal{A}$ , and let $\mathcal{C}$ be an arbitrary reflexive graph. Then, we may define a definitional contravariant lens of reflexive graphs $\mathsf{pp}^+_{\mathcal{A}}(\mathcal{C},\mathcal{B})$ over $\mathcal{A}$ by setting each $(\mathsf{pp}^+_{\mathcal{A}}(\mathcal{C},\mathcal{B}))(x) :\equiv \lvert \mathcal{B}(x) \rvert\pitchfork \mathcal{C}$ . The definitional contravariant lens structure of $\mathsf{pp}^+_{\mathcal{A}}(\mathcal{C},\mathcal{B})$ over $\mathcal{A}$ is defined from the definitional covariant lens structure of $\mathcal{B}$ by precomposition like so:

Construction 85. Let $\mathcal{B}$ be a definitional contravariant lens of reflexive graphs over a reflexive graph $\mathcal{A}$ , and let $\mathcal{C}$ be an arbitrary reflexive graph. Then, we may define a definitional covariant family of path objects $\mathsf{pp}^-_{\mathcal{A}}(\mathcal{C},\mathcal{B})$ over $\mathcal{A}$ by setting each $(\mathsf{pp}^-_{\mathcal{A}}(\mathcal{C},\mathcal{B}))(x) :\equiv \lvert \mathcal{B}(x) \rvert\pitchfork \mathcal{C}$ . The definitional covariant lens structure of $\mathsf{pp}^-_{\mathcal{A}}(\mathcal{C},\mathcal{B})$ over $\mathcal{A}$ is defined from the definitional contravariant lens struture of $\mathcal{B}$ by precomposition:

Definition 86 (Partial products of definitional lenses). Let $\mathcal{B}$ be a definitional covariant lens of reflexive graphs over a reflexive graph $\mathcal{A}$ . Then the covariant partial product of a reflexive graph $\mathcal{C}$ with $\mathcal{B}$ over $\mathcal{A}$ is defined to be the following total reflexive graph:

\[ \mathcal{P}^{+}_{{\mathcal{A}}}({\mathcal{C}, \mathcal{B}}) :\equiv \mathcal{A}.\mathsf{disp}^-_{\mathcal{A}}(\mathsf{pp}^+_{\mathcal{A}}(\mathcal{C},\mathcal{B})) \]

When $\mathcal{B}$ is a definitional contravariant lens, we define the contravariant partial product of $\mathcal{C}$ with $\mathcal{B}$ over $\mathcal{A}$ as follows:

\[ \mathcal{P}^{-}_{\mathcal{A}}(\mathcal{C},\mathcal{B}) :\equiv \mathcal{A}.\mathsf{disp}^+_{\mathcal{A}}(\mathsf{pp}^-_{\mathcal{A}}(\mathcal{C},\mathcal{B})) \]

Computation 87 (Covariant partial product). The covariant partial product of a reflexive graph $\mathcal{C}$ with a definitional covariant lens of reflexive graphs $\mathcal{B}$ over a reflexive graphs $\mathcal{A}$ computes as follows:

Computation 88 (Contravariant partial product). The contravariant partial product of a reflexive graph $\mathcal{C}$ with a definitional contravariant lens of reflexive graphs $\mathcal{B}$ over a reflexive graph $\mathcal{A}$ unravels up to definitional equivlance as follows:

Example 89 (Partial map classifiers). A dominance in the sense of Rosolini (Reference Rosolini1986) is a univalent universe $(\Sigma, T)$ closed under the terminal type and internal sums such that for each $\phi:\Sigma$ , the component $T(\phi)$ is a proposition. In this case, we may equip each $T(\phi)$ with the codiscrete path object structure ${\triangledown}(T(\phi))$ so that $(\Sigma,{\triangledown}{T})$ is a univalent family of path objects and ${\triangledown}{T}$ has definitional covariant and contravariant lens structure over the propositional reflexive graph image $\Sigma/_{-1}T$ .

The partial map classifier of a given type A in the dominance $(\Sigma,T)$ is the polynomial type $\mathbf{P}_{{\Sigma}}({A, T}) \equiv \sum_{(\phi:\Sigma)}\prod_{(p:T(\phi))} A$ . A suitable lifting of the partial map classifiers to path objects $\mathcal{A}$ can be obtained by means of covariant or contravariant partial products:

\[ \mathcal{A}_\bot^{+} :\equiv \mathcal{P}^{+}_{{\Sigma/_{-1}T}}({\mathcal{A}, {\triangledown}{T}}) \qquad \mathcal{A}_\bot^{-} :\equiv \mathcal{P}^{-}_{\Sigma/_{-1}T}(\mathcal{A},{\triangledown}{T}) \]

We compute the above reflexive graph structures as follows:

Example 90 (Lists). A simple example of a polynomial type operator is that of lists: letting F(n) be the standard finite type $\sum_{(i:\mathbb{N})}i<n$ , we may define $\mathsf{List}(A) :\equiv \mathbf{P}_{{\mathbb{N}}}({A, F})$ . This representation of lists lifts to path objects by means of partial products via our account in Section 3.4.2 of the path object $\Delta_{a}$ classifying finite ordinals and the definitional lens $\mathcal{F}$ of elements of a given finite ordinal. In particular, for any path object $\mathcal{A}$ , we consider the following covariant and contravariant partial products:

\[ \mathsf{List}^{+}(\mathcal{A}) :\equiv \mathcal{P}^{+}_{{\Delta_{a}}}({\mathcal{A}, \mathcal{F}}) \qquad \mathsf{List}^{-}(\mathcal{A}) :\equiv \mathcal{P}^{-}_{\Delta_{a}}(\mathcal{A},\mathcal{F}) \]

These reflexive graphs compute as follows:

Buchholtz (Reference Buchholtz2023) has shown how to use the classifying space of the symmetric group on two elements to define several variations on unordered pairs in univalent foundations. In what follows, we assume the existence of propositional truncations $\lVert - \rVert$ .

Example 91 (Homotopy unordered pairs (Buchholtz Reference Buchholtz2023)). Let (U,E) be a univalent universe containing the two-element type $\mathbf{2}$ ; following Buchholtz (Reference Buchholtz2023), we may define the “classifying type” $B\Sigma_2$ to the type of U-small types merely equivalent to $\mathbf{2}$ as follows, i.e. we define $B\Sigma_2 :\equiv \sum_{(X:U)}\lVert \mathsf{Equiv}(E(X),\mathbf{2}) \rVert$ . We define a decoding family $K:B\Sigma_2 \vdash E_2(K):\mathbf{type}$ by setting $E_2(X,h) :\equiv E(X)$ . Then, op. cit. defines the type of homotopy unordered pairs in a type A to be the following polynomial type:

A suitable path object structure on $B\Sigma_2$ is obtained by path object comprehension:

\[ \mathcal{B}\Sigma_2 :\equiv \{{X:U/E}|{\lVert \mathsf{Equiv}(E(X),\mathbf{2}) \rVert}\} \]

Then, $E_2$ lifts discretely to a definitional covariant and contravariant lens $\mathcal{E}_2$ of path objects over $\mathcal{B}\Sigma_2$ by Example 77, setting $\mathcal{E}_2(K) :\equiv {\triangle}(E_2(K))$ . A suitable path object structure on homotopy unordered pairs is then obtained by either covariant or contravariant partial products:

\[ \mathsf{hUP}^{+}(\mathcal{A}) :\equiv \mathcal{P}^{+}_{{\mathcal{B}\Sigma_2}}({\mathcal{A}, \mathcal{E}_2}) \qquad \mathsf{hUP}^{-}(\mathcal{A}) :\equiv \mathcal{P}^{-}_{\mathcal{B}\Sigma_2}(\mathcal{A},\mathcal{E}_2) \]

These reflexive graph structures compute as follows:

3.5 Coherence in the theory of path object lenses

Proof. Let $\mathcal{A}$ be a reflexive graph; a witness that $\mathcal{A}$ is univalent is an element of the following product:

As being a proposition is a proposition, this is a product of propositions. Dependent function extensionality implies the closure of propositions under products.

Proof. Univalence of $\mathcal{B}$ amounts to the following product:

By dependent function extensionality, it suffices to check that each $\mathsf{isUnivalent}\,(\mathcal{B}(x))$ is a proposition; but this is Lemma 92.

By the distributivity of products over sums, the above is equivalent to the following:

By dependent function extensionality, it suffices to assume that for each $x:\lvert \mathcal{A} \rvert$ the following type is a proposition:

Re-associating, the above is equivalent to:

By our assumption that $\mathcal{A}$ is univalent, we know that $\{x\}_{\mathcal{A}}^{-}$ is contractible with centre $(x,\mathsf{rx}_{\mathcal{A}}{x})$ . Therefore, the above is equivalent to the following simplified type:

The above is definitionally equal to the co-fan of the identity function in $\lvert \mathcal{B}(x) \rvert \pitchfork \mathcal{B}(x)$ as we can see below:

This co-fan is indeed a proposition, as we have assumed that each $\mathcal{B}(x)$ is univalent.

Considering the distributivity of products over sums, it suffices by dependent function extensionality to prove that for each $x:\lvert \mathcal{A} \rvert$ , the following type is a proposition:

We re-associate to expose various fans in $\mathcal{A}$ :

As $\mathcal{A}$ is univalent, the fan $\{x\}_{\mathcal{A}}^{+}$ contracts onto $(x,\mathsf{rx}_{\mathcal{A}}{x})$ as reflected below:

The above is definitionally equivalent to the following:

We can re-associate the above as follows:

As $\mathcal{B}(x,x,\mathsf{rx}_{\mathcal{A}}{a})$ is assumed to be univalent, so is its cotensor by $\lvert \mathcal{B}(x,x,\mathsf{rx}_{\mathcal{A}}{x}) \rvert$ ; thus, the fan of $\lambda{u}\mathpunct{.}{u}$ in the cotensor contracts to $(\lambda{u}\mathpunct{.}{u}, \lambda{u}\mathpunct{.}{\mathsf{rx}_{\mathcal{B}(x,x,\mathsf{rx}_{\mathcal{A}})}{u}})$ :

The above is just the co-fan $\{\lambda{u}\mathpunct{.}u\}_{\lvert \mathcal{B}(x,x,\mathsf{rx}_{\mathcal{A}}{x}) \rvert\pitchfork {\mathcal{B}}(x,x,\mathsf{rx}_{\mathcal{A}}{x})n}^{-}$ , which is a proposition because $\mathcal{B}(x,x,\mathsf{rx}_{\mathcal{A}}{x})$ is assumed univalent.

4.1 Classifying reflexive graphs

Computation 98 (A path object classifying graphs). Thus, we obtain a path object classifying U-small graphs by taking the total path object of the display of the oplax covariant lens $\mathsf{GphOn}_U$ over $U/E$ , defining $\mathsf{Gph}_U :\equiv (U/E).\mathsf{disp}^-_{U/E}{\mathsf{GphOn}_U}$ . Unraveling definitions, we see that its underlying reflexive graph structure is precisely the natural one that would arise from invertible graph homomorphisms:

Given $f : \mathcal{G}_0\approx_{\mathsf{Gph}_U}\mathcal{G}_1$ , we will write $\lvert\,f \rvert:\mathsf{Equiv}(\lvert \mathcal{G}_0 \rvert,\lvert \mathcal{G}_1 \rvert)$ for the first component and $f^\approx : \prod_{(x,y:\lvert \mathcal{G}_0 \rvert)}\mathsf{Equiv}(x\approx_{\mathcal{G}_0}y,\lvert\,f \rvert x\approx_{\mathcal{G}_1}\lvert\,f \rvert y)$ for the second component.

In order to classify reflexive graphs, we naturally wish to define a further lens over $\mathsf{Gph}_U$ whose components classify reflexivity data on a given graph. Unfortunately, due to the mixed variance of reflexivity data in the underlying graph, it seems we can find neither a suitable lax contravariant nor oplax covariant lens for reflexivity data. It happens, however, that the desirable displayed path object over $\mathsf{Gph}_U$ arises via Lemma 65 from a particularly simple unbiased dependent lens that we now define straightaway.

When instantiated to the reflexivity datum in $\mathsf{Gph}_U$ on a specific graph $\mathcal{A}$ , we recover the usual notion of reflexivity data up to definitional equality:

We now equip $\mathsf{RxOn}_U$ with the structure of a unbiased dependent lens over $\mathsf{Gph}_U$ :

The laws of the dependent lens hold definitionally, so we are done!

Computation 100 (A path object classifying reflexive graphs). A path object classifying U-small reflexive graphs is obtained from the display of the unbiased dependent lens $\mathsf{RxOn}_U$ over $\mathsf{Gph}_U$ by setting $\mathsf{RxGph}_U :\equiv \mathsf{Gph}_U.\mathsf{disp}^\pm_{\mathsf{Gph}_U}$ . It is not difficult to see that the underlying reflexive graph of $\mathsf{RxGph}_U$ is precisely the desired one:

Given $f : \mathcal{A}_0 \approx_{\mathsf{RxGph}_U}\mathcal{A}_1$ , we will write $f : \mathcal{A}_0\approx_{\mathsf{Gph}_U}\mathcal{A}_1$ for the first component and $f^{\mathsf{rx}} : \prod_{(x:\lvert \mathcal{A}_0 \rvert)}f^\approx\,x\,x\,(\mathsf{rx}_{\mathcal{A}_0}{x}) =_{\lvert f \rvert x\approx_{\mathcal{A}_1}\lvert f \rvert x} \mathsf{rx}_{\mathcal{A}_1}(\lvert\,f \rvert x) $ for the second component.

4.2 Classifying displayed reflexive graphs over a fixed base

In this section, let $\mathcal{A}$ be a fixed reflexive graph.

We equip the family of path objects above with the structure of a definitional contravariant lens over $\lvert \mathcal{A} \rvert\pitchfork U/E$ as follows:

Along the diagonal we recover the desired notion of displayed reflexivity data:

The dependent lens structure is given in such a way that the laws hold definitionally:

Computation 104. From the display of $\mathsf{DRxOn}_U^{\mathcal{A}}$ over $\mathsf{DGph}_U^{\mathcal{A}}$ , we obtain a path object classifying displayed reflexive graphs over $\mathcal{A}$ , defining $\mathsf{DRxGphOver}_U(\mathcal{A})$ as follows:

\[ \mathsf{DRxGphOver}_U(\mathcal{A}) :\equiv \mathsf{DGph}_U^{\mathcal{A}}.\mathsf{disp}^\pm_{\mathsf{DGph}_U^{\mathcal{A}}} \mathsf{DGphOn}_U^{\mathcal{A}} \]

Unraveling definitions, we see that the underlying reflexive graph $\mathsf{DRxGphOver}_U(\mathcal{A})$ is precisely the desirable one classifying displayed reflexive graphs over $\mathcal{A}$ :

4.3 Classifying displayed reflexive graphs with variable base

To define the restricted reflexivity datum, we must transport along the witness that f preserves reflexivity data.

The lens is definitional because $\mathcal{B} \equiv \mathsf{pull}_{\mathsf{DRxGphOver}_U^{}}^{\mathsf{rx}_{\mathsf{RxGph}_U}{\mathcal{A}}}\mathcal{B}$ definitionally.

Computation 106 (A path object classifying small displayed reflexive graphs over a variable base). By computing the display of $\mathsf{DRxGphOver}_U$ over $\mathsf{RxGph}_U$ , we obtain a path object classifying reflexive graphs $\mathcal{A}$ equipped with a displayed reflexive graph $\mathcal{B}$ over $\mathcal{A}$ .

\[ \mathsf{DRxGph}_U :\equiv \mathsf{RxGph}_U.\mathsf{disp}^-_{\mathsf{RxGph}_U}{\mathsf{DRxGphOver}_U^{}} \]

We compute as follows.

4.4 Classifying families and lenses of reflexive graphs

Computation 108 (A path object classifying U-small families of U-small reflexive graphs). Using our theory of partial products of lenses (Section 3.4.4), we can define a path object that classifies pairs $(\mathcal{A}, \mathcal{B})$ where $\mathcal{A}$ is a U-small reflexive graph and $\mathcal{B}$ is a family of U-small reflexive graphs indexed in the vertices of $\mathcal{A}$ .

\[ \mathsf{RxGphFam}_U :\equiv \mathcal{P}^{+}_{{\mathsf{RxGph}_U}}({\mathsf{RxGph}_U, \mathsf{Vtx}_U}) \]

We have the following computation:

It would be desirable to find a good path object classifying lens structures on a given family of reflexive graphs. Unfortunately, there are some limitations on our ability to do this: although the pushforward/pullback data are easy to classify, it is unclear how to define a notion of path between oplax/lax unitors over a homotopy between pushforward/pullback data. Therefore, the best we can do is impose the discrete reflexive graph structure on the first component of a lens, combining it with a family of path object structures classifying unitors by coproduct (Definition 19 and Lemma 49); on the other hand, we can still exhibit a useful displayed reflexive graph structure classifying lenses with variable base by means of a unbiased dependent lens as we shall see in Construction 109 below.

Naturally, $\mathsf{LensStr}^+_{\mathcal{A}}(\mathsf{rx}_{\mathsf{RxGph}_U}{\mathcal{B}})$ is precisely the type of oplax covariant lens structures on family of U-small reflexive graphs $\mathcal{B}$ over $\mathcal{A}$ .

The unit laws of this unbiased dependent lens hold definitionally.

Computation 110 (A path object classifying oplax covariant lenses with fixed base). Finally, we may define a path object classifying oplax covariant lenses over a fixed reflexive graph $\mathcal{A}$ .

\[ \mathsf{Lens}^+_{\mathcal{A}} :\equiv (\lvert \mathcal{A} \rvert\pitchfork \mathsf{RxGph}_U). \mathsf{disp}^\pm_{\lvert \mathcal{A} \rvert\pitchfork \mathsf{RxGph}_U} \mathsf{LensStr}^+_{\mathcal{A}} \]

5.1 Duality involution for fibred reflexive graphs

In this section, we show that covariant and contravariant fibrations are interchangeable via the duality involution for displayed reflexive graphs (Section 2.2).

Lemma 113 allows us to translate results about all covariant fibrations into results about all contravariant fibrations.

5.2 Fibred reflexive graphs and univalence

Fibred reflexive graphs are always univalent, as we see in Lemma 114. Therefore, it is natural to refer to a fibration of reflexive graphs as a fibration of path objects.

The latter is contractible by our assumption that $\mathcal{B}$ is a covariant fibration. Suppose, on the other hand, that $\mathcal{B}$ is a contravariant fibration. To show that $\mathcal{B}$ is univalent, it suffices to show that each of the co-fans $\{u\}_{\mathcal{B}(x)}^{-}$ is a proposition, which we compute below:

The latter is contractible because $\mathcal{B}$ is a contravariant assumption.

A converse to Lemma 114 cannot hold, as we see in Example 115.

We can see that for each $i:\mathbf{2}$ , the component $\mathcal{G}(i)$ is the codiscrete reflexive graph ${\triangledown}(i =_{\mathbf{2}} 1)$ , which is univalent by Lemma 52 as $\mathbf{2}$ is a set. Therefore $\mathcal{G}$ is univalent, and we will see that it can be neither a covariant nor a contravariant fibration.

Suppose that $\mathcal{G}$ were a covariant fibration. We have an edge ${*} : 1 \approx_{{\triangledown}{\mathbf{2}}} 0$ and thus a pushforward map $(*)_* \colon 1 =_{\mathbf{2}} 1\to 0 =_{\mathbf{2}} 1$ . On the other hand, if $\mathcal{G}$ were a contravariant fibration, we would use pullback along ${*}:0 \approx_{{\triangledown}{\mathbf{2}}}1$ to obtain a contradiction.

Example 115 also shows that not every univalent displayed reflexive graph can be obtained from a lens structure on its components.

5.3 Fibred reflexive graphs from lenses of path objects

We will now show that when a lens is valued in path objects, its display is a fibration.

Conversely, a lax contravariant lens $\mathcal{B}$ over a reflexive graph $\mathcal{A}$ is said to have universal pullbacks when for every edge $p:x\approx_{\mathcal{A}}y$ and vertex $v:\lvert \mathcal{B}(y) \rvert$ , the co-fan $\{\mathsf{pull}_{\mathcal{B}}^{p}{u}\}_{\mathcal{B}(x)}^{-}$ is a proposition.

Thus, we may choose a centre of contraction for the latter freely so long as it is a proposition (which we have by assumption).

1. (1) Suppose that $\mathcal{B}$ has universal pushforwards. We want to show that each fan $\{u\}_{\mathcal{B}(x)}^{+}$ is a proposition; using our assumption of universal pushforwards, we can show that $\{u\}_{\mathcal{B}(x)}^{+}$ is a retract of $\{\mathsf{push}_{\mathcal{B}}^{\mathsf{rx}_{\mathcal{A}}{x}}{u}\}_{\mathcal{B}(x)}^{+}$ , which is a proposition because $\mathcal{B}$ has universal pushforwards.

2. (2) Suppose conversely that each component $\mathcal{B}(x)$ is univalent; then all fans in the components of $\mathcal{B}$ are propositions, including $\{\mathsf{push}_{\mathcal{B}}^{p}{u}\}_{\mathcal{B}(y)}^{+}$ for some $p:x\approx_{\mathcal{A}}y$ and $u:\lvert \mathcal{B}(y) \rvert$ .

The case for when $\mathcal{B}$ is a lax contravariant lens is established by duality via Lemma 114 and Lemma 118.

5.4 The underlying lens of a covariant fibration of reflexive graphs

The pushforward operation $p_*^{\mathcal{B}}$ of a covariant fibration gives rise to an oplax covariant lens structure on the components of $\mathcal{B}$ , which we construct and study in this section.

1. (1) We first exhibit a “straightening/unstraightening” equivalence between displayed edges $u\approx_{\mathcal{B}}^{p:x\approx_{\mathcal{A}}y}v$ and vertical edges $\mathsf{push}_{\mathcal{B}}^{p}{u}\approx_{\mathcal{B}(y)} v$ in Construction 120, Construction 121, and Lemma 122.

2. (2) We use the straightening equivalence to exhibit the underlying lens of a displayed reflexive graph in Construction 123.

3. (3) Finally, Theorem 128 exhibits an equivalence of displayed reflexive graphs between any displayed reflexive graph $\mathcal{B}$ and the display of its underlying lens.

5.5 Straightening of displayed edges

First, we define a more general function incorporating a witness of contraction.

Then, we define the straightening function by instantiation with one of the witnesses of contraction of $\sum_{(w:\lvert \mathcal{B} \rvert(y))}u\approx_{\mathcal{B}}^{p}w$ , which we leave nameless.

Then, we define the unstraightening function by instantiation with an anonymous witness that can be obtained from the fact that $\sum_{(w:\lvert \mathcal{B} \rvert(y))}p_*^{\mathcal{B}} u \approx_{\mathcal{B}(y)}w$ is a proposition.

Conversely, we check that straightening is a section of unstraightening by means of the following generalised lemma.

5.6 The underlying lens of a covariant fibration

We are now equipped to construct the underlying lens of a given covariant fibration.

Theorem 124 below shows that if we take the display of the underlying lens of a covariant fibration, we obtain the original displayed reflexive graph.

The displayed reflexivity data are preserved up to definitional equality, as we have:

5.7 Characterisation of fibred reflexive graphs

Let $\mathcal{B}$ be an oplax covariant lens of U-small path objects over a reflexive graph $\mathcal{A}$ . By Lemma 117, the display $\mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}}$ is a covariant fibration. By Construction 123, we obtain an oplax covariant lens $(\mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}})_{\mathsf{rx}}$ that we may compare with $\mathcal{B}$ . We first recall from Computation 58 the underlying reflexive graph of each component $(\mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}})_{\mathsf{rx}}(x)$ :

\begin{align*} \lvert (\mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}})_{\mathsf{rx}}(x) \rvert & \equiv \lvert \mathcal{B}(x) \rvert \\ u\approx_{(\mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}})_{\mathsf{rx}}(x)} v & \equiv \mathsf{push}_{\mathcal{B}}^{\mathsf{rx}_{\mathcal{A}}{x}}{u} \approx_{\mathcal{B}(x)} v \\ \mathsf{rx}_{(\mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}})_{\mathsf{rx}}(x)}{u} & \equiv \mathsf{pushRx}_{\mathcal{B}}^{x}{u} \end{align*}

Computation 125. We compute the lens structure of $(\mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}})_{\mathsf{rx}}$ as follows, recalling Construction 123 and Lemma 117.

\begin{align*} \mathsf{push}_{(\mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}})_{\mathsf{rx}}}^{p}{u} & \equiv p_*^{\mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}}}u \equiv \mathsf{push}_{\mathcal{B}}^{p}{u} \\ \mathsf{pushRx}_{(\mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}})_{\mathsf{rx}}}^{x}{u} & \equiv \mathsf{str}_{\mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}}}^{\mathsf{rx}_{\mathcal{A}}x}{(\mathsf{rx}_{\mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}}}^{x}{u})} \equiv \mathsf{str}_{\mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}}}^{\mathsf{rx}_{\mathcal{A}}x}{(\mathsf{pushRx}_{\mathcal{B}}^{x}{u})} \end{align*}

Construction 126. We have an equivalence $\eta_{\mathcal{B}} : \mathcal{B}\approx_{\lvert \mathcal{A} \rvert\pitchfork\mathsf{RxGph}_U}(\mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}})_{\mathsf{rx}}$ of underlying families of reflexive graphs over $\mathcal{A}$ , which we define below making use of Corollary 119, Construction 37, and Lemma 38.

\begin{align*} \eta_{\mathcal{B}} & : \mathcal{B} \approx_{\lvert \mathcal{A} \rvert\pitchfork\mathsf{RxGph}_U} (\mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}})_{\mathsf{rx}} \\ \lvert \eta_{\mathcal{B}} x \rvert & :\equiv \mathsf{rx}_{U/E}{\lvert \mathcal{B}(x) \rvert} \\ (\eta_{\mathcal{B}} x)^\approx & :\equiv \lambda{u,v}\mathpunct{.} (\mathsf{pushRx}_{\mathcal{B}}^{x}{u}{{\,\centerdot\,}}_{\mathcal{B}(x)}-) \\ (\eta_{\mathcal{B}} x)^{\mathsf{rx}} & :\equiv \lambda{u}\mathpunct{.} \mathsf{runit}_{\mathsf{pushRx}_{\mathcal{B}}^{x}{u}} \end{align*}

Computation 127. Write $\mathsf{lext}_{\mathsf{LensStr}^+_{\mathcal{A}}}^{\eta_{\mathcal{B}}}\mathcal{B} : \mathsf{LensStr}^+_{\mathcal{A}}(\eta_{\mathcal{B}})$ for the extension of the lens structure from $\mathcal{B}$ onto the equivalence $\eta_{\mathcal{B}} : \mathcal{B}\approx_{\lvert \mathcal{A} \rvert\pitchfork\mathsf{RxGph}_U}(\mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}})_{\mathsf{rx}}$ of families of reflexive graphs. Likewise, write $\mathsf{rext}_{\mathsf{LensStr}^+_{\mathcal{A}}}^{\eta_{\mathcal{B}}}(\mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}})_{\mathsf{rx}} : \mathsf{LensStr}^+_{\mathcal{B}}(\eta_{\mathcal{B}})$ for the corresponding extension of the lens structure of $(\mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}})_{\mathsf{rx}}$ onto $\eta_{\mathcal{B}}$ . We have the definitional computations of the lens structures:

\begin{align*} \mathsf{push}_{\mathsf{lext}_{\mathsf{LensStr}^+_{\mathcal{A}}}^{\eta_{\mathcal{B}}}\mathcal{B}}^{p}{u} & \equiv \mathsf{push}_{\mathcal{B}}^{p}{u} \\ \mathsf{pushRx}_{\mathsf{lext}_{\mathsf{LensStr}^+_{\mathcal{A}}}^{\eta_{\mathcal{B}}}\mathcal{B}}^{x}{u} & \equiv \mathsf{pushRx}_{\mathcal{B}}^{x}{ (\mathsf{push}_{\mathcal{B}}^{\mathsf{rx}_{\mathcal{A}}{x}}{u}) } {{\,\centerdot\,}}_{\mathcal{B}(x)} \mathsf{pushRx}_{\mathcal{B}}^{x}{u} \\ \mathsf{push}_{\mathsf{rext}_{\mathsf{LensStr}^+_{\mathcal{A}}}^{\eta_{\mathcal{B}}}(\mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}})_{\mathsf{rx}}}^{p}{u} & \equiv \mathsf{push}_{\mathcal{B}}^{p}{u} \\ \mathsf{pushRx}_{\mathsf{rext}_{\mathsf{LensStr}^+_{\mathcal{A}}}^{\eta_{\mathcal{B}}}(\mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}})_{\mathsf{rx}}}^{x}{u} & \equiv \mathsf{str}_{\mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}}}^{\mathsf{rx}_{\mathcal{A}}x}]{(\mathsf{pushRx}_{\mathcal{B}}^{x}{u})} \end{align*}

Proof. We outline the explicit computations below, unfolding Construction 109, Construction 123, and Computation 125.

Proof. We must construct a displayed equivalence $\mathcal{B}\approx_{\mathsf{disp}^\pm_{\lvert \mathcal{A} \rvert\pitchfork \mathsf{RxGph}_U}{\mathsf{LensStr}^+_{\mathcal{A}}}}^{\eta_{\mathcal{B}}} \!(\mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}})_{\mathsf{rx}}$ of lens structures over $\eta_{\mathcal{B}}$ . This consists of an (ordinary) equivalence of lenses as below:

\[ \mathsf{lext}_{\mathsf{LensStr}^+_{\mathcal{A}}}^{\eta_{\mathcal{B}}}\mathcal{B} \approx_{\mathsf{LensStr}^+_{\mathcal{A}}(\eta_{\mathcal{B}})} \mathsf{rext}_{\mathsf{LensStr}^+_{\mathcal{A}}}^{\eta_{\mathcal{B}}}(\mathsf{disp}^+_{\mathcal{A}}{\mathcal{B}})_{\mathsf{rx}} \]

Unfolding the definition of $\mathsf{Lens}^+_{\mathcal{A}}{\eta_{\mathcal{B}}}$ from Construction 109 and using Computation 127, we see that the pushforward datum for each lens depicted above is definitionally equal to that of $\mathcal{B}$ . Therefore, it suffices to construct an equivalence between oplax unitors as below:

Up to definitional equality, the goal above is a substitution instance of the following more general goal which we establish below.

1. (1) The type of oplax covariant (resp. lax contravariant) lenses of U-small path objects over $\mathcal{A}$ .

2. (2) The type of covariant (resp. contravariant) fibrations of U-small reflexive graphs over $\mathcal{A}$ .

The equivalence between lax contravariant lenses and contravariant fibrations is established from the above by duality (Section 2.2 and Section 5.1).