2.1 Origins in semantic derivatives and chain rules

CHAD starts from the observation that for a differentiable function:

$$f: {\mathbb{R}}^n\to {\mathbb{R}}^m$$

it is useful to pair the primal function value f(x) with f’s derivative Df(x) at x if we want to calculate derivatives in a compositional way (where we underline the spaces $\underline{\mathbb{R}}^n$ of tangent vectors to emphasize their algebraic structure and we write a linear function type for the derivative to indicate its linearity in its tangent vector argument):

\begin{align*}\mathcal{T}{f} : & \ {\mathbb{R}}^n \to {\mathbb{R}}^m\times (\underline{\mathbb{R}}^n\multimap \underline{\mathbb{R}}^m)\\&x\mapsto (f(x),Df(x)).\end{align*}

Indeed, the chain rule for derivatives teaches us that we compute the derivative of a composition $g\circ f$ of functions as follows, where we write $\mathcal{T}_{1}{f}\stackrel {\mathrm{def}}= \pi_1\circ \mathcal{T}{f}$ and $\mathcal{T}_{2}{f}\stackrel {\mathrm{def}}= \pi_2\circ \mathcal{T}{f}$ for the first and second components of $\mathcal{T}{f}$ , respectively:

$$\mathcal{T}\!\!{(g\circ f)}(x) = (\mathcal{T}_{1}{g}(\mathcal{T}_{1}{f}(x)),\mathcal{T}_{2}{g}(\mathcal{T}_{1}{f}(x))\circ \mathcal{T}_{2}{f}(x)).$$

We make two observations:

(1) the derivative of $g\circ f$ does depend not only on the derivatives of g and f but also on the primal value of f;

(2) the primal value of f is used twice: once in the primal value of $g\circ f $ and once in its derivative; we want to share these repeated subcomputations.

Insight 1. This shows that it is wise to pair up computations of primal function values and derivatives and to share computation between both if we want to calculate derivatives of functions compositionally and efficiently.

Similar observations can be made for f’s transposed (adjoint) derivative ${Df}^{t}$ , which propagates not tangent vectors but cotangent vectors and which we can pair up as:

\begin{align*}\mathcal{T}^*f : & \ {\mathbb{R}}^n \to {\mathbb{R}}^m\times (\underline{\mathbb{R}}^m\multimap \underline{\mathbb{R}}^n)\\ &x\mapsto (f(x),{Df}^{t}(x)) \end{align*}

to get the following chain rule:

$$\mathcal{T}^*{(g\circ f)}(x) = (\mathcal{T}^*_{1}{g}(\mathcal{T}^*_{1}{f}(x)),\mathcal{T}^*_{2}{f}(x)\circ\mathcal{T}^*_{2}{g}(\mathcal{T}^*_{1}{f}(x))).$$

CHAD directly implements the operations $\mathcal{T}_{}$ and $\mathcal{T}^*$ as source code transformations $\overrightarrow{\mathcal{D}}$ and $\overleftarrow{\mathcal{D}}$ on a functional language to implement forward- and reverse-mode AD, respectively. These code transformations are defined compositionally through structural induction on the syntax, by making use of the chain rules above.

2.2 CHAD on a first-order functional language

We first discuss what the technique looks like on a standard typed first-order functional language. Despite our different presentation in terms of a $\lambda$ -calculus rather than Elliott’s categorical combinators, this is essentially the algorithm of Elliott (Reference Elliott2018). Types $\tau,\sigma,\rho$ are either statically sized arrays of n real numbers ${\mathbf{real}}^n$ or tuples $\tau\boldsymbol{\mathop{*}}\sigma$ of types $\tau,\sigma$ . We consider programs t of type $\sigma$ in typing context $\Gamma=x_1:\tau_1,\ldots,x_n:\tau_n$ , where $x_i$ are identifiers. We write such a typing judgment for programs in context as $\Gamma\vdash t:\sigma$ . As long as our language has certain primitive operations (which we represent schematically)

$$\frac{\Gamma \vdash t_1 : {\mathbf{real}}^{n_1}\quad\cdots\quad \Gamma \vdash t_k : {\mathbf{real}}^{n_k}}{\Gamma \vdash \mathrm{op}(t_1,\ldots,t_k) : {\mathbf{real}}^m}$$

such as constants (as nullary operations), (elementwise) addition and multiplication of arrays, inner products and certain nonlinear functions like sigmoid functions, we can write complex programs by sequencing together such operations. For example, writing $\mathbf{real}$ for ${\mathbf{real}}^1$ , we can write a program $x_1:\mathbf{real},x_2:\mathbf{real},x_3:\mathbf{real},x_4:\mathbf{real}\vdash s:\mathbf{real}$ by:

\begin{align*}&\mathbf{let}\,{y}=\,{x_1 * x_4 + 2 * x_2 }\,\mathbf{in}\,{}\\&\mathbf{let}\,{z}=\,{y* x_3}\,\mathbf{in}\,{}\\&\mathbf{let}\,w=\,{z+ x_4}\,\mathbf{in}\,{\sin{(w)}},\end{align*}

where we indicate shared subcomputations with $\mathbf{let}$ -bindings.

CHAD observes that we can define for each language type $\tau$ associated types of

• • forward-mode primal values $\overrightarrow{\mathcal{D}}(\tau)_{1}$ ;

we define $\overrightarrow{\mathcal{D}}({\mathbf{real}}^n)={\mathbf{real}}^n$ and $\overrightarrow{\mathcal{D}}(\tau\boldsymbol{\mathop{*}}\sigma)_1=\overrightarrow{\mathcal{D}}(\tau)_1\boldsymbol{\mathop{*}}\overrightarrow{\mathcal{D}}(\sigma)_1$ , that is, for now $\overrightarrow{\mathcal{D}}(\tau)_1=\tau$ ;

• • reverse-mode primal values $\overleftarrow{\mathcal{D}}(\tau)_1$ ;

we define $\overleftarrow{\mathcal{D}}({\mathbf{real}}^n)={\mathbf{real}}^n$ and $\overleftarrow{\mathcal{D}}(\tau)\boldsymbol{\mathop{*}}(\sigma)_1=\overleftarrow{\mathcal{D}}(\tau)_1\boldsymbol{\mathop{*}}\overleftarrow{\mathcal{D}}(\sigma)_1$ ; that is, for now $\overleftarrow{\mathcal{D}}(\tau)_1=\tau$ ;

• • forward-mode tangent values $\overrightarrow{\mathcal{D}}(\tau)_2$ ;

we define $\overrightarrow{\mathcal{D}}({\mathbf{real}}^n)_2=\underline{\mathbf{real}}^n$ and $\overrightarrow{\mathcal{D}}(\tau\boldsymbol{\mathop{*}}\sigma)=\overrightarrow{\mathcal{D}}(\tau)_2\boldsymbol{\mathop{*}}\overrightarrow{\mathcal{D}}(\sigma)_2$ ;

• • reverse-mode cotangent values $\overleftarrow{\mathcal{D}}(\tau)_2$ ;

we define $\overleftarrow{\mathcal{D}}({\mathbf{real}}^n)_2=\underline{\mathbf{real}}^n$ and $\overleftarrow{\mathcal{D}}(\tau\boldsymbol{\mathop{*}}\sigma)=\overleftarrow{\mathcal{D}}(\tau)_2\boldsymbol{\mathop{*}}\overleftarrow{\mathcal{D}}(\sigma)_2$ .

Indeed, the justification for these definitions is the crucial observation that a (co)tangent vector to a product of spaces is precisely a pair of tangent (co)vectors to the two spaces. Put differently, the space $\mathcal{T}_{(x,y)}{(X\times Y)}$ of (co)tangent vectors to $X\times Y$ at a point (x,y) equals the product space $(\mathcal{T}_{x}X) \times (\mathcal{T}_{y} Y)$ (Tu Reference Tu2011).

We write the (co)tangent types associated with ${\mathbf{real}}^n$ as $\underline{\mathbf{real}}^n$ to emphasize that it is a linear type and to distinguish it from the cartesian type ${\mathbf{real}}^n$ . In particular, we will see that tangent and cotangent values are elements of linear types that come equipped with a commutative monoid structure $(\underline{0},+)$ . Indeed, (transposed) derivatives are linear functions: homomorphisms of this monoid structure¹. We extend these operations $\overrightarrow{\mathcal{D}}$ and $\overleftarrow{\mathcal{D}}$ to act on typing contexts $\Gamma$ :

\begin{align*}\overrightarrow{\mathcal{D}}(x_1:\tau_1,\ldots,x_n:\tau_n)_1&=x_1:\overrightarrow{\mathcal{D}}(\tau_1)_1,\ldots, x_n:\overrightarrow{\mathcal{D}}(\tau_n)_1 \\\overleftarrow{\mathcal{D}}(x_1:\tau_1,\ldots,x_n:\tau_n)_1&=x_1:\overleftarrow{\mathcal{D}}(\tau_1)_1,\ldots, x_n:\overleftarrow{\mathcal{D}}(\tau_n)_1 \\\overrightarrow{\mathcal{D}}(x_1:\tau_1,\ldots,x_n:\tau_n)_2&=\overrightarrow{\mathcal{D}}(\tau_1)_2\boldsymbol{\mathop{*}}\cdots\boldsymbol{\mathop{*}}\overrightarrow{\mathcal{D}}(\tau_n)_2\\\overleftarrow{\mathcal{D}}(x_1:\tau_1,\ldots,x_n:\tau_n)_2&=\overleftarrow{\mathcal{D}}(\tau_1)_2\boldsymbol{\mathop{*}}\cdots\boldsymbol{\mathop{*}}\overleftarrow{\mathcal{D}}(\tau_n)_2.\end{align*}

To each program $\Gamma\vdash t:\sigma$ , CHAD associates programs calculating the forward-mode and reverse-mode derivatives $\overrightarrow{\mathcal{D}}_{{\overline{\Gamma}}}(t)$ and $\overleftarrow{\mathcal{D}}_{{\overline{\Gamma}}}(t)$ , which are indexed by the list ${\overline{\Gamma}}$ of identifiers that occur in $\Gamma$ :

\begin{align*}&\overrightarrow{\mathcal{D}}(\Gamma)_1\vdash \overrightarrow{\mathcal{D}}_{\overline{\Gamma}}(t) : \overrightarrow{\mathcal{D}}(\sigma)(\boldsymbol){\mathop{*}} \left( \overrightarrow{\mathcal{D}}(\Gamma)_2\multimap \overrightarrow{\mathcal{D}}(\sigma)(\right)\\&\overleftarrow{\mathcal{D}}(\Gamma)_1\vdash \overleftarrow{\mathcal{D}}_{\overline{\Gamma}}(t) : \overleftarrow{\mathcal{D}}(\sigma)\boldsymbol{\mathop{*}} \left( \overleftarrow{\mathcal{D}}(\sigma)\multimap\overleftarrow{\mathcal{D}}(\Gamma)_2 \right).\end{align*}

Observing that each program t computes a differentiable function $\unicode{x27E6} t\unicode{x27E7}$ between Euclidean spaces, as long as all primitive operations op are differentiable, the key property that we prove for these code transformations is that they actually calculate derivatives:

Theorem A (Correctness of CHAD, Theorem 124). For any well-typed program:

$$x_1:{\mathbf{real}}^{n_1},\ldots,x_k:{\mathbf{real}}^{n_k}\vdash {t}:{\mathbf{real}}^m$$

we have that $\unicode{x27E6} \overrightarrow{\mathcal{D}}_{x_1,\ldots,x_k}(t)\unicode{x27E7}=\mathcal{T}_{\unicode{x27E6} t\unicode{x27E7}}\;\text{ and }\;\unicode{x27E6} \overleftarrow{\mathcal{D}}_{x_1,\ldots,x_k}(t)\unicode{x27E7}=\mathcal{T}^*{\unicode{x27E6} t\unicode{x27E7}}.$

Once we fix the semantics for the source and target languages, we can show that this theorem holds if we define $\overrightarrow{\mathcal{D}}$ and $\overleftarrow{\mathcal{D}}$ on programs using the chain rule. The proof works by plain induction on the syntax. For example, we can correctly define reverse-mode CHAD on a first-order language as follows:

\begin{align*} &\overleftarrow{\mathcal{D}}_{\overline{\Gamma}}(\mathrm{op}(t_1,\ldots,t_k)) \stackrel{\mathrm{def}}{=} && \mathbf{let}\,\langle x_1,x_1'\rangle=\,\overleftarrow{\mathcal{D}}_{\overline{\Gamma}}(t_1)\,\mathbf{in}\,\cdots\\ &&& \mathbf{let}\,\langle x_k,x_k'\rangle=\,\overleftarrow{\mathcal{D}}_{\overline{\Gamma}}(t_k)\,\mathbf{in}\,\\ &&&\langle\mathrm{op}(x_1,\ldots,x_k),\underline{\lambda} \mathsf{v}. \mathbf{let}\,\mathsf{v}=\,{D\mathrm{op}}^{t}(x_1,\ldots,x_k;\mathsf{v})\,\mathbf{in}\,\\ &&&\phantom{\langle \mathrm{op}(x_1,\ldots,x_k),\underline{\lambda} \mathsf{v}.\rangle}x_1'\bullet \mathbf{proj}_{1}\,{\mathsf{v}}+\cdots+x_k'\bullet \mathbf{proj}_{k}\,{\mathsf{v}}\rangle\end{align*}

\begin{align*}&\overleftarrow{\mathcal{D}}_{\overline{\Gamma}}(x) \stackrel{\mathrm{def}}{=} && \langle x,\underline{\lambda} \mathsf{v}. \mathbf{coproj}_{\mathbf{idx}(x; {\overline{\Gamma}})\,}\,(\mathsf{v})\rangle\end{align*}

\begin{align*} &\overleftarrow{\mathcal{D}}_{\overline{\Gamma}}(\mathbf{let}\,x=\,t\,\mathbf{in}\,s)\stackrel{\mathrm{def}}{=} &&\mathbf{let}\,\langle x,x'\rangle=\,\overleftarrow{\mathcal{D}}_{\overline{\Gamma}}(t)\,\mathbf{in}\,\\ &&& \mathbf{let}\,\langle y,y'\rangle=\,\overleftarrow{\mathcal{D}}_{\overline{\Gamma},x}(s)\,\mathbf{in}\,\\ &&& \langle y, \underline{\lambda} \mathsf{v}. \mathbf{let}\,\mathsf{v}=\,y'\bullet \mathsf{v}\,\mathbf{in}\, \mathbf{fst}\,\mathsf{v}+x'\bullet (\mathbf{snd}\, \mathsf{v})\rangle\end{align*}

\begin{align*}&\overleftarrow{\mathcal{D}}_{\overline{\Gamma}}(\langle t, s\rangle) \stackrel {\mathrm{def}}= &&\mathbf{let}\,\langle x,x'\rangle=\,\overleftarrow{\mathcal{D}}_{\overline{\Gamma}}(t)\,\mathbf{in}\, \\ &&&\mathbf{let}\,\langle y,y'\rangle=\,\overleftarrow{\mathcal{D}}_{\overline{\Gamma}}(s)\,\mathbf{in}\,\\ &&&\langle \langle x, y \rangle,\underline{\lambda} \mathsf{v}. x'\bullet (\mathbf{fst}\,\mathsf{v}) + {y'\bullet(\mathbf{snd}\, \mathsf{v})}\rangle\end{align*}

\begin{align*}&\overleftarrow{\mathcal{D}}_{\overline{\Gamma}}(\mathbf{fst}\, t) \stackrel {\mathrm{def}}= &&\mathbf{let}\,\langle x,x'\rangle=\,\overleftarrow{\mathcal{D}}_{\overline{\Gamma}}(t)\,\mathbf{in}\,\langle \mathbf{fst}\, x, \underline{\lambda} \mathsf{v}. x'\bullet \langle\mathsf{v},\underline{0}\rangle\rangle\end{align*}

\begin{align*}&\overleftarrow{\mathcal{D}}_{\overline{\Gamma}}(\mathbf{snd}\, t) \stackrel {\mathrm{def}}= &&\mathbf{let}\,\langle x,x'\rangle=\,\overleftarrow{\mathcal{D}}_{\overline{\Gamma}}(t)\,\mathbf{in}\,\langle\mathbf{snd}\, x,\underline{\lambda} \mathsf{v}. x'\bullet\langle\underline{0},\mathsf{v}\rangle\rangle\end{align*}

Here, we write $\underline{\lambda} \mathsf{v}. t$ for a linear function abstraction (merely a notational convention – it can simply be thought of as a plain function abstraction) and $t\bullet s$ for a linear function application (which again can be thought of as a plain function application). Furthermore, given $\Gamma;\mathsf{v}:\underline{\alpha}\vdash t:\boldsymbol{(}\underline{\sigma}_1 \boldsymbol{\mathop{*}} \cdots \boldsymbol{\mathop{*}} \underline{\sigma}_n\boldsymbol{)}$ , we write $\Gamma;\mathsf{v}:\underline{\alpha}\vdash \mathbf{proj}_{i}\,(t):\underline{\sigma}_i$ for the i-th projection of t. Similarly, given $\Gamma;\mathsf{v}:\underline{\alpha}\vdash t:\underline{\sigma}_i$ , we write the i-th coprojection $\Gamma;\mathsf{v}:\underline{\alpha}\vdash\mathbf{coproj}_{i}\,(t)= \langle \underline{0},\ldots,\underline{0},t,\underline{0},\ldots,\underline{0}\rangle:\boldsymbol{(}\underline{\sigma}_1 \boldsymbol{\mathop{*}} \cdots \boldsymbol{\mathop{*}} \underline{\alpha}_n\boldsymbol{)}$ and we write $\mathbf{idx}(x_i; x_1,\ldots,x_n)\,=i$ for the index of an identifier in a list of identifiers. Finally, ${D\mathrm{op}}^{t}$ here is a linear operation that implements the transposed derivative of the primitive operation op.

Note, in particular, that CHAD pairs up primal and (co)tangent values and shares common subcomputations. We see that what CHAD achieves is a compositional efficient reverse-mode AD algorithm that computes the (transposed) derivatives of a composite program in terms of the (transposed) derivatives ${D\mathrm{op}}^{t}$ of the basic building blocks op.

2.3 CHAD on a higher-order language: a categorical perspective saves the day

So far, this account of CHAD has been smooth sailing: we can simply follow the usual mathematics of (transposed) derivatives of functions ${\mathbb{R}}^n\to {\mathbb{R}}^m$ and implement it in code. A challenge arises when trying to extend the algorithm to more expressive languages with features that do not have an obvious counterpart in multivariate calculus, like higher-order functions.

Vákár and Smeding (Reference Vákár and Smeding2022) and Vákár (Reference Vákár2021) solve this problem by observing that we can understand CHAD through the categorical structure of Grothendieck constructions (aka $\Sigma$ -types of categories). In particular, they observe that the syntactic category of the target language for CHAD, a language with both cartesian and linear types, forms a locally indexed category ${\mathbf{LSyn}}:{\mathbf{CSyn}}^{op}\to \mathbf{Cat}$ , that is, functor to the category of categories and functors for which $\mathrm{obj} \left( {\mathbf{LSyn}}\right)(\tau)=\mathrm{obj} \left( {\mathbf{LSyn}}\right)(\sigma)$ for all $\tau,\sigma\in\mathrm{obj} \left( {\mathbf{CSyn}}\right)$ and ${\mathbf{LSyn}}(\tau\xrightarrow{t}\sigma):{\mathbf{LSyn}}(\sigma)\to{\mathbf{LSyn}}(\tau)$ is identity on objects. Here, ${\mathbf{CSyn}} $ is the syntactic category whose objects are cartesian types $\tau,\sigma,\rho$ and morphisms $\tau\to \sigma$ are programs $x:\tau\vdash t:\sigma$ , up to a standard program equivalence. Similarly, ${\mathbf{LSyn}}(\tau)$ is the syntactic category whose objects are linear types $\underline{\alpha},\underline{\sigma},\underline{\gamma}$ and morphisms $\underline{\alpha}\to\underline{\gamma}$ are programs $x:\tau;\mathsf{v}:\underline{\alpha}\vdash t:\underline{\gamma}$ of type $\underline{\gamma}$ that have a free variable x of cartesian type $\tau$ and a free variable $\mathsf{v}$ of linear type $\underline{\alpha}$ . The key observation then is the following.

Theorem B (CHAD from a universal property, Corollary 69). Forward- and reverse-mode CHAD are the unique structure-preserving functors:

\begin{align*} &\overrightarrow{\mathcal{D}}({-}):\mathbf{Syn}\to \Sigma_{{\mathbf{CSyn}}}{\mathbf{LSyn}}\\ &\overleftarrow{\mathcal{D}}({-}):\mathbf{Syn}\to \Sigma_{{\mathbf{CSyn}}}{\mathbf{LSyn}}^{op}\end{align*}

from the syntactic category $\mathbf{Syn}$ of the source language to (opposite) Grothendieck construction of the target language ${\mathbf{LSyn}}:{\mathbf{CSyn}}^{op}\to \mathbf{Cat}$ that send primitive operations op to their derivative $D\mathrm{op}$ and transposed derivative ${D\mathrm{op}}^{t}$ , respectively.

In particular, they prove that this is true for the unambiguous definitions of CHAD for a source language that is the first-order functional language we have considered above, which we can see as the freely generated category $\mathbf{Syn}$ with finite products, generated by the objects ${\mathbf{real}}^n$ and morphisms op. That is, for this limited language, “structure-preserving functor” should be interpreted as “finite product-preserving functor.”

This leads (Vákár Reference Vákár2021; Vákár and Smeding Reference Vákár and Smeding2022) to the idea to try to use Theorem B as a definition of CHAD on more expressive programming languages. In particular, they consider a higher-order functional source language $\mathbf{Syn}$ , that is, the freely generated cartesian closed category on the objects ${\mathbf{real}}^n$ and morphisms op and try to define $\overrightarrow{\mathcal{D}}(-)$ and $\overleftarrow{\mathcal{D}}(-)$ as the (unique) structure-preserving (meaning: cartesian closed) functors to $\Sigma_{{\mathbf{CSyn}}}{\mathbf{LSyn}}$ and $\Sigma_{{\mathbf{CSyn}}}{\mathbf{LSyn}}^{op}$ for a suitable linear target language ${\mathbf{LSyn}}:{\mathbf{CSyn}}^{op}\to \mathbf{Cat}$ . The main contribution then is to identify conditions on a locally indexed category $\mathcal{L}:\mathcal{C}^{op}\to \mathbf{Cat}$ that guarantee that $\Sigma_{\mathcal{C}}\mathcal{L}$ and $\Sigma_{\mathcal{C}}\mathcal{L}^{op}$ are cartesian closed and to take the target language ${\mathbf{LSyn}}:{\mathbf{CSyn}}^{op}\to \mathbf{Cat}$ as a freely generated such category.

Insight 2. To understand how to perform CHAD on a source language with language feature X (e.g., higher-order functions), we need to understand the categorical semantics of language feature X (e.g., categorical exponentials) in categories of the form $\Sigma_\mathcal{C}\mathcal{L}$ and $\Sigma_\mathcal{C}\mathcal{L}^{op}$ . Giving sufficient conditions on $\mathcal{L}$ for such a semantics to exist yields a suitable target language for CHAD, with the definition of the algorithm falling from the universal property of the source language.

Furthermore, we observe in these papers that Theorem A again holds for this extended definition of CHAD on higher-order languages. However, to prove this, plain induction no longer suffices and we instead need to use a logical relations construction over the semantics (in the form of categorical sconing) that relates differentiable curves to their associated primal and (co)tangent curves. This is necessary because the program t may use higher-order constructions such as $\lambda$ -abstractions and function applications in its definition, even if the input and output types are plain first-order types that implement some Euclidean space.

Insight 3. To obtain a correctness proof of CHAD on source languages with language feature X, it suffices to give a concrete denotational semantics for the source and target languages as well as a categorical semantics of language feature X in a category of logical relations (a scone) over these concrete semantics. The main technical challenge is to analyze logical relations techniques for language feature X.

Finally, these papers observe that the resulting target language can be implemented as a shallowly embedded DSL in standard functional languages, using a module system to implement the required linear types as abstract types, with a reference Haskell implementation available at https://github.com/VMatthijs/CHAD. In fact, Vytiniotis et al. (Reference Vytiniotis, Belov, Wei, Plotkin and Abadi2019) had proposed the same CHAD algorithm for higher-order languages, arriving at it from practical considerations rather than abstract categorical observations.

Insight 4. The code generated by CHAD naturally comes equipped with very precise (e.g., linear) types. These types emphasize the connections to its mathematical foundations and provide scaffolding for its correctness proof. However, they are unnecessary for a practical implementation of the algorithm: CHAD can be made to generate standard functional (e.g., Haskell) code; the type safety can even be rescued by implementing the linear types as abstract types.

2.4 CHAD for sum types: a challenge – (co)tangent spaces of varying dimension

A natural approach, therefore, when extending CHAD to yet more expressive source languages is to try to use Theorem B as a definition. In the case of sum types (aka variant types), therefore, we should consider their categorical equivalent, distributive coproducts, and seek conditions on $\mathcal{L}:\mathcal{C}^{op}\to\mathbf{Cat}$ under which $\Sigma_{\mathcal{C}}\mathcal{L}$ and $\Sigma_{\mathcal{C}}\mathcal{L}^{op}$ have distributive coproducts. The difficulty is that these categories tend not to have coproducts if $\mathcal{L}$ is locally indexed. Instead, the desire to have coproducts in $\Sigma_{\mathcal{C}}\mathcal{L}$ and $\Sigma_{\mathcal{C}}\mathcal{L}^{op}$ naturally leads us to consider more general strictly indexed categories $\mathcal{L}:\mathcal{C}^{op}\to\mathbf{Cat}$ .

In fact, this is compatible with what we know from differential geometry (Tu Reference Tu2011): coproducts allow us to construct spaces with multiple connected components, each of which may have a distinct dimension. To make things concrete, the space $\mathcal{T}_{x}{({\mathbb{R}}^2 \sqcup {\mathbb{R}}^3)}$ of tangent vectors to ${\mathbb{R}}^2 \sqcup {\mathbb{R}}^3$ is either $\underline{\mathbb{R}}^2$ or $\underline{\mathbb{R}}^3$ depending on whether the base point x is chosen in the left or right component of the coproduct. More generally, a differentiable function $f:X\to Y$ between spaces of varying dimension (which can be formalized as manifolds with multiple connected components) induces functions on the spaces of tangent and cotangent vectors²:

\begin{align*}\mathcal{T}{f}&:\Pi_{x\in X}\Sigma_{y\in Y}(\mathcal{T}_{x} X\multimap \mathcal{T}_{y}Y)\\\mathcal{T}^*{f}&:\Pi_{x\in X}\Sigma_{y\in Y}(\mathcal{T}^*_{y} Y\multimap \mathcal{T}^*_{x}X),\end{align*}

whose first component is f itself and whose second component is the action on (co)tangent vectors that f induces.

If the types $\overrightarrow{\mathcal{D}}(\tau)_2$ and $\overleftarrow{\mathcal{D}}(\tau)_2$ are to represent spaces of tangent and cotangent vectors to the spaces that $\overrightarrow{\mathcal{D}}(\tau)_{1}$ and $\overleftarrow{\mathcal{D}}(\tau)_1$ represent, we would expect them to be types that vary with the particular base point (primal) we choose. This leads to a refined view of CHAD: while $\vdash \overrightarrow{\mathcal{D}}(\tau)_1:\mathrm{type} $ and $\vdash\overleftarrow{\mathcal{D}}(\tau)_1:\mathrm{type}$ can remain (closed/nondependent) cartesian types, ${p}:\overrightarrow{\mathcal{D}}(\tau)_1\vdash \overrightarrow{\mathcal{D}}(\tau)_2:\mathrm{ltype}$ and ${p}:\overleftarrow{\mathcal{D}}(\tau)_1\vdash \overleftarrow{\mathcal{D}}(\tau)_2:\mathrm{ltype}$ are, in general, linear dependent types.

Insight 5. To accommodate sum types in CHAD, it is natural to consider a target language with dependent types: this allows the dimension of the spaces of (co)tangent vectors to vary with the chosen primal. In categorical terms, we need to consider general strictly indexed categories $\mathcal{L}:\mathcal{C}^{op}\to \mathbf{Cat}$ instead of merely locally indexed ones.

The CHAD transformations of the program now becomes typed in the following more precise way:

\[\begin{array}{l} \overrightarrow{\mathcal{D}}(\Gamma)_1\vdash \overrightarrow{\mathcal{D}}_{\overline{\Gamma}}(t):\Sigma{{p}:\overrightarrow{\mathcal{D}}(\tau)_1}.{\overrightarrow{\mathcal{D}}(\Gamma)_2\multimap \overrightarrow{\mathcal{D}}(\tau)_2}\\ \overleftarrow{\mathcal{D}}(\Gamma)_1\vdash \overleftarrow{\mathcal{D}}_{\overline{\Gamma}}(t):\Sigma{{p}:\overleftarrow{\mathcal{D}}(\tau)_1}.{\overleftarrow{\mathcal{D}}(\tau)_2\multimap \overleftarrow{\mathcal{D}}(\Gamma)_2},\end{array}\]

where the action of $\overrightarrow{\mathcal{D}}({-})_2$ and $\overleftarrow{\mathcal{D}}(-)_2$ on typing contexts $\Gamma=x_1:\tau_1,\ldots,x_n:\tau_n$ has been refined to

\[ \overrightarrow{\mathcal{D}}(\Gamma)_2\stackrel {\mathrm{def}}= \boldsymbol{(}\overrightarrow{\mathcal{D}}\tau_1)_2[{}^{x_1}\!/\!_{{p}}] \boldsymbol{\mathop{*}} \cdots \boldsymbol{\mathop{*}} \overrightarrow{\mathcal{D}}\tau_n)_2[{}^{x_n}\!/\!_{{p}}]\boldsymbol{)}\qquad\quad \overleftarrow{\mathcal{D}}(\Gamma)_2\stackrel {\mathrm{def}}= \boldsymbol{(}\overleftarrow{\mathcal{D}}(\tau_1)_2[{}^{x_1}\!/\!_{{p}}] \boldsymbol{\mathop{*}} \cdots \boldsymbol{\mathop{*}} \overleftarrow{\mathcal{D}}(\tau_n)_2[{}^{x_n}\!/\!_{{p}}]\boldsymbol{)}.\]

All given definitions remain valid, where we simply reinterpret some tuples as having a $\Sigma$ -type rather than the more limited original tuple type.

We prove the following novel results.

• • (fibered) finite products, if $\mathcal{C}$ has finite coproducts and $\mathcal{L}$ has strictly indexed products and coproducts;

• • (fibered) finite coproducts, if $\mathcal{C}$ has finite coproducts and $\mathcal{L}$ is extensive;

• • exponentials, if $\mathcal{L}$ is a biadditive model of the dependently typed enriched effect calculus (we intentially keep this vague here to aid legibility – the point is that these are relatively standard conditions).

Furthermore, the coproducts in $\Sigma_\mathcal{C} \mathcal{L}$ and $\Sigma_\mathcal{C} \mathcal{L}^{op}$ distribute over the products, as long as those in $\mathcal{C}$ do, even in the absence of exponentials. Notably, the exponentials are not generally fibered over $\mathcal{C}$ .

The crucial notion here is our (novel) notion of extensivity of an indexed category, which generalizes well-known notions of extensive categories. In particular, we call $\mathcal{L}:\mathcal{C}^{op}\to \mathbf{Cat}$ extensive if the canonical functor $\mathcal{L}(\sqcup_{i=1}^n C_i)\to \prod_{i=1}^n \mathcal{L}(C_i)$ is an equivalence. Furthermore, we note that we need to reestablish the product and exponential structures of $\Sigma_\mathcal{C} \mathcal{L}$ and $\Sigma_\mathcal{C} \mathcal{L}^{op}$ due to the generalization from locally indexed to arbitrary strictly indexed categories $\mathcal{L}$ .

Using these results, we construct a suitable target language ${\mathbf{LSyn}}:{\mathbf{CSyn}}^{op}\to\mathbf{Cat}$ for CHAD on a source language with sum types (and tuple and function types) and derive the forward and reverse CHAD algorithms for such a language and reestablish Theorems A and B in this more general context. This target language is a standard dependently typed enriched effect calculus with cartesian sum types and extensive families of linear types (i.e., dependent linear types that can be defined through case distinction). Again, the correctness proof of Theorem A uses the universal property of Theorem B and a logical relations (categorical sconing) construction over the denotational semantics of the source and target languages. This logical relations construction is relatively straightforward and relies on well-known sconing methods for bicartesian closed categories. In particular, we obtain the following formulas for a sum type $\left\{\ell_1\tau_1\mid\cdots\mid \ell_n\tau_n\right\}$ with constructors $\ell_1,\ldots,\ell_n$ that take arguments of type $\tau_1,\ldots,\tau_n$ :

\begin{align*} &\overrightarrow{\mathcal{D}}\left\{\ell_1\tau_1\mid\cdots\mid \ell_n\tau_n\right\})_1 \stackrel {\mathrm{def}}= \left\{\ell_1\overrightarrow{\mathcal{D}}\tau_1)_1\mid \cdots \mid\ell_n\overrightarrow{\mathcal{D}}\tau_n)_1\right\}\\ &\overrightarrow{\mathcal{D}}\left\{\ell_1\tau_1\mid\cdots\mid \ell_n\tau_n\right\})_2\stackrel {\mathrm{def}}= {\mathbf{case}\,{p}\,\mathbf{of}\,\{{\ell_1{p}\to \overrightarrow{\mathcal{D}}\tau_1)_2\mid\cdots\mid \ell_n{p}\to\overrightarrow{\mathcal{D}}\tau_n)_2}\}}\\ &\overleftarrow{\mathcal{D}}(\left\{\ell_1\tau_1\mid\cdots\mid \ell_n\tau_n\right\})_1 \stackrel {\mathrm{def}}= \left\{\ell_1\overleftarrow{\mathcal{D}}(\tau_1)_1\mid \cdots \mid\ell_n\overleftarrow{\mathcal{D}}(\tau_n)_1\right\}\\ &\overleftarrow{\mathcal{D}}(\left\{\ell_1\tau_1\mid\cdots\mid \ell_n\tau_n\right\})_2\stackrel {\mathrm{def}}= {\mathbf{case}\,{p}\,\mathbf{of}\,\{{\ell_1{p}\to \overrightarrow{\mathcal{D}}\tau_1)_2\mid\cdots\mid \ell_n{p}\to\overleftarrow{\mathcal{D}}(\tau_n)_2}\}},\end{align*}

mirroring our intuition that the (co)tangent bundle to a coproduct of spaces decomposes (extensively) into the (co)tangent bundles to the component spaces.

2.5 CHAD for (co)inductive types: where do we begin?

If we are to really push forward the dream of differentiable programming, we need to learn how to perform AD on programs that operate on data types. To this effect, we analyze CHAD for inductive and coinductive types. If we want to follow our previous methodology to find suitable definitions and correctness proofs, we first need a good categorical axiomatization of such types. It is well known that inductive types correspond to initial algebras of functors, while coinductive types are precisely terminal coalgebras. The question, however, is what class of functors to consider. That choice makes the vague notion of (co)inductive types precise.

Following Santocanale (Reference Santocanale2002), we work with the class of $\mu\nu$ -polynomials, a relatively standard choice, that is functors that can be defined inductively through the combination of

• • constants for primitive types ${\mathbf{real}}^n$ ;

• • type variables ${\alpha}$ ;

• • unit and tuple types $\mathbf{1}$ and $\tau\boldsymbol{\mathop{*}}\sigma$ of $\mu\nu$ -polynomials;

• • sum types $\left\{\ell_1\tau_1\mid\cdots\mid \ell_n\tau_n\right\}$ of $\mu\nu$ -polynomials;

• • initial algebras $\mu{\alpha}.\tau$ of $\mu\nu$ -polynomials;

• • terminal coalgebras $\nu{\alpha}.\tau$ of $\mu\nu$ -polynomials.

Notably, we exclude function types, as the non-fibered nature of exponentials in $\Sigma_\mathcal{C} \mathcal{L}$ and $\Sigma_\mathcal{C} \mathcal{L}^{op}$ would significantly complicate the technical development. While this excludes certain examples like the free state monad (which for type $\sigma$ state would be the intial algebra $\mu{\alpha}.\left\{Get (\sigma\to {\alpha})\mid Put (\sigma\boldsymbol{\mathop{*}} {\alpha})\right\}$ ), it still includes the vast majority of examples of eager and lazy types that one uses in practice, for example, lists $\mu{\alpha}.\left\{Empty\,\mathbf{1}\mid Cons (\sigma\boldsymbol{\mathop{*}} {\alpha})\right\}$ , (finitely branching) labeled trees like $\mu{\alpha}.\left\{Leaf\,\mathbf{1}\mid Node (\sigma\boldsymbol{\mathop{*}} {\alpha}\boldsymbol{\mathop{*}} {\alpha})\right\}$ , streams $\nu{\alpha}.\sigma\boldsymbol{\mathop{*}} {\alpha}$ , and many more.

We characterize conditions on a strictly indexed category $\mathcal{L}:\mathcal{C}^{op}\to\mathbf{Cat}$ that guarantee that $\Sigma_\mathcal{C}\mathcal{L}$ and $\Sigma_\mathcal{C} \mathcal{L}^{op}$ have this precise notion of inductive and coinductive types. The first step is to give a characterization of initial algebras and terminal coalgebras of split fibration endofunctors on $\Sigma_\mathcal{C}\mathcal{L}$ and $\Sigma_\mathcal{C} \mathcal{L}^{op}$ . For legibility, we state the results here for simple endofunctors and (co)algebras, but they generalize to parameterized endofunctors and (co)algebras.

We use this result to give sufficient conditions for (fibered) $\mu\nu$ -polynomials (including their fibered initial algebras and terminal coalgebras) to exist in $\Sigma_\mathcal{C}\mathcal{L}$ and $\Sigma_\mathcal{C}\mathcal{L}^{op}$ . In particular, we show that it suffices to extend the target language ${\mathbf{LSyn}}:{\mathbf{CSyn}}^{op}\to \mathbf{Cat}$ with both cartesian and linear inductive and coinductive types to perform CHAD on a source language $\mathbf{Syn}$ with inductive and coinductive types. Again, an equivalent of Theorem B holds.

We write $\mathbf{roll}\,{x}$ for the constructor of inductive types (applied to an identifier x), $\mathbf{unroll}\,x$ for the destructor of coinductive types, and ${\tau}.\mathbf{roll}^{-1}\,x\stackrel {\mathrm{def}}= \mathbf{fold}\,x\,\mathbf{with}\,y\to\tau{}[^{y\vdash \mathbf{roll}_{}\,y}\!/\!_{{\alpha}}]$ , where we write $\tau[{}^{y\vdash \mathbf{roll}_{}\,y}\!/\!_{{\alpha}}]$ for the functorial action of the parameterized type $\tau$ with type parameter ${\alpha}$ on the term $\mathbf{roll}_{}\,y$ in context y. This yields the following formula for spaces of primals and (co)tangent vectors to (co)inductive types where:

\begin{align*}&\overrightarrow{\mathcal{D}}({\alpha})_1\stackrel {\mathrm{def}}= {\alpha} \qquad\qquad & \overrightarrow{\mathcal{D}}({\alpha})_2 = \underline{\alpha}\\&\overrightarrow{\mathcal{D}}\mu{\alpha}.(\tau)_1\stackrel {\mathrm{def}}= \mu{\alpha}.\overrightarrow{\mathcal{D}}(\tau)_1\qquad\qquad&\overrightarrow{\mathcal{D}}\mu{\alpha}.(\tau)_2\stackrel {\mathrm{def}}= \underline{\mu}\underline{\alpha}.\overrightarrow{\mathcal{D}}(\tau)_2[{}^{{\overrightarrow{\mathcal{D}}(\tau)_1}.\mathbf{roll}^{-1}{p}}\!/\!_{{p}}]\\&\overrightarrow{\mathcal{D}}\nu{\alpha}.(\tau)_1\stackrel {\mathrm{def}}= \nu{\alpha}.\overrightarrow{\mathcal{D}}(\tau)_1\qquad\qquad&\overrightarrow{\mathcal{D}}\nu{\alpha}.(\tau)_2\stackrel {\mathrm{def}}= \underline{\nu}\underline{\alpha}.\overrightarrow{\mathcal{D}}(\tau)_2[{}^{\mathbf{unroll}\,{p}}\!/\!_{{p}}]\\&\overleftarrow{\mathcal{D}}({\alpha})_1\stackrel {\mathrm{def}}= {\alpha} \qquad\qquad & \overleftarrow{\mathcal{D}}({\alpha})_2 = \underline{\alpha}\\&\overleftarrow{\mathcal{D}}(\mu{\alpha}.(\tau)_1\stackrel {\mathrm{def}}= \mu{\alpha}.\overleftarrow{\mathcal{D}}(\tau)_1\qquad\qquad&\overleftarrow{\mathcal{D}}(\mu{\alpha}.(\tau)_2\stackrel {\mathrm{def}}= \underline{\nu}\underline{\alpha}.\overleftarrow{\mathcal{D}}(\tau)_2[{}^{{\overrightarrow{\mathcal{D}}(\tau)_1}.\mathbf{roll}^{-1}{p}}\!/\!_{{p}}]\\&\overleftarrow{\mathcal{D}}(\nu{\alpha}.(\tau)_1\stackrel {\mathrm{def}}= \nu{\alpha}.\overleftarrow{\mathcal{D}}(\tau)_1\qquad\qquad&\overleftarrow{\mathcal{D}}(\nu{\alpha}.\tau)_2\stackrel {\mathrm{def}}= \underline{\mu}\underline{\alpha}.\overleftarrow{\mathcal{D}}(\tau)_2[{}^{\mathbf{unroll}\,{p}}\!/\!_{{p}}]\end{align*}

Insight 6. Types of primals to (co)inductive types are (co)inductive types of primals, types of tangents to (co)inductive types are linear (co)inductive types of tangents, and types of cotangents to inductive types are linear coinductive types of cotangents and vice versa.

For example, for a type $\tau=\mu{\alpha}.\left\{Empty\,\mathbf{1}\mid Cons (\sigma\boldsymbol{\mathop{*}} {\alpha})\right\}$ of lists of elements of type $\sigma$ , we have a cotangent space:

\[\overleftarrow{\mathcal{D}}(\tau)_2 = \underline{\nu}\underline{\alpha}.{\mathbf{case}\,{\mathbf{roll}^{-1}\,{{p}}}\,\mathbf{of}\,\{{Empty\,\_\to \underline{\mathbf{1}}\mid Cons\, {p}\to \overleftarrow{\mathcal{D}}(\sigma)_2[{}^{\mathbf{fst}\,{p}}\!/\!_{{p}}]\boldsymbol{\mathop{*}}\underline{\alpha}}\}}\qquad\text{where}\]

$\mathbf{roll}^{-1}\,{{p}}=\mathbf{fold}\,{p}\,\mathbf{with}\,y\to{\mathbf{case}\,y\,\mathbf{of}\,\{{Empty\,y\to Empty\,y\mid Cons \, y \to Cons\langle\mathbf{fst}\, y, \mathbf{roll}\,(\mathbf{snd}\,y)\rangle}\}}\hspace{-40pt}\\[8pt]and,~for~a~type~\tau=\nu{\alpha}.\sigma\boldsymbol{\mathop{*}} {\alpha}$ of streams, we have a cotangent space:

$$\overleftarrow{\mathcal{D}}(\tau)_2 = \underline{\mu}\underline{\alpha}.\overleftarrow{\mathcal{D}}(\sigma)_2[{}^{\mathbf{fst}\,(\mathbf{unroll}\,{p})}\!/\!_{{p}}]\boldsymbol{\mathop{*}}\underline{\alpha}.$$

We demonstrate that the strictly indexed category $\mathbf{FVect}:\mathbf{Set}^{op}\to \mathbf{Cat}$ of families of vector spaces also satisfies our conditions, so it gives a concrete denotational semantics of the target language ${\mathbf{LSyn}}:{\mathbf{CSyn}}^{op}\to\mathbf{Cat}$ , by Theorem B. To reestablish the correctness Theorem A, existing logical relations techniques do not suffice, as far as we are aware. Instead, we achieve it by developing a novel theory of categorical logical relations (sconing) for languages with expressive type systems like our AD source language.

Insight 7. We can obtain powerful logical relations techniques for reasoning about expressive type systems by analyzing when the forgetful functor from a category of logical relations to the underlying category is comonadic and monadic.

In almost all instances, the forgetful functor from a category of logical relations to the underlying category is comonadic and in many instances, including ours, it is even monadic. This gives us the following logical relations techniques for expressive type systems:

As a consequence, we can lift our concrete denotational semantics of all types, including inductive and coinductive types to our categories of logical relations over the semantics.

These logical relations techniques are suffient to yield the correctness Theorem 1. Indeed, as long as derivatives of primitive operations are correctly implemented in the sense that $\unicode{x27E6} D\mathrm{op}\unicode{x27E7}=D\mathrm{op}$ and $\unicode{x27E6} {D\mathrm{op}}^{t}\unicode{x27E7}={D\unicode{x27E6} \mathrm{op}\unicode{x27E7}}^{t}$ , Theorem E tells us that the unique structure-preserving functors:

\begin{align*}&(\unicode{x27E6} -\unicode{x27E7},\unicode{x27E6} \overrightarrow{\mathcal{D}}(-)\unicode{x27E7}):\mathbf{Syn}\to \mathbf{Set}\times \Sigma_\mathbf{Set} \mathbf{FVect}\\ & (\unicode{x27E6} -\unicode{x27E7},\unicode{x27E6} \overleftarrow{\mathcal{D}}(-)\unicode{x27E7}):\mathbf{Syn}\to \mathbf{Set}\times \Sigma_\mathbf{Set}\mathbf{FVect}^{op}\end{align*}

lift to the scones of $\mathrm{Hom}(({\mathbb{R}}^k,({\mathbb{R}}^k,\underline{\mathbb{R}}^k)),-) :\mathbf{Set}\times \Sigma_\mathbf{Set} \mathbf{FVect}\to\mathbf{Set}$ and $\mathrm{Hom}(({\mathbb{R}}^k, ({\mathbb{R}}^k,\underline{\mathbb{R}}^k)),-) :\mathbf{Set}\times \Sigma_\mathbf{Set} \mathbf{FVect}^{op}\to \mathbf{Set}$ where we lift the image of ${\mathbf{real}}^n$ , respectively, to the logical relations:

\begin{align*}&\left\{(f,(g,h))\mid f=g\text{ and } h = Df\phantom{{}^t}\right\}\hookrightarrow (\mathbf{Set}\times \Sigma_\mathbf{Set} \mathbf{FVect}\phantom{{}^{op}})\left(({\mathbb{R}}^k,({\mathbb{R}}^k,\underline{\mathbb{R}}^k)), ({\mathbb{R}}^n,({\mathbb{R}}^n,\underline{\mathbb{R}}^n))\right)\\&\left\{(f,(g,h))\mid f=g\text{ and } h = {Df}^{t}\right\}\hookrightarrow (\mathbf{Set}\times \Sigma_\mathbf{Set} \mathbf{FVect}^{op})\left(({\mathbb{R}}^k,({\mathbb{R}}^k,\underline{\mathbb{R}}^k)), ({\mathbb{R}}^n,({\mathbb{R}}^n,\underline{\mathbb{R}}^n))\right).\end{align*}

We see that $\unicode{x27E6} \overrightarrow{\mathcal{D}}(t)\unicode{x27E7}$ and $\unicode{x27E6} \overleftarrow{\mathcal{D}}(t)\unicode{x27E7}$ propagate derivatives and transposed derivatives of differentiable k-surfaces (differentiable functions ${\mathbb{R}}^k\to \mathrm{dom}\unicode{x27E6} t\unicode{x27E7}$ ) correctly for all programs t. Seeing that $(\mathrm{id},(\mathrm{id},x\mapsto \mathrm{id}))$ is one such k-surface in the logical relation associated with ${\mathbf{real}}^k$ , we see that $(\unicode{x27E6} t\unicode{x27E7},(\pi_1\circ\unicode{x27E6} \overrightarrow{\mathcal{D}}(t)\unicode{x27E7},\pi_2\circ\unicode{x27E6} \overrightarrow{\mathcal{D}}(t)\unicode{x27E7}))$ and $(\unicode{x27E6} t\unicode{x27E7},(\pi_1\circ \unicode{x27E6} \overleftarrow{\mathcal{D}}(t)\unicode{x27E7}),\pi_2\circ \unicode{x27E6} \overleftarrow{\mathcal{D}}(t)\unicode{x27E7}))$ are k-surfaces in the relations as well, for any $x:{\mathbf{real}}^k\vdash t:{\mathbf{real}}^n$ . That is, Theorem A holds.

Our novel logical relations machinery is in no way restricted to the context of CHAD, however. In fact, it is widely applicable for reasoning about total functional languages with expressive type systems.

2.6 Inductive types and derivatives

So far, we have only phrased the CHAD correctness Theorem A only for programs t with domain/codomain isomorphic to some Euclidean space ${\mathbb{R}}^n$ , even if t may make use of any complex types (including variant, inductive, coinductive, and function types) in its computation. The reason for this restriction is that this limited context of functions $f:{\mathbb{R}}^n\to {\mathbb{R}}^m$ is an obvious setting where we have a simple, canonical, unambiguous notion of derivative $\mathcal{T}{f}:{\mathbb{R}}^n\to {\mathbb{R}}^m\times (\underline{\mathbb{R}}^n\multimap \underline{\mathbb{R}}^m)$ , allowing us to phrase an obvious correctness criterion.

More generally, for $f:X\to Y$ where X and Y are manifolds, we also have an unambiguous notion of derivative $\mathcal{T}{f}:\Pi_{x\in X} \Sigma_{y\in Y}\mathcal{T}_{x}X\multimap \mathcal{T}_{y}Y$ , which allows us to strengthen our correctness result. In fact, for our purposes, it suffices to consider the relatively simple context of differentiable functions $f:\coprod\limits_{i\in I}{\mathbb{R}}^{n_i}\to \coprod\limits_{j\in J}{\mathbb{R}}^{m_j}$ between very simple manifolds that arise as disjoint unions of (finite-dimensional) Euclidean spaces. Such functions f decompose uniquely as copairings $f=[\iota_{\phi(i)}\circ g_i]_{i\in I}$ where we write $\iota_k$ for the k-th coprojection and where $\phi:I\to J$ is some function and $g_i:{\mathbb{R}}^{n_i}\to {\mathbb{R}}^{m_{\phi(j)}}$ . That is, f can be understood as the family $(g_i)_{i\in I}$ and its derivative $\mathcal{T}{f}$ decomposes uniquely as the family of plain derivatives $\mathcal{T}_{g_i}$ in the usual sense. We have a similar decomposition for the transposed derivatives $\mathcal{T}^*f$ .

This notion of derivatives of functions between disjoint unions of Euclidean spaces is relevant to our context, as we have the following result.

Consequently, we can strengthen Theorem A in the following form:

Theorem G (Correctness of CHAD (Generalized), Theorem 129). For any well-typed program

$$x_1:\tau_1,\ldots,x_k:\tau_n\vdash {t}:\sigma,$$

where $\tau_i,\sigma$ are all (closed) $\mu$ -polynomials, we have that $\unicode{x27E6} \overrightarrow{\mathcal{D}}_{x_1,\ldots,x_k}(t)\unicode{x27E7}=\mathcal{T}_{\unicode{x27E6} t\unicode{x27E7}}\;\text{ and }\;\unicode{x27E6} \overleftarrow{\mathcal{D}}_{x_1,\ldots,x_k}(t)\unicode{x27E7}=\mathcal{T}^*{\unicode{x27E6} t\unicode{x27E7}}.$

Again, t can make use of coinductive types and function types in the middle of its computation, but they may not occur in the input or output types. The reason is that, as far as we are aware, there is no canonical³ notion of semantic derivative for functions between the sort of infinite-dimensional spaces that co-datatypes such as coinductive types and function types implement. This makes it challenging to even phrase what semantic correctness at such types would mean.

2.7 How does CHAD for expressive types work in practice?

The CHAD code transformations we describe in this papers are well behaved in practical implementations in the sense of the following compile-time complexity result.

We have ensured to pair up the primal and (co)tangent computations in our CHAD transformation and to exploit any possible sharing of common subcomputations, using $\mathbf{let}$ -bindings. However, we leave a formal study of the runtime complexity of our technique to future work.

As formulated in this paper, CHAD generates code with linear dependent types. This seems very hard to implement in practice. However, this is an illusion: we can use the code generated by CHAD and interpret it as less precise types. We sketch how all type dependency can be erased and how all linear types other than the linear (co)inductive types can be implemented as abstract types in a standard functional language like Haskell. In fact, we describe three practical implementation strategies for our treatment of sum types, none of which require linear or dependent types. All three strategies have been shown to work in the CHAD reference implementation. We suggest how linear (co)inductive types might be implemented in practice, based on their concrete denotational semantics, but leave the actual implementation to future work.

3.1 Basics

We use standard definitions from category theory; see, for instance, Mac Lane (Reference Mac Lane1971), Leinster (Reference Leinster2014). A category $\mathcal{C}$ can be seen as a semantics for a typed functional programming language, whose types correspond to objects of $\mathcal{C}$ and whose programs that take an input of type A and produce an output of type B are represented by the homset $\mathcal{C}(A,B)$ . Identity morphisms $\mathrm{id}_{A}$ represent programs that simply return their input (of type A) unchanged as output and composition $g\circ f$ of morphisms f and g represents running the program g after the program f. Notably, the equations that hold between morphisms represent program equivalences that hold for the particular notion of semantics that $\mathcal{C}$ represents. Some of these program equivalences are so fundamental that we demand them as structural equalities that need to hold in any categorical model (such as the associativity law $f\circ (g \circ h)=(f\circ g)\circ h$ ). In programming languages terms, these are known as the $\beta$ - and $\eta$ -equivalences of programs.

3.2 Tuple types

Tuple types represent a mechanism for letting programs take more than one input or produce more than one output. Categorically, a tuple type corresponds to a product $\prod_{i\in I}A_i$ of a finite family of types $\left\{A_i\right\}_{i\in I}$ , which we also write $\mathbb{1}$ or $A_1\times A_2$ in the case of nullary and binary products. For basic aspects of products, we refer the reader to Mac Lane (Reference Mac Lane1971, Chapter III).

We write $\left({f_i}\right)_{i\in I}:C\to \prod_{i\in I}A_i$ for the product pairing of $\left\{f_i:C:A_i\right\}_{i\in I}$ and $\pi_{j}:\prod_{i\in I}A_i\to A_j$ for the j-th product projection, for $j\in I$ . As such, we say that a categorical semantics $\mathcal{C}$ models (finite) tuples if $\mathcal{C}$ has (chosen) finite products.

3.3 Primitive types and operations

We are interested in programming languages that have support for a certain set $\textrm{Ty}$ of ground types such as integers and (floating point) real numbers as well as certain sets $\mathsf{Op}(T_1,\ldots, T_n; S)$ , for $T_1,\ldots,T_n,S\in\textrm{Ty}$ , of operations on these basic types such as addition, multiplication, and sine functions. We model such primitive types and operations categorically by demanding that our category has a distinguished object $C_T$ for each $T\in \textrm{Ty}$ to represent the primitive types and a distinguished morphism $f_{\mathrm{op}}\in \mathcal{C}(C_{T_1}\times \ldots\times C_{T_n}, C_S)$ for all primitive operations $\mathrm{op}\in \mathsf{Op}(T_1,\ldots, T_n; S)$ . For basic aspects of categorical type theory, see, for instance, Crole (Reference Crole1993, Chapters 3&4).

3.4 Function types

Function types let us type popular higher-order programming idioms such as maps and folds, which capture common control flow abstractions. Categorically, a type of functions from A to B is modeled as an exponential $A\Rightarrow B$ . We write $\mathrm{ev}:(A\Rightarrow B)\times A\to B$ (evaluation) for the counit of the adjunction $(-)\times A\dashv A\Rightarrow(-)$ and $\Lambda$ for the Currying natural isomorphism $\mathcal{C}(A\times B, C)\to \mathcal{C}(A,B\Rightarrow C)$ . We say that a categorical semantics $\mathcal{C}$ with tuple types models function types if $\mathcal{C}$ has a chosen right adjoint $(-)\times A\dashv A\Rightarrow(-)$ .

3.5 Sum types (aka variant types)

Sum types (aka variant types) let us model data that exists in multiple different variants and branch in our code on these different possibilities. Categorically, a sum type is modeled as a coproduct $\coprod_{i\in I}A_i$ of a collection of a finite family $\left\{A_i\right\}_{i\in I}$ of types, which we also write ${\mathbb{0}}$ or $A_1\sqcup A_2$ in the case of nullary and binary coproducts. We write $\left[{f_i}\right]_{i\in I}:\coprod_{i\in I}C_i\to A$ for the copairing of $\left\{f_i:C_i\to A\right\}_{i\in I}$ and $\iota_{j}:A_j\to \coprod_{i\in I}A_i$ for the j-th coprojection. In fact, in presence of tuple types, a more useful programming interface is obtained if one restricts to distributive coproducts, that is, coproducts $\coprod_{i\in I}A_i$ such that the map $\left[{\left({\iota_{i}\circ\pi_{1}}\right){\pi_{2}}}\right]_{i\in I}:\coprod_{i\in I}(A_i\times B)\to (\coprod_{i\in I}A_i)\times B$ is an isomorphism; see, for instance, Carboni et al. (Reference Carboni, Lack and Walters1993), Lack (Reference Lack2012). Note that in presence of function types, coproducts are automatically distributive since the left adjoint functors $(-)\times A$ preserve colimits; see, for instance, Leinster (Reference Leinster2014, 6.3). As such, we say that a categorical semantics $\mathcal{C}$ models (finite) sum types if $\mathcal{C}$ has (chosen) finite distributive coproducts.

3.6 Inductive and coinductive types

We employ the usual semantic interpretation of inductive and coinductive types as, respectively, initial algebras and terminal coalgebras of a certain class of functors. We refer the reader, for instance, to Barr and Wells (Reference Barr and Wells2005, Chapter 9), Santocanale (Reference Santocanale2002), and Adamek et al. (Reference Adamek, Milius and Moss2010).

Most of this section is dedicated to describing precisely which class of functors we consider initial algebras and terminal coalgebras, a class we call $\mu\nu$ -polynomials. Roughly speaking, we define $\mu\nu$ -polynomials to be functors that can be constructed from products, coproducts, projections, diagonals, constants, initial algebras, and terminal coalgebras.

To fix terminology and for future reference of the detailed constructions, we recall below basic aspects of parameterized initial algebras and parameterized terminal coalgebras.

Definition 1. (The category of E-algebras). Let $E : \mathcal{D}\to \mathcal{D}$ be an endofunctor. The category of E-algebras, denoted by $E\textrm{-}\mathrm{Alg}$ , is defined as follows. The objects are pairs $(W, \zeta ) $ in which $W\in \mathcal{D} $ and $ \zeta : E(W)\to W $ is a morphism of $\mathcal{D} $ . A morphism between E-algebras $(W, \zeta ) $ and $(Y, \xi) $ is a morphism $g: W\to Y $ of $\mathcal{D} $ such that

(1)

commutes. Dually, we define the category $E\textrm{-}\mathrm{CoAlg}$ of E-coalgebras by:

(2) \begin{equation} E\textrm{-}\mathrm{CoAlg} := \left(E^{\mathrm{op}}\textrm{-}\mathrm{Alg}\right) ^{\mathrm{op}}\end{equation}

in which $E^{\mathrm{op}} : \mathcal{D} ^{\mathrm{op}}\to \mathcal{D} ^{\mathrm{op}} $ is the image of E by $\mathrm{op} :\mathbf{Cat}\to \mathbf{Cat} $ .

Assuming the existence of the initial E-algebra and the terminal E-coalgebra, we denote by:

(3) \begin{equation} \mathrm{fold}_E (Y, \xi): \mu E \to Y, \quad \mathrm{unfold}_E (X, \varrho ): X\to \nu E\end{equation}

the unique morphisms in $\mathcal{D} $ such that

(4)

commute. Whenever it is clear from the context, we denote $ \mathrm{fold}_E (Y, \xi) $ by $\mathrm{fold}_E \xi $ , and $\mathrm{unfold}_E (X, \varrho )$ by $\mathrm{unfold}_E \varrho $ .

Given a functor $ H : \mathcal{D} '\times\mathcal{D} \to \mathcal{D} $ and an object X of $\mathcal{D} ' $ , we denote by $H^X $ the endofunctor:

(5) \begin{equation} H(X, -): \mathcal{D} \to \mathcal{D} .\end{equation}

In this setting, if $\mu H^X$ exists for any object $X\in\mathcal{D}'$ then the universal properties of the initial algebras induce a functor denoted by $\mu H : \mathcal{D}' \to \mathcal{D}$ , called the parameterized initial algebra. In the following, we spell out how to construct parameterized initial algebras and terminal coalgebras.

Proposition 4 ( $\mu$ -operator and $\nu$ -operator). Let $ H:\mathcal{D} '\times \mathcal{D} \to\mathcal{D} $ be a functor. Assume that, for each object $X\in\mathcal{D} '$ , the functor $H^X = H(X,-) $ is such that $\mu H ^X $ exists. In this setting, we have the induced functor:

\begin{eqnarray*} \mu H : \mathcal{D} ' & \to & \mathcal{D}\\ X & \mapsto & \mu H^X\\ \left( f: X\to Y \right) & \mapsto & \mathrm{fold} _{H^X} \left( \mathbf{\mathfrak{in}}_{H^Y}\circ H(f, \mu H ^Y)\right). \end{eqnarray*}

Dually, assuming that, for each object $X\in\mathcal{D} '$ , $\nu H ^X $ exists, we have the induced functor:

\begin{eqnarray*} \nu H : \mathcal{D} ' & \to & \mathcal{D}\\ X & \mapsto & \nu H^X\\ \left( f: X\to Y \right) & \mapsto & \mathrm{unfold} _{H^Y} \left( H(f, \nu H ^X)\circ \mathbf{\mathfrak{out}}_{H^X}\right). \end{eqnarray*}

Given $X\in\mathcal{D} '$ ,

\begin{align*} & \mu H( \mathrm{id} _ X ) \\ & = \mathrm{fold} _{H^X} \left( \mathbf{\mathfrak{in}}_{H^X}\circ H( \mathrm{id} _ X , \mu H ^X)\right) \\ & = \mathrm{fold} _{H^X} \left( \mathbf{\mathfrak{in}}_{H^X} \right) \\ & = \mathrm{id} _{\mu H ^X} . \end{align*}

Moreover, given morphisms $f: X\to Y $ and $g: Y\to Z $ in $\mathcal{D} '$ , we have that

and, hence, the diagram:

commutes. By the universal property of the initial algebra $\left( \mu H^X, \mathbf{\mathfrak{in}} _{H ^X}\right) $ , we conclude that

It is worth noting that in Proposition 4, $\mathcal{D}'$ can be any category. However, in the standard setting of initial algebra semantics, there is a special interest in the case where $\mathcal{D}' = \mathcal{D}^{n-1}$ and $n>1$ , which is described below.

\begin{eqnarray*} \mu H : \mathcal{D} ^{n-1} & \to & \mathcal{D}\\ X & \mapsto & \mu H^X\\ \left( f: X\to Y \right) & \mapsto & \mathrm{fold} _{H^X} \left( \mathbf{\mathfrak{in}}_{H^Y}\circ H(f, \mu H ^Y)\right). \end{eqnarray*}

Dually, if $\nu H ^X $ exists for any $X\in\mathcal{D} ^{n-1}$ , we have the induced functor:

\begin{eqnarray*} \nu H : \mathcal{D}^{n-1} & \to & \mathcal{D}\\ X & \mapsto & \nu H^X\\ \left( f: X\to Y \right) & \mapsto & \mathrm{unfold} _{H^Y} \left( H(f, \nu H ^X)\circ \mathbf{\mathfrak{out}}_{H^X}\right). \end{eqnarray*}

In order to model inductive and coinductive types coming from parameterized types not involving function types, we introduce the following notions.

(O). The objects are defined inductively by:

• (O1) the terminal category $\mathbb{1} $ is an object of $\mu\nu\mathsf{Poly} _ \mathcal{D} $ ;

• (O2) the category $\mathcal{D} $ is an object of $\mu\nu\mathsf{Poly} _ \mathcal{D} $ ;

• (O3) for any pair of objects $\left( \mathcal{D} ', \mathcal{D} '' \right) \in \mu\nu\mathsf{Poly} _ \mathcal{D}\times \mu\nu\mathsf{Poly} _ \mathcal{D} $ , the product $\mathcal{D} '\times \mathcal{D} '' $ is an object of $\mu\nu\mathsf{Poly} _ \mathcal{D} $ .

(M) The morphisms satisfy the following properties:

• (M1) for any object $\mathcal{D} '$ of $\mu\nu\mathsf{Poly} _ \mathcal{D} $ , the unique functor $\mathcal{D} '\to \mathbb{1} $ is a morphism of $\mu\nu\mathsf{Poly} _ \mathcal{D} $ ;

• (M2) for any object $\mathcal{D} '$ of $\mu\nu\mathsf{Poly} _ \mathcal{D} $ , all the functors $\mathbb{1} \to \mathcal{D} ' $ are morphisms of $\mu\nu\mathsf{Poly} _ \mathcal{D} $ ;

• (M3) the binary product $\times : \mathcal{D} \times\mathcal{D} \to \mathcal{D} $ is a morphism of $\mu\nu\mathsf{Poly} _ \mathcal{D} $ ;

• (M4) the binary coproduct $\sqcup : \mathcal{D}\times \mathcal{D} \to \mathcal{D} $ is a morphism of $\mu\nu\mathsf{Poly} _ \mathcal{D} $ ;

• (M5) for any pair of objects $\left( \mathcal{D} ', \mathcal{D} '' \right) \in \mu\nu\mathsf{Poly} _ \mathcal{D}\times \mu\nu\mathsf{Poly} _ \mathcal{D} $ , the projections:

  \begin{equation*} \pi _1 : \mathcal{D} '\times \mathcal{D} '' \to \mathcal{D} ',\qquad \pi _2 : \mathcal{D} '\times \mathcal{D} '' \to \mathcal{D} '' \end{equation*}
  are morphisms of $\mu\nu\mathsf{Poly} _ \mathcal{D} $ ;

• (M6) given objects $ \mathcal{D} ', \mathcal{D} '' , \mathcal{D} '''$ of $\mu\nu\mathsf{Poly} _\mathcal{D} $ , if $E: \mathcal{D} ' \to \mathcal{D} '' $ and $J : \mathcal{D} ' \to \mathcal{D} ''' $ are morphisms of $\mu\nu\mathsf{Poly} _ \mathcal{D} $ , then so is the induced functor $(E,J) :\mathcal{D} ' \to \mathcal{D} '' \times \mathcal{D} ''' $ ;

• (M7) if $\mathcal{D} '$ is an object of $\mu\nu\mathsf{Poly} _ \mathcal{D} $ , $H: \mathcal{D} '\times \mathcal{D} \to\mathcal{D} $ is a morphism of $\mu\nu\mathsf{Poly} _ \mathcal{D} $ and $\mu H : \mathcal{D} ' \to \mathcal{D} $ exists, then $\mu H $ is a morphism of $\mu\nu\mathsf{Poly} _ \mathcal{D} $ ;

• (M8) if $\mathcal{D} '$ is an object of $\mu\nu\mathsf{Poly} _ \mathcal{D} $ , $H: \mathcal{D} '\times \mathcal{D} \to\mathcal{D} $ is a morphism of $\mu\nu\mathsf{Poly} _ \mathcal{D} $ and $\nu H : \mathcal{D} ' \to \mathcal{D} $ exists, then $\nu H $ is a morphism of $\mu\nu\mathsf{Poly} _ \mathcal{D} $ .

We say that $\mathcal{D} $ has $\mu\nu$ -polynomials if $\mathcal{D} $ has finite coproducts and products and, for any endofunctor $ E: \mathcal{D} \to\mathcal{D} $ in $ \mu\nu\mathsf{Poly} _ \mathcal{D} $ , $\mu E $ and $\nu E $ exist. We say that $\mathcal{D} $ has chosen $\mu\nu$ -polynomials if we have additionally made a choice of initial algebras and terminal coalgebras for all $\mu\nu$ -polynomials.

Another suitably equivalent way of defining $\mu\nu\mathsf{Poly} _ \mathcal{D}$ is the following. The category $\mu\nu\mathsf{Poly} _ \mathcal{D}$ is the smallest subcategory of $\mathbf{Cat} $ such that:

• - the inclusion $\mu\nu\mathsf{Poly} _ \mathcal{D}\to\mathbf{Cat} $ creates finite products;

• - $\mathcal{D}$ is an object of the subcategory $\mu\nu\mathsf{Poly} _ \mathcal{D}$ ;

• - for any object $\mathcal{D} '$ of $\mu\nu\mathsf{Poly} _ \mathcal{D}$ , all the functors $\mathbb{1} \to \mathcal{D} ' $ are morphisms of $\mu\nu\mathsf{Poly} _ \mathcal{D}$ ;

• - and the binary product $\times : \mathcal{D} \times\mathcal{D} \to \mathcal{D} $ is a morphism of $\mu\nu\mathsf{Poly} _ \mathcal{D} $ ;

• - the binary coproduct $\sqcup : \mathcal{D}\times \mathcal{D} \to \mathcal{D} $ is a morphism of $\mu\nu\mathsf{Poly} _ \mathcal{D} $ ;

• - if $\mathcal{D} '$ is an object of $\mu\nu\mathsf{Poly} _ \mathcal{D} $ , $H: \mathcal{D} '\times \mathcal{D} \to\mathcal{D} $ is a morphism of $\mu\nu\mathsf{Poly} _ \mathcal{D} $ and $\mu H : \mathcal{D} ' \to \mathcal{D} $ exists, then $\mu H $ is a morphism of $\mu\nu\mathsf{Poly} _ \mathcal{D} $ ;

• - if $\mathcal{D} '$ is an object of $\mu\nu\mathsf{Poly} _ \mathcal{D} $ , $H: \mathcal{D} '\times \mathcal{D} \to\mathcal{D} $ is a morphism of $\mu\nu\mathsf{Poly} _ \mathcal{D} $ and $\nu H : \mathcal{D} '\to \mathcal{D} $ exists, then $\nu H $ is a morphism of $\mu\nu\mathsf{Poly} _ \mathcal{D} $ .

Lemma 8. Let $\mathcal{C}$ be a category with $\mu\nu$ -polynomials. If $\mathcal{D} $ is an object of $\mu\nu\mathsf{Poly} _ \mathcal{C} $ and

$$H: \mathcal{D} \times \mathcal{C} \to\mathcal{C} $$

is a functor in $\mu\nu\mathsf{Poly} _ \mathcal{C} $ , then $\mu H : \mathcal{D}\to\mathcal{C} $ and $\nu H : \mathcal{D}\to\mathcal{C} $ exist (and, hence, they are morphisms of $\mu\nu\mathsf{Poly} _ \mathcal{C} $ ).

Since all the morphisms above are in $ \mu\nu\mathsf{Poly} _ \mathcal{C} $ , we conclude that $H^X $ is an endomorphism of $ \mu\nu\mathsf{Poly} _ \mathcal{C} $ . Therefore, since $\mathcal{C} $ has $\mu\nu$ -polynomials, $\mu H^X $ and $\nu H ^X $ exist.

By Proposition 4, since $\mu H^X $ and $\nu H ^X $ exist for any X in $\mathcal{D} $ , $\mu H $ and $\nu H $ exist.

We say that a categorical semantics $\mathcal{C}$ with (finite) sum and tuple types supports inductive and coinductive types if $\mathcal{C}$ has chosen $\mu\nu$ -polynomials. Note that we do not consider the more general notion of (co)inductive types defined by endofunctors that may contain function types in their construction.

4.1 Preservation, reflection, and creation of initial algebras

We begin by recalling a fundamental result on lifting functors from the base categories to the categories of algebras in Lemma 9. This is actually related to the universal property of the categories of algebras.

\begin{eqnarray*} \check{F}_{\gamma } : & E'\textrm{-}\mathrm{Alg} & \to E\textrm{-}\mathrm{Alg} \\ & \left( X, \zeta \right) & \mapsto \left( F(X), F ( \zeta ) \circ \gamma _X \right)\\ & g & \mapsto F(g). \end{eqnarray*}

which proves that F(g) in fact gives a morphism between the algebras $ \left( F(W), F(\zeta ) \circ \gamma _W \right) $ and $\left( F (Z), F (\xi ) \circ \gamma _Z \right) $ . The functoriality of $\check{F} _\gamma $ follows, then, from that of F.

Dually, we have:

\begin{eqnarray*} \tilde{G}^{\beta } : & E\textrm{-}\mathrm{CoAlg} & \to E'\textrm{-}\mathrm{CoAlg} \\ & (W, \xi ) & \mapsto \left( G(W), \beta _W\circ G ( \xi ) \right)\\ & f & \mapsto G(f). \end{eqnarray*}

Below, whenever we talk about strict preservation, we are assuming that we have chosen initial objects (terminal objects) in the respective categories of (co)algebras.

We can, now, establish the definition of preservation, reflection, and creation of initial algebras using the respective notions for the induced functor. More precisely:

\begin{eqnarray*} \check{F}_ \gamma : & E'\textrm{-}\mathrm{Alg} & \to E\textrm{-}\mathrm{Alg} \\ & \left( X , \zeta \right) & \mapsto \left( F (X), F ( \zeta )\circ \gamma _X \right)\\ & g & \mapsto F (g). \end{eqnarray*}

induced by $\gamma$ strictly) preserves the initial object/reflects the initial object/creates the initial object.

Finally, we say that a functor $F : \mathcal{D}\to \mathcal{C} $ (strictly) preserves initial algebras/reflects initial algebras/creates initial algebras if F (strictly) preserves initial algebras/reflects initial algebras/creates initial algebras of any endofunctor on $\mathcal{D} $ .

Definition 13. (Preservation, reflection, and creation of terminal coalgebras). We say that a functor $G : \mathcal{C}\to \mathcal{D} $ (strictly) preserves the initial algebra/reflects the initial algebra/creates the initial algebra of an endofunctor $E:\mathcal{C}\to\mathcal{C} $ if, for any natural isomorphism $\beta : G\circ E \cong E'\circ G $ (or, in the strict case, $GE = E'G$ ), the functor:

\begin{eqnarray*} \tilde{G}^\beta : & E\textrm{-}\mathrm{CoAlg} & \to E'\textrm{-}\mathrm{CoAlg} \\ & \left( W, \xi \right) & \mapsto \left( G(W), \beta_W\circ G ( \xi ) \right)\\ & f & \mapsto G(f). \end{eqnarray*}

induced by $\beta$ (strictly) preserves the terminal object/reflects the terminal object/creates the terminal object.

Finally, we say that $G : \mathcal{C}\to \mathcal{D}$ (strictly) preserves terminal coalgebras/reflects terminal coalgebras/creates terminal coalgebras if G (strictly) preserves terminal coalgebras/reflects terminal coalgebras/creates terminal coalgebras of any endofunctor on $\mathcal{C} $ .

4.2 $\mu\nu$ -polynomial-preserving functors

Finally, we can introduce the concept of a structure-preserving functor for $\mu\nu$ -polynomials.

6.1 Basics: the categories $\Sigma_\mathcal{C} \mathcal{L}$ and $\Sigma_\mathcal{C} \mathcal{L}^{\mathrm{op}}$

Recall that for any strictly indexed category, that is, a (strict) functor $\mathcal{L}:\mathcal{C}^{\mathrm{op}}\to\mathbf{Cat}$ , we can consider its total category (or Grothendieck construction) $\Sigma_\mathcal{C} \mathcal{L}$ , which is a fibered category over $\mathcal{C}$ (see Johnstone Reference Johnstone2002, Sections A1.1.7, B1.3.1). We can view it as a $\Sigma$ -type of categories, which generalizes the cartesian product. Further, given a strictly indexed category $\mathcal{L}:\mathcal{C}^{\mathrm{op}}\to \mathbf{Cat}$ , we can consider its fiberwise dual category $\mathcal{L}^{\mathrm{op}}:\mathcal{C}^{\mathrm{op}}\to \mathbf{Cat}$ , which is defined as the composition $\mathcal{C}^{\mathrm{op}}\xrightarrow{\mathcal{L}}\mathbf{Cat}\xrightarrow{\mathrm{op}}\mathbf{Cat}$ , where op is defined by $A\mapsto A^{\mathrm{op}} $ . Thus, we can apply the same construction to $\mathcal{L}^{\mathrm{op}}$ to obtain a category $\Sigma_{\mathcal{C}}\mathcal{L}^{\mathrm{op}}$ .

Concretely, $\Sigma_\mathcal{C} \mathcal{L}$ is the following category:

• • objects are pairs (W,w) of an object W of $\mathcal{C}$ and an object w of $\mathcal{L}(W)$ ;

• • morphisms $(W,w)\to (X,x)$ are pairs (f, f’) with $f :W\to X$ in $\mathcal{C}$ and $ {f}' : w \to \mathcal{L}(f)(x)$ in $\mathcal{L}(W)$ ;

• • identities ${\mathrm{id}}_{(W,w)}$ are $({\mathrm{id}}_{W},{\mathrm{id}}_{W})$ ;

• • composition of $(W,w)\xrightarrow{(f, {f}' )}(X,x)$ and $(X,x)\xrightarrow{(g, {g}' )}(Y,y)$ is given by:

$$(g\circ f, \mathcal{L}(f)( {g}' ) \circ {f}' ) .$$

Concretely, $\Sigma_{\mathcal{C}}\mathcal{L}^{\mathrm{op}}$ is the following category:

• • objects are pairs (W, w) of an object W of $\mathcal{C}$ and an object w of $\mathcal{L}(W)$ ;

• • morphisms $(W,w)\to (X,x)$ are pairs (f, f’) with $f :W\to X$ in $\mathcal{C}$ and $ {f}' : \mathcal{L}(f)(x)\to w $ in $\mathcal{L}(W)$ ;

• • identities ${\mathrm{id}}_{(W,w)}$ are $({\mathrm{id}}_{W},{\mathrm{id}}_{W})$ ;

• • composition of $(W,w)\xrightarrow{(f, {f}' )}(X,x)$ and $(X,x)\xrightarrow{(g, {g}' )}(Y,y)$ is given by:

$$(g\circ f, {f}' \circ \mathcal{L}(f)( {g}' ) ) .$$

6.3 Generators

In this section, we establish the obvious sufficient (and necessary) conditions for $\Sigma_\mathcal{C}\mathcal{L}$ and $\Sigma_\mathcal{C}\mathcal{L}^{op}$ to model primitive types and operations in the sense of Section 3. These conditions are an immediate consequence of the structure of $\Sigma_\mathcal{C}\mathcal{L}$ and $\Sigma_\mathcal{C}\mathcal{L}^{op}$ as cartesian categories.

We say that $\mathcal{L}:\mathcal{C}^{op}\to \mathbf{Cat}$ is a $\Sigma$ -bimodel of primitive types $\textrm{Ty}$ and operations $\mathsf{Op}$ if

• • for all $T\in\textrm{Ty}$ , we have a choice of objects $C_T\in \mathrm{obj} \left( \mathcal{C}\right)$ and $L_T,L'_T\in \mathrm{obj} \left( \mathcal{L}\right)(C_T)$ ;

• • for all $\mathrm{op}\in \mathsf{Op}(T_1,\ldots, T_n; S)$ , we have a choice of morphisms:

\begin{align*} &f_{\mathrm{op}}\in \mathcal{C}(C_{T_1}\times \ldots\times C_{T_n}, C_S)\\ &g_{\mathrm{op}}\in \mathcal{L}(C_{T_1}\times \ldots\times C_{T_n})(\mathcal{L}(\pi_1)(L_{T_1})\times\cdots\times \mathcal{L}(\pi_n)(L_{T_n}), \mathcal{L}(f_{\mathrm{op}})(L_S))\\ &g'_\mathrm{op}\in \mathcal{L}(C_{T_1}\times \ldots\times C_{T_n})(\mathcal{L}(f_{\mathrm{op}})(L'_S),\mathcal{L}(\pi_1)(L'_{T_1})\sqcup\cdots\sqcup \mathcal{L}(\pi_n)(L'_{T_n})). \end{align*}

We say that such a model has self-dual primitive types in case $L_T=L'_T$ for all $T\in\textrm{Ty}$ .

6.4 Cartesian closedness of total categories

The question of Cartesian closure of the categories $\Sigma_\mathcal{C} \mathcal{L}$ and $\Sigma_\mathcal{C} \mathcal{L}^{op}$ is a lot more subtle. In particular, the formulas for exponentials tend to involve $\Pi$ - and $\Sigma$ -types; hence, we need to recall some definitions from categorical dependent type theory. As also suggested by Kerjean and Pédrot (Reference Kerjean and Pédrot2021), these formulas relate closely to the Diller–Nahm variant (Diller Reference Diller1974; Hyland Reference Hyland2002; Moss and von Glehn Reference Moss, von Glehn, Dawar and Grädel2018) of the Dialectica interpretation (Gödel Reference Gödel1958) and Altenkirch et al. (Reference Altenkirch, Levy and Staton2010)’s formula for higher-order containers. We plan to explain this connection in detail in future work as it would form a distraction from the point of the current paper.

We use standard definitions from the semantics of dependent type theory and the dependently typed enriched effect calculus. An interested reader can find background on this material in Vákár (Reference Vákár2017, Chapter 5) and Ahman et al. (Reference Ahman, Ghani and Plotkin2016). We briefly recalling some of the usual vocabulary Vákár (Reference Vákár2017, Chapter 5).

Definition 21. Given an indexed category $\mathcal{D}:\mathcal{C}^{op}\to \mathbf{Cat}$ , we say:

• • it satisfies the comprehension axiom if: $\mathcal{C}$ has a chosen terminal object $\mathbb{1}$ ; $\mathcal{D}$ has strictly indexed terminal objects $\mathbb{1}$ (i.e., chosen terminal objects $\mathbb{1}\in\mathcal{D}(X)$ , such that $\mathcal{D}(g)(\mathbb{1})=\mathbb{1}\in\mathcal{D} (W) $ for all $g:W\to X$ in $\mathcal{C}$ ); and, for each object $\left( X, x\right)\in \Sigma_\mathcal{C} \mathcal{D} $ , the functor:

\begin{align*} \mathfrak{re}_{(X,x)} : (\mathcal{C}/X)^{op} &\to \mathbf{Set} \\ \left( W, W\xrightarrow{f}X \right) & \mapsto \mathcal{D}(W)(\mathbb{1}, \mathcal{D}(f)(x)) \end{align*}

are representable by an object $\left( X.x, X.x\xrightarrow{{\mathbf{p}_{X,x}}} X\right) $ of $\mathcal{C}/X$ :

\begin{align*} \mathfrak{re}_{(X,x)} \left( W, W\xrightarrow{f}X \right) = \mathcal{D}(W)(\mathbb{1}, \mathcal{D}(f)(x)) &\cong \mathcal{C}/X\left( \left( W, f \right) ,\left( X.x, {\mathbf{p}_{X,x}}\right) \right)\\ b&\mapsto (f,g). \end{align*}

We write ${\mathbf{v}_{X,x}}$ for the unique element of $\mathcal{D}(X.x)(\mathbb{1}, \mathcal{D}({\mathbf{p}_{X,x}})(x))$ such that $(\mathbf{p}_{X,x}, \mathbf{v}_{X,x})={\mathrm{id}}_{{\mathbf{p}_{X,x}}}$ (the universal element of the representation).

Furthermore, given $f: W\to X $ , we write $\mathbf{q}_{f,b}$ for the unique morphism $(f\circ {\mathbf{p}_{W,\mathcal{D}(f)(x)}}, \mathbf{v}_{W,{\mathcal{D}(f)(x)}})$ making the square below a pullback:

We henceforth call such squares $\mathbf{p}$ -squares;

• • it supports $\Sigma$ -types if we have left adjoint functors $\Sigma_w\dashv \mathcal{D}({\mathbf{p}_{W,w}}):\mathcal{D}(W.w)\leftrightarrows \mathcal{D}(W)$ satisfying the left Beck–Chevalley condition for $\mathbf{p}$ -squares w.r.t. $\mathcal{D} $ (this means that $\mathcal{D}(f) \circ \left( \Sigma_{\mathcal{D}(f)(x)} \to \Sigma_x\right) \circ \mathcal{D}({\mathbf{p}_{f,x}}) $ are the identity);

• • it supports $\Pi$ -types if $\mathcal{D}^{op}$ supports $\Sigma$ -types; explicitly, that is the case iff we have right adjoint functors $\mathcal{D}({\mathbf{p}_{W,w}})\dashv \Pi_w:\mathcal{D}(W)\leftrightarrows \mathcal{D}(W.w)$ satisfying the right Beck–Chevalley condition for $\mathbf{p}$ -squares in the sense that the canonical maps $\Pi_{\mathcal{D}(f)(x)} \circ \left( \mathcal{D}(f)\to\mathcal{D}({\mathbf{p}_{f,x}})\right) \circ \Pi_x$ are the identity.

We can immediately note that our notion of $\Sigma$ -bimodel of function types is also a $\Sigma$ -bimodel of tuple types. Indeed, strong $\Sigma$ -types and comprehension give us, in particular, chosen finite products in $\mathcal{C}$ .

We next show why this name is justified: we show that the Grothendieck construction of a $\Sigma$ -bimodel of function types is cartesian closed.⁴

In the following, we slightly abuse notation to aid legibility:

• • denoting by $!_{W}: W\to\mathbb{1} $ the only morphism, we will sometimes conflate $Z\in\mathrm{obj} \mathcal{C}'(\mathbb{1} )$ and $\mathbb{1} .Z\in\mathrm{obj} \left( \mathcal{C}\right)$ as well as $f\in \mathcal{C}'(W)(\mathbb{1}, \mathcal{C}'(!_{W} )(Z))$ and $(!_{W},f)\in \mathcal{C}(W, \mathbb{1} .Z)$ ); this is justified by the democratic comprehension axiom;

• • we will sometimes simply write z for $\mathcal{D}({\mathbf{p}_{W,w}})(z)$ where the weakening map $\mathcal{D}({\mathbf{p}_{W,w}})$ is clear from context.

Given $X, Y\in \mathcal{C}$ we will write ${\mathrm{ev1}}$ for the obvious $\mathcal{C}$ -morphism

$${\mathrm{ev1}}:\Pi_{X} \Sigma_{Y}Z.X\to Y,$$

that is, the morphism obtained as the composition (where we write $\pi_1$ for the projection $\Sigma_{Y}Z\to Y$ ):

$$\Pi_{X}\Sigma_{Y}Z.X\cong (\Pi_{X}\Sigma_{Y}Z)\times X\xrightarrow{(\Pi_{X}\pi_1)\times X}(\Pi_{X}Y)\times X\cong (X\Rightarrow Y)\times X\xrightarrow{\mathrm{ev}}Y$$

With these notational conventions in place, we can describe the cartesian closed structure of Grothendieck constructions.

Theorem 25 (Exponentials of the total category). For a $\Sigma$ -bimodel $\mathcal{L}$ for function types, $\Sigma_{\mathcal{C}}\mathcal{L}$ has exponential:

$$(X,x)\Rightarrow (Y,y)= (\Pi_{X}\Sigma_{Y} \mathcal{L}(\pi_1)(x)\multimap \mathcal{L}(\pi_2)(y), \Pi_{X} \mathcal{L}({\mathrm{ev1}})(y)).$$

Proof. We have (natural) bijections:

Codually, we have

Theorem 26 For a $\Sigma$ -bimodel $\mathcal{L}$ for function types, $\Sigma_{\mathcal{C}}\mathcal{L}^{\mathrm{op}}$ has exponential:

$$(X,x)\Rightarrow (Y,y)= (\Pi_{X}\Sigma_{Y} \mathcal{L}(\pi_2)(y)\multimap \mathcal{L}(\pi_1)(x), \Sigma_{X} \mathcal{L}({\mathrm{ev1}})(y)).$$

Note that these exponentials are not fibered over $\mathcal{C}$ in the sense that the projection functors $\Sigma_\mathcal{C} \mathcal{L}\to \mathcal{C}$ and $\Sigma_\mathcal{C} \mathcal{L}^{op}\to \mathcal{C}$ are generally not cartesian closed functors. This is in contrast with the interpretation of all other type formers we consider in this paper.

6.6 Extensive indexed categories and coproducts in total categories

We introduce a special property that fits our context well. We call this property extensivity because it generalizes the concept of extensive categories (see Section 6.12 for the notion of extensive category).

As we will show, the property of extensivity is a crucial requirement for our models. One significant advantage of this property is that it allows us to easily construct coproducts in the total categories, even under lenient conditions. We demonstrate this in Theorem 35.

• • We assume that the category $\mathcal{C} $ has finite coproducts. Given $ W, X\in\mathcal{C} $ , we denote by:

(8)

the coproduct (and coprojections) in $\mathcal{C} $ , and by ${\mathbb {0}} $ the initial object of $\mathcal{C} $ .

Definition 31. (Extensive indexed categories). We call an indexed category $\mathcal{L} : \mathcal{C} ^{\mathrm{op}} \to \mathbf{Cat} $ extensive if, for any $(W,X)\in\mathcal{C}\times\mathcal{C} $ , the unique functor:

(9)

induced by the functors:

(10)

is an equivalence. In this case, for each $(W,X)\in\mathcal{C}\times\mathcal{C} $ , we denote by:

(11) \begin{equation} \mathcal{S} ^{(W,X)} : \mathcal{L} (W)\times \mathcal{L} (X)\to \mathcal{L} (W\sqcup X)\end{equation}

an inverse equivalence of ${\left(\mathcal{L} (\iota _ W), \mathcal{L} (\iota _ X)\right)} $ .

Since the products of $\mathcal{C} ^{\mathrm{op}} $ are the coproducts of $\mathcal{C} $ , the extensive condition described above is equivalent to say that the (pseudo)functor $\mathcal{L} : \mathcal{C} ^{\mathrm{op}} \to \mathbf{Cat} $ preserves binary (bicategorical) products (up to equivalence).

Since our cases of interest are strict, this leads us to consider strict extensivity, that is to say, whenever we talk about extensive strictly indexed categories, we are assuming that (9) is invertible. In this case, it is even clearer that extensivity coincides with the well-known notion of preservation of binary products.

Recall that, in general, preservation of binary products implies preservation of preterminal objects; see, for instance, Lucatelli Nunes (Reference Lucatelli Nunes2022, Remark 4.14). Lemma 33 is the appropriate analog of this observation suitably applied to the context of extensive indexed categories. Moreover, Lemma 33 can be seen as a generalization of Carboni et al. (Reference Carboni, Lack and Walters1993, Proposition 2.8).

Lemma 33 (Preservation of terminal objects). Let $ \mathcal{L} : \mathcal{C} ^{\mathrm{op}} \to \mathbf{Cat} $ be an extensive indexed category which is not (naturally isomorphic to the functor) constantly equal to ${\mathbb {0}} $ . The unique functor

(12) \begin{equation} \mathcal{L} ({\mathbb {0}} )\to \mathbb{1} \end{equation}

is an equivalence. If, furthermore, (9) is an isomorphism, then (12) is invertible.

Proof. Firstly, given any $X\in\mathcal{C} $ such that $\mathcal{L} (X) $ is not (isomorphic to) the initial object of $\mathbf{Cat} $ , we have that $\mathcal{L} ( i_X : {\mathbb {0}}\to X ) $ is a functor from $\mathcal{L} (X) $ to $\mathcal{L} ({\mathbb {0}} ) $ . Hence, $\mathcal{L} ({\mathbb {0}} ) $ is not isomorphic to the initial category as well.

Secondly, since $\iota _{{\mathbb {0}} } : {\mathbb {0}} \to {\mathbb {0}}\sqcup {\mathbb {0}} $ is an isomorphism, $\left(\mathcal{L} (\iota _ {{\mathbb {0}}} ), \mathcal{L} (\iota _ {\mathbb {0}} )\right)$ is an equivalence and

(13)

we conclude that $ \pi _{\mathcal{L}({\mathbb {0}} )} $ is an equivalence. This proves that $\mathcal{L} ({\mathbb {0}} ) \to \mathbb{1} $ is an equivalence by Appendix A, Lemma 132.

We proceed to study the cocartesian structure of $\Sigma_\mathcal{C} \mathcal{L} $ (and $\Sigma_\mathcal{C} \mathcal{L} ^{\mathrm{op}} $ ) when $\mathcal{L} $ is extensive. We start by proving in Theorem 34 that, in the case of extensive indexed categories, the hypothesis of Proposition 27 always holds.

Theorem 34. Let $\mathcal{L} : \mathcal{C} ^{\mathrm{op}} \to \mathbf{Cat} $ be an extensive (strictly) indexed category. Assume that X is an object of $\mathcal{C} $ such that $\mathcal{L} (X) $ has initial object ${\mathbb {0}} $ . In this case, for any $W\in \mathcal{C} $ , we have an adjunction:

(14)

in which, by abuse of language, ${\mathbb {0}} : \mathcal{L} (W)\to \mathcal{L} (X) $ is the functor constantly equal to ${\mathbb {0}} $ . Dually, we have an adjunction:

(15)

provided that $\mathcal{L} (X) $ has terminal object $\mathbb{1} $ and, by abuse of language, we denote by $\mathbb{1} : \mathcal{L} (W)\to \mathcal{L} (X) $ the functor constantly equal to $\mathbb{1} $ .

Proof. Assuming that $\mathcal{L} (X) $ has initial object ${\mathbb {0}} $ , we have the adjunction:

(16)

whose unit is the identity and counit is pointwise given by $ \varepsilon _{(w,x)} = ({\mathrm{id}}_w, {\mathbb {0}} \to x ) $ . Therefore, we have the composition of adjunctions:

Proof. In fact, by Proposition 27, we have that $({\mathbb {0}} , {\mathbb {0}} )$ is the initial object of $\Sigma_\mathcal{C} \mathcal{L} $ . Moreover, given $\left( (W,w), (X,x)\right) \in \Sigma_\mathcal{C}\mathcal{L} \times \Sigma_\mathcal{C}\mathcal{L} $ , we have that

\begin{eqnarray*}\mathcal{S} ^{(W,X)}\circ \left({\mathrm{id}}_{\mathcal{L} (W)}, {\mathbb {0}} \right) = \mathcal{L} (\iota _W )! &\dashv &\mathcal{L} (\iota _W )\\\mathcal{S} ^{(W,X)}\circ \left( {\mathbb {0}} , {\mathrm{id}}_{\mathcal{L} (X)}\right) = \mathcal{L} (\iota _X )! &\dashv &\mathcal{L} (\iota _X )\end{eqnarray*}

by Theorem 34. Therefore, we get that

In particular, finite coproducts in $\Sigma_{\mathcal{C}}\mathcal{L}$ are fibered in the sense that the projection functor $\Sigma_{\mathcal{C}}\mathcal{L}\to \mathcal{C}$ preserves them, on the nose.

Codually, we have:

(17) \begin{equation} (W, w)\sqcup (X,x) = \left( W\sqcup X, \mathcal{S} ^{(W,X)} (w,x) \right). \end{equation}

6.7 Distributive property of the total category

We refer the reader to Carboni et al. (Reference Carboni, Lack and Walters1993) and Lack (Reference Lack2012) for the basics on distributive categories.

As we proved, $\Sigma_\mathcal{C} \mathcal{L} $ is bicartesian closed provided that $\mathcal{L} :\mathcal{C} ^{\mathrm{op}}\to\mathbf{Cat} $ is $\Sigma$ -bimodel for function types and sum types. Therefore, in this setting, we get that $\Sigma_\mathcal{C} \mathcal{L} $ is distributive.

However, even without the assumptions concerning closed structures, whenever we have a $\Sigma$ -bimodel for sum types, we can inherit distributivity from $\mathcal{C}$ . Namely, we have Theorem 39.

Recall that a category $\mathcal{C} $ with finite products and coproducts is a distributive category if, for each triple $\left( W, Y, Z\right) $ of objects in $\mathcal{C} $ , the canonical morphism:

(18) \begin{equation}\left< W\times \iota _ Y ^{Y\sqcup Z}, W\times\iota _Z^{Y\sqcup Z} \right> : \left( W\times Y\right) \sqcup \left( W\times Z\right)\rightarrow W\times \left( Y\sqcup Z \right) ,\end{equation}

induced by $W\times \iota _ {Y} $ and $W\times \iota _ {Z} $ is invertible. It should be noted that, in a such a distributive category $\mathcal{C}$ , for any such a triple $\left( W, Y, Z\right) $ of objects in $\mathcal{C} $ , the diagram

commutes. Therefore, we have:

Lemma 38 Let $\mathcal{L} : \mathcal{C} ^{\mathrm{op}}\to\mathbf{Cat} $ be an extensive strictly indexed category, in which $\mathcal{C} $ is a distributive category. For each triple $\left( W, Y, Z\right) $ of objects in $\mathcal{C} $ , the diagrams

(19)

(20)

commute.

Let $\mathcal{D} $ be a category with finite coproducts and products. A category is distributive if the canonical morphisms (18) are invertible. However, by Lack (Reference Lack2012, Theorem 4), the existence of any natural isomorphism $\left( W\times Y\right) \sqcup \left( W\times Z\right)\cong W\times \left( Y\sqcup Z \right)$ implies that $\mathcal{D} $ distributive. Hence, we proceed to prove below that such a natural isomorphism exists in the case of $\Sigma_\mathcal{C} \mathcal{L} $ , leaving the question of canonicity omitted.

We indeed have the natural isomorphisms in $\left( \left( W, w\right) , \left( Y, y\right) , \left( Z, z\right)\right) \in \Sigma_\mathcal{C}\mathcal{L} \times\Sigma_\mathcal{C}\mathcal{L}\times \Sigma_\mathcal{C}\mathcal{L}: $

which, by the distributive property of $\mathcal{C} $ , is (naturally) isomorphic to

(21) \begin{equation}\left( \left(W\times Y\right)\sqcup \left(W\times Z\right), \mathcal{L} \left( \left< W\times \iota _ Y, W\times\iota _Z \right> \right) \left( \mathcal{L} (\pi _W ) (w) \times\mathcal{L} (\pi _{Y\sqcup Z})\mathcal{S} ^{(Y , Z)} (y, z)\right) \right) .\end{equation}

Moreover, we have the natural isomorphisms

which is naturally isomorphic to

(22) \begin{equation}\mathcal{S} ^{(W\times Y , W\times Z)} \left( \mathcal{L} (\pi _W ) (w) \times \mathcal{L} (\pi _Y ) (y) , \mathcal{L} (\pi _W ) (w) \times \mathcal{L} (\pi _Z ) (z) \right) .\end{equation}

since $\mathcal{S} ^{(W\times Y , W\times Z)} $ is invertible. Therefore, we have the natural isomorphisms:

which completes our proof.

Codually, we have:

6.8 Extensive property of the total category

As per the definition provided in Carboni et al. (Reference Carboni, Lack and Walters1993, Definition2.1), a category $\mathcal{C}$ is considered extensive if the basic (codomain) indexed category $\mathcal{C} /- : \mathcal{C} ^{\mathrm{op}} \to \mathbf{Cat}$ is an extensive indexed category as introduced at Definition 31. Recall that every extensive category is distributive (Carboni et al. Reference Carboni, Lack and Walters1993, Proposition 4.5).

The result below also holds for the nonstrict scenario.

The first step is to see that, indeed, $\Sigma_\mathcal{C}\mathcal{L} $ has coproducts by Corollary 35. We, then, note that, for each pair (W,w) and (X,x) of objects in $\Sigma_\mathcal{C}\mathcal{L} $ , we note that, in fact, we have that

(23) \begin{equation} \mathcal{S} ^{\left( (W,w), (X,x) \right)} _ {\Sigma_\mathcal{C}\mathcal{L} /-} : \Sigma_\mathcal{C}\mathcal{L} /(W,w) \times \Sigma_\mathcal{C}\mathcal{L} /(X,x) \to \Sigma_\mathcal{C}\mathcal{L} /\left( (W,w)\sqcup (X,x) \right) \end{equation}

defined by the coproduct of the morphisms is an equivalence. Explicitly, given objects $A= \left((W _0 ,w _0 ) , ( f: W_0\to W, f' : w_0\to \mathcal{L}\left( f \right) w ) \right) $ of $\Sigma_\mathcal{C}\mathcal{L} /(W,w) $ and $B = \left((X _0 ,x _0 ) , ( g: X_0\to X, g' : x_0\to \mathcal{L}\left( g \right) x ) \right) $ of $\Sigma_\mathcal{C}\mathcal{L} /(X,x)$ , $ \mathcal{S} ^{\left( (W,w), (X,x) \right)} _ {\Sigma_\mathcal{C}\mathcal{L} /-}\left( A, B\right) $ is given by:

$$\left( \left( W_0\sqcup X_0, \mathcal{S} ^{\left( W, X \right) } _{\mathcal{L} } (w_0, x_0) \right) , \left( f\sqcup g: W_0\sqcup X_0\to W\sqcup X, \mathcal{S} ^{\left( W,X \right)} _\mathcal{L} \left(f', g'\right) \right) \right) $$

which is clearly an equivalence given that the functor $\left( (W_0, f), (X_0, g) \right)\mapsto \left( W_0\sqcup X_0, f\sqcup g\right) $ is an equivalence $\mathcal{C} / W\times \mathcal{C} / X \to \mathcal{C} / W\sqcup X $ .

It is worth mentioning that free cocompletions under (finite) coproducts are extensive, as shown in Carboni et al. (Reference Carboni, Lack and Walters1993, Proposition 2.4) for the infinite case. This implies that freely generated models on languages featuring variant types are extensive. Therefore, having an extensive base category $\mathcal{C}$ is a common occurrence in our setting.

6.9 Strictly indexed categories and split fibrations

Before we specialize to our setting of $\mu\nu$ -polynomials, we need to establish and prove general results on parameterized initial algebras (and terminal coalgebras) in the total category of a split fibration (see Sections 6.10 and 6.11).

In order to talk about these results, we need to talk about strictly indexed functors and split fibration functors and the one-to-one correspondence between them. For this purpose, we shortly recall the equivalence between strict indexed categories and split fibrations below.

Definition 43. (Strictly indexed functor). Let $\mathcal{L} ' : \mathcal{D} ^{\mathrm{op}}\to \mathbf{Cat} $ and $ \mathcal{L} : \mathcal{C} ^{\mathrm{op}}\to \mathbf{Cat} $ be two strictly indexed categories. A strictly indexed functor between $\mathcal{L} ' $ and $\mathcal{L} $ consists of a pair $(\overline{H}, h ) $ in which $\overline{H} : \mathcal{D} \to \mathcal{C} $ is a functor and

(24) \begin{equation} h: \mathcal{L} ' \longrightarrow \left(\mathcal{L} \circ \overline{H} ^{\mathrm{op}}\right) \end{equation}

is a natural transformation, where $\overline{H} ^{\mathrm{op}} $ denotes the image of $\overline{H} $ by $\mathrm{op} $ . Given two strictly indexed functors $(\overline{E} ,e) :\mathcal{L} ''\to \mathcal{L} ' $ and $(\overline{H} , h) : \mathcal{L} '\to \mathcal{L} $ , the composition is given by:

(25) \begin{equation} \left( \overline{HE}, (h _ {\overline{E} ^{\mathrm{op} }})\cdot e: \mathcal{L} '' \longrightarrow \left(\mathcal{L} \circ \left(\overline{HE}\right) ^{\mathrm{op} }\right) \right) . \end{equation}

Strictly indexed categories and strictly indexed functors do form a category, denoted herein by $\mathfrak{Ind} $ .

It is well known that the Grothendieck construction provides an equivalence between indexed categories and fibrations. Restricting this to our setting, we get the equivalence:

\begin{eqnarray*} \int : & \mathfrak{Ind} &\to \mathrm{S}\mathfrak{pFib}\\ & \mathcal{L} : \mathcal{C} ^{\mathrm{op}}\to \mathbf{Cat} & \mapsto \left(\mathsf{P}_{\mathcal{L} } : \Sigma_\mathcal{C} \mathcal{L}\to \mathcal{C}\right)\\ & (\overline{E}, e) & \mapsto (E, \overline{E} )\end{eqnarray*}

between the category of strictly indexed categories (with strictly indexed functors) and the category of (Grothendieck) split fibrations.

Although not necessary to your work, we refer to Gray (Reference Gray1966) and Johnstone (Reference Johnstone2002, Theorem 1.3.6) for further details. We explicitly state the relevant part of this result below.

\begin{equation*} \left(\overline{H} : \mathcal{D} \to \mathcal{C} , h: \mathcal{L} ' \longrightarrow \left(\mathcal{L} \circ \overline{H} ^{\mathrm{op}}\right)\right) : \mathcal{L} '\to \mathcal{L} \end{equation*}

and pairs $ (H, \overline{H} ) $ in which $H:\Sigma_\mathcal{D} \mathcal{L} ' \to\Sigma_\mathcal{C} \mathcal{L}$ is a functor satisfying the following two conditions.

$\Sigma$ A) The diagram

(26)

commutes.

$\Sigma$ B) For any morphism $(f: X\to Y, {\mathrm{id}}: \mathcal{L} ' (f) (y) \to \mathcal{L} ' (f) (y) ) $ between $(X, \mathcal{L} ' (f) (y) )$ and (Y, y) in $\Sigma_\mathcal{D} \mathcal{L} '$ ,

(27) \begin{equation} H(f, {\mathrm{id}}) = ( \overline{H} (f) , {\mathrm{id}}) : H(X, \mathcal{L} ' (f) (y) )\to H(Y, y ). \end{equation}

For each strictly indexed functor $(\overline{H}, h) : \mathcal{L} ' \to \mathcal{L} $ , we define

(28) \begin{equation} H (f: X\to Y , f': x\to \mathcal{L} '(f) y ) := (\overline{H} (f), h_X (f') ). \end{equation}

Reciprocally, given a pair $(H, \overline{H}) $ satisfying (26) and (27), we define

(29) \begin{equation} h_X (f': w\to x ) := H \left( ({\mathrm{id}}_ X , f') : (X, w)\to (X, x) \right) \end{equation}

for each object $X\in \mathcal{D} $ and each morphism $f': w\to x $ of $\mathcal{L} '(X)$ .

Definition 45. (Split fibration functor). A pair $(H, \overline{H} ): \mathsf{P}_{\mathcal{L} '}\to \mathsf{P}_{\mathcal{L} } $ satisfying (26) and (27) is herein called a split fibration functor. Whenever it is clear from the context, we omit the split fibrations $\mathsf{P}_{\mathcal{L} '}$ , $\mathsf{P}_{\mathcal{L} }$ , and the functor $\overline{H} $ .

Following the above, given a strictly indexed functor $(\overline{H} , h) : \mathcal{L} '\to \mathcal{L} $ , we denote

\begin{eqnarray*} \int \mathcal{L} & = & \left( \mathsf{P}_{\mathcal{L} } : \Sigma_\mathcal{C} \mathcal{L}\to \mathcal{C}\right) \\ \int \left( \overline{H} , h \right) & = & \left( H ,\overline{H} \right)\end{eqnarray*}

in which $H \left( f : X\to Y , f': x\to \mathcal{L} (f) (y) \right) = (\overline{H} (f), h_X (f') ) $ .

Let $\mathcal{L} ': \mathcal{D} ^{\mathrm{op}} \to \mathbf{Cat} $ and $\mathcal{L} : \mathcal{C} ^{\mathrm{op}} \to \mathbf{Cat} $ be strictly indexed categories. We denote by $ \mathcal{L} '\,\underline{\times }\, \mathcal{L} $ the product of the strict indexed categories in $\mathfrak{Ind} $ . Explicitly,

\begin{eqnarray*} \mathcal{L} '\,\underline{\times }\, \mathcal{L} : &(\mathcal{D}\times \mathcal{C} ) ^{\mathrm{op}} & \to \mathbf{Cat}\\ & (X,Y ) & \mapsto \mathcal{L} ' (X) \times \mathcal{L} (Y)\\ & (f, g ) & \mapsto \mathcal{L} ' (f)\times \mathcal{L} (g) .\end{eqnarray*}

It should be noted that

(30) \begin{equation} \left(\int \mathcal{L} '\,\underline{\times }\, \mathcal{L} \right) \cong \left(\int \mathcal{L} '\right)\times \left( \int \mathcal{L} \right) = \left(\mathsf{P}_{\mathcal{L} ' }\times\mathsf{P}_{\mathcal{L} } : \left( \Sigma_\mathcal{D} \mathcal{L} '\right)\times\left(\Sigma_\mathcal{C} \mathcal{L} \right) \to \mathcal{D}\times\mathcal{C} \right) ,\end{equation}

which means that the product in $ \mathrm{S}\mathfrak{pFib} $ coincides with the usual product of functors $\mathsf{P}_{\mathcal{L} }\times\mathsf{P}_{\mathcal{L} '}$ . Moreover, given indexed functors $(\overline{H}, h) : \mathcal{H}\to \mathcal{H}' $ and $(\overline{E}, e) : \mathcal{L}\to \mathcal{L} ' $ , we have that

$$ (\overline{H}, h)\,\underline{\times }\, (\overline{E}, e) = \left( \overline{H}\times \overline{E}, h\times e \right) $$

and, since the product of split fibrations is given by the usual product of functors:

(31) \begin{equation} \int \left( (\overline{H}, h)\,\underline{\times }\, (\overline{E}, e)\right) = \left(\int (\overline{H}, h)\right) \times \left( \int (\overline{E}, e)\right) .\end{equation}

Codually, given a strictly indexed category $ \mathcal{L} : \mathcal{C} ^{\mathrm{op}}\to \mathbf{Cat} $ , we have the Grothendieck codual construction:

in which $H \left( f : X\to Y , f': \mathcal{L} (f) (y)\to x \right) = (\overline{H} (f), h_X (f') ) $ . This construction gives an equivalence between the indexed categories and split op-fibrations (if we consider the opposite of $ \int ^{\mathrm{op}} \mathcal{L} $ ). We, of course, have the codual observations above.

6.10 General result on initial algebras in total categories

In order to study the $\mu\nu$ -polynomials of total categories in our setting in Section 6.12, we start by establishing general results about parameterized initial algebras in the Grothendieck construction of split fibrations. More precisely, in Theorem 47, we investigate when a total category $ \Sigma_\mathcal{C} \mathcal{L} $ has the parameterized initial algebra of a split fibration functor:

(32) \begin{equation}H : \left( \Sigma_\mathcal{D} \mathcal{L} '\right) \times \left( \Sigma_\mathcal{C} \mathcal{L} \right) \to \Sigma_\mathcal{C}\mathcal{L} .\end{equation}

We start by studying initial algebras os strictly indexed endofunctors:

( $\mathfrak{e}$ 1) the initial $\overline{E}$ -algebra $\left( \mu \overline{E},\, \mathbf{\mathfrak{in}} _{\overline{E}} \right) $ exists;

( $\mathfrak{e}$ 2) the initial $\left( \mathcal{L}(\mathbf{\mathfrak{in}} _{\overline{E}} )^{-1} e_{\mu\overline{E}} \right) $ -algebra $\left( \mu \left( \mathcal{L} (\mathbf{\mathfrak{in}} _{\overline{E} } )^{-1} e_{\mu\overline{E}} \right),\, \mathbf{\mathfrak{in}} _{\left(\mathcal{L} (\mathbf{\mathfrak{in}} _{\overline{E} } )^{-1} e_{\mu \overline{E} }\right) } \right) $ exists.

Denoting by $ \underline{e} $ the endofunctor $\mathcal{L} (\mathbf{\mathfrak{in}} _{\overline{E} } )^{-1} e_{\mu \overline{E} } $ on $\mathcal{L} (\mu \overline{E} ) $ , the initial E-algebra exists and is given by:

(33) \begin{equation} \mu E = \left( \mu \overline{E} ,\, \mu \underline{e} \right), \qquad \mathbf{\mathfrak{in}} _E = \left( \mathbf{\mathfrak{in}} _{\overline{E} },\, \mathcal{L} (\mathbf{\mathfrak{in}} _{\overline{E} } ) \left(\mathbf{\mathfrak{in}} _{ \underline{e} } \right) \right). \end{equation}

Moreover, for each E-algebra:

$$\left( (Y,y),\, (\xi ,\xi '): E(Y,y)\to (Y,y) \right) = \left( (Y,y), \left( \xi : \overline{E} (Y)\to Y, \xi ': e_Y(y)\to \mathcal{L} (\xi ) (y) \right) \right), $$

we have that

(34) \begin{equation} \mathrm{fold} _{E} \left( \xi , \xi ' \right) = \left( \mathrm{fold} _ {\overline{E} } \xi , \, \, \mathrm{fold} _{ \underline{e} } \left( \mathcal{L} \left(\overline{E} ( \mathrm{fold} _{\overline{E} }\, \xi )\cdot \mathbf{\mathfrak{in}} _{\overline{E} } ^{-1} \right) (\xi ')\right) \right). \end{equation}

Proof. In fact, under the hypothesis above, given an E-algebra:

\begin{equation*} \left( \xi : \overline{E} (Y)\to Y, \xi ': e_Y(y)\to \mathcal{L} (\xi ) (y) \right) \end{equation*}

on (Y, y), we have that there is a unique morphism:

\begin{equation*} \left(\mathrm{fold} _{ \underline{e} } \, \mathcal{L} \left(\overline{E} ( \mathrm{fold} _{\overline{E} } \xi )\cdot \mathbf{\mathfrak{in}} _{\overline{E} } ^{-1} \right) (\xi ')\right) : \mu \underline{e} \to \mathcal{L}\left( \mathrm{fold} _{\overline{E} } \xi \right)(y) \end{equation*}

in $\mathcal{L} (\mu \overline{E} ) $ such that

commutes. Since $\mathcal{L} (\mathbf{\mathfrak{in}} _{\overline{E}})$ is invertible, this implies that

\begin{equation*} \left( \mathrm{fold} _{ \underline{e} } \, \mathcal{L} \left(\overline{E} ( \mathrm{fold} _{\overline{E} } \xi )\cdot \mathbf{\mathfrak{in}} _{\overline{E} } ^{-1} \right) (\xi ') \right) : \mu \underline{e} \to \mathcal{L}\left( \mathrm{fold} _{\overline{E} } \xi \right)(y) \end{equation*}

is the unique morphism in $\mathcal{L} \left( \overline{E} (\mu \overline{E} ) \right)$ such that

commutes. Finally, by the above and the universal property of $\mathrm{fold} _{\overline{E}} \xi$ , this completes the proof that

(35) \begin{equation} \mathfrak{u} = \left( \mathrm{fold} _{\overline{E}} \xi, \, \left(\mathrm{fold} _{ \underline{e} } \, \mathcal{L} \left(\overline{E} ( \mathrm{fold} _{\overline{E} } \xi )\cdot \mathbf{\mathfrak{in}} _{\overline{E} } ^{-1} \right) (\xi ')\right)\right) \end{equation}

is the unique morphism in $\Sigma_\mathcal{C}\mathcal{L} $ such that

$$ (\xi , \xi ')\circ E( \mathfrak{u} ) = \mathfrak{u} \circ \left( \mathbf{\mathfrak{in}} _{\overline{E} },\, \mathcal{L} (\mathbf{\mathfrak{in}} _{\overline{E} } ) \left(\mathbf{\mathfrak{in}} _{ \underline{e} } \right) \right) . $$

This completes the proof that $\left( (\mu\overline{E} , \mu \underline{e} ), \left( \mathbf{\mathfrak{in}} _{\overline{E} },\, \mathcal{L} (\mathbf{\mathfrak{in}} _{\overline{E} } ) \left(\mathbf{\mathfrak{in}} _{ \underline{e} } \right) \right) \right) $ is the initial object of $E\textrm{-}\mathrm{Alg}$ , and that $\mathrm{fold} _{E} ((Y,y), (\xi, \xi ') ) = \mathfrak{u} $ .

Let $\mathcal{L} : \mathcal{C} ^{\mathrm{op}} \to \mathbf{Cat} $ , $\mathcal{L} ': \mathcal{D} ^{\mathrm{op}} \to \mathbf{Cat} $ be strictly indexed categories as above. We denote by $\mathcal{L} '\,\underline{\times }\, \mathcal{L} : \left( \mathcal{D}\times\mathcal{C} \right) ^{\mathrm{op}} \to \mathbf{Cat} $ the product of the indexed categories (see Section 6.9). An object of $ \Sigma _{\mathcal{D}\times \mathcal{C} } \left( \mathcal{L} '\,\underline{\times }\, \mathcal{L}\right) \cong \left( \Sigma_\mathcal{D} \mathcal{L} '\right) \times \left( \Sigma_\mathcal{C} \mathcal{L} \right) $ can be seen as a quadruple $\left( (X, x) , (W, w)\right) $ in which $x \in\mathcal{L} '(X) $ and $ w\in \mathcal{L} (W) $ . Moreover, a morphism between objects $\left( (X_0 , x _0) , (W _0, w_ 0 )\right) $ and $\left( (X_1 , x_1 ) , (W_1, w _1)\right) $ consists of a quadruple $\left( (f, f') , (g, g') \right) $ in which $ (f, g) : ( X _ 0 , W _ 0 )\to ( X_1 , W_1 ) $ is a morphism in $\mathcal{D}\times \mathcal{C} $ , and $ (f', g') : (x_0, w_0)\to \left( \mathcal{L} ' (f)( x_1 ), \mathcal{L} (g)( w_1 )\right) $ is a morphism in $\mathcal{L} ' ( X_0 )\times \mathcal{L} ( W_0 ) $ .

Given a strictly indexed functor $(\overline{H}, h) : \mathcal{L} '\,\underline{\times }\, \mathcal{L} \to\mathcal{L} $ and an object (X,x) of $\left( \Sigma_\mathcal{D} \mathcal{L} '\right) $ , we can consider the restriction $( \overline{H} ^X, h ^{(X,x)} ) $ in which $\overline{H} ^X = \overline{H} (X,-)$ and $h^{(X,x)} : \mathcal{L}\longrightarrow \left( \mathcal{L}\circ \overline{H} ^X \right) $ is pointwise defined by:

\begin{eqnarray*} h^{(X,x)}_Y : &\mathcal{L} (Y) &\to \mathcal{L}\circ \overline{H} ^X (Y)\\ & f' : y\to z & \mapsto h_{(X,Y)} (x, f' )\end{eqnarray*}

in which we denote by $(X,Y) \in \mathcal{D} \times \mathcal{C} $ . To be consistent with the notation previously introduced (in Proposition 4), we also denote by $h_{(X,Y)}^x $ the morphism $h^{(X,x)}_Y $ above.

As a consequence of Theorem 46, we have that, under suitable conditions, parameterized initial algebras of split fibration functors are split fibration functors, namely, we have:

Theorem 47 (Parameterized initial algebras are split fibration functors). Let $( \overline{H} , h ) $ be a strictly indexed functor from $\mathcal{L} '\,\underline{\times }\, \mathcal{L} : \left( \mathcal{D}\times\mathcal{C} \right) ^{\mathrm{op}} \to \mathbf{Cat} $ to $\mathcal{L} : \mathcal{C} ^{\mathrm{op}}\to\mathbf{Cat} $ , and

$$ H : \left( \Sigma_\mathcal{D} \mathcal{L} '\right) \times \left( \Sigma_\mathcal{C} \mathcal{L} \right) \to \Sigma_\mathcal{C}\mathcal{L} $$

the corresponding split fibration functor. Assume that:

( $\mathfrak{h}$ 1) for each object X of $\mathcal{D} $ , the initial $\overline{H} ^X $ -algebra $\left( \mu \overline{H} ^X,\, \mathbf{\mathfrak{in}} _{\overline{H} ^X} \right) $ exists;

( $\mathfrak{h}$ 2) for each object (X,x) in $\Sigma_\mathcal{D}\mathcal{L} ' $ , denoting by $\underline{h} _X $ the functor:

(36) \begin{equation} \mathcal{L}(\mathbf{\mathfrak{in}} _{\overline{H} ^X} )^{-1} h_{\left(X, \mu\overline{H} ^X\right)} : \mathcal{L} '( X ) \times \mathcal{L}(\mu \overline{H} ^X ) \to\mathcal{L}(\mu \overline{H} ^X ) \end{equation}

is such that the initial $\underline{h} _X ^x $ -algebra $\left( \mu \underline{h} _X^x ,\, \mathbf{\mathfrak{in}} _{\underline{h} _X^x} \right) $ exists;

( $\mathfrak{h}$ 3) for each morphism $ g : X\to Y $ in $\mathcal{D} $ and $ y\in \mathcal{L} ' (Y) $ , Eq. (37) holds

(37) \begin{equation} \mathcal{L} \left( \mu\overline{H} (g) \right) (\mathbf{\mathfrak{in}} _{\underline{h} _ Y^y } ) = \mathbf{\mathfrak{in}} _ {\underline{h} _ X^{\mathcal{L} ' (g) (y) }} \end{equation}

In this setting, the parameterized initial algebra $\mu H: \Sigma_\mathcal{D} \mathcal{L} ' \to \Sigma_\mathcal{C}\mathcal{L} $ exists and is a split fibration functor.