We observe that the existence of sequential and parallel composition supermaps in higher order physics can be formalised using enriched category theory. Encouraged by physically relevant examples such as unitary supermaps and layers within higher order causal categories (HOCCs), we treat the modelling of higher order physical theories with enriched monoidal categories in analogy with the modelling of physical theories are with monoidal categories. We use the enriched monoidal setting to construct a suitable definition of structure preserving map between higher order physical theories via the Grothendieck construction. We then show that the convenient feature of currying in higher order physical theories can be seen as a consequence of combining the primitive assumption of the existence of parallel and sequential composition supermaps with an additional feature of linking. In a second application we use our definition of structure preserving map to show that categories containing infinite towers of enriched monoidal categories with full and faithful structure preserving maps between them inevitably lead to closed monoidal structures. The aim of the proposed definitions is to step towards providing a broad framework for the study and comparison of novel causal structures in quantum theory, and, more broadly, a paradigm of physical theory where static and dynamical features are treated in a unified way.

§ INTRODUCTION

Traditionally,  physical theories have been concerned with the laws governing the evolution of certain physical systems, such as particles or fields.
In the ontology of a theory, the physical systems are regarded as fundamental objects, while their evolution  is  regarded as a tool for predicting relations among objects in different regions of space and time.
Over the past decade,  a series of works in quantum information theory started exploring the idea that processes  themselves could be regarded as objects, which can be acted upon by a kind of higher order physical transformations, known as quantum supermaps   [1, 2, 3, 4, 5, 6, 7].
Quantum supermaps have found a wide range of applications to  quantum information and computation [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23], and to the study of new types of causal structures arising in quantum mechanics [3, 24, 25, 26]. In addition, higher order transformations  provide a broad framework for general physical theories with dynamical causal structure, and, eventually, are expected contribute to the formulation of a complete theory of quantum gravity, as  originally suggested by  Hardy [27]. Complementary to this research direction is the development of programming languages which permit higher order types whilst retaining compatibility with quantum theory (by forbidding cloning [28], the signature of the cartesian monoidal structure underlying the standard lambda calculus), such as linear or quantum lambda calculi  [29, 30, 31, 32, 33, 34].

A compositional foundation for the study of physical theories, including quantum and classical theory, is provided by the process theory framework [35]. The framework is built on the notion of a symmetric monoidal category, which captures some basic structures present in a broad class of physical theories.  Such structures include a notion of system, a notion of processes between systems, and, crucially, a notion of the sequential and parallel composition of processes, diagrammatically represented as

\begin{equation}
\tikzfig{figs/proccomp}
\end{equation}

The tensor unit $I$ represents the trivial system, and, for every object $A$,  the processes of type $I \to A$ are viewed as states  of system $A$.
The process theoretic treatment has contributed new insights into quantum foundations and the general structure of physical theories [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51].

In a recent work [52], a process theoretic framework for supermaps was developed for the purpose of providing a categorical language for causal structures. In this framework, causal structures are represented by the objects of a $*$-autonomous category of higher order processes $\mathbf{Caus(}\mathbf{C}\mathbf{)}$ built on from a “pre-causal” category $\mathbf{C}$.  This construction reveals deep relations between $*$-autonomy and the structure of higher order transformations in quantum theory, in particular producing a convenient type system for reasoning about causal structures. However, there is a sense that the notion of a raw-material pre-causal (and so compact closed) category may be too restrictive a requirement in the study of infinite dimensional systems such as those encountered in quantum field theory, and ultimately quantum theories of gravity.

In this work we aim to to pin down a notion of a higher order theory as a mathematical structure in its own right, independently of the study of causality, and independently of the notion of a raw material category from which a theory might be constructed. Our main motivations for formalisation of structural features of higher order physical theories are the following:

* Quantum resource theories were recently captured in the abstract by construction from the notion of a sub-symmetric monoidal category [53], at the level of identifying quantum theory as forming a symmetric monoidal category, this essentially phrases the definition of a resource theory as simply the definition of a sub-theory. To build on the approaches of [54, 55, 56] and fully extend resource theories in a satisfactory way to higher order quantum theory, we need to develop a notion of higher order sub-theory.

* Identification of structural features of higher order physics will allow for the study of higher order principles for the axiomatization of physical theories. Preliminary investigations in this direction, which build from the results of this paper by beginning from the assumption of closed monodial structure and then imposing causality principles, are presented in [57].

* Once enough structural features are identified it may be possible to characterize supermap definitions in terms of universal properties. Indeed, the generally accepted definition of first-order quantum process can be motivated in this way, as arising from a universal property in the category of affine monoidal categories [58].

The contributions of this paper can be summarised in three parts, first, enrichment is used to formalize in categorical terms the existence of sequential and parallel composition supermaps. Second, suitable structure preserving maps with respect to enrichment are defined and identified as the result of the Grothendieck construction for the change of base for enriched symmetric monodial categories. Third, closed monoidal structure of higher order quantum theories is motivated in terms of extra axioms on top of monoidal enrichment, in doing so the previously developed notion of structure-preserving map is applied. We now expand on each of these points in more detail.

Process Manipulation:
Any symmetric monoidal category $\mathbf{C}$, can be interpreted as theory of processes which can be composed in sequence or in parallel. A feature common to theories of transformations of processes is the existence of higher order transformations which actually perform these composition rules. Namely a key feature is the existence of higher order processes which put processes together in sequence or in parallel
\[  \tikzfig{figs/axiom_seq_a} \quad \quad \quad \quad \tikzfig{figs/axiom_par_1}. \]
This feature of higher order processes can be identified with the categorical definition of a $\mathbf{V}_{\cong}$-smc $\mathbf{C}$. This equips the higher theory $\mathbf{V}$ with types $[A,A']$ representing the space of processes of tpye $A \rightarrow A'$ in a lower order theory $\mathbf{C}$, and a tensor $\otimes_{V}$ so that $[A,A'] \otimes [B,B']$ represents the space of bipartite processes which can be plugged together in either order, being treated as freely manipulable.

Structure Preserving Maps:
Our second contribution is to show that one can define structure preserving maps between theories of higher order transformations. These are defined to be functors $\mathcal{F}^C:\mathbf{C} \rightarrow \mathbf{C}', \mathcal{F}^{V}:\mathbf{V} \rightarrow \mathbf{V}'$ on the lower and higher order parts of the theory along with morphisms of type $\mathcal{F}_{AB}:\mathcal{F}^{V}[A,B] \rightarrow [\mathcal{F}^{C}A,\mathcal{F}^{C}B]$ which encode the preservation  of enriching structure
\[  \tikzfig{figs/inuitive_functor}. \]
These functors are then identified as the result of the Grothendieck construction applied to the change of base functor for symmetric monoidal enriched categories. This introduced notion of structure preserving map allows us to formalise the following, given the statments $(a)$: $\mathbf{C}^3$ is a theory of higher order transformations for $\mathbf{C}^2$ and $(b)$: $\mathbf{C}^2$ is a theory of higher order transformations for $\mathbf{C}^1$, then there is a morphism $\Gamma:(b) \rightarrow (a)$ embedding the latter statement into the former
\[  \tikzfig{figs/inuitive_functor_2}. \]
A key application of identifying structure preserving maps as the result of the Grothendieck construction is that such maps are immediately guaranteed to be composable forming a category of higher-order theories.

Linking and Process Manipulation $\rightarrow$ Closed Monoidal Structure:
The third contribution is to show that the possibility to curry processes, that is, the existence of closed monodial structure, can be derived from appending an additional notion to the above primitive operational principles observed in higher order physical theories. We show that currying can be viewed as a consequence of (i) The possibility to compose processes in sequence, (ii) The possibility to compose processes in parallel, (iii) The possibility to translate between an object $A$ and the space $[I,A]$ of states on $A$. These principles are combined together in the definition of a linked, and faithful, enriched monoidal category, which is shown to be equivalent to the definition of closed monoidal category. The crux of the proof can be conveyed intuitively using the following picture
\[  \tikzfig{figs/axiom_eval}.   \]
Here we see links used to convert between first and second-order systems, combined with sequential composition supermaps to construct an evaluation process of type $A \otimes [A,B] \rightarrow B$. This result allows us to state, a series of reasonable physical principles which motivate working with closed monoidal structure without directly assuming the possibility to curry processes.

By using the introduced notion of structure  preserving map $\Gamma:(b) \rightarrow (a)$ between layers within sequences of higher order theories: we proceed to generalise this result by showing that any infinite sequence of enriched monoidal categories, with well behaved structure preserving maps between the layers of the sequence, leads to closed monoidal structure
\[\tikzfig{figs/hopt_2a}. \]
Closed monoidal structure is a simple to state and easy to interpret mathematical structure on top of monoidal structure, the aim of these results is to show that deductions made by combining other physical principles with closed monoidal structure are likely to be general statements about higher order physical theories, first steps in the direction of combining closed monoidal structure with otherv standard physical principles such as causality and determinism are given in [57].

After laying the ground work we consider iterated theories of supermaps, to provide a formalisation of the notion of a theory of super-super maps and so on.

\begin{equation}
\tikzfig{figs/even_higher}
\end{equation}

We then consider theories which are conveniently infinitely iterated as a consequence of being internal or self-contained, in other words theories with the property that they are a theory of higher order transformations for themselves. We observe that this leads to precisely a characterisation of closed monoidal categories. The characterisation is entirely operationally motivated by two key principles:

* The theory permits the manipulation of all higher order transformations

* There is an equivalence between having access to a generic system type $A$ and the higher order type $[I,A]$ associated to states on $A$ - the processes from the monoidal unit $I$ to type $A$.

We call such categories  Linked enriched monoidal Categories and prove that they are closed symmetric monoidal categories by characterising closed monoidal categories in terms of enriched categories.

A category $\mathbf{C}$ has a Linked faithful self-enrichment if and only if it is a closed symmetric monoidal category.

By using the higher order notion of a enriched monoidal category from the outsety we use the above observation to develop the following corollary:

A category $\mathbf{C}$ is a Linked enriched monoidal Category if and only if it is a closed symmetric dimonoidal category such that the canonical natural transformation
\[  [A,A'] \boxtimes [B,B'] \Rightarrow [A \otimes B,A' \boxtimes B']  \]
is an isomorphism

Where the notion of a closed symmetric dimonoidal category is introduced to be nothing other than a closed symmetric monoidal category equipped with an additional product $\boxtimes$ with respect to which $\otimes$ and $\boxtimes$ form a dimonoidal category.
We finish by proving an analogous structure theorem for towers of higher order theories theories, each level of the tower being a theory of supermaps for the theory bellow it. We provide a suitable notion of a tower of enriched monoidal Categories, with the higher levels of the hierarchy representing physical transformations on the lower levels. Operationally, one can associate higher levels to more powerful agents, who have the ability to manipulate more and more complex physical objects. The notion of a Merger of such a hierarchy is introduced as a candidate for the notion of a theory in which the inhabitants are all-powerful, with the capability to manipulate objects of any level in a hierarchy of super-theories. For this definition, we formally require that a hierarchy of enriched monoidal Categories be embedded into a single symmetric monoidal category, and that the different levels of the hierarchy are linked so that it is possible to transition between the layers of the hierarchy, formally this linking is modelled by the existence of an isomorphism $A\simeq [I,A]$ between each system type $A$ and the higher order type $[I,A]$ representing the space of states on (elements of) $A$. We use the above structure to propose a fundamental object of study in higher order physics,

A Merger for a coherent sequence of enriched monoidal categories $\mathbf{C}^{(i)}$ is a $2$-Cone $(\mathcal{F}_i:\mathbf{C}^{i} \longrightarrow \mathbf{C})$ over the diagram
\[\begin{tikzcd}
\dots & \mathbf{C}_{i-1} \arrow[rr, "{\mathcal{R}_{i-1}^i(-)}"] & & \mathbf{C}^{i} \arrow[rr, "{\mathcal{R}_{i}^{i+1}(-)}"]  &                                               & \mathbf{C}^{i + 1} & \dots
\end{tikzcd}\]
in $\mathbf{ffSymCat}$: the $2$-Category of Symmetric monoidal categories, symmetric monoidal functors, and symmetric monoidal natural transformations, such that the canonical functor
\[\coprod_{i} \mathcal{F}_i: \coprod_{i} \mathbf{C}^{i} \rightarrow \mathbf{C} \]
from the co-product $\coprod_{i} \mathbf{C}^{i}$ in $\mathbf{Cat}$ is essentially surjective

The final technical contribution of this manuscript is to demonstrate that every such higher order theory possesses a simple, convenient, and familiar categorical property, that of being closed.

The apex $\mathbf{C}$ of any Merger of infinite order is a Closed Symmetric Monoidal Category.

\[\tikzfig{figs/hopt_2a}\]

This theorem derives the notion of closed-ness without reference to any physical feature of a theory except the condition that it be an infinite tower of higher order physical theories. Closed monoidal structure is the key notion for linear lambda calculi <cit.> and so our main theorem entails that every complete higher order theory of physics over a base physical theory $\mathbf{C}_{1}$ will permit a notion of $\mathbf{C}_{1}$-Lambda Calculus, in turn this connection along with the results of [52] may aid the cross pollination of results between the development of quantum lambda calculi [29, 30, 31, 32] and the study of causal structure and higher order physics from the perspective of GPTs <cit.> and CPTs <cit.>.

The connections between new <cit.> and old [59] categorical structures uncovered result in a convenient end product for the study of higher order physics, a closed monoidal category. A conceptual takeaway from these results is that the possibility for agents to plug processes together in sequence and in parallel, along with a linking between lower and higher order levels of a theory, are sufficient to operationally justify the convenient mathematical structure of the possibility to curry any process. Just as with the notion of a compact closed category, this categorical notion can easily be exported to the process theoretic framework for physics, now with strong motivation to do so. With a sound framework in place, the imposition of physical principles on top of higher order structure, the development higher order extensions of infinite dimensional quantum, field-theoretic, and post-quantum process theories <cit.>, the characterisation of the notion of higher order measurement <cit.>, the exploration of the consequences for higher order resource theories <cit.>, the relation to open diagrams in terms of the co-end calculus <cit.>, the intersection with probabilistic <cit.>, inferential theories <cit.>, and the extension to operational theories with time symmetry <cit.> present a non-exhaustive list of research avenues that could in principle be supported by this framework.

§ NOTATION AND BASIC DEFINITIONS

We will use the abbreviation SMC for symmetric monoidal category, the definition of which may be found in [59]. An SMC consists of objects $A,B, \dots$ morphisms $f:A \rightarrow B$ and composition rules $(\circ, \otimes)$. A morphism $f:A \rightarrow B$ can be represented by a box with input wire $A$ and output wire $B$. The parallel composition $f \otimes g$ of morphisms is written by placing $f$ next to $g$, the sequential composition $g \circ f$ of $f:A \rightarrow B$ and $g:B \rightarrow C$ is written by connecting boxes along wire $B$ as in the following pictures:

\begin{equation}
\tikzfig{figs/proccomp} .
\end{equation}

There is furthermore a unit object $I$ which is not explicitly written, interpreted as representing only empty space. Similarly associativity of sequential composition and associativity of parallel composition up to natural isomorphism are absorbed into the graphical language, neither being explicitly written. The categorical notion of one monoidal category living inside another is that of a monoidal functor. In general, there may be more than one way of  representing a monoidal category $\mathbf{C}$ inside another monoidal category $\mathcal{D}$. The notion of equivalence between two representations  is that of a monoidal natural isomorphism. We will often refer to a full subcategory of a category $\mathbf{C}$ with objects given by combining all objects in some set $S \subseteq ob(\mathbf{C})$ iteratively using some family of functions $\boxtimes_k :ob(\mathbf{C}) \times ob(\mathbf{C}) \rightarrow ob(\mathbf{C})$. As a shorthand for such a set generated by $S$ and functions $\boxtimes_k$ we use the symbol $S|\boxtimes_1| \dots | \boxtimes_n$.

Closed Monoidal Categories
Closed monoidal structure [59] is a standard categorical structure defined with the purpose of abstracting the notion of currying found in the category $\mathbf{Set}$ of functions between sets. Currying for functions is the property that for every pair of sets $A,B$ there exists a function $\texttt{eval}_{A,B}: A \times \mathbf{Set}(A,B) \rightarrow B$ which on elements is defined by $\texttt{eval}_{A,B}(a,f) : = f(a)$. In the general monoidal setting then closed monoidal structure of a monoidal category $\mathbf{C}$ is given by requiring a co-universal arrow of type $\texttt{eval}_{A,B}:A \otimes [A,B] \rightarrow B$. Explicitly such a co-universal arrow specifies for each morphism $f:A \otimes C \rightarrow B$ a unique morphism $\hat{f}:C \rightarrow [A,B]$ such that
\[\tikzfig{figs/closed_monoidal}. \]
This generalises curring by giving a natural isomorphism between $\mathbf{C}(A \otimes  C,B)$ and $\mathbf{C}(C,[A,B])$. The evaluation morphism can inuitively be understood as an open hole into which a process can be inserted.

Higher Order Quantum Theory
Moving beyond first-order process theories in which objects are interpreted as representing state spaces and processes interpreted as transformations of states is the higher order framework for quantum theory, in which transformations of processes are considered, intuitively represented as
\begin{equation}
\tikzfig{figs/informal}
\end{equation}
. To define iterated higher order transformations a construction is provided in [7] which takes advantage of the Choi isomorphism. In this construction types are first defined from atomic types $A,B, \dots$ combined with bracketing and arrows to form terms such as $(A \Rightarrow B)$ and $(A \Rightarrow B) \Rightarrow (C \Rightarrow D)$. For each atomic type $A$ a raw material Hilbert space $H_{A}$ is assigned along with a deterministic set $T_{1}A := \{ \rho \in \mathcal{L}(H_A) \quad s.t \quad \mathbf{Tr}[\rho] = 1 \wedge \quad \rho \geq 0 \}$ representing deterministic first order objects. For any type $A$ built from atmoic types $A_i$ the linear space $T_{\mathbf{R}}A := \otimes_{i}T_{\mathbf{R}}A_i$ is assigned, deterministic sets are given inductively by

\[  M \in T_{1}(A \Rightarrow B) \iff M \geq 0 \quad \wedge \quad (c^{-1}M)T_{1}A \subseteq T_{1}B  \]

where $c^{-1}M$ represents the inverse of the choi isomorphism applied to $M$. Non-deterministic admissible states are given by

\[  M \in TA \iff M \geq 0 \quad \wedge \quad \exists D \in T_1 A \quad  s.t \quad M \leq D   \]

Defining ${A}^{*} := A \Rightarrow I$ then a “parallel" type constructor $A \otimes B := (A \Rightarrow B^{*})^{*}$. For atomic types $A,B,A',B'$ then $(A \Rightarrow A')$ represents the space of quantum channels from $A$ to $A'$ and $(A \Rightarrow A') \otimes (B \Rightarrow B')$ represents the space of no-signalling channels from $AB$ to $A'B'$. I.E those which forbid signalling from $A$ to $B'$ and from $B$ to $A'$:
\[ \tikzfig{figs/loop_sig_pre}  \]
. All of the above data can be used to construct a monoidal category $\mathbf{HOQT}$ with $\mathbf{HOQT}(A,B) := c^{-1}(\mathbf{T}(A \Rightarrow B))$ and sequential and parallel composition inhertied by sequential and tensor products for linear maps. One can define a series of monoidal subcategories $\mathbf{HOQT}_{i}$ defined inductively by taking $\mathbf{HOQT}_1$ to be the full subcategory of $\mathbf{HOQT}$ with objects given by atmoic types, then defining $\mathbf{HOQT}_{i+1}$ to be the full subcategory with objects given by $A_{i} \Rightarrow B_i|\otimes$ for $A_i$ and $B_i$ types in $\mathbf{HOQT}_i$. For any $A,B,C$ in $\mathbf{HOQT}^2$ there exists a morphism $\circ_{ABC}: (A \Rightarrow B) \otimes (B \Rightarrow C) \rightarrow A \Rightarrow C$ in $\mathbf{HOQT}^2$ given by plugging together wires of non-signalling channels
\[ \tikzfig{figs/loop_sig}  \]
Indeed by the characterisation of deterministic sets one can see that $\circ_{ABC}$ is a deterministic morphism since any deterministic non-signalling transformation factorises as:
\[ \tikzfig{figs/loop_decomp}  \]
and so applying the morphism $\circ_{ABC}$ gives:
\[ \tikzfig{figs/loop_comp}  \]
which is deterministic since first order deterministic processes in $\mathbf{HOQT}$ define symmetric monoidal categories. This observation, that there is a composition process which can be applied to tensor products of types generalises to $\mathbf{HOQT}^i$ and can be seen as a consequence of the fact that $\mathbf{HOQT}$ is a closed monoidal category. In this sense the existence of a composition process can be seen as a generalisation of the non-signalling property of quantum chnanels to higher order processes.

Higher Order Causal Categories
Given any pre-causal (and so compact closed) category $\mathbf{C}$ a category $[\mathbf{C}]$ can be constructed which generalises $\mathbf{HOQT}$ including lower order types $A,B, \dots$ and methods for constructing higher order types given by closed monoidal structure, which assigns to each pair of objects $A,B$ an object $[A,B]$ representing the space of transformations from $A$ to $B$. Higher order processes can then be represented as those acting on higher order types, for instance $S:[A,A'] \rightarrow [B,B']$ represents a quantum supermap from processes of type $A \rightarrow A'$ to processes of type $B \rightarrow B'$. In this paper we will refer to the subcategory $[\mathbf{C}]_1$ as the category with objects given by first order types. For $\mathbf{C} := \mathbf{CPM}[\mathbf{FHilb}]$ this is equivalent to the category of CPTP quantum processes. We similarly refer to objects in $[A_1,B_1]|\otimes| \& $ with $A_1,B_1$ first order types as second order types and define $[\mathbf{C}]_{2}$ to be the full subcategory with objects given by second order types. Iterating this we can define $[\mathbf{C}]_{n+1}$ with objects given by $[A_n,B_n]|\otimes| \& $. Each such category is symmetric monoidal with unit given by $I_{n} := [I,[I,[ \dots ]]$. The same observations for $\mathbf{HOQT}$ apply to $[\mathbf{C}]$ in general, that there exist sequential composition processes $\circ_{ABC}$ for any types $A,B,C$. This can be checked explicitly or seen as a consequence again of closed monoidal structure of $[\mathbf{C}]$. The caus construction can be generalised to include non-deterministic events as are present in $\mathbf{HOQT}$, whenever the pre-causal category $\mathbf{C}$ is a monoidal poset-enriched category. Intuitively a monoidal poset enriched category is one for which every homset $\mathbf{C}(A,B)$ is equipped with a pre-order $\leq$ such that $f \leq f' \wedge g \leq g' $ implies both that $ f \circ g \leq f' \circ g'$ and $f \otimes g \leq f' \otimes g'$. The objects of $\mathbf{Caus}_{\leq}[\mathbf{C}]$ (from now on denoted $[\mathbf{C}]_{\leq}$ for convenience) are taken to be those of $[\mathbf{C}]$ and the morphisms from $(A,c_A)$ to $(B,c_B)$ are taken to be those such that $f \leq f' \in [\mathbf{C}]((A,c_A),(B,c_B))$. Note that $\mathbf{Caus}_{=}[\mathbf{C}] = \mathbf{Caus}[\mathbf{C}]$, we however keep in mind the preorder on positive operators over Hilbert spaces inherted from the natural ordering relation on the positive numbers, in this case the non-deterministic events mirror those in $\mathbf{HOQT}$.

Higher Order Causal Categories
Moving beyond first-order process theories in which objects are interpreted as representing state spaces and processes are interpreted as transformations of states, is the higher order framework for quantum theory, in which transformations of processes are considered, intuitively represented as
\begin{equation}
\tikzfig{figs/informal}.
\end{equation}
To define iterated higher order transformations of quantum processes a construction of higher order quantum theory ($\mathbf{HOQT}$) is provided in [7] which takes advantage of the Choi isomorphism. The Choi isomorphism can be viewed as a particular consequence of compact closure, and using this observation the deterministic part of $\mathbf{HOQT}$ can be generalised to a wider variety or raew-material physical theories. For any pre-causal (and so compact closed) category $\mathbf{C}$ a category $\mathbf{Caus}[\mathbf{C}]$, which we now for notational convenience refer to as $[\mathbf{C}]$, can be constructed. $[\mathbf{C}]$ includes lower order types $A,B, \dots$, two tensor products $(\otimes, \& )$, and methods for constructing higher order types given by closed monoidal structure. To each pair of objects $A,B$ an object $[A,B]$ is specified representing the space of transformations from $A$ to $B$. Higher order processes can then be represented as those acting on higher order types, for instance $S:[A,A'] \rightarrow [B,B']$ represents a quantum supermap from processes of type $A \rightarrow A'$ to processes of type $B \rightarrow B'$. In this paper we will refer to the subcategory $[\mathbf{C}]^1$ as the category with objects given by first order types as defined in [52]. For $\mathbf{C} := \mathbf{CPM}[\mathbf{FHilb}]$ then $[\mathbf{C}]^1$ is equivalent to the category of CPTP quantum processes. We similarly refer to objects in $[A_1,B_1]|\otimes| \& $ with $A_1,B_1$ first order types as second order types and define $[\mathbf{C}]^{2}$ to be the full subcategory with objects given by second order types.  Iterating this we can define $[\mathbf{C}]^{n+1}$ with objects given by $[A_n,B_n]|\otimes| \& $. Each such category is symmetric monoidal with unit given by $I_{n} := [I,[I,[ \dots ]]$.

Inuitively, each of these categories $[\mathbf{C}]^{n+1}$ can be interpreted as a theory of higher order transformations of the processes in $[\mathbf{C}]^{n}$. In this paper we will argue the a key feature of $[\mathbf{C}]^{n+1}$ which allows it to be considered in this way is that it contains morphisms which implement the sequential and parallel composition of morphisms of $[\mathbf{C}]^{n}$. Let us now see how such morphisms can be seen to be present within the layers of HOCCs. For first order types $A,B,A',B'$ then $[A , A'] \otimes [B , B']$ represents the space of no-signalling channels from $AB$ to $A'B'$. I.E those which forbid signalling from $A$ to $B'$ and from $B$ to $A'$:
\[ \tikzfig{figs/loop_sig_pre}.  \]

For any $A,B,C$ in $[\mathbf{C}]^1$ there exists a morphism $\circ_{ABC}: [A,B] \otimes [B,C] \rightarrow [A,C]$ in $[\mathbf{C}]^2$ given by plugging together wires of non-signalling channels
\[ \tikzfig{figs/loop_sig}.  \]
Indeed, one can see that $\circ_{ABC}$ is a morphism by checking it preserves first order processes, this is verified by noting that every deterministic non-signalling transformation factorises as:
\[ \tikzfig{figs/loop_decomp}  \]
and so applying the morphism $\circ_{ABC}$ gives:
\[ \tikzfig{figs/loop_comp}  \]
which is a morphism of $[\mathbf{C}]^1$ since $[\mathbf{C}]^1$ is an smc. This observation, that there is a composition process which can be applied to tensor products of types generalises to $[\mathbf{C}]^i$ and can be seen as a consequence of the fact that $[\mathbf{C}]$ is a closed monoidal category. In this sense the existence of a sequential composition process can be seen as a generalisation of the non-signalling property of quantum channels to higher order processes.

The $\mathbf{Caus}[\mathbf{C}]$ construction can be generalised to $\mathbf{Caus}_{\leq}[\mathbf{C}]$ to include non-deterministic events as are present in $\mathbf{HOQT}$ whenever the pre-causal category $\mathbf{C}$ is a locally posetal monoidal $2$-category. Intuitively a locally posetal monoidal $2$-category is one for which every homset $\mathbf{C}(A,B)$ is equipped with a partial-order $\leq$ such that $f \leq f' \wedge g \leq g' $ implies both that $ f \circ g \leq f' \circ g'$ and $f \otimes g \leq f' \otimes g'$. The objects of $\mathbf{Caus}_{\leq}[\mathbf{C}]$ (from now on denoted $[\mathbf{C}]_{\leq}$ for convenience) are taken to be those of $[\mathbf{C}]$ and the morphisms from $(A,c_A)$ to $(B,c_B)$ are taken to be those $f$'s  such that there exists $f'$ s.t $f \leq f' \in [\mathbf{C}]((A,c_A),(B,c_B))$. Note that $\mathbf{Caus}_{=}[\mathbf{C}] = \mathbf{Caus}[\mathbf{C}]$, we however keep in mind the preorder on positive operators inherted from the natural ordering relation on the positive numbers, in this case the non-deterministic events are those of $\mathbf{HOQT}$. From now on we denote $\mathbf{Caus}_{\leq}[\mathbf{C}]$ by $[\mathbf{C}_{\leq}]$, and identically to the construction given for $[\mathbf{C}]$ one can construct for each $i$ an $i^{th}$-order category $[\mathbf{C}_{\leq}]^i$.

§.§ Linearly Distributive Categories

A linearly distributive category is one which models the kinds of composition permitted in a poly category. Inuitively composition in a polycategory is given by composing one-leg-at-a-time as in the following picture. Two tensor polycategories are a conservative extension  of polycategories and two tensor polycategories categories are in turn equivalent to linearly distributive categories. We will use linearly distributive categories to ensure that our morphisms can be applied in parallel in the sense which is familliar to the quantum foundations community - by means of a monoidal product.

A linearly distributive category is

A functor between linearly distributive categories is taken to be one which is lax in $\otimes$ and colax in $\boxtimes$.

A linearly distributice functor

A linearly distributive natural transformation

A pseudo monoida in a monoidal $2$-category is a pair of morphisms $m: A \otimes A \rightarrow A$ and $u: I \rightarrow A$ which satisfy the defining equations of a monoid up-to-natural isomorphism, with in addition a series of coherence conditions between those isomorphisms. Considering $\mathbf{Cat}$ as a monoidal $2$-category with monoidal product given by the product $\times$ in $\mathbf{Cat}$ then in fact a neat characterisation of monoidal categories can be given, they are pseudomonoids in $\mathbf{Cat}$. the tensor product bifunctor $\otimes: \mathbf{C} \times \mathbf{C} \rightarrow \mathbf{C}$ defines $m$ and the unit object $I$ defines the functor $u:\{ \bullet \} \rightarrow \mathbf{C}$ given by $u(\bullet) := I$. Similarly a $\mathbf{V}$-monoidal category can be efficiently expressed as a pseudomonoid in $\mathbf{VCat}_{\cong}$.

§.§ The arrow $2$ category

From any $2$-category $\mathbf{C}$ one can construct the $2$ category $\mathbf{Arr}[\mathbf{C}]$, given by taking objects of $\mathbf{Arr}[\mathbf{C}]$ to be morphisms of $\mathbf{C}$ with morphisms in $\mathbf{Arr}[\mathbf{C}]$ given by commuting squares. $2$-morphisms are given by commuting cubes in the following sense:

It is known that this kind of construction indeed has the structure of a $2$-category, as noted in this blog post [] (in which instead a twisted arrow construction is defined, with no essential technical difference). A proof that this data indeed gives a $2$-category is given in the appendix for completeness.

§ THE NEED FOR PROCESS MANIPULATION

Higher order processes which exhibit quantum causal structures or even classical mixtures  of causal structures typically involve combining ways of plugging processes together. Indeed the quantum switch consist in a superposition of the following two compositions.

We will need to be able to consider some space of bipartite processes on which both of these compositions are defined. This will be given by requiring that a theory of supermaps enriches the category which it is interpreted as acting on. One may have wondered if in fact one could bypass introduction of a second tensor product by considering single-input $3^{rd}$-order processes. Indee=d in higher order quantum theories one can consider the following map:

Which is the process which maps $f$ to the higher order process which takes theswitch of $f$ with an arbitrary input process. However even this form of the switch is not easilly captured without a second notion of tensor product. In <cit.> the authors introduced basic theories of single-party supermaps for circuit theories, what they found however is that they required maps to be defined on all bipartite processes, this in fact expludes even this second form of the switch.

The solution here is to require that $3^{rd}$-order maps, when said to be defined on the space of bipartite processes, are meant to preserve the spaces of type $[A \otimes B,A' \boxtimes B']$ the largest imaginable space of processes. As a result we find that either way to introduce causal structures into higher order theories requires us to introudce a second tensor product. Even when we try to model indefinite causal structure as $3^{rd}$-order maps.

The rest of this paper is laid out as follows

* We introduce some basic definitions and lemma for the most basic types of categories with two monoidal products

* We then model multi-party $2^{nd}$ supermaps by allowing for two tensor products, one representing full extendaability and one representing process manipulation

* We then  consider $3^{rd}$ and general higher order theories, in which we generalise full extendability to types of the form $[A \otimes B,A' \boxtimes B']$ so that $3^{rd}$-order maps can themmselves account for process manipulation within the category they act on.

Dimonoidal Categories

Two notions of tensor product will be used in this paper to allow us to speak of a parallel composition which constains the space of all possible bipartitie processes and a seperate composition which captures the existence of channels which can be treated as freely manipulable (in physics, non-signalling channels).

A dimonoidal category is a category $\mathbf{C}$ equipped with two monoidal structures $\boxtimes$ and $\otimes$

A dimonidal functor $\mathcal{F}: \mathbf{V}_1 \rightarrow \mathbf{V}_2$ between dimonoidal categories $\mathbf{V}_1$ and $\mathbf{V}_2$ is a functor $\mathcal{F}$ monoidal seperately for $\otimes$ and $\boxtimes$.

Since monoidal functors individually compose, dimonoidal categories and morphisms of dimonoidal categories define a category, furthermore equipped with monoidal natural transformations they define a $2$-category. The definition of a symmetric monoidal category can be packaged neatly by defining pseudo-dimonids in cat, this has the additional feature for helping to prove theorems about change of base when moving to the enriched setting.

A pseudodimonoid on object $A$ in a monoidal $2$-category is a pair of pseudomonoids on $A$, consequently a dimonoidal category is a pseudodimonoid in $\mathbf{Cat}$

From any monoidal $2$-category $\mathbf{X}$ one can construct a $2$-category $\mathbf{diMon}[\mathbf{X}]$ with

* Objects given by pseudodimonoids in $\mathbf{X}$

* Each morphism $f_{d}:(l,r) \rightarrow (l',r')$ given by a morphism $f$ of $\mathbf{X}$ equipped to a pseudomonoid homomorphism $l \rightarrow l'$ and $r \rightarrow r'$.

* Each $2$-morphism given by a $2$-morphism $\eta: f \Rightarrow g$ of $\mathbf{X}$ equipped to a $2$-morphism $\eta_{l}: f_l \Rightarrow g_l$ and $\eta_{c}: f_c \Rightarrow g_c$.

The $2$-category of dimonoidal categories can then be defined as $\mathbf{diMon}[\mathbf{Cat}]$. The above definitions can then be generalised to the enriched setting by upgrading all monoids from $\mathbf{Cat}$ to $\mathbf{VCat}_{\cong}$ for some monoidal category $\mathbf{V}$.

For any symmetric monoidal category $\mathbf{V}$, a $\mathbf{V}_{\cong}$ dimonoidal category $\mathbf{C}$ is a pseudodimonoid in $\mathbf{VCat}_{\cong}$. The $2$-category of $\mathbf{V}_{\cong}$ dimonoidal categories is defined by $\mathbf{diMon}[\mathbf{VCat}_{\cong}]$

Since lax monoidal functors preserve pseudomonoids, they also preserve pseudodimonoids, indeed this result extends to homomorphisms of pseduodimonoids and their $2$-morphisms to give the following result.

There is a monoidal functor $\mathcal{M}_{dimon}(-): \mathbf{MonCat} \rightarrow \mathbf{Cat}$ which assigns to each monoidal functor $\mathcal{F}: \mathbf{V} \rightarrow \mathbf{V}'$ a functor $\mathcal{M}(\mathcal{F}):\mathbf{diMon}_{\cong}[\mathbf{VCat}_{\cong}] \rightarrow \mathbf{diMon}_{\cong}[\mathbf{VCat}_{\cong}]$.

It is already known that for every monoidal functor $\mathcal{F}$ the change of base $\mathcal{M}(\mathcal{F})$ is a monoidal $2$-functor, from which preservation of pseudomonoids and their homomorphisms follows.

§ $2^{ND}$-ORDER PHYSICAL THEORIES

In this section we will refer to the previously defined categorical structures, to define what we consider to be the most prominent structural features of higher order physics. We will refer to two separate primitive principles: Process manipulations and full-extend-ability. We then present a basic definition of higher order theory - A $2^{nd}$-order physical theory is a enriched monoidal category. Formally:

A $\mathbf{V}$-enriched monoidal category $\mathbf{C}$ is a dimonoidal category $\mathbf{V}$ with products $\boxtimes_{\mathbf{V}}$ and $\otimes_{\mathbf{V}}$  equipped with

* Process Manipulation: The product $\otimes_{V}$ is equipped to a $\mathbf{V}$-monoidal category $\mathbf{C}$

* Full Extend-ability: There is a natural isomorphism $p:[A \otimes B,A' \otimes B'] \rightarrow [A,A'] \boxtimes [B,B']$

Such that $p$ satisfies coherence conditions with associators and unitors given in the appendix.

We now unpack this claim, explaining in informal terms the structure captured in each case and giving examples. The proposal of a basic structure for supermaps then allows us to define structure preserving maps which allow us to compare $2^{nd}$-order Theories.

§ MAIN OBSERVATION: MONOIDAL ENRICHMENT

In this section we aim to highlight and formalize a prominent feature of higher order physical theories, the existence of primitive higher order sequential and parallel composition processes. In short, we make the following observation
\[ \textit{Higher Order Theories of Transformations of Processes are Enriched Monoidal Categories}. \]
The use of enrichmment as a semantics for higher-order manipulation of functions has been previously observed in [60], where the higher order theory was taken to be cartesian, we will however be interested in theories with non-trivial correlations between processes being manipulated such as those present between the two halves of a non-signalling channel. When we use the term ”enriched monoidal category" we mean a slight generalisation of it's standard usage, rather than the notion of a $\mathbf{V}$-enriched symmetric monoidal category $\mathbf{C}$ (from now on termed $\mathbf{V}$-smc $\mathbf{C}$ for short) we use the notion of a $\mathbf{V}_{\cong}$- smc $\mathbf{C}$ (defined explicitly in the appendix). In a $\mathbf{V}_{\cong}$- enriched symmetric monoidal category $\mathbf{C}$, for each pair of objects $A,B$ of $\mathbf{C}$ there exists an object $[A,B]$ in $\mathbf{V}$ whose states represent processes in the underlying category $\mathbf{C}$ via a bijection
\[ \kappa: \mathbf{C}(A,B) \cong \mathbf{V}(I,[A,B])  \]
In string diagrams of $\mathbf{V}$ and $\mathbf{C}$ respectively the isomorphism can be represented by:
\begin{equation}
\tikzfig{figs/kappa}
\end{equation}

From now on we omit from explicitly writing $\kappa$ whenever its presence is clear. In the standard definition of a $\mathbf{V}$-smc $\mathbf{C}$ this bijection would be required to be an equality and so would not so easily incorporate standard constructions of higher order physical theories in which variants of the Choi isomorphism are used [61, 7, 52]. For each $[A,B]$ and $[B,C]$ in a $\mathbf{V}_{\cong}$-smc $\mathbf{C}$ there is a morphism in $\mathbf{V}$ which allows to plug their underlying processes together:
\[\bigcirc: [A,B] \otimes [B,C] \rightarrow [A,C]\]
represented formally on the left as a string diagram in $\mathbf{V}$ and informally on the right to show intuitively its action on the underlying category $\mathbf{C}$:

\begin{equation}
\tikzfig{figs/axiom_seq_a}.
\end{equation}

Associativity and unitality of the sequential composition process in $\mathbf{V}$ and the guaruntee that it actually implements sequential composition for $\mathbf{C}$, are represented by:

\begin{equation}
\tikzfig{figs/comp_new}.
\end{equation}

Monoidal enrichment [62] provides furthermore a parallel composition process in $\mathbf{V}$: of type
\[\otimes_{ABA'B'} : [A,A'] \otimes [B,B'] \rightarrow [A \otimes B, A' \otimes B']\]
which can be represented formally and intuitively respectively by:

\begin{equation}
\tikzfig{figs/axiom_par_1}.
\end{equation}

Again in a $\mathbf{V}_{\cong}$-smc $\mathbf{C}$ the following conditions are required, guaranteeing that $\otimes_{ABA'B'}$ really does behave like a parallel composition process:

\begin{equation}
\tikzfig{figs/comppar_new}.
\end{equation}

Which represents, associativity of parallel composition, parallel compositon with empty space having no effect, and finally that the morphism indeed implemements the parallel composition of processes respectively. lastly the condition:

\begin{equation}
\tikzfig{figs/interchange},
\end{equation}

is required, which represents the interchange law between sequential and parallel composition. The above conditions are technically not well typed without introducing the $\mathcal{V}$-natural isomorphisms which carry no operational relevence, the formal conditions which include these manipulations of empty space are taken care of in the appendix.

Full extend-ability
Given two supermaps one ought to be able to apply them to any bipartite process.
\[  \tikzfig{figs/axiom_5_new} \]
We model this by requiring an invertible natural transformation $p:[A \otimes B, A' \otimes B'] \rightarrow [A,A'] \boxtimes [B,B']$ so that any supermaps $S:[A,A'] \rightarrow [B,B']$ and $T:[C,C'] \rightarrow [D,D']$ can be combined  using $p^{-1} \circ (S \boxtimes T) \circ p$ an action of type $[A \otimes C, A' \otimes C'] \rightarrow [B \otimes D,B' \otimes D']$.

An example of a category $\mathbf{V}$ which allows for process manipulation on the processes of $\mathbf{C}$ but which does not clearly allow for full-extend-ability on $\mathbf{C}$ is the category $\mathbf{Set}$. Indeed given that $\mathbf{C}$ is a monoidal category this by definition entails that $\mathbf{C}$ is enriched in $\mathbf{Set}$. However it is not the case that we would expect any function $S:\mathbf{C(A,A')} \rightarrow \mathbf{C}(B,B')$ to represent a valid supermap, and indeed this is in part because it is not clear how to extend any function of thee above type to a function $\mathbf{C}(A \otimes E,A' o\times E') \rightarrow \mathbf{C}(B \otimes E, B' \otimes E')$.

§.§ Examples

We now present a series of examples of theories of supermaps, and observe that each is indeed and example of an enriched monoidal category.

[Higher Order Causal Categories]
For any $i$ there is a $[\mathbf{C}]_{\cong}$-smc $[\mathbf{C}]^{i}$ and similarly for any $i,i+1$ there is a $[\mathbf{C}]^{i+1}_{\cong}$-smc $[\mathbf{C}]^{i}$. In each case enrichment is ensured by the closed monoidal structure of $[\mathbf{C}]$ [52]. The same can be said for $[\mathbf{C}_{\leq}]$ and each $[\mathbf{C}_{\leq}]^{i}$.

[Unitary Supermaps]
For any $\mathbf{M}^1 \subseteq [\mathbf{C}]^{1}$ such that
\[ \forall \phi \in \mathbf{M}^1 \quad s.t \quad \tikzfig{figs/subcat_pre_1}: \quad \quad  \exists L,M \in \mathbf{M}^1 \quad s.t  \quad \tikzfig{figs/subcat_pre_2} ,  \]
one can construct a category $\mathbf{M}^2 \subseteq [\mathbf{C}]$ of completely-$\mathbf{M}$-preserving supermaps and a corresponding $\mathbf{M}^2_{\cong}$-smc $\mathbf{M}^1$. $\mathbf{M}^2$ is defined by taking objects in the algebra $[\mathbf{C}^1,\mathbf{C}^1] | \otimes | \&$ and morphisms given by $S \in \mathbf{M}^{2}([A,A'],[B,B']) \iff  \forall \phi \in \mathbf{M}^{1}(AE,A'E')  \textrm{ then } p^{-1} \circ (S \& I_{E,E'}) \circ \hat{\phi} \in \mathbf{M}^1 (B \otimes E, B' \otimes E')$. $\mathbf{M}^{2}$ inherits monoidal structure from $[\mathbf{C}]$ since for every $S,T$ then $(S \& T \& I) \circ \hat{\phi}$ is in $\mathbf{M}^1$ from the fact that $S \& I \& I$ preserves $\mathbf{M}^{1}$ as does $T \& I \& I$ and the braid. Similarly $(S \otimes T) \& I$ preserves $\mathbf{M}^{1}$ as a consequence of the embedding $X \otimes Y \rightarrow X' \& Y'$ in $[\mathbf{C}]$. To show that what remains is still a monoidal enriched category, one must check that the composition morphisms $\otimes$ and $\bigcirc$ completely preserve unitarity. Indeed this can be checked, first noting that for every type $([A,A'] \otimes [B,B']) \& X$ in the given algebra there is an embedding into $([A,A'] \otimes [B,B']) \& [C,C']$, this entails that every process of the former type embeds as a tripartite process satisfying
\[ \forall \sigma, \rho: \quad \quad  \tikzfig{figs/subcat_rho_1} , \]
and so satisfies
\[   \tikzfig{figs/subcat_1} .  \]
Using the decomposition theorem for non signalling relations then gives:
\[    \tikzfig{figs/subcat_2} .  \]
Finally, noting that the sequential composition morphism of $[\mathbf{C}]$ is constructed from the underlying compact closed structure of $\mathbf{C}$, we find that the sequential composition morphism is a morphism of $\mathbf{M}^{2}$, since it sends morphisms of $\mathbf{M}^1$ to morphisms of $\mathbf{M}^{1}$ in the following sense: $\mathbf{M}^1$:
\[ \tikzfig{figs/subcat_3}. \]
An important special case of subcategories captured by this construction is given by taking $\mathbf{C} := \mathbf{CPM}[\mathbf{FHilb}]$ and taking $\mathbf{M}^1 \subseteq [\mathbf{C}]^1$ to be the category of unitary channels. In this case $\mathbf{M}^2$ is a monoidal category of unitary supermaps, such supermaps have previously been defined and are of particular interest in quantum causal modelling [63]. The monoidal and enrichment structure whilst natural, had not yet been defined to the authors knowledge.

[Idempotent completion preserves enriched monoidal categories]
Any enriched monoidal category can be completed to include idempotents as types, meaning that decoherences in lower order theories can be inherited to construct a higher order theory with classical types. For any $\mathbf{V}$-smc $\mathbf{C}$ the idempotent completions $\mathcal{K}(\mathbf{C})$ of $\mathbf{C}$ and $\mathcal{K}(\mathbf{V})$ of $\mathbf{V}$ define a $\mathcal{K}(\mathbf{V})$-smc $\mathcal{K}(\mathbf{C})$. The idempotent completion of a category has for each object a pair $(X,x)$ of an object $X$ of $\mathbf{C}$ and an idempotent $x:X \rightarrow X$. The enriched structure is given by the functor $[(X,x),(Y,y)] = ([X,Y],[x,y])$ and the corresponding composition morphisms $\bigcirc_{xyz}$ are given by $[x,z] \circ  \bigcirc_{XYZ} \circ ([x,y] \otimes [y,z])$ and simmillarly for the parallel composiiton morphisms. All required coherences follow from coherences of the $\mathbf{V}$-smc $\mathbf{C}$. This second example, when applied to the $[\mathbf{C}]^2_{\cong}$-smc $[\mathbf{C}]^1$ with $\mathbf{C} = \mathbf{CPM}[\mathbf{FHilb}]$ of quantum supermaps over completely positive trace preserving quantum channels, produces a theory which includes classical channels in $\mathbf{C}$ as those of type $(X,dec_{X}) \rightarrow (Y,dec_{Y})$.

For any $\mathbf{V}_{\cong}$-smc $\mathbf{C}$ one can construct the $\mathbf{V}_{\cong}^{comb}$-smc $\mathbf{C}$ which is generated by the structural morphisms $\mathbf{V}(I,[A,B]) \cup \{ \circ_{ABC} \} \cup \{  \otimes_{AA'BB'}  \} \cup \{  \texttt{coherences} \}$ of $\mathbf{V}$, meaning that all parallel composition and sequential composition supermaps are kept along with all states. This category is the category of combs of processes from $\mathbf{C}$, Indeed a comb drawn intuitively as:
\[ \tikzfig{figs/comb}    \]
can be formally represented as:
\[ \tikzfig{figs/comb_2}    \]

Consequently all combs of processes in $\mathbf{C}$ have to exist as processes in $\mathbf{V}$ for any $\mathbf{V}_{\cong}$-smc $\mathbf{C}$.

§.§ Consequences

Some basic important consequences of monoidal enrichment are the following: There always exists a family of morphisms which represent partial insertion and and a family functions which represent usage of the output of a transformation.

Partial Insertion The partial insertion morphism $\Delta: [A,X] \otimes [Y \otimes X,Z] \rightarrow [Y \otimes A,Z]$ takes a valid sub-input of a process and inserts a pre-processing there, leaving the rest of the inputs unchanged. Formally it is defined by:

\begin{equation}
\tikzfig{figs/define_partial_1},
\end{equation}

up to unitors and associators, where $\otimes : [Y,Y] \otimes [A,X] \rightarrow [YA,YX]$ and $\circ : [YA,YX] \otimes [YX,Z] \rightarrow [YA,Z]$. The partial insertion can be intuitively be understood as representing the following picture:

\begin{equation}
\tikzfig{figs/axiom_delta}.
\end{equation}

Crucially for $A = I$ then $\Delta$ satisfies:

\begin{equation}
\tikzfig{figs/define_partial_2}.
\end{equation}
Intuitively the above represents the equality between
\begin{equation}
\tikzfig{figs/axiom_delta_2},
\end{equation}

\begin{equation}
\tikzfig{figs/axiom_delta_3}.
\end{equation}

Usage The usage transformation is a particular natural transformation $\theta:{\mathcal{V}(-,[A,-])} \Longrightarrow  {\mathcal{V}([I,A] \otimes -,[I,-])} $, a family of functions $\theta_{BX}$ given by for each $S:X \rightarrow [A,B]$ taking $\theta_{BX}(S)$ to be:

\begin{equation}
\tikzfig{figs/theta_new}.
\end{equation}

Intuitively, $\theta$ place $S$ into one of the two holes of a sequential composition supermap:
\[ \tikzfig{figs/axiom_usage}.  \]
We will find that injectivity of the function $\theta$ is crucial for results about embeddings between layers of higher order theories.

A $\mathbf{V}_{\cong}$-smc $\mathbf{C}$ will be called faithful if the usage transformation
\[\theta: {\mathcal{V}(-,[A,-])} \Longrightarrow  {\mathcal{V}([I,A] \otimes -,[I,-])}\]
is a monomorphism in the functor category $\mathbf{Cat}(\mathcal{V}^{op} \otimes \mathbf{C},\mathbf{Set})$.

Faithful usage when present says that two higher order processes $S,T: X \rightarrow [A,B]$ should only be distinguishable if they are distinguishable when their outputs are applied to the space of states on $A$. Stated formally faithful usage is the requirement of injectivity, that for all $I,A,B$ the composition process $\bigcirc_{IAB}$ satisfies

\begin{equation}
\tikzfig{figs/complete_inj}.
\end{equation}

This has the additional consequence of entailing that the functor $[I,-]$ be faithful. All of the previously stated examples have faithful usage transformations.

§.§ Structure preserving maps

The standard given notion of subcategory given by the definition of an enriched monoidal category is that of a faithful monoidal $\mathbf{V}$ enriched functor, which leaves the higher order category $\mathbf{V}$ unchanged. Our goal is to develop a language which allows us to compare theories of supermaps, the ready cooked notion of morphism of enriched monoidal categories will not be fit for our purposes since an enriched monoidal functor is one which allow comparisons of the following type:
\[ \begin{tikzcd}
{(\mathbf{V},\mathbf{C})} \arrow[r] & {(\mathbf{V},\mathbf{C}')},
\end{tikzcd}    \]
that is, those in which the enriching category is left untouched. Even in the basic case of the inclusions between unitary combs unitary supermaps quantum combs and quantum supermaps, functors which which fix the enriching category are unsuitable, it is after-all the enriching categories that we are varying:
\[\begin{tikzcd}
{(\mathbf{U}^{2,comb},\mathbf{U}^{1})} \arrow[rrdd] \arrow[d] &                                          \\
{(\mathbf{U}^2,\mathbf{U}^1)} \arrow[rrdd]            &                                          \\
& & {([\mathbf{C}]^{2,comb},[\mathbf{C}]^1)} \arrow[d] \\
& & {([\mathbf{C}]^{2},[\mathbf{C}]^1)}
\end{tikzcd}\]
In fact these inclusions can be seen as instances of the change of base for enriched monodial categories. Change of base alone however, will be insufficient for our purposes, we will find under additional conditions of faithful usage, that for any sequence in which $\mathbf{C}^3$ enriches $\mathbf{C}^2$ which in turn enriches $\mathbf{C}^1$ then there exists a notion of embedding
\[  \begin{tikzcd}
{(\mathbf{C}^2,\mathbf{C}^1)} \arrow[rd] &                               \\
& {(\mathbf{C}^3,\mathbf{C}^2)}
\end{tikzcd} \]
which cannot be understood as purely a change of base. An explicit example is the embedding of the $[\mathbf{C}]^2_{\cong}$-smc $[\mathbf{C}]^1$ into the $[\mathbf{C}]^3_{\cong}$-smc $[\mathbf{C}]^2$ for any pre-causal category $\mathbf{C}$. To address these problems and provide a suitable notion of structure preservating map, we will construct a notion of functor which allows us to vary both the lower and higher categories at the same time using combinations of (vertical) change of base functors and (horizontal) enriched monoidal functors:
\[  \begin{tikzcd}
{(\mathbf{C}^2,\mathbf{C}^1)} \arrow[rd] \arrow[d, "\texttt{Change of base}"'] &                               \\
{} \arrow[r, "\texttt{Enriched Functor}"']                                     & {(\mathbf{C}^3,\mathbf{C}^2)}
\end{tikzcd} \]

This pair of functors along with their expected compatibilities, can be viewed as an instance of what we call a pm-functor, a morphism in $\mathbf{PM}:= \mathcal{G}(\mathcal{M}_{smc})$.  Here $\mathcal{G}(\mathcal{M}_{smc})$ denotes the result of the Grothendieck construction of the symmetric monoidal change of base functor $\mathcal{M}_{sym}$, the functor $\mathcal{M}_{sym}$ is  given formally in the appendix building on the results of [64]. The key components of a pm functor are laid out explicitly here to demonstrate how they incorporate combinations of functors $\mathcal{F}^{V}:\mathbf{V} \rightarrow \mathbf{V}'$ and functors $\mathcal{F}^{C}:\mathbf{C} \rightarrow \mathbf{C}'$ along with compatibilities between them.

A pm-functor is a morphism in $\mathbf{PM} := \mathcal{G}(\mathcal{M}_{sym})$, that is:

* A symmetric monoidal functor $\mathcal{F}^{V}:\mathbf{V} \rightarrow \mathbf{V}'$

* A symmetric monoidal functor $\mathcal{F}^C: \mathbf{C} \rightarrow \mathbf{C}'$

* A family of morphisms $\mathcal{F}_{AB}:\mathcal{F}^V [A,B] \rightarrow [\mathcal{F}^CA,\mathcal{F}^C B]$

Which together satisfy:
\[   \tikzfig{figs/smsc_monhom_r_nf_simp}  \]
where for readability any structural isomorphisms and symmetry conditions for monoidal functors $\mathcal{F}^{C,V}$ have been omitted and any coherences required for an enriched symmetric monoidal functor from $\mathcal{F}^{V}[-,-]$ to $[\mathcal{F}^C - , \mathcal{F}^C -]$ have been omitted as if strict.

We will refer to a morphism $\mathcal{F}$ of $\mathbf{PM}$ as fully faithful if $\mathcal{F}^V,\mathcal{F}^C$ are full and faithful and $\mathcal{F}_{A,B}$ are isomorphisms. Indeed as a consequence we have constructed a notion of sub-enriched monoidal category, as an embedding which provides a faithful pm-morphism.

There are pm-functors between the examples of unitary combs, general unitary supermaps, $2^{nd}$-order combs and $2^{nd}$ order transformations built from $[\mathbf{C}]$, as depicted in the preamble to this section. In each case $\mathcal{F}^C$ and $\mathcal{F}^V$ are given by inclusions and each $\mathcal{F}_{AB}$ can as a result be taken as the identity.

Another example with trivial $\mathcal{F}_{AB}$ is given by embedding into the Karoubi envelope.

There is a pm-functor from any enriched monoidal category to its Karoubi envelope given by the embeddings $\mathbf{C} \rightarrow \mathcal{K}[\mathbf{C}]$ and $\mathbf{V} \rightarrow \mathcal{K}[\mathbf{V}]$. Taking again $\mathcal{F}_{AB}$ to be the identity the embeddings directly preserve all structural morphisms. This functor is faithful, meaning that it represents an instance of the invariant notion of subcategory.

An example with non-trivial $\mathcal{F}_{AB}$ is given by considering embeddings between different layers $[\mathbf{C}]^i$ of $[\mathbf{C}]$. From now on we will refer to the functor $[I,-]:\mathbf{C} \rightarrow \mathbf{V}$ as the raising functor $\mathcal{R}_{C}^{V}$.

Given any $\mathbf{C}^3_{\cong}$-smc $\mathbf{C}^2$ and $\mathbf{C}^2_{\cong}$-smc $\mathbf{C}^1$ there exists a morphism $\Gamma$ between them in $\mathbf{PM}$ given by given by $\Gamma := (\mathcal{R}_{2}^{3}, \mathcal{R}_{1}^{2}, \gamma)$ where $\gamma_{A,B}:\mathcal{R}_{2}^{3}([A,B]^2) \rightarrow [\mathcal{R}_{1}^2(A),\mathcal{R}_1^2 (B)]^3 $ is given by
\begin{equation}
\tikzfig{figs/smsc_monhom_2}
\end{equation}

that is, applying partial insertion in $\mathbf{C}^3$ to the composition map in $\mathbf{C}^2$.

All the required coherences for pm-functors are here inherited by the coherence's of $\mathbf{C}^3$ and $\mathbf{C}^2$, an explicit proof is given in the appendix.

Note that here $\bigcirc_{IAB}: [I,A] \otimes_{2} [A,B] \rightarrow [I,B]$ so that $\kappa(\bigcirc_{IAB}):I \rightarrow [[I,A] \otimes_{2} [A,B]]$ and we omit unitors so that $\Delta$ has type $\Delta: [I,[A,B]] \otimes_3 [[I,A] \otimes_2 [A,B], [I,B]] \rightarrow [[I,A] \otimes_{2} I,[I,B]] \cong [[I,A],[I,B]]$.

There is a morphism in $\mathbf{PM}$ from the $[\mathbf{C}]^{i+1}_{\cong}$-smc $[\mathbf{C}]^{i}$ to the $[\mathbf{C}]^{i+2}_{\cong}$-smc $[\mathbf{C}]^{i+1}$ for every $i$. The same story holds for $[\mathbf{C}_{\leq}]$.

Before giving some concrete examples we identify a feature of iterated higher order theories.

Let $\mathbf{C}^1  \triangle \mathbf{C}^2$ and $\mathbf{C}^2  \triangle \mathbf{C}^3$ be monoidal enrichments then there exists a morphism of monoidal enrichments
\[\Gamma: \mathbf{C}^1  \triangle \mathbf{C}^{2} \rightarrow \mathbf{C}^{2}  \triangle \mathbf{C}^{3} \]
given by $\Gamma := (\mathcal{R}_{1}^{2}, \mathcal{R}_{2}^{3}, \gamma)$ where in turn $\gamma$ is given by
\begin{equation}
\tikzfig{figs/smsc_monhom_2}
\end{equation}

Given in the appendix

Indeed as a consequence, little extra work is required to show that sequences of enriched monoidal categories admit enriched monoidal functors between them:

Let $\mathbf{C}^1  s \mathbf{C}^2$ and $ \mathbf{C}^2 s \mathbf{C}^3$ be enriched monoidal Categories then there exists a enriched monoidal functor
\[\Gamma: \mathbf{C}^{1}  s \mathbf{C}^{2} \rightarrow \mathbf{C}^{2}  s \mathbf{C}^{3} \]

A proof that this example is a enriched monoidal functor is given in the appendix.

Often we will aim to compare theories of supermaps over the same underlying category $\mathbf{C}$. To do this we restrict to a category $\mathbf{SuperMon}[\mathbf{C}]$, with each object given by a $\mathbf{V}$-enriched monoidal category $\mathbf{C}$ for some $\mathbf{C}$.

$\mathbf{SuperMon}[\mathbf{C}]$ is the full subcategory $\mathbf{SuperMon}[\mathbf{C}] \subseteq \mathcal{G}(\mathcal{M})$ with

* Each object given by a $\mathbf{V}$-enriched monoidal category $\mathbf{C}$

* Each morphism having $\mathcal{F}_{\mathbf{C}} = \mathcal{I}_{\mathbf{C}}:\mathbf{C} \rightarrow \mathbf{C}$.

Again subcategories provide basic examples by inclusion, for instance the inclusion between the category of quantum combs and the category of general quantum supermaps. As a general point, since all combs have to exist in supermonoidal categories, the free category of combs is actually initial in the category $\mathbf{SuperMon}[\mathbf{C}]$.

The category $\mathbf{cwpComb}[\mathbf{C}]$ is the $2$-initla object of $\mathbf{SuperMon}[\mathbf{C}]$.

§ SELF-CONTAINED HIGHER ORDER THEORIES

Current examples of higher-order quantum-like theories have a further common feature of self-containment. We mean by this the idea that all processes, no matter their higher or lower order status,
exist in the same theory, that is, the same category. Explicit examples of theories which are self-contained in this sense are higher order quantum theory and higher order causal categories both of which are conveniently closed monoidal categories. Those familiar with classical category theory may imagine that closed monoidal structure is the appropriate mathematical formalization of the idea of self-containment. In this section rather than taking as an axiom that currying (the key feature of closed monoidal categories) is the appropriate formalization of a self-contained theory of higher order transformations, we will treat this as a statement to be proven, by combining enrichment with a more relaxed formalization of self-containment. The three operational features which we will use to derive currying and closed monoidal structure will at the intuitive level be the following:

* All processes in $\mathbf{C}$ have higher order representations in $\mathbf{C}$ (Self-enrichment).

* There is an equivalence $A \cong [I,A]$ between $A$ and the higher order system $[I,A]$ representing the states of $A$ (linking).

* The usage transformation is faithful

Conceptually, the first condition models the assumption that it is the same agents that can perform processes, super-processes, and so on. We model this with the notion of a $\mathbf{C}$-enriched monoidal Category $\mathbf{C}$. The second condition is captured by the following:

A Linked Monoidal Category is a $\mathbf{C}$ Monoidal Category $\mathbf{C}$ equipped with a monoidal natural isomorphism $\eta_A: A \rightarrow [I,A]$.

We furthermore say that a linked monoidal category $\mathbf{C}$ is faithful if it is faithful as a $\mathbf{C}_{\cong}$-smc $\mathbf{C}$. Intuitively, linked categories have enough structure to define canonical evaluation morphisms (the structural feature of closed monoidal categories), mixed-order morphisms of type $\texttt{eval}_{AB}:A \otimes [A,B] \rightarrow B$ which apply processes to lower order objects, by using link-morphisms and sequential composition morphisms:
\[  \tikzfig{figs/axiom_eval}   \]
Intuitively in the above diagram the available inputs are the bottom wire $A$ and the dotted process input of type $[A,B]$, the output wire is the top wire of type $B$. Indeed linked categories will turn out to be closed monoidal if they satisfy one additional condition - that they have faithful usage, this is proven by constructing evaluation morphisms in the above way.

A category $\mathbf{C}$ is closed smc if and only if $\mathbf{C}$ is a faithful linked monoidal category.

A full proof is given in the appendix, here we show for reference the formal construction of evaluations analogous to the above intuitive picture:

\begin{equation}
\texttt{eval} : = \tikzfig{figs/main_thm_1_a}
\end{equation}

the requirement of being faithful ensures uniqueness/universality.

In this section we considered theories which are from the start assumed to be self contained, in the next we generalise this result to infinite towers of enriched monoidal categories, using along the way the developed notion of structure [preserving map as $pm$-functor. Given that linked faithful categories are exactly closed symmetric monoidal categories they give an alternative way to view a familiar categorical structure. For instance the closed monoidal structure of $\mathbf{Set}$ can be viewed as a consequence of the fact that (i) $\mathbf{Set}$ is monoidal (ii) Trivially $\mathbf{Set}$ is monoidally enriched in $\mathbf{Set}$ (iii) The bijection $\kappa$ for enrichment provides a function $\eta : A \rightarrow [I,A]$ which is monoidal (iv) The composition function is faithful. As noted in the preliminary section $[\mathbf{C}]$ has closed monoidal structure for any $\mathbf{C}$.

§ TOWERS OF HIGHER ORDER THEORIES

In a second application of our framework we show that theories resulting from the gluing together of a suitably well-behaved tower of higher order theories, are again inevitably closed monoidal. The essence of the proof will be that of the previous section, the formal tools used will be the notions of enriched monodial category along with properties of previously constructed pm-functors (structure preserving maps) within towers of enriched categories. We begin by presenting the notion of a tower of theories over a base theory, each a theory of supermaps over the theory that precedes it. The base theory represents a given physical theory, such as quantum or classical probability theory. The second layer represents a theory of supermaps, the third layer a theory of super-supermaps, and so on:

\begin{equation}
\tikzfig{figs/hopt_3}
\end{equation}

The ultimate goal of introducing this construction is for the specification of a unified higher order theory, into which such a sequence will embed. Mathematically, a hierarchy of higher order physical processes is represented by an ascending sequence of enriched monoidal Categories

An ascending sequence of enriched monoidal Categories $\mathbf{C}^1 ,  \mathbf{C}^2 ,  \dots  ,  \mathbf{C}^N$ is a specification for each $i \in [\mathbf{N-1}]$ of a $\mathbf{C}^{i + 1}_{\cong}$-smc $\mathbf{C}^i$.

In any such sequence, the category ${\cal C}_i$ is  “encoded” into the higher level ${\cal C}_{i+1}$  by the monoidal raising functor $\mathcal{R}_{i}^{i+1}(-):= [I_{i},-]^{i+1}$. It will be convenient to define the following compact notation for the induced encoding (full faithful braided monoidal functor) from level $i$ to level $j>1$.

\[\mathcal{R}_i^j: \mathbf{C}^i \longrightarrow \mathbf{C}^j \quad \quad \mathcal{R}_i^j := \mathcal{R}_{j-1}^{j} \circ \mathcal{R}_i^{j-1}\]

There is a sense in which the agents inhabiting layer $\mathbf{C}_{j}$ are strictly more powerful than the inhabitants of $\mathbf{C}_{i < j}$, in the sense that each $\mathbf{C}_{i<j}$ may be embedded into $\mathbf{C}_{j}$. For any finite sequence $\mathbf{C}^{i}$ of length $n$, the final category $\mathbf{C}_{n}$ may be seen as the arena in which agents may manipulate processes from any category in the preceding sequence. In fact as observed in section $3$ the embeddings can be run in parallel and phrased as pm-morphisms.

Let $\mathbf{C}^1 ,  \mathbf{C}^2,   \dots   , \mathbf{C}^N$ be a an ascending sequence of monoidal enriched categories: then for every $0<i<N-1$ there exists a morphism $ \Gamma_i $ from the $\mathbf{C}^{i+1}$-smc $\mathbf{C}^i$ to the $\mathbf{C}^{i+2}_{\cong}$-smc $\mathbf{C}^{i+1}$ in $\mathbf{PM}$.

Direct consequence of lemma $1$.

We will find that closed monoidal structure arises when the $\Gamma_i$ are fully faithful.

An ascending sequence of monoidal enrichments is fully faithful if each $\mathbf{C}^{i+1}_{\cong}$-smc $\mathbf{C}^i$ is faithful and each $\Gamma_i$ is fully faithful.

Furthermore sequences of enriched monoidal categories admit enriched monoidal functors between their layers, again proven in the previous section:

Let $\mathbf{C}^1   \mathbf{C}^2,   \dots    \mathbf{C}^N$ be an ascending sequence of enriched monoidal r-Categories then for every $0<i<N-1$ there exists a enriched monoidal functor
\[\Gamma^{i}: \mathbf{C}^{i}   \mathbf{C}^{i+1} \rightarrow \mathbf{C}^{i+1}   \mathbf{C}^{i+2} \]
an ascending sequence will be referred to as fully coherent if every $\gamma^i$ is full and faithful

We will discover that when fully coherent sequences surjectively embed into a monoidal category, a merger for the sequence, it is guaranteed that the merger will in turn be closed monoidal.

Theories consisting of fully faithful sequences are closed monoidal
We now present operational conditions on a theory $\mathbf{C}$ in terms of an embedded ascending sequence $\mathbf{C}^{(i)}$ which will lead to closed monoidal structure for $\mathbf{C}$. Intuitively the conditions are the following

* $\mathbf{C}$ contains nothing more and nothing less than a fully faithful sequence $\mathbf{C}^{i}$ of enriched monoidal categories

* $\mathbf{C}$ provides links between the layers of $\mathbf{C}^{i}$
Such an embedding for a generic sequence $\mathbf{C}^{(i)}$ is captured categorically by a sequence of full and faithful functors $\mathcal{F}_{i}: \mathbf{C}^{(i)} \rightarrow \mathbf{C}$. To capture that there is nothing more in $\mathbf{C}$ we require a further condition of essential surjectivity on the union (co-product) functor
\[\coprod_{i} \mathcal{F}_i : \mathbf{C}^{(i)} \rightarrow \mathbf{C} .\]

Finally we impose the condition that there be a link between layers of the theory. The most basic notion of a linking between levels is via an isomorphism $A \cong [I,A]$. Formally this equivalence when consistent with the monoidal embeddings, is captured by the existence of a monoidal natural isomorphism

\begin{equation} \eta_{i-1}^{i} : \mathcal{F}_{i-1}(-)  \longrightarrow \mathcal{F}_{i} \circ \mathcal{R}_{i-1}^{i} (-)  \, ,\end{equation}

In short, $\eta_{i-1}^{i}$ provides a witness for the equivalence between $A$ and $[I,A]$ inside $\mathbf{C}$. For ease of notation we will denote the inverse $(\eta^{i}_{i-1})^{-1}$ by $\eta^{i-1}_{i}$ when needed. The existence of a natural isomorphism $\eta_{i-1}^{i}$ for each $i$ is can be concisely phrased in the language of $2$-Categories, it is precisely the requirement that $\mathbf{C}$ be a $2$-Cone in the $2$-Category $\mathbf{ffSymCat}$ of

* Symmetric monoidal categories

* Full faithful strong symmetric monoidal functors

* Symmetric monoidal natural transfomations

For a diagram $\mathcal{D}$ in $\mathbf{ffSymCat}$ given by a fully faithful sequence of enriched monoidal categories, and the monoidal functors $\mathcal{R}_{i}^{i+1}: \mathbf{C}^i \longrightarrow \mathbf{C}^{i + 1}$ between them, a cone over $\mathcal{D}$ is an “apex" category $\mathbf{C}$ equipped with a family of functors $\mathcal{F}_i:\mathbf{C}^i \rightarrow \mathbf{C}$ such that each of the following triangles commutes up to a monoidal natural isomorphism $\eta_{i}^{i+1}$.

\[\begin{tikzcd}
\mathbf{C}^i \arrow[rr, "{\mathcal{R}_{i}^{i+1}}"] \arrow[rrdd, "\mathcal{F}_{i}"'] &                                               & \mathbf{C}^{i + 1} \arrow[dd, "\mathcal{F}_{i+1}"] \\
& {} \arrow[ru, "\eta" description, Rightarrow] &                                                   \\
&                                               & \mathbf{C}
\end{tikzcd}\]

The above discussion culminates in the following definition, that of a Merger.

A Merger for a fully faithful ascending sequence of enriched monoidal categories (“merger" for short) $\mathbf{C}^{(i)}$ is a $2$-Cone $(\mathcal{F}_i: \mathbf{C}^i \longrightarrow \mathbf{C})$ over the diagram
\[\begin{tikzcd}
\dots & \mathbf{C}_{i-1} \arrow[rr, "{\mathcal{R}_{i-1}^{i}}"] & & \mathbf{C}^i \arrow[rr, "{\mathcal{R}_i^{i+1}}"]  &                                               & \mathbf{C}^{i + 1} & \dots
\end{tikzcd}\]
in $\mathbf{ffSymCat}$ such that

* $\coprod_{i} \mathcal{F}_i$ is essentially surjective

A Merger is furthermore termed “$n$-th order" if the sequence has length $n$.

For any sequence of finite order the notion of a merger is essentially trivial, given a sequence of order $n$ one can simply construct a cone of the above type by taking $\mathbf{C} = \mathbf{C}^N$ and taking $\mathcal{F}_{i}:= \mathcal{R}_{i}^{n}$. The primary technical contribution of this manuscript is the observation that the apex of any $\infty$-Order Merger possesses a simple categorical property, it must be a closed monoidal category.

The apex $\mathbf{C}$ of any Merger of infinite order is a closed symmetric monoidal category.

This result provides an operational justification for using closed symmetric monoidal categories to study higher order physics. From this position the consequences of basic physical principles in higher order physics can be explored within a simple mathematical addition to symmetric monoidal categories. First steps in this direction of research are taken in [65], in which an interaction between the strength of spatial correlations, determinism, and the possibility of signalling between parties is observed.

§ CONCLUSION

Presented in this manuscript is a proposed beginning of a mathematical framework for higher order physical theories, centred around monoidal enrichment. This framework is analogous in attitude to the process theory framework for standard physics based on the notion of a symmetric monoidal category. The definitions proposed are easily iterated to define towers of theories, after-which currying in higher order theories is understood through two results: Linked faithful enriched categories are exactly closed monoidal categories, and categories into which infinite towers of higher order theories are suitably embedded, are always similarly always closed monoidal. Many open questions then follow from this framework:

* Do quantum supermaps, quantum combs, higher order quantum theory, and higher order causal categories, satisfy universal properties with respect to the above defined structure preserving maps, in analogy to the universal properties satisfied by CPTP maps [66]?

* How can we freely construct theories of supermaps over arbitrary circuit theories?

* What other properties should be expected of theories of supermaps, in particular, how can another key feature of local-applicability [67, 68], be combined with enriched monoidal structures.

* How can we model supermaps on constrained spaces such as one-way signalling channels and routed quantum supermaps [69, 70, 71]?

* What does the view of higher order physics as enrcihed monoidal structure have to say about resource theories of higher order processes [54, 55, 56]?

This paper aims to lay a basic starting point from which the above questions can be formulated and answered.

MW would like to thank A Vanrietvelde, J Hefford, V Wang and G Boisseau for useful conversations, in particular MW is grateful to J Hefford for numerous helpful suggestions based on an earlier draft. This work is supported by the Hong Kong Research Grant Council through grant 17300918 and though the Senior Research Fellowship Scheme SRFS2021-
7S02, by the Croucher Foundation, by the John Templeton Foundation through grant 61466, The Quantum Information Structure of Spacetime (qiss.fr). Research at the Perimeter Institute is supported by
the Government of Canada through the Department of Innovation, Science and Economic Development
Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science. The
opinions expressed in this publication are those of the authors and do not necessarily reflect the views of
the John Templeton Foundation. MW gratefully acknowledges support by University College London
and the EPSRC Doctoral Training Centre for Delivering Quantum Technologies.

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§ ENRICHED MONOIDAL CATEGORIES

A key notion for the description of supermaps will be that of an enriched category, we begin with a brief overview of the basics of enriched category theory relevant to this paper, and then expand in detail.

Summary of Enriched Categories
In the data of a category is a family of “composition" functions $\circ_{A,B,C}:\mathbf{C}(A,B) \times \mathbf{C}(B,C) \rightarrow \mathbf{C}(A,C)$, in other words a family of morphisms in the category $\mathbf{Set}$ of functions between sets. A key notion for the description of supermaps will be that of an enriched category [72], which generalises the notion of a $\mathbf{Set}$-category $\mathbf{C}$ to a $\mathbf{V}$-category $\mathbf{C}$ in which composition is specified as a morphism $\circ_{A,B,C}:[A,B] \otimes_{V} [B,C] \rightarrow [A,C]$ in a category $\mathbf{V}$ rather than some function in $\mathbf{SET}$. We relax the standard definition of $\mathbf{V}$-enriched category $\mathbf{C}$ to a $\mathbf{V}_{\cong}$ category $\mathbf{C}$. In a $\mathbf{V}_{\cong}$-category $\mathbf{C}$ from each pair of objects $A,B$ of $\mathbf{C}$ an object $[A,B]$ is given in $\mathbf{V}$ intended to represent the space of processes $\mathbf{C}(A,B)$ as a set of states $\mathbf{V}(I,[A,B])$ in $\mathbf{V}$. The meaning of $\cong$ is to express that the representation of $ f \in \mathbf{C}(A,B)$ as a state of $\mathbf{V}$ is not exact but up to a bijection $\kappa: \mathbf{C}(A,B) \rightarrow \mathbf{V}(I,[A,B])$. A category $\mathbf{V}_{\cong}\mathbf{Cat}$ can be constructed with each object a $\mathbf{V}_{\cong}$-category $\mathbf{C}$ and each morphism termed a $\mathbf{V}_{\cong}$-functor, given by a functor $\mathcal{F}^{C}:\mathbf{C} \rightarrow \mathbf{C}'$ and a family of morphisms $\mathcal{F}_{A,B}:[A,B] \rightarrow [\mathcal{F}A,\mathcal{F}B]$ which represent on the states of $\mathbf{V}$ the action of the functions $\mathcal{F}^{C}:\mathbf{C}(A,B) \rightarrow \mathbf{C}(\mathcal{F}A,\mathcal{F}B)$ induced by the functor $\mathcal{F}^C$. A functor $\mathcal{M}:\mathbf{MonCat} \rightarrow \mathbf{2Cat}$ can be constructed which assigns to each monoidal functor $\mathcal{F}:\mathbf{V} \rightarrow \mathbf{V}'$ a new “change-of-base" $2$-functor $\mathcal{M}(\mathcal{F}):\mathbf{VCat}_{\cong} \rightarrow \mathbf{V'Cat}_{\cong}$.

A generalisation of the standard notion of a ($\mathbf{Set}$) monoidal category $\mathbf{C}$ with its parallel composition functions $\otimes_{A,A',B,B'}:\mathbf{C}(A,A') \times \mathbf{C}(B,B') \rightarrow \mathbf{C}(A \otimes B,A' \otimes B')$ is given by a $\mathbf{V}_{\cong}$-monoidal category $\mathbf{C}$ with morphisms $\otimes_{A,A',B,B'}:[A,A'] \times [B,B'] \rightarrow [A \otimes B,A' \otimes B']$. Furthermore one can construct a category $\mathbf{VMonCat}_{\cong}$ of $\mathbf{V}_{\cong}$-monoidal categories and $\mathbf{V}_{\cong}$-monoidal functors, along with a functor $\mathcal{M}_{mon}:\mathbf{MonCat} \rightarrow \mathbf{2Cat}$ which assigns to each monoidal functor $\mathcal{F}:\mathbf{V} \rightarrow \mathbf{V}'$ a change-of-base $2$-functor $\mathcal{M}(\mathcal{F}):\mathbf{VMonCat}_{\cong} \rightarrow \mathbf{V'MonCat}_{\cong}$.

Enriched Categories - Formally
Note that from any object $B$ of a monoidal category $\mathbf{V}$ and functor $[-,-]: \mathbf{C}^{op} \times \mathbf{C} \rightarrow \mathbf{V}$ a new functor $[-,B] \otimes [B,-]: \mathbf{C}^{op} \times \mathbf{C} \rightarrow \mathbf{V}$ can be constructed.

Let $\mathbf{C}$ be a category and $\mathbf{V}$ be a symmetric monoidal category, a $\mathbf{V}_{\cong}$ enriched category $\mathbf{C}$ is a specification of

* A hom-functor $[A,B]: \mathbf{C} \times \mathbf{C} \rightarrow \mathbf{V}$

* A natural transformation  ${\circ}_{ABC} : [A,B] \otimes [B,C] \rightarrow [A,C]$ (with a minor abuse of notation in the use of the symbol $\circ$)

* A natural isomorphism $\kappa: \mathbf{C}(A,B) \cong \mathbf{V}(I,[A,B]])$

* Such that the following unitality and associativity conditions for the natural transformation $\circ_{ABC}$ hold:

\begin{equation}
\tikzfig{figs/smsc_asso1}
\end{equation}

From the definition of $\mathbf{V}$-categories one can define $\mathbf{V}_{\cong}$-functors and $\mathbf{V}_{\cong}$-natural transformations.

A $\mathbf{V}_{\cong}$-functor is a functor $\mathcal{F}^C:\mathbf{C} \rightarrow \mathbf{C}'$ and a natural transformation $\mathcal{F}_{A,B}:[A,B] \rightarrow [\mathcal{F}^C A, \mathcal{F}^C B]$ satisfying:
\[  \tikzfigscale{0.7}{figs/vfunct_1}  \]

A $\mathbf{V}_{\cong}$-natural transformation $\eta : \mathcal{F} \Rightarrow \mathcal{G}$ is a natural transformation $\eta^C: \mathcal{F}^C \Rightarrow \mathcal{G}^{C}$ satisfying:
\[  \tikzfigscale{0.7}{figs/vnat_1}  \]

Indeed all together $\mathbf{V}_{\cong}\mathbf{Cat}$ defines a monoidal $2$-category, the monoidal product is given by taking $(\mathbf{V},\mathbf{C}) \times (\mathbf{V},\mathbf{D}) = (\mathbf{V},\mathbf{E})$ where $\mathbf{E}$ is given by $\mathbf{E}((A,B),(A',B')) := \mathbf{V}(I,[A,A']_{\mathbf{C}} \otimes [B,B']_{\mathbf{D}})$ and all structure defined pairwise. The unit object is given by $(\mathbf{V},\mathbf{I})$ with $\mathbf{I}:= \{ \bullet \}$ (the singleton category) with enrichment given by $[\bullet,\bullet] := I$. The notion of monoidal enrichment can then be immediately generalised to $\mathbf{VCat}_{\cong}$.

A $\mathbf{V}_{\cong}$-monoidal category $\mathbf{C}$ is a pseudomonoid in $\mathbf{V}_{\cong}$. Explicitly this gives the category $\mathbf{C}$ monoidal structure and the category $\mathbf{V}$ an associative family of maps $\otimes_{AA'BB'}:[A,A'] \otimes [B,B'] \rightarrow [A \otimes_{\mathbf{C}} B, A' \otimes_{C} B']$ in $\mathbf{V}$ which implement the monoidal structure of $\mathbf{C}$.

A $2$-category can immediately be defined, the category $\mathbf{VMonCat}_{\cong}$ given by the free construction of a $2$-category of pseudomonoids from any monoidal $2$-category. For every lax monoidal functor $\mathcal{F}:\mathbf{V} \rightarrow \mathbf{V}'$ one can construct the change-of-base $2$-functor $\mathcal{M}(\mathcal{F}): \mathbf{VCat}_{\cong} \rightarrow \mathbf{V'Cat}_{\cong}$. This functor is given by sending $(\mathbf{V},\mathbf{C})$ to $(\mathbf{V}', \mathbf{U})$ where $\mathbf{U}$ is the underlying category of the standard change of base of enriched categories.

For every monoidal functor $\mathcal{F}$ the functor $\mathcal{M}(\mathcal{F})$ is a monoidal $2$-functor

In [64] it is shown that the change of base functor for standard enriched categories $\mathcal{M}_{*}(\mathcal{F}):\mathbf{VCat} \rightarrow \mathbf{V'Cat}$ is a monoidal $2$-functor. $\mathcal{M}$ is given by $\mathcal{Y}_{\mathbf{V}'} \circ  \mathcal{M}_{*} \circ \mathcal{X}_{\mathbf{V}}$, where $\mathcal{X}_{\mathbf{V}}:\mathbf{VCat}_{\cong} \rightarrow \mathbf{VCat}$ is the monoidal $2$-functor which replaces each category $\mathbf{C}$ with the underlying category $\mathbf{U}$ from enrichment and $\mathcal{Y}_{\mathbf{V}}:\mathbf{VCat} \rightarrow \mathbf{VCat}_{\cong}$ is the monoidal $2$-functor which views each $\mathbf{V}$ monoidal category $\mathbf{U}$ as a $\mathbf{V}_{\cong}$ monoidal category $\mathbf{U}$ using the identity function as bijection.

We can extend the definition of a $\mathbf{V}_{\cong}$-monoidal category to a $\mathbf{V}_{\cong}$ symmetric monoidal category using symmetric pseudomonoids [73, 74] rather than pseudomonoids.

A $\mathbf{V}_{\cong}$-smc $\mathbf{C}$ is a symmetric pseudomonoid [74] on $\mathbf{C}$ in $\mathbf{VCat}_{\cong}$

One can construct freely construct an entire $2$-category of symmetric pseudomonoids [74, 75] $\mathbf{SymPsMon}[\mathbf{VCat}_{\cong}]$, whilst keeping change-of-base theorems. This provides a higher level way to see that pm-functors as written explicitly in the main text can be defined and composed without explicitly checking all coherences.

There is a functor $\mathcal{M}_{smc}:\mathbf{SymMonCat} \rightarrow \mathbf{2Cat}$ which assigns to each $\mathcal{F}:\mathbf{V} \rightarrow \mathbf{V}'$ a $2$-functor $\mathcal{M}_{smc}(\mathcal{F}):\mathbf{SymPsMon}[\mathbf{VCat}_{\cong}] \rightarrow \mathbf{SymPsMon}[\mathbf{V'Cat}_{\cong}]$

When $\mathbf{V}$ is a symmetric monoidal category and $\mathcal{F}:\mathbf{V} \rightarrow \mathbf{V}'$ is a symmetric monoidal functor the change of base functor $\mathcal{M}:\mathbf{SymMonCat} \rightarrow \mathbf{SM2Cat}$ assigns to each $\mathcal{F}:\mathbf{V} \rightarrow \mathbf{V}'$ a symmetric monoidal $2$-functor $\mathcal{M}(\mathcal{F}):\mathbf{VCat}_{\cong} \rightarrow \mathbf{V'Cat}_{\cong}$ indeed the braiding coherence required for $\mathcal{M}(\mathcal{F})$ to be a symmetric monoidal $2$-functor is directly inherited from the coherence for $\mathcal{F}$. The $2$-functor $\mathbf{SymPsMon}:\mathbf{SM2Cat} \rightarrow  \mathbf{2Cat}$ constructed in [75] can be applied to return $\mathcal{M}_{smc}(\mathcal{F}) := \mathbf{SymPsMon}[\mathcal{M}(\mathcal{F})]:\mathbf{SymPsMon}[\mathbf{VCat}_{\cong}] \rightarrow \mathbf{SymPsMon}[\mathbf{V'Cat}_{\cong}]$.

Given a functor $\mathcal{M}:\mathbf{D} \rightarrow \mathbf{Cat}$ its Grothendieck construction $\mathcal{G}(\mathcal{M})$ has as objects pairs $(d,c)$ where $x$ is an object of $\mathbf{D}$ and $c$ is an object of $\mathbf{M}(d)$. Morphisms $(d_1,c_1) \rightarrow d_2,c_2)$ are given by pairs $(f,g)$ of morphisms $f:d_1 \rightarrow d_2$ and $g:\mathcal{M}(f)(c_1) \rightarrow c_2$. The Grothendieck construction $\mathbf{PM}:=\mathcal{G}(\mathcal{M}_{smc})$ of the functor $\mathcal{M}_{smc}$ consequently has for each object a pair of an smc $\mathbf{V}$ and a $\mathbf{V}_{\cong}$-smc $\mathbf{C}$ and for each morphism a pair of a symmetric monoidal functor $\mathcal{F}^V : \mathbf{V} \rightarrow \mathbf{V}'$ and an enriched symmetric monoidal functor from $\mathcal{M}(\mathcal{F}^V)(\mathbf{V},\mathbf{C})$ to $ (\mathbf{V}',\mathbf{C}')$ explicitly meaning a pair of a symmetric monoidal functor $\mathcal{F}^{C}: \mathbf{C} \rightarrow \mathbf{C}'$ and a family of natural transformations $\mathcal{F}_{A,B}:\mathcal{F}[A,A'] \rightarrow [\mathcal{F}^C A, \mathcal{F}^C A']$.

§ PM MORPHISMS BETWEEN LAYERS OF THE GROTHENDIECK CONSTRUCTION

Given any $\mathbf{C}^3_{\cong}$ monoidal category $\mathbf{C}^2$ and $\mathbf{C}^2_{\cong}$ monoidal category $\mathbf{C}^1$ there exists a morphism $\Gamma$ between them in $\mathbf{PM}$ given by given by $\Gamma := (\mathcal{R}_{2}^{3}, \mathcal{R}_{1}^{2}, \gamma^i)$ where $\gamma^{i}_{A,B}:\mathcal{R}_{2}^{3}([A,B]^2) \rightarrow [\mathcal{R}_{1}^2(A),\mathcal{R}_1^2 (B)]^3 $ is given by
\begin{equation}
\tikzfig{figs/smsc_monhom_2}
\end{equation}

Including the isomorphisms associated to the monoidal functors in the definition of a pm-morphism (which were omitted in the main text for readability) gives the following three conditions:
\[   \tikzfig{figs/smsc_monhom_r_nf}  \]

For readability of the proofs we take all categories $\mathbf{C}^i$ to be strict. The generic non-strict case is proven identically up to additional book-keeping of associators and unitors. The first state preservation condition is immediate so we begin by checking the second condition for pm-functors: preservation of sequential composition.

\begin{equation}
\tikzfig{figs/prove_enrich_2a}
\end{equation}

Then the third condition, for preservation of parallel composition:

\begin{equation}
\tikzfig{figs/prove_enrich_par}.
\end{equation}

\begin{equation}
\tikzfig{figs/prove_enrich_1a}
\end{equation}

\begin{equation}
\tikzfig{figs/prove_enrich_1b}
\end{equation}

\begin{equation}
\tikzfig{figs/prove_enrich_1c}
\end{equation}

Then doing the same to the right and side of the condition.

\begin{equation}
\tikzfig{figs/prove_enrich_2a}
\end{equation}

\begin{equation}
\tikzfig{figs/prove_enrich_2b}
\end{equation}

Next we confirm that the parallel composition preservation condition is satisfied, deriving equality between the following two terms, firstly:

\begin{equation}
\tikzfig{figs/prove_enrich_3a}
\end{equation}

\begin{equation}
\tikzfig{figs/prove_enrich_3b}
\end{equation}

and then secondly:

\begin{equation}
\tikzfig{figs/prove_enrich_4}
\end{equation}

Finally the state-based condition can be confirmed:

\begin{equation}
\tikzfig{figs/prove_enrich_6}
\end{equation}

§ LINKED FAITHFUL CATEGORIES ARE CLOSED MONOIDAL

A category $\mathbf{C}$ is closed symmetric monoidal if and only if $\mathbf{C}$ is a symmetric monoidal and furthermore

* $\mathbf{C}$ is linked

* $\mathbf{C}$ has faithful usage

We begin by showing that the above two bullet points give closed monoidal structure. Let the three bullet points be true for $\mathbf{C}$, then to each pair $A,B \in \mathbf{C}$ assign the candidate for evaluation

\begin{equation}
\texttt{eval} : = \tikzfig{figs/main_thm_1_a}
\end{equation}

Since every $\circ$ is completely injective by assumption, so is every $\texttt{eval}$. Since $\eta : A \rightarrow [I,A]$ is a natural isomorphism for any $f \in \mathbf{C}(A,B)$ there exists a morphism $\kappa{f}$ such that

\begin{equation}
\texttt{eval} : = \tikzfig{figs/main_thm_2_a}
\end{equation}

One can apply the isomorphism $\eta$ to the partial insertion operation to generate a partial insertion using a lower level type $Y$ as opposed to the higher level type $[I,Y]$.

\begin{equation}
\tikzfig{figs/smsc_partial_1}
\end{equation}

This partial insertion operation can be used to construct the curried version of any process $f$ from its static version $\kappa(f)$,  since

\begin{equation}
\tikzfig{figs/smsc_partial_2}
\end{equation}

It follows that for every process $f$ its curried version exists, that is, the co-universal arrow definition of a closed symmetric monoidal category is satisfied.

Now we demonstrate the converse. Let $\mathbf{C}$ be a closed SMC, then there exist sequential and parallel composition morphisms defined as adjuncts to circuits of evaluation morphisms. Concretely the definition of closed monoidal categoryu enforces that there must exist processes $\otimes$ and $\circ$ satisfying,

\begin{equation}
\tikzfig{figs/smsc_comppar}
\end{equation}

which satisfy the coherence conditions for a symmetric monoidal category. The uniqueness property for co-universal arrows lifts to faithful usage for each sequential composition maps. Finally a monoidal natural isomorphism $A \cong [I,A]$ for the induced functor $[I,-]$ must be constructed. Indeed, up to unitor the inverse $\eta$ of $\texttt{eval}_{I \Rightarrow A}$, being an isomorphism by assumption, is such a candidate. $\texttt{eval}_{I \Rightarrow A}$ is natural for any closed monoidal category, so $\eta$ being its inverse is immediately also natural. Furthermore $\eta$ is easily checked to be monoidal.

\begin{equation}
\tikzfig{figs/smsc_prove_monoidal}
\end{equation}

This completes the proof.

§ THE APEX OF A MERGER IS CLOSED MONOIDAL

Here we prove our main technical result. As a recap, from a series of enriched monoidal categories $\mathbf{C}_{i-1}, \mathbf{C}_{i}, \mathbf{C}_{i+1}$ a chain of raising functors $\mathcal{R}_{i-1}^{i}: \mathbf{C}_{i-1} \longrightarrow \mathbf{C}_{i}$ can be written down with $\mathcal{R}_{i-1}^{i}(-) = [I_{i-1},-]$.

In any merger of infinite order, the following condition holds for the isomorphism $\mu_i^{i+1} := \mathcal{F}^{i+1}(\gamma) \circ \eta_i^{i+1}$

\begin{equation}
\tikzfig{figs/eta_commute}.
\end{equation}

\begin{equation}
\tikzfig{figs/eta_commute_2b}
\end{equation}

Indeed the above property  is the key ingredient in the construction of our main result. We work with the following definition of a closed symmetric monoidal category
An SMC  $\mathbf{C}$ is closed if for every $A,B \in Ob(\mathbf{C})$ there exists an object $A \Rightarrow B$ and a morphism $\texttt{eval}_{A \Rightarrow B} : A \otimes A \Rightarrow B \rightarrow B$, called the evaluation morphism, such that for all $f: A \otimes C \rightarrow B$ there exists  a unique $\bar{f}: C \rightarrow (A \Rightarrow B)$ such that $\texttt{eval}_{A \Rightarrow B} \circ (id \otimes \bar{f}) = f$.

The apex $\mathbf{C}$ of any Merger of infinite order is a closed symmetric monoidal category

Since the coproduct $\coprod_{i} \mathcal{F}_i$ is essentially surjective, each object $A$ can be assigned an object $X_A$ an “index" $l_A$ and an isomorphism $L_A$ such that $L_A:A \rightarrow \mathcal{F}_{l_A}(X_A)$. A compact notation can be introduced for combinations of functors of the form $[I_i,-]$.

* $\mathcal{R}_{i}^{i+1} := [I_i,-]$

* $\mathcal{R}_{i}^{j} := \mathcal{R}_{j-1}^{j} \circ \mathcal{R}_{i}^{j-1}$

furthermore the function $l:ob(\mathbf{C}) \rightarrow \mathbb{N}$ can be extended to lists by
\[l_{AB} := \texttt{max}(l_A,l_A)\]

After which one can define the object representing the space of morphisms from $A$ to $B$ by
\[A \Rightarrow B := \mathcal{F}_{l_{AB}+1}[\mathcal{R}_{l_A}^{l_{AB}}(X_A),\mathcal{R}_{l_{B}}^{l_{AB}}(X_B)]\]

This is the object representing the lifting of both $A$ and $B$ in to the $C^{l_{AB}}$ which contains them both, and then using the process object in the next category $C^{l_{AB}+1}$ to represent the processes between them. For each $A$,$B$ an evaluation $e_{A \Rightarrow B}: A \otimes (A \Rightarrow B) \rightarrow B$ can be defined by,

\begin{equation}
\tikzfig{figs/hopt_eval_1f}
\end{equation}

For $\mathbf{C}$ to be closed monoidal one must show that for every $A,B,C$ and for every $f:A \otimes C \rightarrow B$ there exists a unique $\bar{f}:C \rightarrow (A \Rightarrow B)$ such that,

\begin{equation}
\tikzfig{figs/hopt_eval_3}
\end{equation}

Indeed such a map $\bar{f}$ can be constructed. Firstly defining $g$ such that

\begin{equation}
\tikzfig{figs/hopt_eval_7f}
\end{equation}

Such a $g$ must exist since each functor $\mathcal{F}_i$ is full. In terms of this $g$ define $\bar{f}$ by

\begin{equation}
\tikzfig{figs/hopt_eval_4}
\end{equation}

Then to prove the required identity first requires repeated application of  lemma (16),

\begin{equation}
\tikzfig{figs/hopt_eval_2f}
\end{equation}

and then using the defining identity for the partial insertion operation $\Delta$.

\begin{equation}
\tikzfig{figs/hopt_eval_5f}
\end{equation}

and finally using monoidal naturality of the transformation $\eta_{l_{ABC}+1}^{l_{{ABC}}}$.

\begin{equation}
\tikzfig{figs/hopt_eval_6f}
\end{equation}

The morphism $\bar{f}$ satisfying $e \circ (A \otimes \bar{f}) = f$ must be demonstrated to be unique. Every $\mu_{i}^{j}$ is an isomorphism by fully faithful-ness of the sequence of enriched monoidal categories, as a result every morphism $h: C \rightarrow A \Rightarrow B$ can be written in the form

\begin{equation}
\tikzfig{figs/hopt_eval_8}
\end{equation}

Where in the last line fullness of each $\mathcal{F}_i$ is used. Assuming $h$ and $h'$ have decomposition in terms of $m$ and $m'$ respectively both evaluate to the same morphism $e \circ (A \otimes h) = e \circ (A \otimes h')$:

\begin{equation}
\tikzfig{figs/hopt_eval_9f},
\end{equation}

which in turn implies

\begin{equation}
\tikzfig{figs/hopt_eval_10f}.
\end{equation}

Since each $\eta$ and $L$ is an isomorphism, and each composition morphism $\bigcirc$ is part of the structure of a faithful monoidal enrichment, and each $\mathcal{F}_i$ is faithful this entails that $m = m'$ and as a result that $\bar{f} = \bar{f}'$. It follows that $\bar{f}$ is the unique morphism satisfying the evaluation condition for $f$.

The remaining condition to prove is that each $e_{I \Rightarrow A}$ is an isomprphism, firstly each $e_{I \Rightarrow A}$ takes the following form,

\begin{equation}
\tikzfig{figs/hopt_eval_11f}
\end{equation}

Two properties of the natural transformations $\eta$ can now be leveraged, firstly a family of commuting squares involving $I_{\mathbf{C}}$ which indeed commute since $\eta$ is monoidal,

\[\begin{tikzcd}
& I_C \arrow[ld, "\mathcal{V}_{n-1}"'] \arrow[rd, "\mathcal{V}_n"] &                                                                                \\
\mathcal{F}_{n-1}(I_{n-1}) \arrow[rd, "\eta_{I_{n-1}}"'] &                                                                  & \mathcal{F}_n(I_n) \arrow[ld, "\mathcal{F}_{n}(\kappa_{n-1}^{n}(i_{I_{n-1}}))"] \\
& \mathcal{F}_{n}(\mathcal{R}_{n-1}^{n}(I_{n-1}))                  &
\end{tikzcd}\]

secondly a family of squares which commute since $\eta$ is natural,

\[\begin{tikzcd}
\mathcal{F}_{n-1}(A) \arrow[d, "\mathcal{F}_{n-1}(f)"'] \arrow[r, "\eta_{n-1}^{n}"] & \mathcal{F}_{n}(\mathcal{R}_{n-1}^{n}(A)) \arrow[d, "\mathcal{F}_{n}(\mathcal{R}_{n-1}^{n}(f))"] \\
\mathcal{F}_{n-1}(B) \arrow[r, "\eta_{n-1}^{n}"']                                   & \mathcal{F}_{n}(\mathcal{R}_{n-1}^{n}(B))
\end{tikzcd}\]

The above two famillies may be used to show that for each $k$ the term $\eta_{1}^k \circ \nu_1$ may be replaced by
\[\eta_{1}^{k} \circ \nu_1 = \mathcal{F}_k(R_2^k(i_2)) \circ \mathcal{F}_k(R_3^k(i_3)) \circ \dots \circ \mathcal{F}_{k}(i_k) \circ \nu_k \]

using the following commutative diagram built from the above families

\[
\rotatebox{270}{%
\begin{tikzcd}[ampersand replacement=\&]
\&                                                                                \&                                                                                \& I_{\mathbf{C}} \arrow[lllddddddd, "\nu_1"'] \arrow[lldddddd, "\nu_2"'] \arrow[lddddd, "\nu_3"'] \arrow[rddd, "\nu_{k-2}"'] \arrow[rrdd, "\nu_{k-1}"'] \arrow[rrrd, "\nu_{k}"'] \&                                                           \&                                                                                \&                                                                                \\
\&                                                                                \&                                                                                \&                                                                                \&                                                           \&                                                                                \& \mathcal{F}_{k}(I_{k}) \arrow[d, "\mathcal{F}_k(\kappa_{k-1}^{k}{i_{k-1}})"]                             \\
\&                                                                                \&                                                                                \&                                                                                \&                                                           \& \mathcal{F}_{k-1}(I_{k-1}) \arrow[r, "\eta_{k-1}^{k}"'] \arrow[d, "\mathcal{F}_{k-1}(\kappa_{k-2}^{k-1}(i_{k-2}))"']               \& \mathcal{F}_{k}(R_{k-1}^k(I_{k-1})) \arrow[d, "\mathcal{F}_k(R_{k-1}^k \kappa_{k-2}^{k-1}(i_{k-2}))"] \\
\&                                                                                \&                                                                                \&                                                                                \& \mathcal{F}_{k-2}(I_{k-2}) \arrow[r, "\eta_{k-2}^{k-1}"'] \& \mathcal{F}_{k-1}(R_{k-2}^{k-1}(I_{k-2})) \arrow[r, "\eta_{k-1}^{k}"', dashed] \arrow[d, dotted]               \& \mathcal{F}_{k}(R_{k-2}^k(I_{k-2})) \arrow[d, dotted]                              \\
\&                                                                                \&                                                                                \&                                                                                \&                                                           \& {} \arrow[d, dotted]                                                                                \& {} \arrow[d, dotted]                                                               \\
\&                                                                                \& \mathcal{F}_3(I_3) \arrow[r, "\eta_{3}^{4}"'] \arrow[d, "\mathcal{F}_3(\kappa_{2}^{3}(i_2))"']              \& {} \arrow[r, dotted]                                                                                \& {} \arrow[r, "\eta_{k-2}^{k-1}"']                         \& \mathcal{F}_{k-1}(R_3^{k-1}(I_3)) \arrow[d, "\mathcal{F}_{k-1}(R_3^{k-1}(\kappa_{2}^{3}(i_2)))"'] \arrow[r, "\eta_{k-1}^{k}"'] \& \mathcal{F}_k(R_3^k(I_3)) \arrow[d, "\mathcal{F}_k(R_3^k(\kappa_{2}^{3}(i_2)))"]                   \\
\& \mathcal{F}_2(I_2) \arrow[r, "\eta_{2}^{3}"'] \arrow[d, "\mathcal{F}_2(\kappa_{1}^{2}(i_1))"'] \& \mathcal{F}_3(R_2^3(I_2)) \arrow[r, "\eta_{3}^{4}"'] \arrow[d, "\mathcal{F}_3(R_2^3(\kappa_1^2 (i_1)))"] \& {} \arrow[r, dotted]                                                                                \& {} \arrow[r, "\eta_{k-2}^{k-1}"']                         \& \mathcal{F}_{k-1}(R_2^{k-1}(I_2)) \arrow[d, "\mathcal{F}_{k-1}(R_2^{k-1}(\kappa_1^2(i_2)))"'] \arrow[r, "\eta_{k-1}^{k}"'] \& \mathcal{F}_k(R_2^k(I_2)) \arrow[d, "\mathcal{F}_k(R_2^k(\kappa_{1}^{2}(i_1)))"]                   \\
\mathcal{F}_1(I_1) \arrow[r, "\eta_{1}^{2}"'] \& \mathcal{F}_2(R_1^2(I_1)) \arrow[r, "\eta_{2}^{3}"']                           \& \mathcal{F}_3(R_1^3(I_1)) \arrow[r, "\eta_{3}^{4}"']                                        \& {} \arrow[r, dotted]                                                                                \& {} \arrow[r, "\eta_{k-2}^{k-1}"']                         \& \mathcal{F}_{k-1}(R_1^{k-1}(I_1)) \arrow[r, "\eta_{k-1}^{k}"']                                                 \& \mathcal{F}_k(R_1^k(I_1))
\end{tikzcd}
\]

In particular the above identity is given by the two routes along the circumference of this commutative diagram. Using the derived identity the evaluation $e_{I \Rightarrow A}$ is consequently equal the following diagram

\begin{equation}
\tikzfig{figs/hopt_eval_12f}
\end{equation}

Using the definition of the functor $\mathcal{R}_{l_A}^{l_A + 1}$

\begin{equation}
\tikzfig{figs/hopt_eval_13f}
\end{equation}

The above is a circuit built from the functor $\mathcal{F}_{L_A + 1}$ applied to a morphism $k$ along with post composition by an isomorphism. Since the functor $\mathcal{F}_{L_A + 1}$ will preserve isomorphism, all that remains is to show that the morphism $k$ on which the functor is applied, is also an isomorphism. To demonstrate this we note that the mapping
\[\tikzfig{figs/hopt_eval_16}\]
is a contravariant functor, and therefore preserves isomorphism. Furthermore since each raising functor $([I_{i},\phi_{i+1},\kappa_{i}^{i+1}(i_{I_{i}})],)$ is strong monoidal by assumption it follows that each $\kappa_{k}^{k+1}(i_{k})$ is an isomorphism, each $\mathcal{R}_i^j$ is a functor an so preserves isomorphism, entailing all together that every morphism in the circuit $k$ is an isomorphism.