This paper has two purposes. The first is to extend the theory of linearly distributive categories by considering the structures that emerge in a special case: the normal duoidal category $(\Poly,\yon,\otimes,\tri)$ of polynomial functors under Dirichlet and substitution product. This is an isomix LDC which is neither $*$-autonomous nor fully symmetric. The additional structures of interest here are a closure for $\otimes$ and a co-closure for $\tri$, making $\Poly$ a bi-closed LDC, which is a notion we introduce in this paper.

The second purpose is to use $\Poly$ as a source of examples and intuition about various structures that can occur in the setting of LDCs, including duals, cores, linear monoids, and others, as well as how these generalize to the non-symmetric setting. To that end, we characterize the linearly dual objects in $\Poly$: every linear polynomial has a right dual which is a representable. It turns out that the linear and representable polynomials also form the left and right cores of $\Poly$. Finally, we provide examples of linear monoids, linear comonoids, and linear bialgebras in $\Poly$.

§ INTRODUCTION

Cockett and Seely introduced weak distributive categories, renamed to linear distributive categories (LDCs), as a categorical semantics for multiplicative linear logic <cit.>. Adding negation to the semantics produces $*$-autonomous categories. The primary contribution of this paper is to present a detailed study of a reasonably approachable yet a rich example of an LDC, one which is neither symmetric nor $*$-autonomous, namely the category $\Poly$ of polynomial functors and natural transformations. While $\Poly$ is a well-known category with a vast body of literature, to the best of our knowledge this paper is the first to explore $\Poly$ purely from the perspective of LDCs.

Indeed, though this project is focused on the category $\Poly$ as an LDC, our definitions and results are more general. We first prove a straightforward result, that normal duoidal categories are isomix LDCs—there is a faithful functor as we'll see in <ref>—thereby providing a family of examples for isomix LDCs, one of which is $\Poly$. These results are presented in <ref>.

The category $\Poly$ is not $*$-autonomous: not every object in $\Poly$ has a dual. However, there are certain pairs of polynomials which are duals: for any set $A$, the linear polynomial $A \yon$ is left dual to the representable $\yon^A$, and these are the only polynomials with duals.
[We will explain this notation in <ref>.]
We prove these results in more generality by introducing the notion of biclosed LDCs which are LDCs with a closure for the $\ox$ operation and a coclosure for the $\tri$ operation, i.e. LDCs for which $(- \ox a)$ has a right adjoint $[a, -]$, and $(- \tri a)$ has a left adjoint $\coclose{-}{a}$. In $\Poly$, the former $[-,-]$ is the closure for the Dirichlet product and the latter $\coclose{-}{-}$ is the left Kan extension. We characterize the dual objects in biclosed LDCs, generalizing the above results for $\Poly$; all this is shown in <ref>.

Next we explore the notion of core in mix LDCs within our context. Mix LDCs are equipped with a mix map $\m: \bot \to \top$, which induces a natural transformation $\indep_{a,b}: a \ox b \to a \tri b$ for all objects $a, b$. An isomix LDC is a mix LDC in which the mix map $\m: \bot \cong \top$ is an isomorphism. The core of a symmetric (iso)mix LDC is the full subcategory spanned by those objects for which the $\indep_{a,-}$ is a natural isomorphism. That is, an object $a$ is in the core if the map $\indep_{a,x}: a \ox x \cong a \tri x$ is an isomorphism, for all $x$; from this it follows from the symmetry that $\indep_{x,a}: x \ox a \cong x \tri a$ is also an isomorphism.

The notion of core in mix LDCs has been studied only for symmetric LDCs. Inspired by $\Poly$, this paper explores the notion of core in non-symmetric settings. Without the assumption of symmetry, one may define the left core (resp. right core) to be the full subcategory spanned by the objects $a$ for which $\indep_{a,-}$ (resp. $\indep_{-,a}$) is a natural isomorphism. We show that in $\Poly$, the linear polynomials ($A\yon$) comprise the left core and the representables ($\yon^A$) comprise the right core. The cores and the duals thus coincide: a polynomial is in the left core iff it is a left dual iff it is linear, and it is in the right core iff it is a right dual iff it is representable.

For a mix LDC $\X$, we say that the left and the right cores are opposing when $\lCore(\X) \cong \rCore(\X)^{\op}$. We will show that $\Poly$ has opposing cores, which are equivalent to $\Set$ and $\Set^\op$, respectively. Another example of mix LDCs with opposing cores are compact closed categories.  <ref> provides the above definitions and proves relevant results.

Linear monoids and linear comonoids generalize Frobenius algebras from monoidal categories to LDCs. We show that $\Poly$ has non-trivial linear-monoid-like structures: every monoid in $\Set$ defines a left linear monoid and every set induces a right linear monoid in $\Poly$; these definitions are new. Such linear monoids and linear comonoids in $\Poly$ interact to produce left and right linear bialgebras. <ref> contains these results.

The following are the main technical contributions of this paper:

* We prove that all normal duoidal categories are isomix LDCs. In particular, the category $\Poly$ is an isomix LDC.

* We define the notion of biclosed LDCs.

* We characterize the linear duals in isomix biclosed LDCs.

* We prove that a polynomial $q$ is left dual of $p$, i.e. $q\dashvv p$ iff $q = A \yon$ and $p = \yon^A$ for some set $A$.

* We define the left and the right cores of non-symmetric LDCs. In $\Poly$, we show that the left core consists of the linear polynomials ($A\yon$) and the right core consists of the representable polynomials ($\yon^A$).

* We show that for every monoid in $\Set$, we get a left linear monoid and a right linear comonoid in $\Poly$. Moreover, every set induces a left linear comonoid and a right linear monoid in $\Poly$. These structures interact to produce right and left linear bialgebras which are the notion we introduce in this paper.

In <ref> we recollect the preliminary definitions and results used in this paper.

§.§ Acknowledgments

We thank Harrison Grodin and Reed Mullanix for their conjecture, that normal duoidal categories are linear distributive categories, which spawned the present paper.

This material is based upon work supported by the Air Force Office of Scientific Research under award number FA9550-23-1-0376.

§ PRELIMINARIES

For preliminaries on $\Poly$ see <ref>; for preliminaries on LDCs see <ref>.

§.§ The category Poly

While there are many equivalent definitions of polynomial functors <cit.>,
[For a friendly and very non-technical introduction to polynomial functors, see <https://topos.site/blog/2022-01-19-poly-makes-me-happy/> or <https://topos.site/blog/2023-11-13-solving-problem-solving/>.]
in this paper we view them as coproducts of representables.

Given an object $A : \C$, a functor $F: \C \to \Set$ is represented by $A$ if there is an isomorphism $F \cong \C(A, -)$; in this case we also say that $F$ is representable. For functors $\Set\to\Set$, we denote the functor $\Set(A,-)$ by $\yon^A$.

A polynomial functor is a functor $p: \Set \to \Set$ that is isomorphic to a coproduct of representables, i.e.
\[ p \cong \sum_{i : I} \yon^{A_i} \]

for some indexing set $I:\Set$ and sets $A_i:\Set$. We refer to the elements of $I$ as the positions of $p$, and we refer to elements of $A_i$ as the directions of $p$ at position $i:I$.

The following are a few examples of polynomial functors:

* $\yon^A$ is a representable polynomial, for any $A:\Set$.

* $\sum_{i : I} \yon \cong I \yon$ is a linear polynomial, for any $I:\Set$.

* $I \yon^0$ is a constant polynomial; it sends $X\mapsto I$ for all $X:\Set$.

* $\yon^3 + \yon^2$ where $2$ and $3$ are 2-element and 3-element sets respectively. For this polynomial, the index set $I = 2$, $A_{1} = 3$, and $A_2 = 2$.

Corolla forests are an intuitive visualization tool for polynomials <cit.>.
For example, the polynomial $\yon^3 + \yon^2$, can be drawn as a corolla forest as follows:

\[  \includegraphics[scale=0.05]{figures/corolla.jpeg} \]

For a polynomial $p \cong \sum_{i : I} \yon^{A_i}$, the elements of $I$ are referred to as the positions of $p$. Thus, each corolla in a corolla forest corresponds to a position of its polynomial. For each $i:I$, the elements of $A_i$ are referred to as the directions at that position. The directions at position $i$ are denoted as the branches of the corolla corresponding to that position.

The positions of a polynomial $p$ can be found by evaluating the polynomial functor at the single element set $1$:

\[ p(1) = \sum_{i:I} \yon^{A_i} (1) \cong \sum_{i:I} 1^{A_i} \cong \sum_{i:I} 1 \cong I \]

Hence, writing $p[P]\coloneqq A_P$ for any position $P:p(1)$, we can denote a polynomial succinctly as follows:

\begin{equation}\label{eqn.succinct_poly}
p \cong \sum_{P:p(1)} \yon^{p[P]}
\end{equation}

We define $\Poly$ to be the category of polynomial functors $\Set\to\Set$ and the natural transformations between them.

We can give a fully set-theoretic account of the natural transformations between any two polynomial functors by using the universal property of coproducts and the Yoneda Lemma. Indeed, for any two polynomials $p = \sum_{P: p(1)} \yon^{p[P]}$ and $q = \sum_{Q: q(1)} \yon^{q[Q]}$,

\begin{align}
\nonumber
\Poly(p,q) &= \Poly\left(\sum_{i:P} \yon^{p[P]}, q\right) \\\nonumber
&\cong \prod_{P:p(1)} \Poly\left(\yon^{p[P]}, q\right) & \text{Universal property of coproduct}\\ \nonumber
&\cong \prod_{P:p(1)} q(p[P]) &  \text{Yoneda Lemma} \\  \nonumber
& \cong \prod_{P:p(1)} \sum_{Q: q(1)} p[P]^{q[Q]} \\\label{eqn.poly_map}
& = \prod_{P:p(1)} \sum_{Q: q(1)} \prod_{e: q[Q]} \sum_{d:p[P]} 1
\end{align}

Since products distribute over sums, the expression in (<ref>) tells us that a natural transformation $\varphi: p \to q$ can be identified with:

* a function $\varphi_1: p(1) \to q(1)$, from the positions of $p$ to the the positions of $q$, and

* for each $P:p(1)$, a function $\varphi_i^\sharp : q[\varphi_1(P)] \to p[P]$, from the directions of $q$ at $f_1(P)$ to directions of $p$ at $P$.

A map $\varphi$ of polynomials is called cartesian if, for each $P:p(1)$, the map $\varphi_P^\sharp$ is a bijection.
[In general, a natural transformation $\varphi$ is called cartesian if its naturality squares are pullbacks. A map $\varphi$ of polynomials is cartesian in this sense iff it satisfies the above condition, that $\varphi_P^\sharp$ is a bijection.]

As an example of polynomial maps, let us consider the polynomials $p = \yon^3 + \yon^2$ and $q = \yon + \yon^2$. The following depicts a natural transformation $f: p \to q$:

\[  \includegraphics[scale=0.15]{figures/corolla-map.jpeg} \]

One can count that there are $(3^1+3^2)\times(2^1+2^2)=72$ natural transformations $p\to q$.

Here are two useful functors between $\Poly$ and $\Set$, based on the positions and the directions of polynomials:

* the functor $\Poly(\yon, -): \Poly \to \Set$, which we denote by $-(1)$. For any polynomial $p$, an element of $p(1)$ is a position of $p$.

* the functor $\Poly(-,\yon): \Poly \to \Set^\op$, which we denote by $\Gamma_-$. For any polynomial $p$, an element of $\Gamma_p$ is a global section: it picks a direction for each position of $p$:

\begin{equation}\label{eqn.gamma}
\Gamma_p \cong \prod_{P: p(1)} p[P]
\end{equation}

While $\Poly$ has many monoidal structures, in this article we are concerned with only two: the tensor (Dirichlet) product and the substitution product <cit.>. The tensor product $\ox$ is given by the Day convolution of Cartesian product $\times$ in $\Set$, and the substitution product $\tri$ is given by functor composition. They have straightforward formulas:

\begin{align}\label{eqn.dirichlet}
p \ox q = \sum_{P: p(1)} \yon^{p[P]} \ox \sum_{Q: q(1)} \yon^{q[Q]} &:= \sum_{(P,Q): p(1) \times q(1)} \yon^{p[P] \times q[Q]} \\\nonumber
p \tri q = \sum_{P: p(1)} \yon^{p[P]} \tri \sum_{Q: q(1)} \yon^{q[Q]} &:=
\sum_{P: p(1)} \Big(\sum_{Q: q(1)} \yon^{q[Q]} \Big)^{p[P]}
= \sum_{P:p(1)} \prod_{d:p[P]} \sum_{Q:q(1)} \prod_{e:q[Q]} \yon
%= \sum_{P: p(1)} \sum_{Q': p[P] \to q(1)} \prod_{d: p[P]} \prod_{e: q[Q'(P)]} \yon
\end{align}

We leave the proofs of the following two propositions as exercises for the reader.

* The functor $A\mapsto A\yon$ is strong monoidal $(\Set,1,\times)\to(\Poly,\yon,\otimes)$.

* The functor $p\mapsto p(1)$ is strong monoidal $(\Poly,\yon,\otimes)\to(\Set,1,\times)$.

* The functor $A\mapsto\yon^A$ is strong monoidal $(\Set^\op,1,\times)\to(\Poly,\yon,\otimes)$.

* The functor $p\mapsto\Gamma_p$ is colax monoidal $(\Poly,\yon,\tri)\to(\Set^\op,1,\times)$.

§.§ Linearly distributive categories (LDCs)

Here we discuss various flavors of LDCs and the appropriate sorts of functors between them.

§.§.§ Flavors of LDCs

A linearly distributive category (LDC) is a category $\X$, equipped with two monoidal structures[The operation $a \tri b$ is usually written as $a \oa b$ in the literature.]

$$(\X, \ox, \top)~~~\mbox{ and } ~~~ (\X, \tri, \bot)$$

linked by natural transformations

\begin{equation}\label{eqn.lin_dist}
\begin{aligned}
&\partial^L_L: a \ox (b \tri c) \rightarrow  (a \ox b) \tri c \\
& \partial^R_R: (b \tri c) \ox a \rightarrow b \tri (c \ox a)
\end{aligned}
\end{equation}

called  linear distributors,
[The maps $\partial^L$ and $\partial^R$ are called linear  distributors because they do not copy $a$ as in distributive categories:

\[ A \times (B + C)  \cong (A \times B) + (A \times C) \]

In this article, we use the word linear for two different but firmly-established notions. First, a polynomial is linear if it has the form $A\yon$ for some set $A$. Second, the terminology of linear distributive categories often appends the term “linear” to refer to various notions in that context, e.g. linear duals. Hopefully, this name-space collision will not cause confusion.
such that the the associators and unitors of the $\ox$ and the $\tri$ products interact coherently with the linear distributors <cit.>.

Every monoidal category is an LDC in which the two tensor products coincide.

For more examples, see <cit.>.

A symmetric LDC is an LDC in which both monoidal structures are symmetric,  i.e. having symmetry maps $\sigma_\ox$ and $\sigma_\tri$ such that the following diagram commutes:

\[ \begin{tikzcd}
(A \tri B) \ox C \ar[d, "\partial^R_R"'] \ar[r, "c_\ox" ]
& C \ox (A \tri B) \ar[r, "c_\tri"]
& C \ox (B \tri A) \ar[d, "\partial^L_L"] \\
A \tri (B \ox C)
& A \tri (C \ox B)  \ar[l, "\sigma_\tri"]
& A \tri (B \ox C) \ar[l, "\sigma_\ox"]
\end{tikzcd}\]

In a symmetric LDC, the symmetry maps induce the following permuting distributivity maps:

\begin{equation}\label{eqn.other_dists}
\begin{aligned}
\partial^L_R &: a \otimes (b \tri c) \to b \tri (a \otimes c) \\
\partial^R_L &: (b \tri c) \otimes a \to (b \otimes a) \tri c
\end{aligned}
\end{equation}

An LDC which is equipped with all the four linear distributivity maps is said to be non-planar <cit.>.

Motivated by $\Poly$, we consider LDCs in which only the tensor $\ox$ is symmetric.

A $\otimes$-symmetric LDC $(\X, \otimes, \top, \tri, \bot)$ is an LDC such that:

* $(\X, \otimes, \top)$ is a symmetric monoidal category (parallel  structure), and

* $(\X, \tri, \bot)$ is a (not-necessarily-symmetric) monoidal category (sequential structure).

All $\ox$-symmetric LDCs are non-planar.

In a $\otimes$-symmetric LDC, the symmetry of the tensor product $\otimes$ induces the two permuting distributivity maps as follows:

\[ \partial^L_R := \left( a \otimes (b \tri c) \to^{c_\ox} (b \tri c) \ox a \to^{\delta^R_R} b \tri (c \ox a) \to^{b \tri c_\ox} b \tri (a \ox c) \right) \]

The other permuting map $\delta^R_L$, is constructed similarly.

An LDC satisfies the mix inference rule in linear logic if it has a map $\bot \to \top$ satisfying certain coherences:

A mix LDC is an LDC with a map $\m: \bot \to \top$, called the mix map. The mix map induces, for any $a,b:\X$, a natural map $\indep_{a,b}: a \ox b \to a \tri b$, as shown as the dotted arrow in the following commutative diagram:

\begin{equation}
\label{eqn: mix coherence}
\begin{tikzcd}[column sep=50pt, ampersand replacement=\&]
a \ox b \ar[r, "\id \ox u_\tri^{L^{-1}}"] \ar[d, "(u_\tri^R)^{-1}\ox \id"'] \ar[ddrr, dotted, "\indep_{a,b}"] \&
a \ox (\bot \tri b) \ar[r, "\id \ox (\m \tri \id)"] \&
a \ox ( \top \tri b) \ar[d, "\partial^L"] \\
(a \tri \bot) \ox b \ar[d, "\partial^R"] \&
\&
( a \ox \top ) \tri b  \ar[d, "u_\ox^R \tri \id"'] \\
a \tri (\bot \ox b) \ar[r, "\id \tri (\m \ox \id)"'] \&
a \tri (\top \ox b) \ar[r, "\id \tri u_\ox^L"'] \&
a \tri b
\end{tikzcd}
\end{equation}

An isomix LDC is a mix LDC in which the mix map $\top \to^{\m}_{\raisebox{1pt}{$\cong$}} \bot$ is an isomorphism.

When $\m: \bot \to \top$ is an isomorphism, the coherence requirement in eqn (<ref>) is automatically satisfied (see <cit.>).

In the LDC literature, the $\indep$ map is usually referred to as the mixor and is written as mx. In this article, we use indep, because of its semantics in $\Poly$; see below (<ref>).

A compact LDC is an isomix LDC in which for all $a,b$, the map $\indep_{a,b}: a \ox b \to^{\cong} a \tri b$ is an isomorphism. In fact, any compact LDC is linearly equivalent to a monoidal category <cit.>. The term `compact' in the context of LDCs refers to the fact that the two monoidal products are isomorphic via the $\indep$ map.

A monoidal category is a compact LDC in which for all $a,b$, the map $\indep_{a,b} = \id_{a \ox b}$ is the identity natural transformation.

The following schematic diagram summarizes some relevant flavors of LDCs:

\[
\begin{tikzpicture}[scale=1.8]
\begin{pgfonlayer}{nodelayer}
\node [style=circle, scale=2, color=black, fill=red] (0) at (-5.75, 2.75) {};
\node [style=circle, scale=2, color=black, fill=red!70] (1) at (-3.5, 2.75) {};
\node [style=circle, scale=2, color=black, fill=red!60] (2) at (-1, 2.75) {};
\node [style=circle, scale=2, color=black, fill=red!40] (3) at (1.75, 2.75) {};
\node [style=circle, scale=2, color=black, fill=red!20] (5) at (4, 2.75) {};
\node [style=none] (4) at (-7.75, 2.75) {};
\node [style=none] (6) at (6, 2.75) {};
\node [style=none] (7) at (-5.75, 2) {LDC};
\node [style=none] (8) at (-3.5, 4) {Mix LDC};
\node [style=none] (9) at (-3.5, 3.5) {$\m: \bot \to \top$};
\node [style=none] (10) at (-1, 2) {Isomix LDC};
\node [style=none] (11) at (1.75, 4.25) {Compact LDC};
\node [style=none] (12) at (1.75, 3.65) {$a \ox b \to^{\indep}_{\cong} a \tri b$};
\node [style=none] (13) at (4, 2) {Monoidal category};
\node [style=none] (14) at (-1, 1.4) {$\bot \to^{\m}_{\cong} \top$};
\node [style=none] (15) at (4, 1.5) {$\ox = \tri$, $\top=\bot$};
\node [style=none] (16) at (-5.75, 1.5) {$(\X, \ox, \top)$};
\node [style=none] (17) at (-5.75, 1) {$(\X, \tri, \bot)$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [dotted] (4.center) to (0);
\draw (0) to (1);
\draw (1) to (2);
\draw (2) to (3);
\draw (3) to (5);
\draw [dotted] (5) to (6.center);
\end{pgfonlayer}
\end{tikzpicture}
\]

We will be focused mainly on the center dot: we will see that $\Poly$ is an isomix LDC.

Notation. In this article, we write $\partial^L$ and $\partial^R$ for $\partial^L_L$ and $\partial^R_R$ respectively, and we will have very little use for the other two maps (<ref>). For the rest of the paper, the term LDC will refer to non-symmetric LDCs unless otherwise specified.

§.§.§ Functors between LDCs

Linear functors <cit.> are the most general notion of maps between LDCs. In this article we use a simpler notion: that of Frobenius functors <cit.>.

Suppose $\X$ and $\Y$ are LDCs. A Frobenius functor consists of a functor $F: \X \to \Y$ equipped with:

* a lax monoidal structure $(F, m_\ox, m_I): (\X, \ox, \top) \to (\Y, \otimes, \top)$  and

* a colax monoidal structure $(F, n_\tri, n_I): (\X, \tri, \bot) \to (\Y, \tri, \bot)$

such that  the laxors $m_\ox$ and $n_\tri$ and the distributivity maps interact as follows:

\[ {\bf [F.1]}\quad  \begin{tikzcd}[column sep=large]
F(a) \otimes F(b \tri c) \ar[r, "\id \otimes n_\tri"] \ar[d, "m_\otimes"'] &
F(a) \otimes (F(b) \tri F(c)) \ar[d, "\delta^L"] \\
F(a \otimes (b \tri c)) \ar[d, "F(\delta^L)"']
& (F(a) \otimes F(b)) \tri F(c) \ar[d, "m_\otimes \tri \id"] \\
F((a \ox b) \tri c) \ar[r, "n_\tri"'] & F(a \tri b) \tri F(c)
\end{tikzcd} \]

\[
\quad {\bf [F.2]}\quad \begin{tikzcd}[column sep=large]
F(a \tri b) \otimes F(c) \ar[r, "n_\tri \otimes  \id"]  \ar[d, "m_\otimes"']  &
(F(a) \tri F(b)) \otimes F(c) \ar[d, "\delta^R"] \\
F((a \tri b) \ox c) \ar[d, "F(\delta^R)"'] & F(a) \tri (F(b) \otimes F(c)) \ar[d, "\id \tri m_\ox"] \\
F(a \tri (b \otimes c)) \ar[r, "n_\tri"'] & F(a) \tri F(b \otimes c)
\end{tikzcd} \]

Suppose $\X$ and $\Y$ are mix LDCs. A mix functor $F: \X \to \Y$ is a Frobenius functor making the following diagram commute:

\[
\begin{tikzcd}
F(\bot) \ar[r, "n_I"] \ar[d, "F(\m)"'] &
\bot \ar[d, "\m"] \\
F(\top) \ar[r, <-, "m_I"'] &
\top
\end{tikzcd}
\]

A mix functor $F$ is isomix if $m_I: \top \to F(\top)$ and $n_I: F(\bot) \to \bot$ are isomorphisms.

§ POLY IS AN ISOMIX LDC

In this section, we prove that $\Poly$ is an isomix LDC. This result follows from the general observation that there is a faithful functor from the category of normal duoidal categories and functors to that of (non-planar) isomix LDCs and isomix Frobenius functors.

A duoidal category is a category $\X$ with two monoidal structures $(\X, \ox, \top)$ and $(\X, \tri, \bot)$ along with a natural transformation:

\[ \duo: ~(a \tri b) \ox (c \tri d) \to (a \ox c) \tri (b \ox d) \]

called the interchange law, and morphisms:

\begin{align*}
e_\top:& ~\top \to \top \tri \top \\
e_\bot:& ~\bot \otimes \bot \to \bot
\end{align*}

such that the functors $\tri: \X \times \X \to \X$ and $\bot: \mathbbm{1} \to \X$ are $\ox$-lax monoidal, and the assosciativity and unitor natural isomorphisms of $(\tri, \bot)$ are $\ox$-monoidal natural transformations.
The above data generates a map $k: \top \to \bot$ as follows:

\[ \top \to^{\cong}  \top \ox \top \to^{\cong}  (\top \tri \bot) \ox (\bot \tri \top) \to^{\duo} (\top \ox \bot) \tri (\top \ox \bot) \to^{\cong} \bot \tri \bot \to^{\cong} \bot \]

We say that a duoidal category is normal if the above composite is an isomorphism.

A bilax duoidal functor $D: (\X, \ox, \top, \tri, \bot) \to (\Y, \ox, \top, \tri, \bot)$ consists of a functor $F: \X \to \Y$ that is $\ox$-lax monoidal, $\tri$-colax monoidal, and such that the following diagrams commute:

\[
{\bf [D.1]} \xymatrix{
F((a \tri b) \otimes (c \tri d)) \ar[d]_{F(\duo)} &F(a \tri b) \otimes F(c \tri d) \ar[r]^{\hspace{-20pt}n_\tri \otimes n_\tri} \ar[l]_{m_\otimes} & (F(a) \tri F(b)) \otimes (F(c) \tri F(d)) \ar[d]^{\duo} \\
F((a \otimes c) \tri (b \otimes d)) \ar[r]_{n_\tri} &F(a \otimes c) \tri F(b \otimes d) & (F(a) \otimes F(c)) \tri (F(b) \otimes F(d)) \ar[l]^{\hspace{-20pt}m_\otimes \tri m_\otimes}
\]

\[
{\bf [D.2]} \xymatrix{
\top \ar[r]^{m_I} \ar[d]_{e_\top} &
F(\top) \ar[r]^{F(e_\top)} &
F(\top \tri \top) \ar[d]^{n_\tri} \\
\top \tri \top \ar[rr]_{m_I \tri m_I} & &
F(\top) \tri F(\top)
} ~~~~~~~~
{\bf [D.3]} \xymatrix{
F(\bot) \otimes F(\bot) \ar[d]_{m_\otimes} \ar[rr]^{n_I \otimes n_I} &  & \bot \otimes \bot \ar[d]^{e_\bot} \\
F(\bot \otimes \bot) \ar[r]_{F(e_\bot)} & F(\bot) \ar[r]_{n_I} &  \bot
\]

\[ {\bf [D.4]}~~
\begin{tikzcd}
\top \ar[r, "m_I"] \ar[d, "k"' ] & F(\top) \ar[d, "F(k)"] \\
\bot \ar[r, <- , "n_I"'] & F(\bot)
\end{tikzcd}
\]

A bilax duoidal functor $F$ between normal duoidal categories is normal if the unit laxors are isomorphisms:

\[
m_I: \top \to^{\cong} F(\top) ~~~~ \text{ and } ~~~~~ n_I: F(\bot) \to^{\cong} \bot
\]

[column sep=small, ampersand replacement=&]
F(a) F(c d) [llldd, "m_"'] [rrrdd, "𝕀n_"] [dd, "F(u_^-1) 𝕀", "≅"'] &

F( a ( c d)) [ddd, "F(δ)"'] [drr, "F(u_^-1 𝕀)", "≅"' ] [rrr, phantom, "=nat."] &
F(a ) (F(c d)) [dl, "m_"'] [dr, "n_n_" ] &
F(a) (F(c) F(d)) [ddd, "δ^L"] [dl, "F(u_^-1) 𝕀"']

F((a )(c d)) [d, "F()"'] &
[d, phantom, "=(D.1)", description]  &
(F(a) F()) (F(c) F(d)) [d, ""] [r, "(𝕀n_I) 𝕀"] [dr, phantom, "=nat."] &
(F(a) ) (F(c) F(d)) [d, ""] &

[rr, phantom, ":=", description] &
F((a c) (d)) [d, "F(𝕀(k^-1 𝕀))"'] [dr, "n_"]
. &
(F(a) F(c) (F() F(d)) [ld, "m_m_"'] [r, "𝕀(n_I 𝕀)"] [d, "𝕀(F(k^-1) ⊗𝕀"'] [dr, phantom, "=(D.1)", description] &
(F(a) F(c)) (F(d)) [d, "𝕀k^-1 ⊗𝕀"] [r, phantom, "=:"] &

F((a c) d) [ddrrr, "n_"'] [rr, "F(𝕀u_^-1)"'] &
F((a c) (⊤d)) [dr, "n_"']  &
F(a c) F(⊤d) [d, "𝕀F(k 𝕀)"]  &
F(a) F(c) F(⊤) F(d) [ld, "m_m_"] &
F(a) F(c) ⊤F(d) [l, "𝕀m_I 𝕀"] [r, "F(𝕀u_)"] &
(F(a) F(c)) F(d)  [ddlll, "n_"]

F(a c) F(⊤d) [d, "𝕀F(u_)"] &

F(a c) F(d) &

Normal duoidal functor is also isomix Frobenius

Let $\nD$ be the category of normal duoidal categories and bilax normal duoidal functors. Let $\Iso$ be the category of isomix categories and isomix functors.

There is a faithful functor $H: \nD \to \Iso$.

Suppose $(\X, \ox, \top, \tri, \bot)$ is a normal duoidal category. We show that it can be given an isomix LDC structure by defining the linear distributivity and the mix maps. The linear distributivity maps are constructed from the duoidal map and the isomorphism $\top \to^{k}_{\cong} \bot$ by introducing and eliminating monoidal units. The construction of the left distributor $\partial^L$ is shown in eqn (<ref>).

\begin{equation}
\label{eqn: left distributor}
\begin{tikzcd}[column sep = large]
a \ox (b \tri c)
\ar[rr, "{u_\tri^{-1} \ox \id}", "\cong"' ]
\ar[d, "\partial^L"']
\ar[drr, phantom, "\partial^L:="]
&& (a \tri \bot) \ox (b \tri c)
\ar[d, "\duo" ]
\\
(a \ox b) \tri c
& (\bot \ox b) \tri (a \ox c)
\ar[l, "\id \tri u_\ox"', "\cong" ]
& (a \ox b) \tri (\top \ox c)
\ar[l, "k^{-1}"', "\cong" ]
\end{tikzcd}
\end{equation}

The mix map $\m: \bot \to \top$ is defined to be $k^{-1}$. The right distributivity map $\partial^R$ and the two permuting distributivity maps $\partial^L_R$, $\partial^R_L$ are constructed similarly.

Suppose $\X$ and $\Y$ are normal duoidal categories, and $F: \X \to \Y$ is a normal duoidal functor. Then,  $H(F) := F$ is a Frobenius functor, see <ref> for the proof that [F.1] holds. In <ref>, the inner squares shown without label commute because $F$ is a lax monoidal functor or a co(lax)monoidal functor. Similary, it can be proven that [F.2] holds.

The normal duoidal functor $F$ is also an isomix functor because $\top \to^{k} \bot$ is an isomorphism, and [D.4] holds.

The category $\Poly$ of polynomial functors and natural transformations is normal duoidal:

The category $({\sf Poly}, \otimes, \tri, \yon)$ is normal duoidal.

For any symmetric monoidal structure $\cdot$ on $\Set$, there is a symmetric monoidal structure $\odot$ on Poly which is the Day convolution of $\cdot$. The Cartesian product on Set induces the $\ox$ structure on Poly given in <ref>.
Thus, for $p,q: \Poly$, and  $A,B: \Set$:

\[ p(A) \times q(B) \to^{\sf Day} (p \otimes q) (A \times B) \]

natural in $A,B : \Set$, where $p(A) := p \tri A$ denotes functor application. The above map satisfies the following universal property: for all functors $r\colon\Set\to\Set$ and functions $\phi: p(A) \times q(B) \to r(A \times B)$, there exists a unique (dotted) map which makes the following diagram commute:

\begin{equation}
\label{eq.dayUniversalProp}
\xymatrix{
p(A) \times q(B) \ar[r]^{\sf Day} \ar[dr]_{\phi} & (p \otimes q) (A \times B<EMAIL_ADDRESS>\\
& r(A \times B)
\end{equation}

With this data, one can prove that $({\sf Poly}, \otimes, y, \tri, y)$ is normal duoidal. Indeed, to give the duoidal map $d$ for $p_1, p_2, q_1, q_2: {\sf Poly}$,

\[ d: (p_1 \tri p_2) \otimes (q_1 \tri q_2) \to (p_1 \otimes q_1) \tri (p_2 \otimes q_2) \]

it suffices to give, naturally for all $A, B: {\sf Set}$, a map of the form

\[ d_{A,B}: \big((p_1 \tri p_2) \otimes (q_1 \tri q_2)\big) (A \times B) \to (p_1 \otimes q_1) \tri (p_2 \otimes q_2) (A \times B) \]

And by the universal property of the Day convolution product, in order to give $d_{A,B}$, it is enough to provide a map of the form

\[ \phi:  (p_1 \tri p_2)(A) \times (q_1 \tri q_2)(B) \to (p_1 \otimes q_1) \tri (p_2 \otimes q_2) (A \times B) \]

which can be constructed as follows:

\begin{align*}
(p_1 \tri p_2)(A) \times (q_1 \tri q_2)(B)
&= (p_1 \tri (p_2 (A)))\times (q_1 \tri (q_2(B))) \\
&\to^{\sf Day} (p_1 \otimes q_1) (p_2(A) \times q_2(B)) \\
&\to^{\id\tri {\sf Day}} (p_1 \otimes q_1)\big((p_2 \otimes q_2)(A \times B)\big) \\
&= (p_1 \otimes q_1) \tri (p_2 \otimes q_2) (A \times B)
\end{align*}

Then, one can define the duoidal map as the unique map given by the universal property of the Day convolution product as in <ref>.

The category $({\sf Poly}, \otimes, \tri, \yon)$ is an isomix non-planar linearly distributive category.

By <ref>, the category $\Poly$ is isomix LDC. By <ref>, it is non-planar because it is $\ox$-symmetric.

Now that we know that $\Poly$ is an isomix LDC, let us examine its mix and $\indep$ maps. In $\Poly$, the units of the $\ox$- and the $\tri$-products coincide: $\bot = \top = \yon$. Hence, the mix map $\m$ can be taken to be the identity map. For any two polynomials $p$ and $q$, the $\indep$ map $p \ox q \to^{\indep} p \tri q$ is defined as follows:

\begin{align}
\label{eq.indep}
\sum_{P:p(1)} \sum_{Q:q(1)} \prod_{d:p[P]} \prod_{b: q[Q]} \yon &\to^{\indep} \nonumber
\sum_{P:p(1)} \prod_{d:p[P]} \sum_{Q:q(1)} \prod_{b: q[Q]} \yon \\
\indep_1: (P,Q) &\mapsto (P, \textit{const} Q: p[P] \to q(1); d \mapsto Q)   \\
\indep_{P,Q}^\sharp:  (d,b) &\mapsto (d,b): p[P] \times q[Q] \label{eqn.indep_2}
\end{align}

The fact that, under this map, $Q$ arrives in the codomain as a constant, i.e. that it is independent of $d$, is the reason for the name $\indep$.
In passing, we note that for each $P:p(1)$ and $Q:q(1)$, the function $\indep_{P,Q}^\sharp$ is a bijection, and hence we record the following.

For any $p,q:\Poly$, the map $\indep_{p,q}$ is cartesian.

§ LINEAR DUALS

In this section, we introduce biclosed LDCs and characterize the dual objects in these categories. The category $\Poly$ is a biclosed LDC. We apply the general results on biclosed LDCs to identify the dual objects in $\Poly$.

§.§ Linear duals in LDCs

Let us recall the definition of dual objects in LDCs.

Suppose $\X$ is a LDC and $a,b : \X$ are objects. Then $b$ is left dual to $a$—and $a$ is right dual to $b$—written as
\[ (\eta, \epsilon): b \dual a \]
if there exists a unit map $\eta: \top \to a \tri b$, and a counit map $\epsilon: b \otimes a \to \bot$ such that the following diagrams commute:

\begin{equation}\label{eqn.linduals}
\textbf{[dual.1]}
\begin{tikzcd}
a \ar[r, "u_\ox^{-1}"] \ar[d, equals]
&[-5pt] \top \ox a  \ar[r,"\eta \ox \id"]
& (a \tri b) \ox a \ar[d, "\delta^R"]
\\
a \ar[r, <-,  "u_\tri"']
& a \tri \bot \ar[r, <-,  "1 \tri \epsilon"']
& a \tri (b \ox a)
\end{tikzcd}
\textbf{[dual.2]}
\begin{tikzcd}
b \ar[r, "u_\ox^{-1}"] \ar[d, equals]
&[-5pt] b \ox \top  \ar[r,"\id \ox \eta "]
& b \ox (a \tri b) \ar[d, "\partial^L"]
\\
b \ar[r, <-,  "u_\tri"']
& a \tri \bot \ar[r, <-,  "\epsilon \tri 1"']
& (b \ox a) \tri b
\end{tikzcd}
\end{equation}

Note that the above coherence conditions involve mainly $\eta$, $\epsilon$ and the linear distributivity maps. For easy understanding of the coherences, the $\eta$ and the $\epsilon$ can be drawn as a cap and a cup respectively (by hiding the units), even though they refer to different monoidal products. The two coherence diagrams [dual.1] and [dual.2] can then be drawn as snake diagrams, as shown below.

\[ \includegraphics[scale=0.09]{figures/duals.jpeg}\]

Notation. All string diagrams in this article are to be read top-to-bottom, i.e. in the direction of gravity.

A morphism of linear duals, $(f,g): (\eta,  \epsilon) \to (\eta', \epsilon')$, is given by a pair of maps

\[ \xymatrix{ b \ar[d]_f \ar@{-||}[r]^{(\eta,  \epsilon)}  &  a \\ b' \ar@{-||}[r]_{(\eta', \epsilon')} &  a' \ar[u]_{g}} \]

such that the following equations hold:
\[ (i) ~~~  \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-1, 2) {};
\node [style=none] (1) at (0.5, 3) {};
\node [style=none] (2) at (-1, 3) {};
\node [style=none] (3) at (-0.25, 4) {$\eta'$};
\node [style=none] (4) at (0.5, 2) {};
\node [style=none] (5) at (-1.25, 2.25) {$b'$};
\node [style=none] (6) at (1, 3.5) {$a'$};
\node [style=circle, scale=1.5] (7) at (0.5, 2.75) {};
\node [style=none] (8) at (0.5, 2.75) {$g$};
\node [style=none] (9) at (1, 2.25) {$a$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (4.center) to (7);
\draw (1.center) to (7);
\draw (2.center) to (0.center);
\draw [bend left=90, looseness=1.75] (2.center) to (1.center);
\end{pgfonlayer}
\end{tikzpicture}
= \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0.5, 2) {};
\node [style=none] (1) at (-1, 3) {};
\node [style=none] (2) at (0.5, 3) {};
\node [style=none] (3) at (-0.25, 4) {$\eta$};
\node [style=none] (4) at (-1, 2) {};
\node [style=none] (5) at (0.75, 2.25) {$a$};
\node [style=none] (6) at (-1.5, 3.5) {$b$};
\node [style=circle, scale=1.5] (7) at (-1, 2.75) {};
\node [style=none] (8) at (-1, 2.75) {$f$};
\node [style=none] (9) at (-1.5, 2.25) {$b'$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (4.center) to (7);
\draw (1.center) to (7);
\draw (2.center) to (0.center);
\draw [bend right=90, looseness=1.75] (2.center) to (1.center);
\end{pgfonlayer}
\end{tikzpicture}
~~~~~~(ii)~~~ \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (0.5, 3.75) {};
\node [style=none] (1) at (-1, 2.75) {};
\node [style=none] (2) at (0.5, 2.75) {};
\node [style=none] (3) at (-0.25, 1.75) {$\epsilon'$};
\node [style=none] (4) at (-1, 3.75) {};
\node [style=none] (5) at (0.75, 3.5) {$a'$};
\node [style=none] (6) at (-1.5, 2.25) {$b'$};
\node [style=circle, scale=1.5] (7) at (-1, 3) {};
\node [style=none] (8) at (-1, 3) {$f$};
\node [style=none] (9) at (-1.5, 3.5) {$b$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (4.center) to (7);
\draw (1.center) to (7);
\draw (2.center) to (0.center);
\draw [bend right=90, looseness=1.75] (1.center) to (2.center);
\end{pgfonlayer}
\end{tikzpicture}
=  \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (-1, 3.75) {};
\node [style=none] (1) at (0.5, 2.75) {};
\node [style=none] (2) at (-1, 2.75) {};
\node [style=none] (3) at (-0.25, 1.75) {$\epsilon$};
\node [style=none] (4) at (0.5, 3.75) {};
\node [style=none] (5) at (-1.25, 3.5) {$b$};
\node [style=none] (6) at (1, 2.25) {$a$};
\node [style=circle, scale=1.5] (7) at (0.5, 3) {};
\node [style=none] (8) at (0.5, 3) {$g$};
\node [style=none] (9) at (1, 3.5) {$a'$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (4.center) to (7);
\draw (1.center) to (7);
\draw (2.center) to (0.center);
\draw [bend right=90, looseness=1.75] (2.center) to (1.center);
\end{pgfonlayer}
\end{tikzpicture} \]

Notice that a morphism of duals is determined completely by either of the pair of maps ($f$
completely determines $g$, and vice versa), and are referred to as Australian mates <cit.>. :

\[ g := \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle, scale=2] (0) at (0, 2) {};
\node [style=none] (1) at (0, 2) {$f$};
\node [style=none] (2) at (0, 2.5) {};
\node [style=none] (3) at (1, 2.5) {};
\node [style=none] (4) at (1, 0.75) {};
\node [style=none] (5) at (0, 1.5) {};
\node [style=none] (6) at (-1, 1.5) {};
\node [style=none] (7) at (-1, 3.5) {};
\node [style=none] (8) at (-1.25, 3) {$a'$};
\node [style=none] (9) at (1.25, 1.25) {$a$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (2.center) to (0);
\draw (0) to (5.center);
\draw [bend left=90, looseness=1.75] (5.center) to (6.center);
\draw (6.center) to (7.center);
\draw [bend left=90, looseness=2.00] (2.center) to (3.center);
\draw (3.center) to (4.center);
\end{pgfonlayer}
\end{tikzpicture}
\]

The map $f$ is an isomorphism if and only if $g$ is an isomorphism. Also, notice that
if $f$ or $g$ is an isomorphism, equations $(i)$ and $(ii)$ imply one another in the
definition of morphism of duals.

In a mix LDC, we have the following equations:

\[ \includegraphics[scale=0.07]{figures/fig1.jpeg}\]

Equivalently, the following squares commute:

\[ (i) ~~~ \begin{tikzcd}
b \ox a \ar[r, "\epsilon"] \ar[d, "u_\ox^{-1}"']
& \bot \ar[r, "\m"]
& \top \ar[r, "\eta"]
& a \tri b \ar[dd, equals]
\\
\top \ox (b \ox a) \ar[d, "\eta \ox \id"']
\\
(a \tri b) \ox (b \ox a) \ar[r, "\id \ox \epsilon"']
& (a \tri b) \ox \bot \ar[r, "\id \ox \m"']
& (a \tri b) \ox \top \ar[r, "u_\ox"']
& a \tri b
\end{tikzcd} \]

\[ (ii) ~~~ \begin{tikzcd}
b \ox a \ar[r, "\epsilon"] \ar[d, "u_\ox^{-1}"']
& \bot \ar[r, "\m"]
& \top \ar[r, "\eta"]
& a \tri b \ar[dd, equals]
\\
(b \ox a) \ox \top \ar[d, "\id \ox \eta"']
\\
(b \ox a) \ox (a \tri b) \ar[r, "\epsilon \ox \id"']
& \bot \ox (a \tri b) \ar[r,  "\m \ox \id"']
& \top \ox (a \tri b)  \ar[r, "u_\ox"']
& a \tri b
\end{tikzcd}
\]

We prove below that $(i)$ commutes. The proof that $(ii)$ commutes follows similarly.

\[
\begin{tikzcd}[column sep=large]
b \ox a \ar[r, "\epsilon"] \ar[d, "u_\ox^{-1}"'] \ar[dr, phantom, "\text{=nat.}"]
& \bot \ar[r, "\m"] \ar[d, "u_\ox^{-1}"] \ar[dr, phantom, "\text{=nat.}"]
& \top  \ar[r, "\eta"] \ar[d, "u_\ox^{-1}"] \ar[dr, phantom, "\text{=inv.}"]
&  a \tri b  \ar[dd, bend left, equals]
\\
\top \ox (b \ox a) \ar[r, "\id \ox \epsilon"] \ar[d, "\eta \ox \id"'] \ar[dr, phantom, "=\ox\text{-bifunctor}"]
& \top \ox \bot \ar[r, "\id \ox \m"]  \ar[d, "\eta \ox \id"] \ar[dr, phantom, "=\ox\text{-bifunctor}"]
& \top \ox \top  \ar[r, "u_\ox"] \ar[d, "\eta \ox \id"] \ar[dr, phantom, "\text{=nat.}"]
& \top \ar[d, "\eta"]
\\
(a \tri b) \ox (b \ox a) \ar[r, "\id \ox \epsilon"']
& (a \tri b) \ox \bot \ar[r, "\id \ox \m"']
& (a \tri b) \ox \top  \ar[r, "u_\ox"']
& (a \tri b)
\end{tikzcd}
\qedhere
\]

§.§ Duals in biclosed LDCs

In this section, we discuss what happens when the parallel structure ($\otimes$) has a left closure and the sequential structure ($\tri$) has a right coclosure. These are related to dual objects in LDCs.

We define a linearly distributive category $(\X, \ox, \top, \tri, \bot)$ to be:

* $\ox$-closed if for all $a:\X$, the functor $ - \otimes a : \X \to \X$  has a right adjoint, denoted $ a \lollipop ~ -$,

* $\tri$-coclosed if for all $a: \X$, the functor $ - \tri a: \X \to \X$ has a left adjoint, denoted $ a \slash-$,

* biclosed if $\X$ is both $\ox$-closed and $\tri$-coclosed.

Let $a$ be any object in a $\ox$-closed LDC. Then, by the adjunction, for all $b$, and $c$, we have an isomorphism:

\begin{equation}
\label{eqn: closure}
\X(a \ox b, c) \cong \X(b, a \lollipop c)
\end{equation}

We refer to the unit of the adjunction as the eval map:

\begin{equation}
\label{eqn:eval}
\eval: a \ox (a \lollipop b) \to b
\end{equation}

Let $a$ be any object in a $\tri$-coclosed LDC. Then, by the adjunction, for all $b$, and $c$, we have an isomorphism:

\begin{equation}
\label{eqn: coclosure}
\X \left(a \slash c, b \right) \cong \X(a, b \tri c)  \end{equation}

We call the counit of the adjunction as the coeval map:

\begin{equation}
\label{eqn:coeval}
\coeval: a \to (a \slash b) \tri b   % a \slash b is b over a co-closure
\end{equation}

[<cit.>] The category $\Poly$ is a biclosed LDC where, for any two polynomials $p, q$:

\begin{align}
p \lollipop q := [p,q] & =  \prod_{P : p(1)} \sum_{Q : q(1)} \prod_{d : q[Q]} \sum_{d' : p[P]} \yon
\label{eqn: Poly-close}
\\
p \slash q := \coclose{q}{p} & = \sum_{P : p(1)} \yon^{ q \tri p[P]} = \sum_{P : p(1)} \prod_{Q: q(1)}\prod_{d:q[Q]\to p[P]}\yon
\label{eqn: Poly-coclose}
\end{align}

Although we will not need it, the coclosure $\coclose{q}{p}$ is the left Kan extension of $p$ along $q$:

\[
\begin{tikzcd}
\Set\ar[r, "p", ""' name=p]\ar[d, "q"']&
\Set\\
\Set\ar[ru, bend right=25pt, "\coclose{q}{p}"', "" name=cc]
\ar[from=p, to=cc-|p, xshift=-17pt, yshift=-5pt, shorten=2pt, Rightarrow, "\;\;\coeval"]
\end{tikzcd}
\]

In the rest of the paper, without loss of generality, we will assume $\top = \bot$ for isomix LDCs. For an isomix LDC, we will denote the monoidal unit by $\yon$. Since we are mainly concerned with $\Poly$ category, in the rest of the paper, we will use the notation $[a, b]$ for $a \lollipop b$, and $\coclose{b}{a}$ for $a \slash b$ for readability and uniformity.

For any object $a:\X$ in a biclosed isomix LDC, there are two sorts of “double dual” maps, $\Phi_a$ and $\Psi_a$, each of which uses both the $\otimes$-closure and the $\tri$-coclosure. They are defined as the following composites:
[For brevity, we suppress identities and the unitors $u_\ox$ and $u_\tri$ in <ref>. We follow this convention in the rest of the document when writing proofs and composites]

\begin{equation}
\label{eqn: phip}
\begin{tikzcd}
a \ar[r, "\coeval"] \ar[rrr, bend right = 15pt, "\Phi_a"'] &[10pt]  \left(\coclose{[a, \yon]}{\yon} \tri [a,\yon] \right) \ox a \ar[r, "\partial^R"] &   \coclose{[a, \yon]}{\yon} \tri ([a,\yon]) \ox a) \ar[r, "\eval"] & \coclose{[a,\yon]}{\yon}
\end{tikzcd}
\end{equation}

\begin{equation}
\label{eqn: psip}
\begin{tikzcd}
\left[ \coclose{a}{\yon}, \yon \right] \ar[r, "\coeval"] \ar[rrr, bend right = 15pt, "\Psi_a"']
& \left[ \coclose{a}{\yon}, \yon \right] \ox \left( \coclose{a}{\yon} \tri a \right) \ar[r, "\partial^L"]
& \left( \left[ \coclose{a}{\yon}, \yon \right] \ox \coclose{a}{\yon} \right)
\tri a \ar[r, "\eval"]
& a
\end{tikzcd}
\end{equation}

For an object $a$ in a biclosed isomix LDC, the following are equivalent:

* $\Phi_a$ has a retraction,

\[ a \to^{\Phi_a} \coclose{[a,\yon]}{\yon}
\to^{\chi_a} a \]

and the following composite is the identity on $[a,\yon]$,

\begin{multline*}
[a,\yon] \to^{\coeval} [a,\yon] \otimes \left(\coclose{[a,\yon]}{\yon} \tri [a,\yon] \right) \to^ {\partial^R}
\left( [a,\yon] \otimes \coclose{[a,\yon]}{\yon} \right) \tri [a,\yon] \\
\to^{\chi_a}
([a,\yon] \ox a) \tri [a,\yon]
\to^{\eval}   [a,\yon]
\end{multline*}

* $[a,\yon] \dashvv a$, with $\epsilon=\eval$ as maps $[a, \yon] \ox a \to \yon$.

(i) $\Rightarrow$ (ii): Given that $(i)$ holds, we must show that $[a, \yon] \dashvv a$ with $\epsilon=\eval$. Define $\eta$ as follows:

\begin{align}
\label{eqn.eta}
\eta := \left(\yon \to^{\coeval} \coclose{[a,\yon]}{\yon} \tri [a, \yon] \to^{\chi_a} a \tri [a,\yon]\right)
\end{align}

The following diagram proves that [dual.1] from <ref> holds:

\[ \begin{tikzcd}
a \ar[r, "\coeval"] \ar[dd, equals] \ar[rr, bend left = 25 pt, "\eta"] \ar[dr, phantom, "\text{=assumption}"]
& \left( \coclose{[a,\yon]}{\yon} \tri [a,\yon] \right) \ox a \ar[r, "\chi_a"] \ar[d, "\partial^R"]
& (a \tri \left[ a,\yon \right]) \ox a \ar[ddl, bend left = 10pt, "\partial^R"] \ar[dl, phantom, "\text{=nat.}"] \\
& \coclose{ \left[ a,\yon \right]}{a} \tri (\left[ a,\yon \right] \ox a) \ar[d, "\chi_a"]
\\
a \ar[r, <-, "\epsilon = \eval"']
& a \tri (\left[a,\yon \right] \ox a)
\end{tikzcd}\]

The proof that [dual.2] holds is similar.

(ii) $\Rightarrow$ (i): Given that $[a, \yon] \dashvv a$ with $\epsilon = \eval$, we must prove that (i) holds.
Since the LDC is $\tri$-coclosed, we know that that $ - \tri [a, \yon]$ has a left adjoint, <ref>. Define $\chi_a\colon\coclose{[a,\yon]}{\yon}\to a$ to be the universal map induced by $\eta$:

\[ \begin{tikzcd}[column sep=large]
\yon \ar[d, "\coeval"' ] \ar[dr, "\eta"] & \\
\coclose{[a,\yon]}{\yon} \tri [a,\yon] \ar[r, dotted, "\chi_a \tri \id"'] &  a \tri [a,\yon]
\end{tikzcd} \]

We first prove that the specified composite is indeed the identity on $[a,\yon]$:

\[
\begin{tikzcd}
{[}a,\yon{]} \ar[r, "\coeval"] \ar[d, equals] \ar[dr, phantom, "\text{=univ. prop.}"', description]
& {[}a,\yon] \ox \left( \coclose{[a,\yon]}{\yon} \tri [a,\yon] \right)
\ar[d, "\chi_a"] \ar[r, "\partial^L"] \ar[dr, phantom, "\text{=nat.}"]
& \left( {[}a,\yon] \ox \coclose{[a,\yon]}{\yon} \right) \ar[d, "\chi_a"]
\\
{[}a, \yon] \ar[r, "\eta"] \ar[drr, equals, bend right = 10pt]
& {[}a, \yon] \ox ([a,\yon] \tri a) \ar[r, "\partial^L"] \ar[dr, phantom, "\text{=[dual.2]}"]
& \left( [a,\yon] \ox a \right) \tri a \ar[d, "\eval"]
\\
& {[}a,\yon]
\end{tikzcd}
\]

Next we show that $\Phi_a \then \chi_a$ is a retract, completing the proof:

\[
\begin{tikzcd}
a \ar[r, "\coeval"] \ar[d, equals] \ar[dr, phantom, "\text{=couniv.~prop.}"] \ar[rrr, bend left = 12 pt, shift left=6pt, "\Phi_a"]
& \left( \coclose{[a,\yon]}{\yon} \tri [a,\yon] \right) \ox a \ar[r, "\partial^L", ""' name = d]
\ar[d, "\chi_a"] \ar[dr, phantom, "\text{=nat.}"]
& \coclose{[a,\yon]}{\yon} \tri ([a,\yon] \ox a) \ar[r, "\eval"] \ar[d, "\chi_a"] \ar[dr, phantom, "\text{=nat.}"]
& \coclose{[a,\yon]}{\yon} \ar[d, "\chi_a"]
\\
a \ar[r, "\eta"'] \ar[rrr, equals, bend right = 15 pt, shift right=5pt, ""' name = i]
& (a \tri [a,\yon]) \ox a \ar[r, "\partial^L"']
& a \tri ([a,\yon] \ox a) \ar[r, "\eval"']
& a %\ar[from=d, to=i, phantom, "[dual-1]"]
\end{tikzcd}
\qedhere
\]

For an object $b$ in a biclosed isomix LDC, the following are equivalent:

* $\Psi_b$ has a section,

\[ b \to^{\Omega_b} \left[ \coclose{b}{\yon}, \yon \right] \to^{\Psi_b} b \]

and the following composite is the identity on $\coclose{b}{\yon}$,

\begin{multline*}
\coclose{b}{\yon} \to^{\coeval}  \left( \coclose{b}{\yon} \tri b \right) \ox \coclose{b}{\yon}
\to^{\partial}  \coclose{b}{\yon} \tri  \left( b  \ox \coclose{b}{\yon} \right) \\
\to^{\Omega_b}  \coclose{b}{\yon} \tri  \left( \left[ \coclose{b}{\yon}, \yon \right]  \ox \coclose{b}{\yon} \right)
\to^{\eval} \coclose{b}{\yon}
\end{multline*}

* $b \dashvv \coclose{b}{\yon}$ with $\eta = \coeval$ maps $\yon \to \coclose{b}{\yon} \tri b$

(i) $\Rightarrow$ (ii): Given that $(i)$ is true, we must show that $b \dashvv \coclose{b}{\yon}$ with $\eta = \coeval$. Define $\epsilon$ as follows:

\begin{align}
\label{eqn.epsilon}
\epsilon := \left( b \ox \coclose{b}{\yon} \to^{\Omega_b}  \left[ \coclose{b}{\yon}, \yon \right] \ox \coclose{b}{\yon} \to^{\eval} \yon \right)
\end{align}

(ii) $\Rightarrow$ (i):
$b \dashvv \coclose{b}{\yon}$ with $\eta = \coeval$, we must show that $(i)$ holds. Since the LDC is $\ox$-closed, we know that that $ - \ox \coclose{b}{\yon}$ has a right adjoint, see <ref>. Define $\Omega_b\colon b\to\left[ \coclose{b}{\yon}, \yon \right]$ to be the universal map induced by $\epsilon$:

\[ \begin{tikzcd}[column sep=large]
b \ox \coclose{b}{\yon} \ar[r, dotted, "\Omega_b \ox \id"] \ar[dr, "\epsilon"']
& \left[ \coclose{b}{\yon}, \yon \right] \ox \coclose{b}{\yon} \ar[d, "\eval"]
\\
\yon
\end{tikzcd} \]

The rest follows similarly to the proof of <ref>.

In a biclosed isomix LDC, suppose there are maps $\yon \to^{\eta} a \tri b$ and $b \ox a \to^{\epsilon} \yon$ defining a linear duality $(\eta, \epsilon): b \dashvv a$. Then adjoint of $\eta$ which $\eta'\colon\coclose{b}{\yon} \to a$ is a retraction and the adjoint of $\epsilon$ which is $\epsilon'\colon b \to [a, \yon]$ is a section.

Given that $(\eta, \epsilon): b \dual a$, we first prove that $\coclose{b}{\yon} \to^{\eta'} a$ is a retraction. Define the map $a \to^{\varphi} \coclose{b}{\yon}$ as the following composite:

\begin{equation}\label{eqn.sectionofeta'} \varphi := \left(a \to^{\coeval } \left( \coclose{b}{\yon} \tri b \right) \ox a \to^{\partial^R}
\coclose{b}{\yon} \tri \left( b \ox a \right)
\to^{\epsilon} \coclose{b}{\yon}
\right)
\end{equation}

The following commuting diagram shows that $\varphi \then \eta' = \id_a$ making $\eta'$ a retraction.

\[
{ \footnotesize \begin{tikzcd}[column sep=large]
\ar[dd, equals]
\ar[ddr, phantom, "\text{=Defn. }\varphi"]
\ar[r, "\coeval"]
\ar[rr, bend left = 30pt, "\eta"]
& \left( \coclose{b}{\yon} \tri b \right) \ox a \ar[d, "\partial^R"]
\ar[r, "\eta'"]
\ar[dr, phantom, "\text{=nat.}"]
& (a \tri b) \ox a \ar[d, "\partial^R"]
\\
&  \coclose{b}{\yon} \tri \left( b  \ox a \right)
\ar[r, "\eta'"']
\ar[d, "\epsilon"]
\ar[dr, phantom, "=\tri- \text{bifunctor~ and nat. of } u_\tri"]
& a \tri (b \ox a)
\ar[d, "\epsilon"]
\\
a \ar[r, "\varphi"']
\ar[rr, equals, bend right]
& \coclose{b}{\yon}
\ar[r, "\eta'"']
& a
\end{tikzcd}} \]

Indeed, the large left square commutes by the definition of $\varphi$. The outer diagram commutes by [dual.1]. Thus, bottom triangle commutes, making $\eta'$ a retraction.

Next we prove that $b \to^{\epsilon'} [a,\yon]$ is a section. Define the map $[a, \yon] \to^{\psi} b$ to be the following composite:

\[ \psi := \left([a, \yon] \to^{\eta} [a, \yon] \ox (a \tri b) \to^{\partial^R} ([a,\yon] \ox a) \tri b
\to^{\eval} b\right) \]

The following commuting diagram shows that $\epsilon' \then \psi = \id_b$, making $\epsilon'$ a section. The outer diagram commutes because of [dual.2]. The large right square is the definition of the $\psi$ map.

\[ { \begin{tikzcd}[column sep=large]
\ar[r, "\epsilon'"] \ar[rr, equals, bend left]
\ar[d, "\eta"']
\ar[dr, phantom, "=\ox\text{-bifunctor and nat. of } u_\ox^{-1}" ]
& {\left[a, \yon \right]} \ar[r, "\psi"]
\ar[d, "\eta"']
\ar[ddr, phantom, "\psi :="]
& b
\ar[dd, equals]
\\
b \ox (a \tri b)
\ar[r, "\epsilon'"]
\ar[d, "\partial^L"'] \ar[dr, phantom, "\text{=nat.}" ]
& {[a,\yon]} \ox (a \tri b)
\ar[d, "\partial^L"]
\\
(b \ox a) \tri b
\ar[r, "\epsilon'"]
\ar[rr, bend right = 25pt, "\epsilon"']
& ({[a,\yon]} \ox a) \tri b
\ar[r, "\eval"]
& b
\end{tikzcd} } \]

§.§ Duals in $\Poly$

In this section, we characterize the dual objects of $\Poly$ using its left closure and right coclosure properties. We begin with some simple lemmas.

For any polynomial $p$, there is a natural ismorphism $\coclose{p}{\yon} \cong \yon^{p(1)}$

By <ref>

\[ \coclose{p}{\yon}\cong\coclose{p}{\yon^1} \cong \yon^{p \tri 1} \cong \yon^{p(1)} \qedhere \]

For any set $A$, there are natural isomorphisms in $\Poly$:

* $[A \yon, \yon] \cong \yon^A$

* $[\yon^A, \yon] \cong A \yon$

By <ref>,

\begin{align*}
\textit{(i):} \quad [A \yon, \yon] & \cong \prod_{a:A} \sum_1 \prod_1 \sum_1 \yon \cong \prod_{a:A} \yon \cong \yon^A  \\
\textit{(ii):} \quad [\yon^A, \yon] &\cong  \prod_1 \sum_1 \prod_1 \sum_{a:A} \yon \cong \sum_{a:A} \yon \cong A\yon
\qedhere
\end{align*}

Let us examine the coeval map (<ref>) in $\Poly$,

\[ \sum_{P:p(1)} \yon^{p[P]}=p \to^{\;\;\coeval\;\;} \coclose{q}{p} \tri q  =\sum_{P: p(1)} \yon^{q \tri p[P]} \tri q  \]

We can write this out using only $\sum$ and $\prod$:

\[
\sum_{P:p(1)}\prod_{d:p[P]}\yon\to\sum\limits_{P: p(1)}  \prod\limits_{Q:q(1)} \prod\limits_{d: q[Q] \to p[P]}\sum\limits_{Q': q(1)}  \prod_{b:q[Q']}\yon
\]

The coeval map is defined in $\Poly$ as follows:

\begin{equation}
\label{eqn: coeval defn}
\coeval_1: P \mapsto (P, (Q, d) \mapsto Q)
\qqand
\coeval_P^\sharp: (Q,d,b) \mapsto d(b).
\end{equation}

For any polynomial $p:\Poly$, the following are equivalent:

* $p = \yon^A$ for some $A: \Set$,

* $\Phi_p: p \to \coclose{[p,\yon]}{\yon}$ is the identity,

* The map $\Phi_p$ from (<ref>) has a retraction,

\[ p \to^{\Phi_p} \coclose{[p,\yon]}{\yon}
\to^{\chi_p} p \]

If one (and hence all) of (i), (ii), and (iii) holds, then it is also the case that the following composite is the identity on $[p,\yon]$,

\begin{multline}
\label{eq.PolyComposite}
[p,\yon] \to^{\coeval} [p,\yon] \otimes \left(\coclose{[p,\yon]}{\yon} \tri [p,\yon] \right) \to^ {\partial^L}
\left( [p,\yon] \otimes \coclose{[p,\yon]}{\yon} \right) \tri [p,\yon] \\
\to^{\chi_p}
([p,\yon] \ox p) \tri [p,\yon]
\to^{\eval}   [p,\yon]
\end{multline}

(i)$\Rightarrow$ (ii): Given that $p = \yon^A$ for some $A: \Set$, using <ref> we have that $\Phi_p$ is

\[
\yon^A\to(\yon^A\tri A\yon)\otimes \yon^A\to\yon^A\tri (A\yon\otimes \yon^A)\to\yon^A
\]

and one can check directly that this is the identity:

\begin{align*}
(\Phi_p)_1 &: ! \mapsto ((a \mapsto a), !) \mapsto (a \mapsto (a, !)) \mapsto ! \\
(\Phi_p)^\sharp &: a' \mapsto (a' \mapsto a') \mapsto (a' \mapsto a') \mapsto a'
\end{align*}

(ii) $\Rightarrow$ (iii) :  $\Phi_p$ is the identity, hence is also a retract.

(iii) $\Rightarrow$ (i): Given that $\Phi_p \then \chi_p$ is a retract, since functors preserve retracts, we know that $\Phi_p(1) \then \chi_p(1)$ is a retract:

\[ \begin{tikzcd} p(1) \ar[r, "\Phi_p(1)"] \ar[rr, bend right = 20pt, equals] & \yon^{[p,\yon] (1)} (1) \ar[r, "\chi_p(1)"]  & p(1) \end{tikzcd} \]

However $\yon^{[p,\yon] (1)} (1) = 1$. Hence, we have that $p(1) = 1$. Thus $p = \yon^A$ for some $A:\Set$.

Finally, assuming that (i) holds, use <ref> to prove that the composite $\Phi_p \then \chi_p$ is the identity map.

Using the above result, we can deduce that in $\Poly$, each linear polynomial is left dual to its corresponding representable, as shown in the following corollary.

For any set $A$, there is a linear dual $A \yon \dashvv \yon^A$ in $\Poly$, with $\eta = \coeval$ and $\epsilon = \eval$.

Using <ref> we have that the map $\Phi_{\yon^A}$ and therefore the map $\chi_{\yon^A}$ is the identity.
Then, from <ref>, it follows that $[\yon^A, \yon] \dual \yon^A$ with $\epsilon = \eval$ and $ \eta = \coeval \then \chi_{\yon^A} = \coeval$.
Finally, from <ref>-$(ii)$, we know that $[\yon^A, \yon] = A \yon$.

For any polynomial $p:\Poly$, the following are equivalent:

* $q = A \yon$ for some $A: \Set$

* $\Psi_q: q \to \left[ \coclose{q}{\yon}, \yon \right] $ is an isomorphism

* The map $\Psi_q$ from (<ref>) has a section,

\[ q \to^{\Omega_q} \left[ \coclose{q}{\yon}, \yon \right] \to^{\Psi_q} q \]

If one (and hence all) of $(i)$, $(ii)$, and $(iii)$ holds, it is also the case that the following composite is the identity on $\coclose{q}{\yon}$.

\begin{multline}
\label{eq.PolyComposite_2}
\coclose{q}{\yon} \to^{\coeval}  \left( \coclose{q}{\yon} \tri q \right) \ox \coclose{q}{\yon}
\to^{\partial^L}  \coclose{q}{\yon} \tri  \left( q  \ox \coclose{q}{\yon} \right)
\to^{\Omega_q}  \coclose{q}{\yon} \tri  \left( \left[ \coclose{q}{\yon}, \yon \right]  \ox \coclose{q}{\yon} \right)
\to^{\eval} \coclose{q}{\yon}
\end{multline}

(i) $\Rightarrow$ (ii) and  (ii) $\Rightarrow$ (iii) are straightforward, similar to the proof of <ref>. For (iii) $\Rightarrow$ (i), we have by <ref> that

\[ \left[ \coclose{q}{\yon}, \yon \right] =   \left[ \yon^{q(1)}, \yon \right] = q(1) \yon \]

Since functors preserve retracts, applying $\Gamma_{(-)}: \Poly \to \Set^{\op}$ from (<ref>) to the retract in $(iii)$, we get the following composite to be a retract in $\Set$:

\[ \begin{tikzcd}
\Gamma_q \ar[r, "\Gamma_{\Psi_q}" ]
& \Gamma_{q(1) \yon} \ar[r, "\Gamma_{\Omega_q}"]
\end{tikzcd} \]

Since $\Gamma_{q(1)\yon}=1$, it follows that $\Gamma_q=1$ and hence that $q[Q]=1$ for all $Q:q(1)$. So $q = q(1) \yon$.

Next we show that the linear polynomials and the representables are the only objects with duals in $\Poly$:

In the category $\Poly$, if $(\eta, \epsilon): q \dashvv p$, then $p = \yon^A$ and $q = A\yon$, where $A : {\sf Set}$.

Given that $q \dashvv p$ is a linear dual, we first show that for $A,B : \Set$, $p = \yon^A$ and $q = B \yon$, and then that $A \cong B$.

We know from <ref> the following composite is a retract:

\begin{align}
\label{eqn: retract1}
\begin{tikzcd}[ampersand replacement=\&]
p \ar[r, "\varphi"] \& \coclose{q}{\yon} \ar[r, "\eta'"] \& p \end{tikzcd}
\end{align}

where $\varphi$ is as in (<ref>).
By <ref>, we have $\coclose{q}{\yon} \cong \yon^{q(1)}$, and hence $\coclose{q}{\yon}(1)=1$. Applying the functor $-(1): \Poly \to \Set$ to the retract in <ref>, we have the following retract:

\[ p(1) \to^{\varphi_1} 1 \to^{\eta'_1} p(1) \]

so $p(1)=1$, and hence $p=\yon^A$ for some $A:\Set.$

From <ref>, we have the other retract:

\begin{align}
\label{eqn: retract2}
q \to^{\epsilon'} [p,\yon] \to^{\psi} q
\end{align}

From the previous step we know that $p = \yon^A$. Substituting, we get $[p,\yon] = [\yon^A, \yon] = A\yon$. Hence, we have the following retracts:

\[q \to^{\epsilon'} A\yon \to^{\psi} q \]

Again using $\Gamma$ (as in the proof of <ref>), we find that $q=B\yon$ for some $B:\Set$.

So we get $\coclose{q}{\yon} = \coclose{B\yon}{\yon} = \yon^B$. Substituting in <ref>, we get the retracts:

\begin{align*}
\infer{ A \to^{\eta'^\sharp} B \to^{\varphi^\sharp} A}{ \yon^A \to^{\varphi}  \yon^B \to^{\eta'} \yon^A }
\end{align*}

Similarly, using the fact that $p=\yon^A$ for some $A: \Set$, we have that $[p, \yon] = [\yon^A, \yon] = A \yon$. Substituting in  <ref>, we get the following retracts:

\begin{align*}
\infer{ B \to^{\epsilon'_1} A \to^{\psi_1} B }{ B \yon \to^{\epsilon'} A\yon \to^{\psi} B \yon }
\end{align*}

These two retracts together form the isomorphism $A \cong B$.

In $\Poly$, if $p\dual q$ and $q \dual p$ then $p = q = \yon$.

This follows directly from <ref>.

§ THE CORE OF POLY

Central to mix LDCs is the notion of core. An important way in which compact LDCs (for all objects $a,b$, an isomorphism $a \ox b \to^{\indep}_{\cong} a \tri b$) arise is as the core of a mix category. The core of a mix LDC is defined as follows.

An object $u$ is in the core of a mix category if and only if
the following natural transformations are isomorphisms:

\[ u \ox - \to^{\indep_{u,-}} u \tri - ~~~~\mbox{and}~~~~
- \ox  u \to^{\indep_{-,u}} - \tri u \]

To the best of our knowledge, the notion of core has been studied only within symmetric LDCs. In this article, motivated the category $\Poly$, we consider a more general notion of core, suitable in (possibly non-symmetric) LDCs.

For a mix ($\ox$-symmetric) LDC $(\X, \otimes, \top, \tri, \bot)$,

* its left core is defined to be the full sub-category of objects $a: \X$ such that, for all $x:\X$, the map

\[ {\sf [core_l]} ~~~~ a \otimes x \to^{\indep_{a,x}}_{\cong} a \tri x \]

is an isomorphism.

* its right core of $\X$ is the full sub-category of objects $a: \x$ such that, for all $x:\X$, the map

\[ {\sf [core_r]} ~~~~ x \otimes a \to^{\indep_{x,a}}_{\cong} x \tri a \]

is an isomorphism.

One may have empty left and right cores in a mix LDC. However, in an isomix category, the monoidal unit belongs to both the left and the right core.

If $\X$ is an isomix LDC with unit $\yon$, then $\yon \in \lCore(\X)$ and $\yon \in \rCore(\X)$.

In an isomix LDC, both the left and the right cores are compact LDCs: that is, for all objects $a, b :\lCore$, the map $a \ox b \to^{\indep}_{\cong} a \tri b$ is an isomorphism, and same for $\rCore$. Hence, by Corollary <cit.> both the left and right cores are linearly equivalent to monoidal categories.

In a symmetric mix LDC $\X$, the left and the right cores coincide. The left and right cores of any isomix LDC may be related to each other as follows.

A mix LDC $(\X, \otimes, \top, \tri, \bot)$ is said to have opposing cores if there exists an isomorphism,

\[ \star: \rCore(\X)^{\op} \to^{\cong} \lCore(\X) \]

Compact closed categories have opposing cores which coincide (the entire category is the core) and the $\star$ is the same as the contravariant involution.

We will show that $\Poly$ has opposing cores in <ref>.

The following theorem generalizes the result <cit.> that for any linear dual in a symmetric mix LDC, if the left dual is in the core then the right dual is also in the core, and vice versa. We prove an analogous result in non-symmetric isomix LDCs.

Let $(\eta, \epsilon): b \dashvv a$ be a linear dual  in an isomix LDC $\X$. Then $b :\rCore(\X)$ iff $a :\lCore(\X)$.

We have $\eta\colon\yon\to a\tri b$ and $\epsilon\colon (b\otimes a)\to\yon$ satisfying the diagrams in <ref>.

($\Leftarrow$): Suppose $b :\rCore(\X)$. We want to prove that $a :\lCore(\X)$, i.e. that the map $\indep_{a,x} : a \ox x \to a \tri x$ is an isomorphism for all $x: \X$.

Define a tentative inverse map $a \tri x \to a \ox x$ as follows:

\begin{multline*}
a \tri x  \to^{\eta}  (a \tri b) \ox (a \tri x)
\to^{\indep_{a,b}^{-1}} (a \ox b) \ox (a \tri x)  \\
\to^{a_\ox}_{\cong} a \ox (b \ox (a \tri x))
\to^{\partial^L} a \ox ((b \ox a) \tri x)
\to^{\epsilon} a \ox x
\end{multline*}

The second step uses the assumption that $b :\rCore(\X)$. The proof that the tentative inverse map defined as above is indeed the inverse of $\indep_{a,x}$ follows similarly to <cit.> and uses the inverse of the mix map $\m$.

($\Rightarrow$): [Sketch of proof] Suppose $a :\lCore(\X)$. We want to prove that $b:\rCore(\X)$, that is, the map $\indep_{x,b}: x \ox b \to x \tri b$ is an isomorphism for all $x:\X$.

One checks that a map $x \tri b \to x \ox b$ inverse to $\indep_{b,x}$ can be defined as follows:

\begin{multline*}
x \tri b  \to^{\eta}  (x \tri b) \ox (a \tri b)
\to^{\indep_{a,b}^{-1}} (x \ox b) \ox (a \tri b) \\
\to^{a_\ox}_{\cong} x \ox (b \ox (a \tri b))
\to^{\partial^L} x \ox ((b \ox a) \tri b)
\to^{\epsilon} x \ox b
\end{multline*}

The next result shows that in $\Poly$ only the linear polynomials reside in the left core, and only the representables reside in the right core.

In the category Poly, the following statements hold for any polynomial $p$:

* $p\in\lCore(\Poly)$ iff $p \cong A\yon$ for some $A:\Set$, and

* $q\in\rCore(\Poly)$ iff $q \cong \yon^A$ for some $A:\Set$.

($\Leftarrow$): Suppose $p \cong A\yon$ and $r$ is any polynomial. Then,

\begin{align*}
A\yon \ox r \cong Ar \cong A\yon \tri r
\end{align*}

Thus, $A\yon \in \lCore(\Poly)$.

($\Rightarrow$): Suppose $p :\lCore(\Poly)$. So, for all $r: \Poly$, we have that $\indep: p \ox r \to p \tri r$ is an isomorphism.

Let $r = B$ for some set $B$. We have

\begin{align*}
p \ox r &= p \ox B = \sum_{P:p(1)} B \\
p \tri r &= p \tri B = \sum_{P :p(1)} B^{p[P]}
\end{align*}

and $\indep$ is an isomorphism iff $p[P]=1$ for all $P:p(1)$.

($\Leftarrow$): Suppose $q \cong \yon^A$, we must prove that for all $p: \Poly$, the map $p \ox \yon^A \to^{\indep_{p, \yon^A}} p \tri \yon^A$ is an isomorphism. Using <ref>, the map $\indep_{q,\yon^A}$ is defined as follows:

\begin{align*}
p \ox \yon^A = \sum_{P:p(1)} \prod_{d:p[P]} \prod_{a: A} \yon &\to^{\indep_{p,\yon^A}} \nonumber
\sum_{P:p(1)} \prod_{d:p[P]} \prod_{a: A} \yon = p \tri \yon^A \\
\indep_1: (P,!) &\mapsto (P, \textit{const}~!: p[P] \to \{!\}; d \mapsto !)   \\
\indep_{P,Q}^\sharp:  (d,a) &\mapsto (d,a): p[P] \times q[Q]
\end{align*}

Thus, the map $\indep_1$ is an isomorphism, and $\indep^\sharp$ is always an isomorphism by <ref>. Thus, we have that $p : \rCore(\Poly)$.

($\Rightarrow$): Suppose $q :\rCore(\Poly)$, so for all $x :\Poly$, the following map is an isomorphism:

\[ \indep_{x,p}: x \ox p \to^{\cong} x \tri p \]

Then taking $x=1$ we have $1\ox p\cong 1\tri p\cong 1$. But from (<ref>) we also have $1\ox p\cong p(1)$. Hence $p(1)\cong 1$, making $p$ a representable.

The category $\Poly$ has opposing cores.

For any $A: \Set$, the object  $\left(\yon^A \right)^\star := A\yon$.

For any $A,B: \Set$ and a map $\varphi: \yon^B \to \yon^A$, the map $\varphi^\star: A \yon \to B\yon$ is given by $\varphi$, where $(\varphi^\star)_1 := \varphi^\sharp$.

It also follows from <ref> that the left duals in $\Poly$ are precisely the left core of $\Poly$ and the right duals are the right core.

In the category ${\sf Poly}$, for any two polynomials $p$ and $q$, we have $q \dashvv p$ iff $q : \lCore(\Poly)$ and $p : \rCore(\Poly)$.

Suppose that $q\dashvv p$, then by <ref>, we have that $q=A \yon$ and $p = \yon^A$ for some set $A$. The rest of the proof is the direct consequence of <ref>. For the converse, apply <ref>.

§ LINEAR MONOIDS, LINEAR COMONOIDS, LINEAR BIALGEBRAS IN $\POLY$

When an object in a cyclic linear dual ($b \dual a$ and $a \dual b$) is equipped with a $\ox$-monoid structure, it is called a linear monoid. Similarly, if such an object is equipped with a $\ox$-comonoid structure, it is called a linear comonoid. Linear monoids and linear comonoids in LDCs are analogous to Frobenius algebras in monoidal categories. Linear monoids and linear comonoids were studied in the context of quantum observables in <cit.>.

In $\Poly$, the only cyclic linear dual is $a=b=\yon$ by <ref>. However, we can consider the notions of left and right linear monoids and comonoids on duals that are not cyclic. This section, focuses on these structures in the category $\Poly$.

In what follows, we are aiming to define a notion of right and left linear monoids and comonoids: four cases. Each is about a linear dual $b\dual a$ such that both sides carry additional structure. The word left and right will refer to the side that carries the $\otimes$-structure, and in this case the other side will (implicitly) carry a complementary $\tri$-structure. For example, a left linear comonoid is a dual $b\dual a$ such that the left side, $b$ carries the $\otimes$-comonoid structure, and the terminology leaves implicit the fact that $a$ carries a $\tri$-monoid structure.

§.§ Linear monoids

Before defining left and right linear monoids, let's consider the standard definition that we are generalizing. In an LDC, a linear monoid consists of linear duals $a \dual b$ and $b \dual a$, with either $a$ or $b$ carrying a $\ox$-monoid structure and satisfying certain coherences <cit.>. Thus, a linear monoid requires cyclic duals, that is, each object is both the left and the right dual of the other.

In any LDC, a linear monoid, $b  \linmonw a$, consists of linear duals $(\eta_L, \epsilon_L): b \dual a$ and $(\eta_R, \epsilon_R): a \dual b$,  and a $\ox$-monoid $(b, \mu, \nu)$ such that the $\tri$-comonoid structures induced on $a$ by the dualities coincide:

\begin{equation}
\label{eq.linmon}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (-2.75, 0.75) {};
\node [style=none] (1) at (-3.25, 1.25) {};
\node [style=none] (2) at (-2.75, 0.5) {};
\node [style=none] (3) at (-2.25, 1.25) {};
\node [style=none] (4) at (-1.5, 1.25) {};
\node [style=none] (5) at (-1.5, -0) {};
\node [style=none] (6) at (-3.25, 1.5) {};
\node [style=none] (7) at (-1, 1.5) {};
\node [style=none] (8) at (-1, -0) {};
\node [style=none] (9) at (-3.75, 0.5) {};
\node [style=none] (10) at (-3.75, 2.25) {};
\node [style=none] (11) at (-4, 2) {$a$};
\node [style=none] (12) at (-1.75, 0.25) {$a$};
\node [style=none] (13) at (-0.75, 0.25) {$a$};
\node [style=none] (14) at (-2.25, 2.5) {$\eta_R$};
\node [style=none] (15) at (-2, 1.75) {$\eta_R$};
\node [style=none] (16) at (-3.25, -0.25) {$\epsilon_R$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [in=150, out=-90, looseness=1.00] (1.center) to (0);
\draw [in=-90, out=30, looseness=1.00] (0) to (3.center);
\draw (0) to (2.center);
\draw [bend left=90, looseness=1.50] (2.center) to (9.center);
\draw (9.center) to (10.center);
\draw (6.center) to (1.center);
\draw [bend left=90, looseness=1.25] (6.center) to (7.center);
\draw (7.center) to (8.center);
\draw (4.center) to (5.center);
\draw [bend right=90, looseness=1.25] (4.center) to (3.center);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (-2, 0.75) {};
\node [style=none] (1) at (-1.5, 1.25) {};
\node [style=none] (2) at (-2, 0.5) {};
\node [style=none] (3) at (-2.5, 1.25) {};
\node [style=none] (4) at (-3.25, 1.25) {};
\node [style=none] (5) at (-3.25, -0) {};
\node [style=none] (6) at (-1.5, 1.5) {};
\node [style=none] (7) at (-3.75, 1.5) {};
\node [style=none] (8) at (-3.75, -0) {};
\node [style=none] (9) at (-1, 0.5) {};
\node [style=none] (10) at (-1, 2.25) {};
\node [style=none] (11) at (-0.75, 2) {$a$};
\node [style=none] (12) at (-3, 0.25) {$a$};
\node [style=none] (13) at (-4, 0.25) {$a$};
\node [style=none] (14) at (-2.5, 2.5) {$\eta_L$};
\node [style=none] (15) at (-2.75, 1.75) {$\eta_L$};
\node [style=none] (16) at (-1.5, -0.25) {$\epsilon_L$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [in=30, out=-90, looseness=1.00] (1.center) to (0);
\draw [in=-90, out=150, looseness=1.00] (0) to (3.center);
\draw (0) to (2.center);
\draw [bend right=90, looseness=1.50] (2.center) to (9.center);
\draw (9.center) to (10.center);
\draw (6.center) to (1.center);
\draw [bend right=90, looseness=1.25] (6.center) to (7.center);
\draw (7.center) to (8.center);
\draw (4.center) to (5.center);
\draw [bend left=90, looseness=1.25] (4.center) to (3.center);
\end{pgfonlayer}
\end{tikzpicture}
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (-0.75, 1.5) {};
\node [style=none] (1) at (-0.75, 0.5) {};
\node [style=none] (2) at (-1.75, 0.5) {};
\node [style=none] (3) at (-1.75, 2.25) {};
\node [style=none] (4) at (-2, 2) {$a$};
\node [style=none] (5) at (-1.25, -0.25) {$\epsilon_R$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (1.center);
\draw [bend left=90, looseness=1.50] (1.center) to (2.center);
\draw (2.center) to (3.center);
\end{pgfonlayer}
\end{tikzpicture} =
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (-2, 1.5) {};
\node [style=none] (1) at (-2, 0.5) {};
\node [style=none] (2) at (-1, 0.5) {};
\node [style=none] (3) at (-1, 2.25) {};
\node [style=none] (4) at (-0.75, 2) {$a$};
\node [style=none] (5) at (-1.5, -0.25) {$\epsilon_L$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (1.center);
\draw [bend right=90, looseness=1.50] (1.center) to (2.center);
\draw (2.center) to (3.center);
\end{pgfonlayer}
\end{tikzpicture}
\end{equation}

We now fine-grain this notion of linear monoids by considering linear duals which are non-cyclic. We separate the left and right notions as <ref> so that we can include diagrams that explain the implicit $\tri$-comonoid structures.

In an LDC, a left linear monoid $(\eta_L, \epsilon_L): b \linmonwl a$, consists of a linear dual $(\eta_L, \epsilon_L): b \dashvv a$ equipped with a $\ox$-monoid structure $(b, \mu: b \ox b \to b, \nu: \top \to b)$ on the left dual.

The reason for the “$\tri$” near the $a$ in our notation for a left linear monoid $b \linmonwl a$ is that the $\ox$-monoid structure on $b$ induces a $\tri$-comonoid structure $(a, \delta, \gamma)$ on $a$, as can be seen from the following diagram:

\begin{equation*}
(i)~~~~\delta = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (12) at (1.75, 0.25) {$a$};
\node [style=none] (13) at (0.25, 0.25) {$a$};
\node [style=circle] (17) at (1, 1.25) {};
\node [style=none] (18) at (1, 2.5) {};
\node [style=none] (19) at (0.5, 0) {};
\node [style=none] (20) at (1.5, 0) {};
\node [style=none] (21) at (1.25, 2.25) {$a$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [in=-150, out=90, looseness=1.25] (19.center) to (17);
\draw [in=90, out=-30, looseness=1.25] (17) to (20.center);
\draw (17) to (18.center);
\end{pgfonlayer}
\end{tikzpicture} :=
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (-2, 0.75) {};
\node [style=none] (1) at (-1.5, 1.25) {};
\node [style=none] (2) at (-2, 0.5) {};
\node [style=none] (3) at (-2.5, 1.25) {};
\node [style=none] (4) at (-3.25, 1.25) {};
\node [style=none] (5) at (-3.25, -0) {};
\node [style=none] (6) at (-1.5, 1.5) {};
\node [style=none] (7) at (-3.75, 1.5) {};
\node [style=none] (8) at (-3.75, -0) {};
\node [style=none] (9) at (-1, 0.5) {};
\node [style=none] (10) at (-1, 2.25) {};
\node [style=none] (11) at (-0.75, 2) {$a$};
\node [style=none] (12) at (-3, 0.25) {$a$};
\node [style=none] (13) at (-4, 0.25) {$a$};
\node [style=none] (14) at (-2.5, 2.5) {$\eta_L$};
\node [style=none] (15) at (-2.75, 1.75) {$\eta_L$};
\node [style=none] (16) at (-1.5, -0.25) {$\epsilon_L$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [in=30, out=-90, looseness=1.00] (1.center) to (0);
\draw [in=-90, out=150, looseness=1.00] (0) to (3.center);
\draw (0) to (2.center);
\draw [bend right=90, looseness=1.50] (2.center) to (9.center);
\draw (9.center) to (10.center);
\draw (6.center) to (1.center);
\draw [bend right=90, looseness=1.25] (6.center) to (7.center);
\draw (7.center) to (8.center);
\draw (4.center) to (5.center);
\draw [bend left=90, looseness=1.25] (4.center) to (3.center);
\end{pgfonlayer}
\end{tikzpicture}
(ii)~~~~\gamma = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (17) at (1, 0.25) {};
\node [style=none] (18) at (1, 2.5) {};
\node [style=none] (21) at (1.25, 2.25) {$a$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (17) to (18.center);
\end{pgfonlayer}
\end{tikzpicture} :=
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (-2, 1.5) {};
\node [style=none] (1) at (-2, 0.5) {};
\node [style=none] (2) at (-1, 0.5) {};
\node [style=none] (3) at (-1, 2.25) {};
\node [style=none] (4) at (-0.75, 2) {$a$};
\node [style=none] (5) at (-1.5, -0.25) {$\epsilon_L$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (1.center);
\draw [bend right=90, looseness=1.50] (1.center) to (2.center);
\draw (2.center) to (3.center);
\end{pgfonlayer}
\end{tikzpicture}
\end{equation*}

In an LDC, a right linear monoid $(\eta_R, \epsilon_R): b \linmonwr a$ consists of a linear dual  $(\eta_R, \epsilon_R): b \dashvv a$, equipped with a $\ox$-monoid structure $(a, \mu: a \ox a \to a, \nu: \top \to a)$ on the right dual.

The $\ox$-monoid structure on $b$ induces a $\tri$-comonoid structure on $(a,\delta, \gamma)$ via the duality as follows:

\begin{equation*}
(i)~~~\delta = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (12) at (1.75, 0.25) {$b$};
\node [style=none] (13) at (0.25, 0.25) {$b$};
\node [style=circle] (17) at (1, 1.25) {};
\node [style=none] (18) at (1, 2.5) {};
\node [style=none] (19) at (0.5, 0) {};
\node [style=none] (20) at (1.5, 0) {};
\node [style=none] (21) at (1.25, 2.25) {$b$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [in=-150, out=90, looseness=1.25] (19.center) to (17);
\draw [in=90, out=-30, looseness=1.25] (17) to (20.center);
\draw (17) to (18.center);
\end{pgfonlayer}
\end{tikzpicture} :=
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (-2.75, 0.75) {};
\node [style=none] (1) at (-3.25, 1.25) {};
\node [style=none] (2) at (-2.75, 0.5) {};
\node [style=none] (3) at (-2.25, 1.25) {};
\node [style=none] (4) at (-1.5, 1.25) {};
\node [style=none] (5) at (-1.5, -0) {};
\node [style=none] (6) at (-3.25, 1.5) {};
\node [style=none] (7) at (-1, 1.5) {};
\node [style=none] (8) at (-1, -0) {};
\node [style=none] (9) at (-3.75, 0.5) {};
\node [style=none] (10) at (-3.75, 2.25) {};
\node [style=none] (11) at (-4, 2) {$b$};
\node [style=none] (12) at (-1.75, 0.25) {$b$};
\node [style=none] (13) at (-0.75, 0.25) {$b$};
\node [style=none] (14) at (-2.25, 2.5) {$\eta_R$};
\node [style=none] (15) at (-2, 1.75) {$\eta_R$};
\node [style=none] (16) at (-3.25, -0.25) {$\epsilon_R$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [in=150, out=-90, looseness=1.00] (1.center) to (0);
\draw [in=-90, out=30, looseness=1.00] (0) to (3.center);
\draw (0) to (2.center);
\draw [bend left=90, looseness=1.50] (2.center) to (9.center);
\draw (9.center) to (10.center);
\draw (6.center) to (1.center);
\draw [bend left=90, looseness=1.25] (6.center) to (7.center);
\draw (7.center) to (8.center);
\draw (4.center) to (5.center);
\draw [bend right=90, looseness=1.25] (4.center) to (3.center);
\end{pgfonlayer}
\end{tikzpicture}
(ii)~~~\gamma = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (17) at (1, 0.25) {};
\node [style=none] (18) at (1, 2.5) {};
\node [style=none] (21) at (1.25, 2.25) {$b$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (17) to (18.center);
\end{pgfonlayer}
\end{tikzpicture} :=
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (-0.75, 1.5) {};
\node [style=none] (1) at (-0.75, 0.5) {};
\node [style=none] (2) at (-1.75, 0.5) {};
\node [style=none] (3) at (-1.75, 2.25) {};
\node [style=none] (4) at (-2, 2) {$b$};
\node [style=none] (5) at (-1.25, -0.25) {$\epsilon_R$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (0) to (1.center);
\draw [bend left=90, looseness=1.50] (1.center) to (2.center);
\draw (2.center) to (3.center);
\end{pgfonlayer}
\end{tikzpicture}
\end{equation*}

A linear monoid $b  \linmonw a$ in an LDC can be equivalently defined to consist of a left linear monoid $(\eta_L, \epsilon_L): b \linmonwl a$, and a right linear monoid $(\eta_R, \epsilon_R): a \linmonwr b$ such that the monoid structures on $b$ coincide, i.e. <ref> holds.

The monoids in $\Set$ produce left linear monoids in $\Poly$ as follows.

If $(M, *, u) : \Set$ is a monoid, then $M \yon \linmonwl \yon^M$ has the structure of a left linear monoid where the linear duality $(\coeval,\eval)\colon M\yon\dual\yon^M$ is given as in <ref>.

The functor $M\mapsto M\yon$ is strong monoidal by <ref>, so it preserves monoids.

Explicitly, the $\ox$-monoid structure on $M \yon$ induced by $M$ is as follows.

\begin{align}
\mu_1 &:= (a_1, a_2) \mapsto a_1 * a_2 : M \times M \to M \label{eqn.1} \\
\nu_1 &:= * \mapsto e : \{ * \} \to M \label{eqn.2}
\end{align}

The following Lemma shows that in $\Poly$, every representable $\yon^A$ is a right linear monoid.

In $\Poly$, for any set $A$, there is a right linear monoid $A \yon \linmonwr \yon^A$ where the linear duality $(\coeval,\eval)\colon A\yon\dual\yon^A$ is given as in <ref>.

The functor $A\mapsto \yon^A$ is strong monoidal by <ref>, so it preserves comonoids, and every set is a comonoid in a unique way, making $\yon^A$ a $\ox$-monoid.

Explicitly, the $\ox$-monoid structure on $\yon^A$ induced by the unique comonoid structure on $A$ is as follows.

\begin{align}
\mu^\sharp_! &:= a \mapsto (a,a) : A \to A \times A \label{eqn.3} \\
\nu^\sharp_! &:= a \mapsto * : A \to \{ * \} \label{eqn.4}
\end{align}

§.§ Linear comonoids

In this section, we simply run the same story except with $\otimes$-comonoids in place of $\otimes$-monoids, and implicitly also $\tri$-monoids in place of $\tri$-comonoids. If the reader does not need this detail, they can safely skip to <ref>.

<cit.> In an LDC, a linear comonoid $b \lincomonw  a$ consists of linear duals $(\eta_L, \epsilon_L): b \dual a$ and $(\eta_R, \epsilon_R): a \dual b$, and a $\ox$-comonoid $(b, \delta, \epsilon)$, such that the $\tri$-comonoid structures induced on $a$ by the two dualities coincide:

\begin{equation}
\label{eq.lincomon}
{\it (i)}~~~ \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (1.7, 2.75) {};
\node [style=none] (1) at (1.2, 2) {};
\node [style=none] (2) at (2.2, 2) {};
\node [style=none] (3) at (1.7, 3.5) {};
\node [style=none] (4) at (1.95, 3.25) {$b$};
\node [style=none] (5) at (0.7, 2) {};
\node [style=none] (6) at (-0.3, 2) {};
\node [style=none] (7) at (2.7, 3.5) {};
\node [style=none] (8) at (2.7, 1.25) {};
\node [style=none] (9) at (2.95, 1.75) {$a$};
\node [style=oa] (10) at (0.2, 3) {};
\node [style=none] (11) at (0.2, 3.75) {};
\node [style=none] (12) at (-0.25, 3.65) {$a \tri a$};
\node [style=none] (13) at (2.2, 4.15) {$\eta_R$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [in=-165, out=90, looseness=1.25] (1.center) to (0);
\draw [in=90, out=-15, looseness=1.25] (0) to (2.center);
\draw (0) to (3.center);
\draw [bend left=90, looseness=1.75] (1.center) to (5.center);
\draw [bend right=90] (6.center) to (2.center);
\draw [bend right=90, looseness=1.25] (7.center) to (3.center);
\draw (7.center) to (8.center);
\draw [in=90, out=-45] (10) to (5.center);
\draw [in=90, out=-135, looseness=1.25] (10) to (6.center);
\draw (10) to (11.center);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (0.75, 2.75) {};
\node [style=none] (1) at (1.25, 2) {};
\node [style=none] (2) at (0.25, 2) {};
\node [style=none] (3) at (0.75, 3.5) {};
\node [style=none] (4) at (0.5, 3.25) {$b$};
\node [style=none] (5) at (1.75, 2) {};
\node [style=none] (6) at (2.75, 2) {};
\node [style=none] (7) at (-0.25, 3.5) {};
\node [style=none] (8) at (-0.25, 1.25) {};
\node [style=none] (9) at (-0.5, 1.75) {$a$};
\node [style=oa] (10) at (2.25, 3) {};
\node [style=none] (11) at (2.25, 3.75) {};
\node [style=none] (12) at (2.75, 3.65) {$a \tri a$};
\node [style=none] (13) at (0.25, 4.15) {$\eta_L$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [in=-15, out=90, looseness=1.25] (1.center) to (0);
\draw [in=90, out=-165, looseness=1.25] (0) to (2.center);
\draw (0) to (3.center);
\draw [bend right=90, looseness=1.75] (1.center) to (5.center);
\draw [bend left=90] (6.center) to (2.center);
\draw [bend left=90, looseness=1.25] (7.center) to (3.center);
\draw (7.center) to (8.center);
\draw [in=90, out=-135] (10) to (5.center);
\draw [in=90, out=-45, looseness=1.25] (10) to (6.center);
\draw (10) to (11.center);
\end{pgfonlayer}
\end{tikzpicture}
{\it (ii)} ~~~
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (1.5, 3.5) {};
\node [style=none] (2) at (2.5, 3.5) {};
\node [style=none] (3) at (2.5, 1) {};
\node [style=none] (4) at (2.75, 1.75) {$a$};
\node [style=circle, scale=1.5] (5) at (0.75, 1.25) {};
\node [style=none] (6) at (0.75, 1.25) {$\bot$};
\node [style=none] (7) at (0.75, 4.25) {};
\node [style=circle] (8) at (0.75, 2) {};
\node [style=none] (9) at (1.5, 2.75) {};
\node [style=none] (13) at (2, 4.25) {$\eta_R$};
\node [style=circle] (14) at (1.5, 2.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right=90, looseness=1.50] (2.center) to (0.center);
\draw (2.center) to (3.center);
\draw (5) to (7.center);
\draw [dotted, in=-90, out=0, looseness=1.25] (8) to (9.center);
\draw (0.center) to (14);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (2, 3.5) {};
\node [style=none] (2) at (1, 3.5) {};
\node [style=none] (3) at (1, 1) {};
\node [style=none] (4) at (0.75, 1.75) {$a$};
\node [style=circle, scale=1.5] (5) at (2.75, 1.25) {};
\node [style=none] (6) at (2.75, 1.25) {$\bot$};
\node [style=none] (7) at (2.75, 4.25) {};
\node [style=circle] (8) at (2.75, 2) {};
\node [style=none] (9) at (2, 2.75) {};
\node [style=none] (13) at (1.5, 4.25) {$\eta_L$};
\node [style=circle] (14) at (2, 2.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend left=90, looseness=1.50] (2.center) to (0.center);
\draw (2.center) to (3.center);
\draw (5) to (7.center);
\draw [dotted, in=-90, out=180, looseness=1.25] (8) to (9.center);
\draw (0.center) to (14);
\end{pgfonlayer}
\end{tikzpicture}
\end{equation}

Similar to the definition of linear monoids, we can fine-grain the definition of linear comonoids to non-cyclic duals.

In an LDC, a left linear comonoid, $(\eta_L, \epsilon_L): b \lincomonwl a$, consists of a linear dual $(\eta_L, \epsilon_L): b \dashvv a$ equipped with a $\ox$-comonoid structure $(b, \delta: b \ox b \to b, \gamma: b \to \yon)$ on the left dual.

The $\ox$-comonoid structure on $b$ induces a $\tri$-monoid structure $(a, \mu, \nu)$ as follows:

\begin{equation*}
{\it (i)} ~~~ \mu = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (19) at (4.5, 1.25) {};
\node [style=none] (20) at (4.75, 1.5) {$a$};
\node [style=oa] (26) at (4.4, 3.5) {};
\node [style=none] (27) at (4.4, 4.25) {};
\node [style=none] (28) at (3.75, 4.1) {$a \tri a$};
\node [style=circle] (29) at (4.5, 2.25) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (26) to (27.center);
\draw [bend left=60] (26) to (29);
\draw [bend right=60] (26) to (29);
\draw (29) to (19.center);
\end{pgfonlayer}
\end{tikzpicture} := \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (0.75, 2.75) {};
\node [style=none] (1) at (1.25, 2) {};
\node [style=none] (2) at (0.25, 2) {};
\node [style=none] (3) at (0.75, 3.5) {};
\node [style=none] (5) at (1.75, 2) {};
\node [style=none] (6) at (2.75, 2) {};
\node [style=none] (7) at (-0.25, 3.5) {};
\node [style=none] (8) at (-0.25, 1.25) {};
\node [style=none] (9) at (-0.5, 1.75) {$a$};
\node [style=oa] (10) at (2.25, 3) {};
\node [style=none] (11) at (2.25, 3.75) {};
\node [style=none] (12) at (2.95, 3.65) {$a \tri a$};
\node [style=none] (13) at (0.25, 4.15) {$\eta_L$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [in=-15, out=90, looseness=1.25] (1.center) to (0);
\draw [in=90, out=-165, looseness=1.25] (0) to (2.center);
\draw (0) to (3.center);
\draw [bend right=90, looseness=1.75] (1.center) to (5.center);
\draw [bend left=90] (6.center) to (2.center);
\draw [bend left=90, looseness=1.25] (7.center) to (3.center);
\draw (7.center) to (8.center);
\draw [in=90, out=-135] (10) to (5.center);
\draw [in=90, out=-45, looseness=1.25] (10) to (6.center);
\draw (10) to (11.center);
\end{pgfonlayer}
\end{tikzpicture}
{\it (ii)} ~~~ \nu = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (19) at (4.5, 1.25) {};
\node [style=none] (20) at (5, 1.75) {$a$};
\node [style=none] (24) at (4.5, 4) {};
\node [style=none] (26) at (5, 4) {$\bot$};
\node [style=circle] (27) at (4.5, 3.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (19.center) to (27);
\draw (27) to (24.center);
\end{pgfonlayer}
\end{tikzpicture} :=  \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (2, 3.5) {};
\node [style=none] (2) at (1, 3.5) {};
\node [style=none] (3) at (1, 1) {};
\node [style=none] (4) at (0.75, 1.75) {$a$};
\node [style=circle, scale=1.5] (5) at (2.75, 1.25) {};
\node [style=none] (6) at (2.75, 1.25) {$\bot$};
\node [style=none] (7) at (2.75, 4.25) {};
\node [style=circle] (8) at (2.75, 2) {};
\node [style=none] (9) at (2, 2.75) {};
\node [style=none] (13) at (1.5, 4.25) {$\eta_L$};
\node [style=circle] (14) at (2, 2.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend left=90, looseness=1.50] (2.center) to (0.center);
\draw (2.center) to (3.center);
\draw (5) to (7.center);
\draw [dotted, in=-90, out=180, looseness=1.25] (8) to (9.center);
\draw (0.center) to (14);
\end{pgfonlayer}
\end{tikzpicture}
\end{equation*}

In an LDC, a right linear comonoid $(\eta_R, \epsilon_R): b \lincomonwr a$ consists of a linear dual $(\eta_R, \epsilon_R): b \dashvv a$ equipped with a $\ox$-comonoid structure $(a, \delta: a \to a \ox a, \gamma: a \to \yon)$ on the right dual.

The $\ox$-comonoid structure on $a$ induces a $\tri$-monoid structure $(b, \mu, \nu)$ as follows:

\begin{equation*}
{\it (i)} ~~~ \mu = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (19) at (4.5, 1.25) {};
\node [style=none] (20) at (4.75, 1.5) {$b$};
\node [style=oa] (26) at (4.4, 3.5) {};
\node [style=none] (27) at (4.4, 4.25) {};
\node [style=none] (28) at (3.75, 4.1) {$b \tri b$};
\node [style=circle] (29) at (4.5, 2.25) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (26) to (27.center);
\draw [bend left=60] (26) to (29);
\draw [bend right=60] (26) to (29);
\draw (29) to (19.center);
\end{pgfonlayer}
\end{tikzpicture} :=\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (1.7, 2.75) {};
\node [style=none] (1) at (1.2, 2) {};
\node [style=none] (2) at (2.2, 2) {};
\node [style=none] (3) at (1.7, 3.5) {};
\node [style=none] (4) at (1.95, 3.25) {$a$};
\node [style=none] (5) at (0.7, 2) {};
\node [style=none] (6) at (-0.3, 2) {};
\node [style=none] (7) at (2.7, 3.5) {};
\node [style=none] (8) at (2.7, 1.25) {};
\node [style=none] (9) at (2.95, 1.75) {$b$};
\node [style=oa] (10) at (0.2, 3) {};
\node [style=none] (11) at (0.2, 3.75) {};
\node [style=none] (12) at (-0.5, 3.65) {$b \tri b$};
\node [style=none] (13) at (2.2, 4.15) {$\eta_R$};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [in=-165, out=90, looseness=1.25] (1.center) to (0);
\draw [in=90, out=-15, looseness=1.25] (0) to (2.center);
\draw (0) to (3.center);
\draw [bend left=90, looseness=1.75] (1.center) to (5.center);
\draw [bend right=90] (6.center) to (2.center);
\draw [bend right=90, looseness=1.25] (7.center) to (3.center);
\draw (7.center) to (8.center);
\draw [in=90, out=-45] (10) to (5.center);
\draw [in=90, out=-135, looseness=1.25] (10) to (6.center);
\draw (10) to (11.center);
\end{pgfonlayer}
\end{tikzpicture}     ~~~~~~~~
{\it (ii)} ~~~ \nu = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (19) at (4.5, 1.25) {};
\node [style=none] (20) at (5, 1.75) {$b$};
\node [style=none] (24) at (4.5, 4) {};
\node [style=none] (26) at (5, 4) {$\bot$};
\node [style=circle] (27) at (4.5, 3.5) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (19.center) to (27);
\draw (27) to (24.center);
\end{pgfonlayer}
\end{tikzpicture} :=
\begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (0) at (1.5, 3.5) {};
\node [style=none] (1) at (1.25, 3.5) {$a$};
\node [style=none] (2) at (2.5, 3.5) {};
\node [style=none] (3) at (2.5, 1) {};
\node [style=none] (4) at (2.75, 1.75) {$b$};
\node [style=circle, scale=1.5] (5) at (0.75, 1.25) {};
\node [style=none] (6) at (0.75, 1.25) {$\bot$};
\node [style=none] (7) at (0.75, 4.25) {};
\node [style=circle] (8) at (0.75, 2) {};
\node [style=none] (9) at (1.5, 2.75) {};
\node [style=none] (13) at (2, 4.25) {$\eta_R$};
\node [style=circle] (14) at (1.5, 2.75) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right=90, looseness=1.50] (2.center) to (0.center);
\draw (2.center) to (3.center);
\draw (5) to (7.center);
\draw [dotted, in=-90, out=0, looseness=1.25] (8) to (9.center);
\draw (0.center) to (14);
\end{pgfonlayer}
\end{tikzpicture}
\end{equation*}

A linear comonoid $b \lincomonw  a$  in any LDC can be equivalently defined to consist of a left linear comonoid $(\eta_L, \epsilon_L): b \lincomonwl a$ and a right linear comonoid $(\eta_R, \epsilon_R): a \lincomonwr b$ such that the $\ox$-comonoid structures on $b$ coincide and <ref> holds.

The following Lemma shows that in $\Poly$, every linear polynomial $M \yon$ is a left linear comonoid:

In $\Poly$, for any set $A$, there is a left linear comonoid $A \yon \lincomonwl \yon^A$ where the linear duality $(\coeval,\eval)\colon A\yon\dual\yon^A$ is given as in <ref>.

The functor $A\mapsto A\yon$ is strong monoidal by <ref>, so it preserves comonoids, and every set is a comonoid in a unique way.

Explicitly, the $\ox$-comonoid structure on $A \yon$ is as follows:

\begin{align}
\delta_1 &:= a \mapsto (a,a) : A \to A \times A \label{eqn.5} \\
\gamma_1 &:= a \mapsto * : A \to \{ * \} \label{eqn.6}
\end{align}

The monoids in $\Set$ produce right linear comonoids in $\Poly$ as follows.

If $(M, *, u) : \Set$ is a monoid, then $M \yon \lincomonwl \yon^M$ has the structure of a right linear comonoid where the linear duality $(\coeval,\eval)\colon M\yon\dual\yon^M$ is given as in <ref>.

The functor $M\mapsto M\yon$ is strong monoidal by <ref>, so it preserves monoids.

§.§ Linear bialgebras

In LDCs, a linear bialgebra on $b\dual a$ arises from a linear monoid structure interacting bialgebraically with a linear comonoid structure. In this section, we again define a left and right versions of linear bialgebra in $\otimes$-symmetric LDCs, and then we proceed to identify them in $\Poly$. We will see that they can be identified with monoids in $\Set$.

Let us first recall the definition of a bialgebra in a symmetric monoidal category (SMC).

In a SMC, a bialgebra $\bialg{a}$ consists of a monoid $\monoid{a}$ and a comonoid $\comonoid{a}$
satisfying the following equations:

\begin{equation}
\label{eqn.bialg}
\textit{(i)}~~~  \begin{tikzpicture}[yscale=-1]
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (0, 4) {};
\node [style=circle] (1) at (0, 5) {};
\node [style=none] (2) at (-0.5, 3) {};
\node [style=none] (3) at (0.5, 3) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right, looseness=1.25] (0) to (2.center);
\draw [bend left, looseness=1.25] (0) to (3.center);
\draw (1) to (0);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}[yscale=-1]
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (1) at (0, 5) {};
\node [style=none] (3) at (0, 3) {};
\node [style=circle] (4) at (0.75, 5) {};
\node [style=none] (5) at (0.75, 3) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (3.center) to (1);
\draw (5.center) to (4);
\end{pgfonlayer}
\end{tikzpicture} ~~~~~~~~~~~~ \textit{(ii)} ~~~ \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (0, 4) {};
\node [style=circle] (1) at (0, 5) {};
\node [style=none] (2) at (-0.5, 3) {};
\node [style=none] (3) at (0.5, 3) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [bend right, looseness=1.25] (0) to (2.center);
\draw [bend left, looseness=1.25] (0) to (3.center);
\draw (1) to (0);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (1) at (0, 5) {};
\node [style=none] (3) at (0, 3) {};
\node [style=circle] (4) at (0.75, 5) {};
\node [style=none] (5) at (0.75, 3) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (3.center) to (1);
\draw (5.center) to (4);
\end{pgfonlayer}
\end{tikzpicture} ~~~~~~~~~~~ \textit{(iii)} ~~~ \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (1) at (0, 5) {};
\node [style=circle] (2) at (0, 3) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1) to (2);
\end{pgfonlayer}
\end{tikzpicture} = \id_I  ~~~~~~~~~~~~ \textit{(iv)} ~~~ \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=circle] (0) at (0, 5.25) {};
\node [style=circle] (1) at (1.25, 5.25) {};
\node [style=circle] (2) at (0, 3.75) {};
\node [style=circle] (3) at (1.25, 3.75) {};
\node [style=none] (4) at (0, 3) {};
\node [style=none] (5) at (1.25, 3) {};
\node [style=none] (6) at (0, 6) {};
\node [style=none] (7) at (1.25, 6) {};
\node [style=none] (8) at (1.25, 3) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw (1) to (2);
\draw (0) to (3);
\draw [bend right=45] (0) to (2);
\draw [bend left=45] (1) to (3);
\draw (3) to (8.center);
\draw (2) to (4.center);
\draw (6.center) to (0);
\draw (7.center) to (1);
\end{pgfonlayer}
\end{tikzpicture} = \begin{tikzpicture}
\begin{pgfonlayer}{nodelayer}
\node [style=none] (4) at (0, 3) {};
\node [style=none] (6) at (0, 6) {};
\node [style=none] (7) at (1.5, 6) {};
\node [style=none] (8) at (1.5, 3) {};
\node [style=circle] (9) at (0.75, 4) {};
\node [style=circle] (10) at (0.75, 5.25) {};
\end{pgfonlayer}
\begin{pgfonlayer}{edgelayer}
\draw [in=90, out=-165] (9) to (4.center);
\draw [in=90, out=-15] (9) to (8.center);
\draw (9) to (10);
\draw [in=255, out=15] (10) to (7.center);
\draw [in=285, out=165] (10) to (6.center);
\end{pgfonlayer}
\end{tikzpicture}
\end{equation}

The final equation is often referred to as the bialgebra rule.
<ref> $(i)$ – $(iv)$ can equivalently be seen as the multiplication map acting as a comonoid morphism for the
$\ox$-comonoid $A \ox A$ or the comultiplication acts as a monoid morphism for the
$\ox$-monoid $A \ox A$ <cit.>.

Linear bialgebras were introduced as structures suitable to study complementary quantum observables in isomix LDCs.

In a symmetric LDC, a linear bialgebra, $\frac{(\eta,\epsilon)}{(\eta',\epsilon')}\!\!:\! b \linbialgwb a$, consists of:

* a linear monoid, $(\eta,\epsilon)\!\!:\! b \linmonw a$, and

* a linear comonoid, $(\eta',\epsilon')\!\!:\! b \lincomonb a$,

such that:

* $\bialg{b}$ is a $\ox$-bialgebra, and

* $\bialgb{a}$ is a $\tri$-bialgebra.

As before we fine-grain the definition of linear bialgebra to non-cyclic duals.

A left linear bialgebra, $\frac{(\eta_L,\epsilon_L)}{(\eta'_L,\epsilon'_L)}\!: b\! \linbialgwl a$ in a $\ox$-symmetric LDC consists of:

* a left linear monoid, $(\eta_L,\epsilon_L)\!\!:\! b \linmonwl a$, denoted below by white $\circ$, and

* a left linear comonoid, $(\eta'_L,\epsilon'_L)\!\!:\! b \lincomonwl a$, denoted below by black $\bullet$,

such that $\bialg{b}$ is a $\ox$-bialgebra, that is, the left dual is a bialgebra via the left linear monoid and comonoid.

A right linear bialgebra, $\frac{(\eta_R,\epsilon_R)}{(\eta'_R,\epsilon'_R)}\!\!:\! b \linbialgwr a$ in a $\ox$-symmetric LDC consists of:

* a right linear monoid, $(\eta_R,\epsilon_R)\!\!:\! b \linmonwr a$, denoted below by white $\circ$, and

* a right linear comonoid, $(\eta'_R,\epsilon'_R)\!\!:\! b \lincomonwr a$, denoted below by black $\bullet$,

such that $\bialg{a}$  is a $\ox$-bialgebra, that is, the right dual is a bialgebra via the right linear monoid and comonoid.

$\Poly$ is a $\ox$-symmetric LDC, and it has a left linear bialgebra and a right linear bialgebra for every monoid $M: \Set$.

Suppose $(M, *, e)$ is a monoid in $\Set$.

* The left linear monoid $(\coeval, \eval): M \yon \linmonwl \yon^M$ as defined in <ref>, and left linear comonoid $(\coeval, \eval): M \yon \lincomonwl \yon^M$ as defined in <ref>  together produce a left linear bialgebra.

* The right linear monoid $(\coeval, \eval): M \yon \linmonwr \yon^M$ as defined in <ref>, and right linear comonoid $(\coeval, \eval): M \yon \lincomonwr \yon^M$ as defined in <ref>  together produce a right linear bialgebra.

Using the $\ox$-monoid structure specified in <ref> and the $\ox$-comonoid structure specified in <ref> to prove that $M\yon$ is a $\ox$-bialgebra. The same idea applies to proving that the right linear monoid and comonoid gives a $\ox$-bialgebra on $\yon^M$.

In $\Poly$, we have that $M \yon \linbialgwl \yon^M$ is a left linear bialgebra and a right linear bialgebra iff $M$ is a monoid in $\Set$.

Direct consequence of <ref>.

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

titleMonoidal functors, species and Hopf algebras

journaltitleJournal of Pure and Applied Algebra
titleFeedback for linearly distributive categories: traces and fixpoints

journaltitleJournal of Pure and Applied Algebra
titleNatural deduction and coherence for weakly distributive categories

Cambridge University Press

journaltitleMathematical Structures in Computer Science
titleIntroduction to linear bicategories

journaltitleTheory and Applications of categories
titleProof theory for full intuitionistic linear logic, bilinear logic, and mix categories

journaltitleJournal of Pure and Applied Algebra
titleWeakly distributive categories

journaltitleJournal of Pure and Applied Algebra
titleLinearly distributive functors

journaltitleNew York J. Math
titleNote on Frobenius monoidal functors

journaltitlearXiv e-prints
titlePolynomial Functors: A Mathematical Theory of Interaction

journaltitleQuaestiones Mathematicae
titleOn categories of monoids, comonoids, and bimonoids

journaltitlearXiv preprint arXiv:2202.00534
titleA reference for categorical structures on Poly

University of Calgary

Calgary, AB

noteAvailable at   <https://arXiv:2303.14231>
titleDagger linear logic and categorical quantum mechanics
typePhD thesis