Linearly distributive categories were introduced to model the tensor/par
fragment of linear logic, without resorting to the use of negation.
Linear bicategories are the bicategorical version of
linearly distributive categories. Essentially, a linear bicategory has two forms of composition, each
determining the structure of a bicategory, and the two compositions are related by a linear distribution.

After the initial paper on the subject, there was little further work as there seemed to be a lack of examples. The main goal of this paper is to demonstrate that there are in fact a great many examples, which are mainly obtained by extending familiar constructions from the (ordinary) bicategorical setting.

While it is standard in the field of monoidal topology that the category of quantale-valued relations is
a bicategory, we begin by showing that if the quantale is a Girard quantale, we obtain a linear bicategory. We
further show that  for $Q$ a unital quantale is a Girard quantaloid if and only if $Q$ is a Girard
quantale. The tropical and arctic semiring structures fit together into a Girard quantale, so this construction is likely to have multiple
applications. More generally, we define -quantales, which are sup-lattices with two quantale structures
related by a linear distribution, and show that  is a linear bicategory if $Q$ is an -quantale.

We then consider several standard constructions from bicategory theory, and show that these lift to the linear bicategory setting and produce new
examples of linear bicategories. In particular, we consider quantaloids. We first define the notion of a linear quantaloid  and then consider
linear -categories and linear monads in , where  is a linear quantaloid. Every linear quantaloid is a linear bicategory.

We want to develop non-locally posetal examples as well. We consider $\biloc$, the bicategory whose objects are locales, 1-cells are bimodules
and two-cells are bimodule homomorphisms. This bicategory turns out to be what we call a Girard bicategory, which are in essence a closed
version of linear bicategories.

§ INTRODUCTION

The idea behind the theory of linearly distributive categories as introduced by Cockett and
Seely [4] is that (the multiplicative fragment of) linear logic is best modelled by taking the
two multiplicative connectives, $\ox$ (tensor) and $\op$ (par), as primitive. One then
obtains a category with two
monoidal structures, related by linear distributions. A linear distribution is a natural transformation, not an
isomorphism typically, of the following form or a symmetric equivalent:

\[A\ox (B\op C)\rarr (A\ox B)\op C\]

This approach to the model theory of linear logic differs from the original approach, using the $*$-autonomous categories of Barr [2], of taking tensor and negation as primitive and then defining
the par by de Morgan duality.

Just as monoidal categories can be viewed as one-object bicategories, one can ask for the
bicategorical version of linearly distributive categories. These are the linear bicategories
of Cockett, Koslowski and Seely [6], defined below.
The primary goal of this paper is to give new classes of linear bicategories arising from several
different sources.

Before diving into the formal definitions, we describe the example that led to the more general
discussion below. The structures we consider are two ordered semiring [11] structures on the extended
integers $\ZZP=\ZZ\cup\{+\infty,-\infty\}$. (One could just as well consider the extended reals.) This set
in fact has two semiring structures and these are typically called the tropical and arctic
semirings. They are of great use in the theory of synchronization as considered in [1].
See [8, 9] for how extensively these structures arise.

In both, multiplication is given by the usual addition of integers. But we must be careful in defining $
\infty+-\infty$.  In the first structure, we define $-\infty+_1\infty=-\infty=\infty+_1-\infty$. The addition for
this structure is given by max. This gives  the structure of an ordered semiring with equipped with its usual order.

For the second structure, we again have that the $\otimes$ is given by addition. But analogously
we now must have $-\infty+_2\infty=\infty=\infty+_2-\infty$. The addition for this structure is given by
min. This gives  the structure of an ordered semiring with equipped with the opposite of its usual order.

If $X$ and $Y$ are sets, we
will define a -relation from $X$ to $Y$ to be a function $R\colon X\times Y\rarr\ZZP$. As in the
category of relations, we consider this as a morphism $R\colon X\tov Y$. Then the
two semiring constructions above allow us to define two distinct relational compositions.

Explicitly, given

\[X\stackrel{A}{\tov}Y \stackrel{B}{\tov}Z\]

we define

\[A\ox B(x,z)=\bigvee_{y\in Y}(A(x,y)+_1B(y,z)) \mbox{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}A\girpar B(x,z)=\bigwedge_{y\in

We are of course using the fact that both $\ZZP$ and $\ZZP^{op}$ are not just semirings with the above structures, but are in
fact quantales. We are thus following the program of monoidal topology as described in [14].

The current project began with the observation that the two compositions described above are related
by a linear
distribution in the sense of [4] and in fact determine a locally posetal linear bicategory
in the sense of Cockett, Koslowski and Seely [6]. On the other hand, we have the observation
that  with the above operations is a Girard quantale [23, 18] and the two structures are related by
the Girard duality. Our first result is that the category  is a locally posetal linear bicategory
whenever $Q$ is a Girard quantale. It is furthermore closed in the sense that each 1-cell has a linear
adjunct. We also show that, for $Q$ a unital quantale,  is a Girard quantaloid, i.e.
a quantaloid with linear logic-like duality,  if and only if $Q$ is a Girard quantale. Following Rosenthal [19], given a Girard quantaloid , we
introduce a bicategory whose 0-cells are -cats (defined below), whose 1-cells are bimodules and 2-cells are pointwise inequalities.
We show that this is a
Girard quantaloid and furthermore demonstrate that  is Girard if and only if  is Girard. Similar results hold for the sub-bicategory of
monads in .

We then introduce the quantale analogue of linearly distributive categories which we call -quantales. All Girard
quantales are  and we prove that -quantales lead to linear bicategories. Analogously we introduce the
notion of linear quantaloid, and develop similar results at this level.

We want to develop non-locally posetal examples as well. This is done beginning in Section <ref>. We begin with an
example, which we use to illustrate a more general notion. We consider $\biloc$, the bicategory whose objects are locales, 1-cells are bimodules
and two-cells are bimodule homomorphisms. This bicategory turns out to be what we call a Girard bicategory.
The authors of [6] refer to this notion as a biclosed bicategory with cyclic dualizing 1-cells. Girard bicategories
are best seen as the lifting of the notion of symmetric monoidal closed category with a dualizing object to the bicategorical
setting. Every Girard bicategory is a linear bicategory and we show that a number of
classic examples of bicategories fit into this framework and thus give examples of linear bicategories. In particular, various bicategories of
bimodules and matrices are Girard bicategories. The notion of what we call a Girard
bicategory is found in Section 3.4 of [6], but we deveop the theory in detail as we see it as a rich source of

There are unfortunate notational conflicts between linear logic notation and the usual notation for ordered
as well as within the linear logic community.  We now give our choice for notation for the remainder of the paper.

* For partially ordered sets, we will denote the top element by $\qone$ and the bottom element by $\qzero$, if they exist.

* We will use $\top$ for the tensor unit in a monoidal category with
tensor $\ox$, as well as the unit for the multiplication $\ox$ in a quantale.  If we are using $\ox$ to denote the composition in a
bicategory, then we will use $\top_{X}$ to denote the identity 1-cell on $X$.

* In a $*$-autonomous or linearly distributive category, there is a second monoidal structure, which we will denote by $\op$
rather than the $\bindnasrepma$ of [10].
This is in line with the series of papers beginning with [7]. Similarly in a Girard
or -quantale (defined below), and in a Girard or linear quantaloid, there is a second multiplication, which we also denote by $\op$.

* When we use $\op$ to denote the second composition in a linear bicategory, we will denote the identity 1-cells by $
\perp_X$. We will denote the tensor unit for the second monoidal structure by $\perp$ as well as the unit for the second
multiplication in a Girard or -quantale. Note that this $\perp$ is also the (cyclic) dualizing element or object in a Girard quantale or quantaloid.

* There are also issues with the notation and terminology for closed structures, which we will address in Remark <ref>.

* We will denote quantales by $Q$ and quantaloids by .

§ PRELIMINARIES

§.§ Quantales and quantaloids

See [14, 17, 18, 19, 23] for the following standard definitions and results.

* A quantale is a partially ordered set $Q$ with all suprema and an associative multiplication
$\ox\colon Q\times Q\rarr Q$ with unit $\top$ such that for all subsets
$P\subseteq Q$ and all elements $a\in Q$, we have

\[\big(\bigvee P\big)\ox a=\bigvee_{p\in P}p\ox a\mbox{\,\,\,\,\,\,\,\,\, and \,\,\,\,\,\,\,\,\,\,}a\ox\big(\bigvee
P\big)=\bigvee_{p\in P}a\ox p\]

Note that $Q$ necessarily satisfies
$a\ox{\qzero=\qzero=\qzero}\ox a$, where $\qzero$ is  the bottom element.

* Since for all $a\in Q$ the operations $(-)\ox a$ and $a\ox(-)$ preserve all sups, they have right
adjoints. We denote them by $(-)\lTo a$ and $a\rTo(-)$.

* An element $\bot\in Q$ is a cyclic dualizing element if for all $a\in Q$, we have

\[   \bot\lTo a=a\rTo \bot    \mbox{\,\,\,\,\,\,\,\,\,\,\, and \,\,\,\,\,\,\,\,\,\, }(a\rTo \bot)\rTo \bot=a\]

A Girard quantale is a pair $(Q,\bot)$ where $Q$ is a quantale and $\bot$ is a chosen
cyclic dualizing object. We denote $a\rTo \bot$ as $a^\perp$. Note that $\top^\bot=\bot$.

* If $Q$ is a Girard quantale, it has a second multiplication defined by the linear logic version of de Morgan duality.

Let $(Q,\bot)$ be a Girard quantale. The operation $(-)^\perp$ is a contravariant
isomorphism. So $Q^{op}$ is also a quantale. We will denote its multiplication by

\[a\op b=(b^\perp\ox a^\perp)^\perp\]

Evidently this operation satisfies:

\[(\bigwedge P)\op a=\bigwedge_{p\in P} p\op a \mbox{\,\,\,\,\,\,\,\,\, and \,\,\,\,\,\,\,\,\,\,}a\op(\bigwedge
P)=\bigwedge_{p\in P} a\op p \]

* A quantaloid  is a category enriched over the category of complete
lattices and sup-preserving maps. In particular, we note that a one-object quantaloid is the same
as a quantale.

* A family of 1-cells $\cD=\{ d_a\colon a\to<125>a\mid a\in\QQ\}$ is a cyclic family if $f-\!\!\!\!-\!\!\circ d_a= d_b  \circ\!\!-\!\!\!\!- f$, for all $f$, and let $f^\perp$  denote
their common value. Then  is called a  cyclic dualizing family if $f^{\perp\perp}=f$, for all  $f$.

A Girard quantaloid is a quantaloid $\QQ$ together with a cyclic dualizing family .  Note that
a Girard quantaloid with one object is a Girard quantale.

* If $\QQ$ is Girard, it has a second composition $\op$ given by the linear logic version of the Morgan duality, such that $\cQ^{co}$ is also a quantaloid. See [19] for further details.

§.§ The category

We assume the reader is familiar with the theory of bicategories, introduced by Benabou. Appropriate references are [3, 16].
Many of our examples will be locally posetal bicategories, i.e. bicategories where each $Hom$-category is a poset.
The standard example of a locally posetal bicategory is the category of sets and relations, , where the hom-sets are
ordered under inclusion.

* If $Q$ is a quantale, we can form the category  whose objects are sets and arrows
$f\colon X\tov Y$ are functions $f\colon X\times Y\rarr Q$. Given $f\colon X\tov Y$
and $g\colon Y\tov Z$, composition is defined by

\[ f\ox g(x,z)=\bigvee_{y\in Y} f(x,y)\ox g(y,z)\]

Note that the use of $\ox$ on the left refers to composition and on the right refers to multiplication in $Q$.

Identities are given by

\[id_X(x,x')=\begin{cases} \qzero &\mbox{if } x\neq x'\\
\top& \mbox{if } x=x'  \end{cases}\]

See [14] for further details.

* If $Q$ is a Girard quantale, there is a second categorical structure, using the fact that $Q^{op}$ is also a quantale.
Given $f\colon X\tov Y$
and $g\colon Y\tov Z$, composition is given by

\[ f\op g(x,z)=\bigwedge_{y\in Y} f(x,y)\op g(y,z)\]

and obvious identities.

If $Q$ is a quantale, then  is a locally posetal bicategory, under pointwise order.
If $Q$ is a Girard quantale, it has two locally posetal bicategory structures as described above.

It is the case that  is more than just a locally posetal bicategory, it is a quantaloid [19].

Suppose $Q$ is a unital quantale. Then  is a quantaloid. Furthermore  is a Girard quantaloid with  cyclic dualizing family
$D=\{d_X\colon X\tov X\}$ if and only if
$Q$ is a Girard quantale with cyclic dualizing element $d$ corresponding to $d_1\colon 1\tov1$ and

\[d_X(x,x')=\begin{cases}d  & x=x' \\ \qone & x\ne x'
\end{cases}\]

-.5in $(*)$

Suppose is a Girard quantaloid. Given  $a\in Q$, let $r_a\colon 1\tov1$ denote the $a$-valued constant map.
Since  $ r_a -\!\!\!\!-\!\!\circ  d_1=d_1  \circ\!\!-\!\!\!\!-   r_a $ and $r_a^{\perp\perp}=r_a$, it follows that
$d$ is a cyclic dualizing element for $Q$, and so $Q$ is a Girard quantale.

Let $x'\in X$ and define $r\colon 1\to X$ by

\[r(x)=\begin{cases} e & x=x' \cr \qone & x\ne x'\end{cases}\]

Since $(d_X  \circ\!\!-\!\!\!\!-  r)(x)=\displaystyle\bigwedge_{\bar x} d_X(x,\bar x)   \circ\!\!-\!\!\!\!-  r =

\[(r  -\!\!\!\!-\!\!\circ  d_1)(x)=r(x) -\!\!\!\!-\!\!\circ d_1=
\begin{cases}  d& x=x' \\
\qone & x\ne x'\end{cases}\]

since $e-\!\!\!\!-\!\!\circ d=d$.

\[d_X(x,x')=\begin{cases} d  & x=x' \\  \qone & x\ne x'\end{cases}\]

Suppose $Q$ is a Girard quantale satisfying $(*)$, and let $r\colon X\tov Y$. Then
$$(r  -\!\!\!\!-\!\!\circ
d_X)(y,x) =  \displaystyle \bigwedge_{\bar x} r(\bar x,y)  -\!\!\!\!-\!\!\circ d_X(\bar x,x)=r(x,y)  -\!\!\!\!-\!\!\circ

$$(d_Y  \circ\!\!-\!\!\!\!-  r)(y,x) = \displaystyle \bigwedge_{\bar y} d_Y(y,\bar y)  \circ\!\!-\!\!\!\!-  r(x,
\bar y)= d \circ\!\!-\!\!\!\!-  r(x,y)$$
and so   is a cyclic dualizing family for , since $r  -\!\!\!\!-\!\!
\circ d_X = d_Y  \circ\!\!-\!\!\!\!-  r$ and $r^{\perp\perp}=r$.

§.§ -categories and monads in a quantaloid

We introduce two standard constructions in bicategory theory. We will show that they generalize to the linear bicategory setting and
produce new examples of linear bicategories.  The following definition is a special case of a category enriched in a bicategory, as described for example in [21, 22].

Let  be a quantaloid. A -category is a pair $M=(X,\rho)$ where:

* $X$ is a set.

* $\rho$ is a function $\rho:X\rightarrow \textnormal{ob}\cQ$.

* We have a function, called the enrichment assigning to each pair $(x,x')\in X\times X$
a morphism $M(x,x'):\rho(x)\rightarrow\rho(x')$ such that

* $\top_{\rho(x)}\leq M(x,x)~ \forall x\in X$ (Recall that we are using $\top$ to denote identity 1-cells.)

* $M(x,x')\ot M(x',x'')\leq M(x,x'')~ \forall x, x', x''\in X$

Alternatively, a -category is a lax bifunctor from set $X$ to , when $X$ is viewed as an indiscrete bicategory.

Let  be a quantaloid, $M=(X,\rho_M)$ and $N=(Y,\rho_N)$ be -categories. A -bimodule $\Theta:M\tov N$ consists of an assignment to every pair $(x,y)\in X\times Y$
a morphism $\Theta(x,y): \rho_M(x)\rightarrow\rho_N(y)$ such that

* $\Theta(x,y) \ot N(y,y')\leq \Theta(x,y')~\forall x\in X, y, y'\in Y$

* $M(x,x')\ot\Theta(x',y) \leq \Theta(x,y)~\forall x, x'\in X, y\in Y$

* Let  be a quantaloid, then  is the locally posetal bicategory whose 0-cells are -categories, 1-cells are -bimodules, and 2-cells are pointwise inequalities. Given $M\stackrel{\Theta}{\tov}N \stackrel{\Pi}{\tov}P$, composition $\Theta\ot\Pi:M\tov P$ is defined by
$$(\Theta\ot\Pi)(x,z)=\bigvee_{y\in Y}\Theta(x,y)\ot\Pi(y,z)$$

Identity 1-cells are $\iota_M:M\tov M$ defined by $\iota_M(x,x')=M(x,x')$.
Note that the use of $\ox$ on the left refers to composition in  and on the right refers to composition in .

* If  is a Girard quantaloid,  has a second bicategorical structure. Given $M\stackrel{\Theta}{\tov}N \stackrel{\Pi}{\tov}P$, composition $\Theta\op\Pi:M\tov P$ is defined by
$$(\Theta\op\Pi)(x,z)=\bigwedge_{y\in Y}\Theta(x,y)\op\Pi(y,z)$$
with identity 1-cells $\delta_M\colon M\tov M$, described below.

We have chosen to denote the bicategory of -categories and -bimodules as  in agreement
with [5], while it is called Bim() in [19].

Suppose  is a quantaloid. Then  is a quantaloid. Furthermore  is a Girard quantaloid with cyclic dualizing family  $\{d_a\colon a\rarr a\}$ if and only if  is a Girard quantaloid with cyclic dualizing family $\{\delta_M\colon M\tov M\}$, where
$$\delta_M(x,x')=M(x',x)-\!\!\!\!-\!\!\circ d_{\rho(x')}$$

We only need to provide the backwards direction, the other results are proven in ([19], Proposition 2.3.1, Theorem 2.4.2).

Suppose  is a Girard quantaloid with cyclic dualizing family $\{\delta_M\colon M\tov M\}$. Given an object $a$ in , define the  -category $M_a=(\{x_a\},\rho_a)$ by $\rho_a(x_a)=a$. Further, given a morphism $f\colon a\rarr b$, define the -bimodule $\Theta_f\colon M_a\tov M_b$ by $\Theta_f(x_a, x_b)=f$. Consider the family of morphisms =$\{d_a\colon a\rarr a\}$, where $d_a=\delta_{M_a}(x_a, x_a)$.

is a cyclic family in :

\begin{align*}
f-\!\!\!\!-\!\!\circ d_a &= \Theta_f(x_a, x_b)-\!\!\!\!-\!\!\circ \delta_{M_a}(x_a, x_a) = \bigwedge_{x\in \{x_a\}} \Theta_f(x, x_b)-\!\!\!\!-\!\!\circ \delta_{M_a}(x, x_a) \\
& = (\Theta_f -\!\!\!\!-\!\!\circ \delta_{M_a})(x_b, x_a) = (\delta_{M_b} \circ\!\!-\!\!\!\!-  \Theta_f)(x_b, x_a) \\
& = \bigwedge_{x\in \{x_b\}} \delta_{M_b}(x_b, x) \circ\!\!-\!\!\!\!-  \Theta_f(x_a, x) = \delta_{M_b}(x_b, x_b) \circ\!\!-\!\!\!\!- \Theta_f(x_a, x_b) \\
& = d_b  \circ\!\!-\!\!\!\!-  f
\end{align*}

is a dualizing family in :

\begin{align*}
d_a  \circ\!\!-\!\!\!\!-  (f-\!\!\!\!-\!\!\circ d_a) &= \delta_{M_a}(x_a, x_a) \circ\!\!-\!\!\!\!-  (\Theta_f(x_a, x_b)-\!\!\!\!-\!\!\circ \delta_{M_a}(x_a, x_a)) \\
&= \delta_{M_a}(x_a, x_a) \circ\!\!-\!\!\!\!- \bigwedge_{x\in\{x_a\}}(\Theta_f(x, x_b)-\!\!\!\!-\!\!\circ \delta_{M_a}(x, x_a)) = \delta_{M_a}(x_a, x_a) \circ\!\!-\!\!\!\!- (\Theta_f-\!\!\!\!-\!\!\circ \delta_{M_a})(x_b, x_a) \\
&= \bigwedge_{x\in \{x_a\}} \delta_{M_a}(x_a, x) \circ\!\!-\!\!\!\!- (\Theta_f-\!\!\!\!-\!\!\circ \delta_{M_a})(x_b, x) = (\delta_{M_a} \circ\!\!-\!\!\!\!- (\Theta_f-\!\!\!\!-\!\!\circ \delta_{M_a}))(x_a, x_b) \\
&= \Theta_f(x_a, x_b) = f
\end{align*}

We next consider the bicategorical notion of monad in the context of quantaloids. This is an instance of a more general construction we give in the appendix, but this case is of sufficient interest that we give it separately.

Let  be a quantaloid. A monad $(a,m)$ in  is an object $a$ equipped with a morphism $m:a\rightarrow a$ such that $\top_a\leq m$ and $m \ot m \leq m$.  An $(m,n)$-bimodule $f:(a,m)\tov(b,n)$ is a morphism $f:a\rightarrow b$ in  such that $m\ot f\leq f$ and $f\ot n\leq f$.

Alternatively, a monad in  is a lax bifunctor from the terminal bicategory 1 to .

* Let  be a quantaloid, then  is the locally posetal bicategory whose 0-cells are monads in  and 1-cells are monad bimodules. The 2-cells and composition are directly inherited from . The identity 1-cells are $\top_{a,m}:(a,m)\tov(a,m)$ is $m:a\rightarrow a$.

* If  is a Girard quantaloid,  has a second bicategorical structure, inherited from $\cQ^{co}$.

Suppose  is a quantaloid. Then  is a quantaloid. Furthermore  is a Girard quantaloid with cyclic dualizing family $\{d_a\colon a\rarr a\}$ if and only if  is a Girard quantaloid with cyclic dualizing family $\{\delta_{(a,m)}\colon (a,m)\rarr(a,m)\}$, where
$$\delta_{(a,m)}=m-\!\!\!\!-\!\!\circ d_a$$

We need only provide the backwards direction. The other statements are proven in ([19], Proposition 2.3.3, Theorem 2.4.4).

There is an embedding of  into  by mapping objects $a\in\cQ$ to the trivial monad $(a, 1_a)\in\MonQ$. Then, it is immediate that the family of morphisms $=\{d_a\colon a\rarr a\}$, where $d_a$ is $\delta_{(a, 1_a)}\colon (a,1_a)\rarr (a,1_a)$ when viewed as a morphism in , is cyclic and dualizing.

§ LINEAR BICATEGORIES

We introduce the theory of linear bicategories as defined by Cockett, Koslowski and Seely
[6]. The material in this subsection is entirely from that paper.

As usual with bicategories, one begins with a class of 0-cells which we will denote $\cB_0=\{X,Y,Z,
\ldots\}$. Then for every pair of 0-cells, one has a category $\cB(X,Y)$. The objects
of $\cB(X,Y)$ are called 1-cells and the arrows are called 2-cells. But now we have two
composition functors which we denote by $\ox$ and $\girpar$:

\[\ox,\op\colon \cB(X,Y)\times\cB(Y,Z)\lrarr\cB(X,Z)\]

such that each of these compositions gives a bicategory structure. Thus for each composition
we have all of the morphisms and coherence that this entails. In particular
we must have identity 1-cells for each of the two compositions:

\[ \top_X\colon X\rarr X \mbox{\,\,\,\,\,\,\,\,\,\,and \,\,\,\,\,\,\,\,\,\,}  \bot_X\colon X\rarr X\]

These two bicategory structures are related by a linear distribution as follows. Given:

\[X-\!\!\!-\!\!\!-\!\!\!\lrarr Y-\!\!\!-\!\!\!-\!\!\!\lrarr Z-\!\!\!-\!\!\!-\!\!\!\lrarr W\]

we have two functors:

\[-\!\!\!-\ox(-\!\!\!-\girpar -\!\!\!-),\,\,\,\, (-\!\!\!-\ox -\!\!\!-)\girpar -\!\!\!-:\cB(X,Y)\times\cB(Y,Z)\times\cB(Z,W)-\!
\!\!-\!\!\!-\!\!\!\lrarr\cB(X,W)\]

and we require a natural transformation between them, which is not necessarily an

\[-\!\!\!-\ox(-\!\!\!-\op -\!\!\!-) \Longrightarrow (-\!\!\!-\ox -\!\!\!-)\op -\!\!\!-.\]

Symmetrically, we also need a natural transformation

\[(-\!\!\!-\op -\!\!\!-)\ox-\!\!\!- \Longrightarrow -\!\!\!-\op(-\!\!\!-\ox -\!\!\!-).\]

All of this structure must satisfy the usual coherence requirements. See [6] for
details as well as for a number of examples.

We do mention here the quintessential
example of the linear bicategorical structure on , as it will be the example on which ours are based.
It is standard that  is a locally posetal bicategory. Its first composition is the
standard one. If we have $R\colon X\rarr Y$ and $S\colon Y\rarr Z$ then define:

\[(x,z)\in S\ox R \mbox{\,\,\,\,\,\, if and only if\,\,\,\,}
\exists y \mbox{\,\,\,\,\,}(x,y)\in R\,\, \mbox{\,and\,} \,\,
(y,z)\in S\]

But we have a second composition given by

\[(x,z)\in S\op R \mbox{\,\,\,\,\,\, if and only if\,\,\,\,} \forall y \mbox{\,\,\,\,\,\,}(x,y)\in R\,\,
\mbox{\, or\,} \,\,(y,z)\in S\]

The above determines a linear bicategory structure on .

§.§ Structures in linear bicategories

For a discussion of the following types of structure in (ordinary) bicategories, see the Appendix.
Following [6]:

Given a linear bicategory , a linear adjunction consists of:

* 1-cells $A\colon X\rarr Y$ and $B\colon Y\rarr X$.

* 2-cells $\tau\colon\top_X\rarr A\girpar B$ and $\gamma\colon B\ox A\rarr\bot_Y$

These must satisfy standard coherence.

We say $A$ is left linear adjoint to $B$.

By adapting the usual bicategorical arguments to the linear setting, the authors show:
The following are equivalent:

* $A$ is left linear adjoint to $B$

* For all 0-cells Z, the functor $(-)\ox A\colon Hom(Z,X)\rarr Hom(Z,Y)$ is left adjoint to $(-)\girpar

* For all 0-cells Z, the functor $B\ox (-)\colon Hom(X,Z)\rarr Hom(Y,Z)$ is left adjoint to $A\girpar (-)

We first remind the reader of a standard notion in ordinary bicategory theory. But we do remark that, following [6], we
use notation and terminology in terms of left and right homs, rather than the more traditional notions of extensions and liftings.

The authors prove:

A linear bicategory is closed if and only if every 1-cell has both left and right linear adjunctions.

Let  be a linear bicategory.

* Suppose that $A\colon X\rarr Y$. Then $A\multimap \perp_X\colon Y\rarr X$ will be denoted $A^\perp$

* $^\perp A$ is defined dually using the left hom

* Using this notation, one may identify $A\multimap B$ with $A^\perp\op B$

* We similarly have isomorphisms:

\[^\perp\!(A^\perp\ox B^\perp)\cong (^\perp\!\! A\ox^\perp\!\! B)^\perp\cong B\op A\]

See [6] for further details.

§ , , AND  AS A LINEAR BICATEGORIES

We now demonstrate that if  $Q$ is a Girard quantale, then  determines a linear bicategory, with the locally posetal bicategorical
structures introduced in Section 2. Recall that our 0-cells are sets. If $X$ and $Y$
are sets, then we define:

\[\cB(X,Y)=\{f\colon X\times Y \rarr Q | f\mbox{\, is a function}\}\]

As above, this is a partial order under pointwise ordering. Next we require two notions of

\[\ox,\op\colon \cB(X,Y)\times\cB(Y,Z)\lrarr\cB(X,Z)\]

Given $f\colon X\tov Y$ and $g\colon Y\tov Z$, we define

\[f\ox g(x,z)=\bigvee_{y\in Y}(f(x,y)\ox g(y,z)) \mbox{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}f\girpar g(x,z)=\bigwedge_{y\in
Y}(f(x,y)\op g(y,z))\]

The identity 1-cells are given by:

\[\top_X(x,x')=\begin{cases} \qzero &\mbox{if } x\neq x'\\
\top& \mbox{if } x=x'  \end{cases}\mbox{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}
\bot_X(x,x')=\begin{cases} \qone &\mbox{if } x\neq x'\\
\bot& \mbox{if } x=x'  \end{cases}\]

Note that we are using the same symbols, $\ox$ and $\op$ for both the multiplication in $Q$ and the
corresponding composition in the category, but this should cause no confusion.

In summary, we have:

The above determines   as a linear bicategory.

This will follow from a more general result in the next section.

In the same vein, if  is a Girard quantaloid, then  determine a linear bicategory.

* The 0-cells are -category $M=(X,\rho)$. Notice that each -category $M$ has a second enrichment assigning to each pair $(x,x')\in X$ a morphism from $M(x',x)^\bot\colon\rho(x)\rightarrow\rho(x')$ which satisfies

* $M(x,x)^\bot\leq\bot_{\rho(x)}~ \forall x\in X$

* $M(x,x'')^\bot\leq M(x',x'')^\bot\op M(x,x')^\bot~ \forall x, x', x''\in X$

Further, the second enrichment is a -bimodule as it satisfies the relevant inequalities as well as their versions under the contravariant $(-)^\bot$ isomorphism: $ \forall x, x', x''\in X$,

* $M(x',x)^\bot\ot M(x',x'')\leq M(x'',x)^\bot$ and $M(x'',x)\leq M(x',x'')^\bot\op M(x',x)$

* $M(x'',x)\ot M(x',x)^\bot\leq M(x',x'')^\bot$ and $M(x'',x)\leq M(x'',x')\op M(x,x')^\bot$

* The 1-cells are -bimodules $\Theta:M\tov N$ are also now endowed with a second assignment to each pair $(x,y)\in X\times Y$ a morphism  $\Theta(x,y)^\bot: \rho_N(y)\rightarrow\rho_M(x)$ which satisfies $ \forall x, x'\in X, y, y'\in Y$,

* $\Theta(x,y')^\bot\leq N(y,y')^\bot\op\Theta(x,y)^\bot$

* $\Theta(x,y)^\bot\leq\Theta(x',y)^\bot\op M(x,x')^\bot$

Further, this enrichment is itself -bimodule in its own right as it satisfies the necessary inequalities and their versions under the $(-)^\bot$ isomorphism, $ \forall x, x'\in X, y, y'\in Y$,

* $\Theta(x,y)^\bot \ot M(x,x')\leq \Theta(x,y')^\bot$ and $\Theta(x,y')\leq M(x,x')^\bot\op\Theta(x,y)$

* $N(y,y')\ot\Theta(x,y')^\bot \leq \Theta(x,y)^\bot$ and $\Theta(x,y)\leq \Theta(x,y')\op N(y,y')^\bot$

* The two compositions are defined as follows. Given $M\stackrel{\Theta}{\tov}N \stackrel{\Pi}{\tov}P$, $\Theta\ot\Pi:M\tov P$, $\Theta\ot\Pi:M\tov P$ and $\Theta\op\Pi:M\tov P$ are defined by
$$(\Theta\ot\Pi)(x,z)=\bigvee_{y\in Y}\Theta(x,y)\ot\Pi(y,z)~\textnormal{and}~(\Theta\op\Pi)(x,z)=\bigwedge_{y\in Y}\Theta(x,y)\op\Pi(y,z)$$

As above, there is a second enrichment for these -bimodules:

$$(\Theta\ot\Pi)(x,z)^\bot=\bigwedge_{y\in Y}\Pi(y,z)^\bot\op\Theta(x,y)^\bot~\textnormal{and}~(\Theta\op\Pi)(x,z)^\bot=\bigvee_{y\in Y}\Pi(y,z)^\bot\ot\Theta(x,y)^\bot$$

* The identities 1-cells for  $\ot$ and $\op$ are given by $\iota_M$ and $\iota_M^\bot$ respectively, where

$$ \iota_M(x,x')= M(x,x')~\textnormal{and}~ \iota_M^\bot(x,x')=M(x',x)^\bot$$

The above determines  as a linear bicategory.

This will follow from a more general result in the next section.

Similarly, if  is Girard,  is a linear bicategory.

§.§ -quantales and linear quantaloids

Girard quantales are $*$-autonomous posets. In particular, the second multiplication
is defined as the dual of the first. An alternate approach to the model theory of linear logic
is to take the tensor and par as primitive. This leads to the notion of linearly distributive category,
due to Cockett and Seely [4].
There is a canonical way to add negation to this structure and the result are
$*$-autonomous categories as expected. See [4] for details.

One can ask what the analogous structure would be for quantales. These are the -quantales as
defined below.

An LD-quantale is a complete lattice $Q$ with operations $\ot$ and $\op$ and elements $\top$
and $\perp$ such that

* $(Q,\ot, \top)$ and $(Q^{op},\op,\bot)$ are quantales.

*   $a\ot(b\op c)\le (a\ot b)\op    c\mbox{\,\,\,\,\,\,\, and\,\,\,\,\,\,\,\,}(b\op c)\ot a\leq b\op(c\ot a)$

Clearly a Girard quantale is a LD-quantale.

In the same way that LD-quantales are the linearly distributive category version of quantales. The following linear quantaloids are the appropriate notion for quantaloids and linear bicategories.

A linear quantaloid is a locally small category  whose hom-sets are complete lattices with binary operations $\ot$ and $\op$, and families of distinguished morphisms $\{\top_a~|~a\in \textnormal{ob}\cQ\}$ and $\{\bot_a~|~a\in \textnormal{ob}\cQ\}$ such that

* $(\cQ,\ot,\top_a)$ and $(\cQ^{co},\op,\bot_a)$ are quantaloids,

* $f\ot(g\op h)\leq(f\ot g)\op h$ and $(f\op g)\ot h\leq f\op (g\ot h) ~\forall~\textnormal{composable}~f,g,h\in\textnormal{mor}\cQ$

Every Girard quantaloid is a linear quantaloid. Moreover if $Q$ is an -quantale, $Q$-$\sf Rel$ is in fact a linear quantaloid.

We have the following obvious observation.

A linear quantaloid is a linear bicategory.

Suppose $(Q,\ot, \op)$ is an -quantale, and  $X\hbox{$\to<150>^R$\hskip -.2in \raisebox{1.8pt}{\tiny$\bullet$} \hskip .1in} Y\hbox{$\to<150>^S$\hskip -.2in \raisebox{1.8pt}{\tiny$\bullet$} \hskip .1in} Z$ are $Q$-valued relations.

\[R\otimes S(x,z)=\bigvee_{y\in Y} (R(x,y)\ot S(y,z)) \mbox{\,\,\,\,\,\,\,\,\,}
R\op S(x,z)=\bigwedge_{y\in Y}(R(x,y)\op S(y,z)) \,\,\,\,\,\,\,\,\,\, (*) \]

The identity 1-cells are given by:

\[\top_X(x,x')=\begin{cases} \qzero &\mbox{if } x\neq x'\\
\top& \mbox{if } x=x'  \end{cases}\mbox{\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}
\bot_X(x,x')=\begin{cases} \qone &\mbox{if } x\neq x'\\
\bot& \mbox{if } x=x'  \end{cases}\]

$(Q,\ot, \op)$  is an -quantale if and only if  ($Q$-$\sf Rel$,$\otimes,\girpar$),
where $\otimes$ and $\girpar$ are defined as in $(*)$, is a linear

Suppose  $($Q$,\ot, \op)$  is an -quantale. Then $(Q,\otimes)$ and $(Q,\oplus)$ are clearly bicategories. Given

we have
$R\otimes(S\girpar T) \le (R\otimes S)\girpar T$ if and only if, for all $x,z$,
[R(w,x)\ot(S\girpar T)(x,z)] \le \bigwedge_y [(R\otimes S)(w,y)\op    T(y,z)]$$

if and only if  $R(w,x)\ot(S\girpar T)(x,z) \le (R\otimes S)(w,y)\op    T(y,z)$, for all $w,x,y,z $. But,

\begin{eqnarray*}
R(w,x)\ot(S\girpar T)(x,z)
& = &R(w,x)\ot \bigwedge_ y [S(x,y) \op T(y,z)] \\
& \le & R(w,x)\ot[S(x,y) \op T(y,z)] \\
& \le & [R(w,x)\ot S(x,y)] \op T(y,z) \\
& \le &\bigvee_x[R(w,x)\ot S(x,y)] \op T(y,z) \\
& \le &(R\otimes S)(w,y)\op T(y,z)
\end{eqnarray*}

The other inequality follows similarly. Then note $(\QRel,\ox)$ and $(\QRel^{op},\op)$ inherit their quantaloid structure from $Q$ pointwise and one concludes that   is a linear quantaloid.

Conversely, suppose $Q$-$\sf Rel$ is a linear quantaloid. Then $a,b,c$ in $Q$ induces $1\!\hbox{$\to<125>^{R_a}$\hskip -.2in \raisebox{1.8pt}{\tiny$\bullet$} \hskip .1in}\!1\!\hbox{$\to<125>^{R_b}$\hskip -.2in \raisebox{1.8pt}{\tiny$\bullet$} \hskip .1in}\!1\hbox{$\to<125>^{R_c}$\hskip -.2in \raisebox{1.8pt}{\tiny$\bullet$} \hskip .1in}\!1$ in $Q$-$\rm Rel$, and so the multiplications $\otimes$ and $\oplus$ in $Q$ are associative. Also, $a\ot(b\op c)\le (a\ot b)\op c$, since $R_a\otimes(R_b\girpar R_c) \le (R_a\otimes R_b)\girpar R_c$.
Then note that $Q$ and $Q^{op}$ inherit their quantale structure from . Thus $Q$ is an -quantale.

Note that when $Q$ is the unique 2-element quantale, we get that $Q$-$\sf Rel$ is equivalent to the linear bicategory structure defined on $\sf Rel$ described above and in [6].

§.§ Shift monoids

In [7], the authors introduce the notion of a shift monoid. These provide examples
of discrete linearly distributive categories which are not $*$-autonomous.

* A shift monoid consists of a 4-tuple $\cM=(M,+,\top,a)$ where $(M,+,\top)$ is a commutative monoid and $a$ is an
invertible element of $M$.

* If  is a shift monoid, define a second multiplication by

\[x\cdot y=x+y-a\]

Then this is a second monoid structure on $M$ with unit given by $a$.

It is trivial to see that $x\cdot(y+z)=(x\cdot y)+z$ and so every shift monoid is a discrete linearly distributive category. The authors
generalize this observation as follows.

* Suppose $(\cC,\ox,\top)$ is a monoidal category, with an invertible object $\bot$. One can define a second monoidal
structure on  by

\[X\op Y=X\ox\bot^{-1}\ox Y\]

with unit $\bot$. We call this the shifted tensor.

* Any monoidal category with an object $\bot$ with a tensor inverse has a shifted tensor which together with the original
monoidal structure forms a linearly distributive category in which the linear distributions are natural isomorphisms.

* Any linearly distributive category whose linear distributions are natural isomorphisms has an invertible object $\bot$ whose
shift tensor is naturally equivalent to the second tensor.

We modify the notion of shift monoid as follows in order to construct -quantales. Recall that a commutative monoid $M$ is
cancellative if for all $a,b,c\in\ M$, one has
$$a+b=a+c\Rightarrow b=c$$

* Let $M$ be a cancellative commutative monoid.
We view $M$ as a discrete poset and then add top and bottom
elements which we denote  and . Denote this set as $M^+$, We then extend the addition on $M$:

\[\qone+b=\qone\mbox{    if $b\in M$ or $b=\qone$} \,\,\,\,\,\,\,\,\,\,\,\, \qzero+b=\qzero\mbox{     for all $b\in M^+$}\]

Then $(M^+,+)$ is a commutative quantale.

* If  is furthermore a cancellative commutative shift monoid, then extend the second operation in the dual way:

\[\qzero\cdot b=\qzero\mbox{    if $b\in M$ or $b=\qzero$} \,\,\,\,\,\,\,\,\,\,\,\, \qone\cdot b=\qone\mbox{     for all $b\in M^+$}\]

Then  is an -quantale. Thus by Theorem <ref>, -Rel is a linear quantaloid.

§.§ More examples of linear quantaloids

We will show that the standard bicategorical structures we described previously can be lifted to the linear setting, and produce
linear quantaloids, and hence linear bicategories.

Let  be a linear quantaloid. A linear -category is a pair $M = (X, \rho)$ where:

* X is a set.

* $\rho$ is a function $X\rightarrow ob\cQ$,

* We have a function, called the $\ot$-enrichment, assigning to each pair $(x,x')\in X\times X$
a morphism $M_\ot(x,x'):~\rho(x)\rightarrow\rho(x')$ in  such that

* $\top_{\rho(x)}\leq M_\ot(x,x)~ \forall x\in X$,

* $M_\ot(x,x')\ot M_\ot(x',x'')\leq M_\ot(x,x'')~ \forall x, x', x''\in X$,

* We have a function, called the $\op$-enrichment, assigning to each pair $(x,x')\in X\times X$
a morphism $M_\op(x,x'):~\rho(x)\rightarrow\rho(x')$ in  such that

* $M_\op(x,x)\leq\bot_{\rho(x)}~ \forall x\in X$,

* $M_\op(x,x'')\leq M_\op(x,x')\op M_\op(x',x'') ~ \forall x, x', x''\in X$,

* satisfying the following inequalities $\forall x, x', x''\in X$,

* $M_\ot(x,x'')\leq M_\op(x,x')\op M_\ot(x',x''),$

* $M_\ot(x,x'')\leq M_\ot(x,x')\op M_\op(x',x''),$

* $M_\ot(x,x')\ot M_\op(x',x'')\leq M_\op(x,x''),$

* $M_\op(x,x')\ot M_\ot(x',x'')\leq M_\op(x,x'').$

A linear -category is in fact a linear functor from $X$ to , where $X$ is viewed as a degenerate indiscrete linear bicategory.

Let  be a linear quantaloid, $M=(X, \rho_M)$ and $N=(Y, \rho_N)$ be -categories. A linear -bimodule $\Theta:M\tov N$ consists of two assignments to every pair $(x,y)\in X\times Y$

* a morphism $\Theta_\ot(x,y): \rho_M(x)\rightarrow\rho_N(y)$ such that $\forall x, x'\in X, y, y'\in Y,$

* $\Theta_\ot(x,y) \ot N_\ot(y,y')\leq \Theta_\ot(x,y')$,

* $M_\ot(x,x')\ot\Theta_\ot(x',y) \leq \Theta_\ot(x,y)$,

* $\Theta_\ot(x,y)\leq M_\op(x,x')\op\Theta_\ot(x',y)$,

* $\Theta_\ot(x,y)\leq\Theta_\ot(x,y')\op N_\op(y',y)$,

* a morphism $\Theta_\op(y,x): \rho_N(y)\rightarrow\rho_M(x)$ such that $\forall x, x'\in X, y, y'\in Y,$

* $\Theta_\op(y,x)\leq N_\op(y,y')\op\Theta_\op(y',x)$,

* $\Theta_\op(y,x)\leq\Theta_\op(y,x')\op M_\op(x',x)$,

* $N_\ot(y,y')\ot\Theta_\op(y',x)\leq\Theta_\op(y,x)$,

* $\Theta_\op(y,x')\ot M_\ot(x',x)\leq\Theta_\op(y,x)$.

We can now define a linear bicategory structure attached to the above data.

Let  be a linear quantaloid, then we define two distinct bicategorical structures:

* (, $\ot$, $\iota_M$) is a locally posetal bicategory whose 0-cells are linear -categories, 1-cells are linear -categories, and 2-cells are pointwise inequalities, in particular
$$\Theta\leq\Pi\Leftrightarrow~\forall (x,y)\in X\times Y, \Theta_\ot(x,y)\leq\Pi_\ot(x,y)~\textnormal{and}~\Pi_\op(y,x)\leq\Theta_\op(y,x)$$

Given $M\stackrel{\Theta}{\tov}N \stackrel{\Pi}{\tov}P$, composition $\Theta\ot\Pi:M\tov P$ is defined by

$$(\Theta\ot\Pi)_\ot(x,z)=\bigvee_{y\in Y}\Theta_\ot(x,y)\ot\Pi_\ot(y,z)~\textnormal{and}~(\Theta\ot\Pi)_\op(z,x)=\bigwedge_{y\in Y}\Pi_\op(z,y)\op\Theta_\op(y,x)$$

$\ot$-identity 1-cells are $\iota_M:M\tov M$ defined by

* ($\QMod^{co}, \op, \delta_M$) is also a locally posetal bicategory whose 0-cells are linear -categories, 1-cells are linear -categories, and 2-cells are pointwise inequalities. in particular
$$\Pi\leq\Theta\Leftrightarrow~\forall (x,y)\in X\times Y, \Theta_\ot(x,y)\leq\Pi_\ot(x,y)~\textnormal{and}~\Pi_\op(y,x)\leq\Theta_\op(y,x)$$

Given $M\stackrel{\Theta}{\tov}N \stackrel{\Pi}{\tov}P$, composition $\Theta\op\Pi:M\tov P$ is defined by

$$(\Theta\op\Pi)_\ot(x,z)=\bigwedge_{y\in Y}\Theta_\ot(x,y)\op\Pi_\ot(y,z)~\textnormal{and}~(\Theta\op\Pi)_\op(z,x)=\bigvee_{y\in Y}\Pi_\op(z,y)\ot\Theta_\op(y,x)$$

$\op$-identity 1-cells are $\delta_M:M\tov M$ defined by

We will break the demonstration that this determines a linear bicategory into several pieces.

(, $\ot$, $\iota_X$) as defined above is a quantaloid.

Since we are only considering the $\ot$ structure, this is a standard calculation.

$(\QMod^{co}, \op, \delta_X$) as defined above is a quantaloid.

And finally we can show:

$(\cQ,\ot,\op)$ is a linear quantaloid if and only if $(\QMod, \ot, \op)$ is a linear quantaloid and therefore a linear bicategory.

Suppose $(\cQ,\ot,\op)$ is a linear quantaloid. Consider -categories $L=(W, \rho_L),~ M=(X, \rho_M), ~N=(Y, \rho_N), ~P=(Z, \rho_P)$, and -bimodules $L\stackrel{\Psi}{\tov}M\stackrel{\Theta}{\tov}N \stackrel{\Pi}{\tov}P$.

We have the linear distributivity $\Psi\ot(\Theta\op\Pi)\leq(\Psi\ot\Theta)\op\Pi$ because:

\begin{align*}
(\Psi\ot(\Theta\op\Pi))_\ot(w,z) &=\bigvee_{x\in X}\Psi_\ot(w,x)\ot(\Theta\op\Pi)_\ot(x,z) \\
&= \bigvee_{x\in X}\Psi_\ot(w,x)\ot(\bigwedge_{y\in Y}\Theta_\ot(x,y)\op\Pi_\ot(y,z))
\end{align*}

\begin{align*}
((\Psi\ot\Theta)\op\Pi)_\ot(w,z) &=\bigwedge_{y\in Y}(\Psi\ot\Theta)_\ot(w,y)\op\Theta_\ot(y,z) \\
&= \bigwedge_{y\in Y}(\bigvee_{x\in X}\Psi_\ot(w,x)\ot\Theta_\ot(x,y))\op\Theta_\ot(y,z)
\end{align*}

Therefore $(\Psi\ot(\Theta\op\Pi))_\ot(w,z)\leq((\Psi\ot\Theta)\op\Pi)_\ot(w,z) \forall w,z$ if and only if $\forall w,x,y,z,$

$$ \Psi_\ot(w,x)\ot(\bigwedge_{y\in Y}\Theta_\ot(x,y)\op\Pi_\ot(y,z))\leq (\bigvee_{x\in X}\Psi_\ot(w,x)\ot\Theta_\ot(x,y))\op\Theta_\ot(y,z)$$

which holds as

\begin{align*}
\Psi_\ot(w,x)\ot(\bigwedge_{y\in Y}\Theta_\ot(x,y)\op\Pi_\ot(y,z))&\leq \Psi_\ot(w,x)\ot(\Theta_\ot(x,y)\op\Pi_\ot(y,z)) \\
&\leq (\Psi_\ot(w,x)\ot\Theta_\ot(x,y))\op\Pi_\ot(y,z) \\
&\leq (\bigvee_{x\in X}\Psi_\ot(w,x)\ot\Theta_\ot(x,y))\op\Theta_\ot(y,z)
\end{align*}

While, in the case of $\op$,

\begin{align*}
((\Psi\ot\Theta)\op\Pi)_\op(z,w) &=\bigvee_{y\in Y}\Pi_\op(z,y)\ot(\Psi\ot\Theta)_\op(y,w) \\
&=\bigvee_{y\in Y}\Pi_\op(z,y)\ot(\bigwedge_{x\in X}\Theta_\op(y,x)\op\Psi_\op(x,w))
\end{align*}

\begin{align*}
(\Psi\ot(\Theta\op\Pi))_\op(z,w) &=\bigwedge_{x\in X}(\Theta\op\Pi)_\op(z,x)\op\Psi_\op(x,w) \\
&=\bigwedge_{x\in X}(\bigvee_{y\in Y}\Pi_\op(z,y)\ot\Theta_\op(y,x))\op\Psi_\op(x,w)
\end{align*}

So $((\Psi\ot\Theta)\op\Pi)_\op(z,w)\leq(\Psi\ot(\Theta\op\Pi))_\op(z,w), \forall w,x$ if and only if $\forall w,x,y,z$,

$$\Pi_\op(z,y)\ot(\bigwedge_{x\in X}\Theta_\op(y,x)\op\Psi_\op(x,w))\leq(\bigvee_{y\in Y}\Pi_\op(z,y)\ot\Theta_\op(y,x))\op\Psi_\op(x,w) $$

which is true as

\begin{align*}
\Pi_\op(z,y)\ot(\bigwedge_{x\in X}\Theta_\op(y,x)\op\Psi_\op(x,w))&\leq \Pi_\op(z,y)\ot(\Theta_\op(y,x)\op\Psi_\op(x,w)) \\
&\leq ( \Pi_\op(z,y)\ot\Theta_\op(y,x))\op\Psi_\op(x,w) \\
&\leq (\bigvee_{y\in Y}\Pi_\op(z,y)\ot\Theta_\op(y,x))\op\Psi_\op(x,w)
\end{align*}

A similar calculation gives us the other linear distributivity $(\Psi\op\Theta)\ot\Pi\leq\Psi\op(\Theta\ot\Pi).$

The backwards direction follows from the next main result (<ref>).

We now introduce the notion of linear monad as a way of providing further examples.

* Let  be a linear quantaloid. A linear monad $(a,m)$ in  is an object $a$ in  equipped with compatible $\ot$-monad $(a, m_\ot)$ and $\op$-comonad $(a, m_\op)$. Specifically, there is a morphism $m_\ot:a\rightarrow a$ such that $\top_a\leq m_\ot$ and $m_\ot \ot m_\ot \leq m_\ot$, and a morphism $m_\op:a\rightarrow a$ such that $m_\op\leq\bot_a$ and $m_\op\leq m_\op \op m_\op$, satisfying the following inequalities:

* $m_\ot \leq m_\op \op m_\ot$,

* $m_\ot \leq m_\ot \op m_\op$,

* $m_\ot \ot m_\op \leq m_\op$,

* $m_\op \ot m_\ot \leq m_\op$.

* Let $(a,m)$ and $(b, n)$ be linear monads in . A linear $(m,n)$-bimodule (or monad bimodule) $f:(a,m)\tov(b,n)$ is a pair of morphisms $(f_\ot, f_\op)$ where $f_\ot:a\rightarrow b$ satisfies $\forall x, x'\in X, y, y'\in Y$,

* $f_\ot \ot n_\ot\leq f_\ot$,

* $m_\ot\ot f_\ot\leq f_\ot$

* $f_\ot\leq m_\op\op f_\ot$,

* $f_\ot \leq f_\ot \op n_\op$,

while $f_\op:b\rightarrow a$ satisfies $\forall x, x'\in X, y, y'\in Y$,

* $f_\op\leq n_\op\op f_\op$,

* $f_\op\leq f_\op \op m_\op$,

* $n_\ot \ot f_\op\leq f_\op$,

* $f_\op \ot m_\ot\leq f_\op$.

A linear monad $(a,m)$ in  is a linear functor from $1$ to , where $1$ is the suspended one-object linearly distributive category.

We can define a linear bicategory similarly using the above in the same way as with .

Let  be a linear quantaloid, then we define two distinct bicategorical structures:

* ($\MonQ, \ot, \top_{(a, m)}$) is a locally posetal bicategory whose 0-cells are linear monads in , 1-cells are linear monad bimodules in , and 2-cells are pointwise inequalities, in particular
$$f\leq g\iff f_\ot\leq g_\ot~\textnormal{and}~f_\op\leq g_\op$$

Given $(a, m)\stackrel{f}{\tov}(b, n) \stackrel{g}{\tov}(c, p)$, composition $f\ot g:(a,m)\tov(c,p)$ is defined by

$$(f\ot g)_\ot=f_\ot \ot g_\ot~\textnormal{and}~(f\ot g)_\op=f_\op \op g_\op$$

$\ot$-identity 1-cells are $\top_{(a,m)}:(a,m)\tov (a,m)$ defined by

* ($\MonQ^{co}, \op, \bot_{(a,m)}$) has the same 0-cells and 1-cells as given above, and 2-cells are pointwise inequalities, in particular
$$g\leq f\iff f_\ot\leq g_\ot~\textnormal{and}~f_\op\leq g_\op$$

Given $(a, m)\stackrel{f}{\tov}(b, n) \stackrel{g}{\tov}(c, p)$, composition $f\op g:(a,m)\tov(c,p)$ is defined by

$$(f\op g)_\ot=f_\ot \op g_\ot~\textnormal{and}~(f\op g)_\op=f_\op \ot g_\op$$

$\ot$-identity 1-cells are $\bot_{(a,m)}:(a,m)\tov (a,m)$ defined by

We have:

$(\MonQ, \ot,\top_{(a,m)})$ is a sub-bicategory of (, $\ot$, $\iota_X$) and a quantaloid.
$(\MonQ^{co},\op,\bot_{(a,m)})$ is a sub-bicategory of ($\QMod^{co}, \op, \delta_X$) and a quantaloid.

By construction, the objects of  are the subset of singleton set linear -categories, i.e. they are a subset of the objects of . Indeed, consider any singleton set linear -category, say $M=(\{x_0\}, \rho_M)$ This defines a linear monad $(a, m)$ where $a=\rho_M(x_0), m_\ot=M_\ot(x_0,x_0)$ and $m_\op=M_\op(x_0,x_0)$. Further, given two singleton set linear -categories $M=(\{x_0\}, \rho_M)$ and $N=(\{y_0\}, \rho_Y)$, any linear -bimodule $\Theta:M\tov N$ is a linear monad bimodule $f:(a,m)\tov(b,n)$, where $\Theta_\ot(x_0,y_0) = f_\ot$ and $\Theta_\op(y_0,x_0)=f_\op$. Thus, $\MonQ((a,m),(b,n))=\QMod(M,N)$. This implies that $(\MonQ\ \ot, \top_{(a,m)})$ is a sub-bicategory of $(\QMod, \ot, \iota_M)$ and further that $(\MonQ\ \ot, \top_{(a,m)})$ is a quantaloid, and similarly for the $\op$ bicategorical structure.

Then we get this result:

$(\cQ,\ot,\op)$ is a linear quantaloid if and only if $(\MonQ, \ot, \op)$ is a linear quantaloid and therefore a linear bicategory.

The forward direction is immediate from the above observations.

Suppose $(\MonQ, \ot, \op)$ is a linear quantaloid. For a 0-cell $a$ in , consider $(a, 1_a)$ in  where $1_a$ represents a pair of morphisms $1_{a\ot}=\top_a$ and $1_{a\op}=\bot_a$ in . It is immediate that this is a linear monad in . Consider 1-cell $f\colon a\rightarrow b$ in . Then, the pair of morphisms $F=(f, f -\!\!\!\!-\!\!\circ \top_a)\colon (a,1_a)\tov (b, 1_b)$ defines a linear $(1_a, 1_b)$-bimodule. Note that the choice $f-\!\!\!\!-\!\!\circ\top_a$ is by no means unique as the conditions for a linear monad bimodule are satisfied trivially.

$\ot$ is associative in , as they are in :

\begin{align*}
f\ot(g\ot h) = (f\ot g)\ot h &\iff  F_\ot\ot(G_\ot\ot H_\ot)=(F_\ot\ot G_\ot)\ot H_\ot \\
&\iff (F\ot(G\ot H))_\ot = ((F\ot G)\ot H)_\ot
\end{align*}

Similarly, $\op$ in  is associative.

$(\cQ, \ox, \top_a)$ is a quantaloid as $(\MonQ, \ot,\top_{(a,m)})$ is one:

\begin{align*}
f\ot\bigvee_i g_i &= F_\ot\ot\bigvee_i G_{i\ot} = F_\ot\ot(\bigvee_i G_i)_\ot = (F\ot\bigvee_i G_i)_\ot \\
&= \bigvee_i (F\ot G_i)_\ot = \bigvee_i F_\ot\ot G_{i\ot} = \bigvee_i f\ot g_i
\end{align*}

It also follows that $(\cQ^{co},\op,\bot_a)$ is also a quantaloid.

Finally,  we have:

\begin{align*}
f\ot(g\op h)\leq(f\ot g)\op h &\iff F_\ot\ot(G_\ot\op H_\ot)\leq(F_\ot\ot G_\ot)\op H_\ot \\
&\iff F_\ot\ot(G\op H)_\ot\leq(F\ot G)_\ot\op H_\ot \\
&\iff (F\ot (G\op H))_\ot\leq ((F\ot G)\op H))_\ot
\end{align*}

The latter holds as  is a linear quantaloid. Thus the first linear distributivity holds in   as does the second similarly. Therefore $(\cQ,\ot,\op)$ is a linear quantaloid.

§.§ Structures in ,  and

It is evident that a 1-cell $A\colon X\tov Y$ in  has a linear right adjoint $B\colon Y\tov X$, as defined in Definition <ref>,
if and only if

* $\forall x,y \,\, \top_X\leq A(x,y)\op B(y,x)$

* $\forall x,y \,\, A(x,y)\ox B(y,x)\leq \perp_X$

From this, we readily conclude

If $Q$ is a Girard quantale, all 1-cells $A(x,y)$ in $Q$ have right and left adjoints, given by

The category  is a closed linear bicategory when $Q$ is a Girard quantale.

For more general -quantales, this need not be the case.

In   $\Theta\colon M\tov N$ is the left linear adjoint of $\Pi\colon M\tov N$ if and only if

* $\forall x, x'\in X, y\in Y \,\, M_\ot(x,x')\leq \Theta_\ot(x, y)\op \Pi_\ot(y,x')$ and $\Pi_\op(x',y)\ot\Theta_\op(y,x)\leq M_\op(x',x)$

* $\forall y, y'\in Y, \,\exists x, x'\in X$    $\Pi_\ot(y,x)\ot\Theta_\ot(x,y')\leq N_\op(y',y)$ and $N_\ot(y,y')\leq\Theta_\op(y, x')\op\Pi_\op(x',y')$

As above, we can see that

If  is a Girard quantaloid, all 1-cells $\Theta\colon M\tov N$ in  have a linear right adjoint $\Theta^\perp\colon N\tov M$. Thus  is a closed linear bicategory.

Consider a linear -bimodule $\Theta\colon M\tov N$, then define $\Theta^\perp\colon N\tov M$ by

It is immediate that this is a well-defined linear -bimodule from the observations at the start of Section 4. Further, notice that $M_\op(x',x)=M_\ot(x,x')^\perp$.

\begin{align*}
M_\ot(x,x')\leq \Theta_\ot(x, y)\op \Theta^\perp_\ot(y,x') &\iff M_\ot(x,x')\leq \Theta_\ot(x,y)^\perp -\!\!\!\!-\!\!\circ \Theta_\ot(x',y)^\perp \\
&\iff \Theta_\ot(x,y)^\perp \ot M_\ot(x,x')\leq \Theta_\ot(x',y)^\perp \\
\end{align*}

The last inequality holds as $\Theta(x,y)^\perp$ defines a standard -bimodule.

\begin{align*}
\Theta^\perp_\op(x',y)\ot\Theta_\op(y,x)\leq M_\op(x',x) & \iff \Theta_\ot(x',y)\ot \Theta_\ot(x,y)^\perp\leq M_\ot(x,x')^\perp \\
&\iff M_\ot(x,x')\leq \Theta_\ot(x,y)\op\Theta_\ot(x',y)^\perp
\end{align*}

This holds by above.

The other conditions follow similarly.

If  is a Girard quantaloid, all 1-cells $f\colon (a,m)\tov (b,n)$ have a linear right adjoint $f^\perp\colon (b,n)\tov (a,m)$. Thus  is a closed linear bicategory.

and  need not be closed for other linear quantaloids.

§ NON-LOCALLY POSETAL EXAMPLES

In this section, we present examples of linear bicategories which are not locally partially ordered. It is a special case of a
general result (Theorem <ref>)  yielding examples of non-posetal linear bicategories. We generalize this result in the next
section. We present these results in terms of locales as that is the setting in which they were originally introduced in citeJT. But essentailly everything here works in the broader setting of quantales.

If $X$ and $Y$ are locales, then an $(X,Y)$-bimodule is a sup lattice $A$ which is a left $X$-module and a right
$Y$-module satisfying $x(ay)=(xa)y$, and often written as $A\colon X\tov Y$. Note that if $B$ is a $(W,Y)$-module, then the sup
lattice ${\rm Mod}Y(A,B)$ of right $Y$-module homomorphisms becomes a $(W,X)$-module via  $(wf)(a)=wf(a)$ and $(fx)
(a)=f(xa)$. Dually,  $X{\rm Mod}(A,C)$ is  a $(Y,Z)$-module, if $C$ is a left $(X,Z)$-bimodule. A function $f\colon A\to<125>A'$
is a bimodule homomorphism if it is a left  $X$-module and right $Y$-module homomorphism.

Suppose $A$ is an $(X,Y)$-bimodule. Then the opposite lattice $A^\circ$ becomes a $(Y,X)$-bimodule as follows. Given $x\in
X$, the function $x\cdot - \colon A\to<125> A$ is sup-preserving map, and hence, has a right adjoint $- /x$. Thus, $A^\circ$
becomes a right $X$-module via $(a,x)\mapsto a/x$ and a left $Y$-module via $(y,a)\mapsto r\backslash a$, where $y\backslash
-$ is right adjoint to $-\cdot y$, and $(A^\circ)^\circ\cong A$.

There is a bicategory whose objects are locales, 1-cells bimodules, and 2-cells bimodule
homomorphisms. Composition of  $A\colon X\tov Y$ and $B\colon Y\tov Z$ is given by

$$A\ox B=A\otimes_Y B\cong {\rm Mod}Y(A,B^\circ)^\circ\cong  Y{\rm Mod}(B,A^\circ)^\circ$$

with identity 1-cells $X\colon X\tov X$. We denote this bicategory by $\biloc$.

Now, we note that $\biloc$ admits another composition of 1-cells

$$A\parr B=(B^\circ \ox A^\circ)^\circ = (B^\circ\otimes_Y A^\circ)^\circ\cong Y{\rm Mod}(A^\circ,B)$$

with identity 1-cells $X^\circ\colon X\tov X$,  since $X^\circ\parr A  \cong A\cong A\parr Y^\circ$ and

$$A\parr (B\parr C) =  ((C^\circ\otimes B^\circ)\otimes  A^\circ)^\circ
\cong  (C^\circ\otimes (B^\circ\otimes  A^\circ))^\circ= (A\parr B)\parr C$$

for all $C\colon Z\tov W$. To see that $\biloc$ is a linear bicategory, we will define

$$A\otimes_Y(B\parr C)\to<125> (A\otimes_YB)\parr C$$

Since $A\otimes_Y(B\parr C)\cong A\otimes_Y Z{\rm Mod}(B^\circ,C)$ and $(A\otimes_YB)\parr C \cong  Z{\rm Mod}({\rm Mod}
Y(A,B^\circ),C)$, it suffices to define a 2-cell

$$A\otimes_Y Z{\rm Mod}(B^\circ,C)\to<125>Z{\rm Mod}({\rm Mod}Y(A,B^\circ),C)$$

or equivalently,

$$({\rm Mod}Y(A,B^\circ)\otimes_XA)\otimes_Y Z{\rm Mod}(B^\circ,C)\to<125>C$$

For this, we can use the evaluation maps
$$({\rm Mod}Y(A,B^\circ)\otimes_XA)\otimes_Y Z{\rm Mod}(B^\circ,C)\to<125> B^\circ \otimes_Y Z{\rm Mod}(B^\circ,C)\to<125>C$$

Similarly, we get a 2-cell $(A\parr B)\otimes_Z C\to<125> A\parr (B\otimes_ZC)$, as desired.

Under the above operations, $\biloc$ is a linear bicategory.

§.§ Girard bicategories

See the appendix for examples  and bicategorical structures discussed in this section.

The notion of a $*$-autonomous category was originally introduced independently of linear logic in  [2] by M. Barr, who
was trying to capture some of the dualities present in various categories of topological vector spaces. It was only later seen
to be appropriate to model the multiplicative fragment of linear logic. Barr's definition of $*$-autonomous category was as a
symmetric monoidal closed category with a dualizing object. But linear logic suggested an equivalent definition as a linearly
distributive category with negation, see [4]. We choose to work here with the bicategorical version of Barr's original
definition, and we will thus define a Girard bicategory to be a biclosed bicategory with a dualizing family of 1-cells.
This is also in line with the definition of Girard quantaloid to be found in [19].

Suppose  ${\cal D}=\{{D_X}\colon X\to<125> X \, \vert \, X \in \cal B\}$ is a family of 1-cells in a  biclosed bicategory $\cal B$.
Given $A\colon X\to<125> Y$, applying Definition <ref>, we get

$$A  \to<220>^{\delta_{A,Y}} Y{\rm Mod}({\rm Mod}Y(A,D_Y),D_Y) \quad\quad {\rm and} \quad \quad A \to<220>^{\delta_{X,A}}
{\rm Mod}X(X{\rm Mod}(A,D_X),D_X)$$

which we also write as

$$A\,  \to<220>^{\delta_{A,Y}} \, (D_Y \circ\!\!-\!\!\!\!- A)  -\!\!\!\!-\!\!\circ D_Y \quad\quad {\rm and} \quad \quad A \,
\to<220>^{\delta_{X,A}} \, D_X \circ\!\!-\!\!\!\!-  (A  -\!\!\!\!-\!\!\circ D_X)$$

The following concept was introduced in [6] as cyclic $*$-autonomous bicategory. We choose the shorter
name Girard bicategory in line with previously defined notions.

A family ${\cal D}=\{{D_X}\colon X\to<125> X \, \vert \, X \in \cal B\}$   is called  dualizing  if the 2-cell
\to<220>^{\delta_{A,Y}}(D_Y \circ\!\!-\!\!\!\!- A)  -\!\!\!\!-\!\!\circ D_Y$$
is invertible,  for all $A\colon X\to<125>Y$.
A dualizing family  ${\cal D}$   is called cyclic  if there are invertible 2-cells $\theta_A\colon (D_Y  \circ\!\!-\!\!\!\!-  A) \to<125> (A -\!
\!\!\!-\!\!\circ  D_X)$, natural in  $A\colon X\to<125>Y$,  such that  the following diagram commutes

\Ctriangle/<-``->/<600,300>[\qquad \qquad D_X \circ\!\!-\!\!\!\!-  (A  -\!\!\!\!-\!\!\circ D_X)`A`\qquad\qquad(D_Y \circ\!\!-\!\!\!\!- A)  -\!\!
\!\!-\!\!\circ D_Y;\delta_{X,A}``\delta_{A,Y}]
\morphism(600,600)|r|<0,-300>[\qquad \qquad D_X \circ\!\!-\!\!\!\!-  (A  -\!\!\!\!-\!\!\circ D_X)`\qquad \qquad D_X \circ\!\!-\!\!\!\!-
(D_Y \circ\!\!-\!\!\!\!-  A);D_X  \circ\!\!-\!\!\!\!-\theta_A]
\morphism(600,300)|r|<0,-300>[\qquad \qquad D_X \circ\!\!-\!\!\!\!-  (D_Y \circ\!\!-\!\!\!\!-  A)`\qquad\qquad(D_Y \circ\!\!-\!\!\!\!- A)  -\!
\!\!\!-\!\!\circ D_Y;\theta_{D_Y \circ\!\!-\!\!\!\!- A}]
\efig$$

In this case, we let $A^\perp=A -\!\!\!\!-\!\!\circ  D_X$.

$\cal B$ is a Girard bicategory if it admits a cyclic dualizing family.

We now give several general examples of Girard bicategories to illustrate the application of the definition.

Suppose $\cal V$ is a $*$-autonomous category,
one can show that  ${\cal D}=\{ \perp\colon\star \to<125> \star \}$ is a cyclic dualizing family for $\cal B(V)$, since $(A  -\!\!\!\!-
\!\!\circ \perp) \cong A^\perp\cong (\perp\circ\!\!-\!\!\!\!- A)$ and $A\cong (A^\perp)^\perp$, for all $A$, and so
$\cal B(V)$  is a  Girard
In particular, when  ${\cal V}$ is the category of sup lattices,  ${\cal B}({\rm Sup})$   is a  Girard bicategory with cyclic dualizing
${\cal D}=\{\mathsf{1}\colon\star \to<125> \star \}$. Moreover, $( A-\!\!\!\!-\!\!\circ \mathsf{1} ) \cong A^\circ\cong
(\mathsf{1} \circ\!\!-\!\!\!\!- A)$  $A\cong (A^\circ)^\circ$, for all $A$.

Consider $\vmat$, where $\cal V$ is a star-autonomous category with set-indexed products and coproducts. Given a set $X$, define
$D_X\colon X\tov X$  by

\begin{cases}
\bot & x=x' \\
\top & \text{otherwise}\\
\end{cases}$$

Taking $A^\perp(y,x)=A(x,y)^\perp$, for $A\colon X\tov Y$, one can show that $(D_Y  -\!\!\!\!-\!\!\circ A) \cong A^\perp\cong (A\circ\!\!-\!\!\!\!- D_X)$ and $A\cong (A^\perp)^\perp$,  and so $\vmat$ is a Girard bicategory with   cyclic dualizing family ${\cal D}=\{D_X\colon X\tov X \}$.

Suppose $\cal B$ is a Girard bicategory with local equalizers and coequalizers stable under composition. Then $\bimod(\cal B)$ is a Girard bicategory.

Suppose ${\cal D}=\{{D_X}\colon X\to<125> X \, \vert \, X \in \cal B\}$  is acyclic dualizing family for $\cal B$, and let
$${\cal D}^\perp=\{\, X{\hbox{$\to<150>^{Q^\perp}$\hskip -.2in \raisebox{1.8pt}{\tiny$\bullet$} \hskip .1in}}  X \ \vert \, Q \in \bimod(\cal B)\}$$
where $Q\colon  X\tov X$ and $Q^\perp=Q -\!\!\!\!-\!\!\circ  D_X=X{\rm Mod}(Q,D_X)$.

Given $A\colon Q\tov R$, applying Proposition <ref>, we see that
$$Q{\rm Mod}(A,Q^\perp) = Q{\rm Mod}(A,X{\rm Mod}(Q,D_X))\cong X{\rm Mod}(A,D_X)=A^\perp$$
$${\rm Mod}R(A,R^\perp) \cong {\rm Mod}R(A,{\rm Mod}Y(R,D_Y))\cong {\rm Mod}Y (A,D_Y)\cong A^\perp$$

Since all these 2-cells are natural, $\cal D^\perp$ is a cyclic dualizing family for $\bimod(\cal B)$.

Applying Proposition <ref> to Examples <ref>, <ref>, <ref>, <ref>, we see that:

If $\cal V$ is a star-autonomous category with equalizers and coequalizers stable under composition,
then $\bimod({\cal V})$ is a Girard bicategory. In particular, $\biquant$ and $\biloc$ are.

If $\cal V$ is a complete and cocomplete  star-autonomous category, then $ \cal V$-$\prof$ is a  Girard  bicategory. In particular,  $\qtld$ is.

Following an argument analogous to the one for $\biloc$ in Section <ref>, we will see that every Girard bicategory is inear. To do so, we use the following lemma.
If  $\cal B$  is a Girard bicategory, then there is an invertible 2-cell
$$Y{\rm Mod}(A^\perp,B)\to<125> (B^\perp\!\otimes A^\perp)^\perp$$
for all $A\colon X\to<125> Y$ and $B\colon Y\to<125>Z$.

Suppose ${\cal D}=\{{D_X}\colon X\to<125> X \,\}$  is a cyclic dualizing family for $\cal B$.  Then composition with $\delta_{B,Z}$ induces an invertible 2-cell

$$Y{\rm Mod}(A^\perp,B) \to<125> Y{\rm Mod}(A^\perp,(D_Z \circ\!\!-\!\!\!\!- B)  -\!\!\!\!-\!\!\circ D_Z)$$

Since $D_Z \circ\!\!-\!\!\!\!- B \cong  B ^\perp$, it follows that
$$Y{\rm Mod}(A^\perp,(D_Z \circ\!\!-\!\!\!\!- B)  -\!\!\!\!-\!\!\circ D_Z)\cong Y{\rm Mod}(B^\perp\!\otimes A^\perp,D_Z)\cong (B^\perp\!\otimes A^\perp)^\perp$$
and the desired result follows.

Every Girard bicategory is a linear bicategory.

Suppose ${\cal D}=\{{D_X}\colon X\to<125> X \,\}$  is a cyclic dualizing family for $\cal B$. Given  $A\colon X\to<125> Y$ and $B\colon Y\to<125>Z$, define  $A\parr B= (B^\perp\otimes A^\perp)^\perp$.

Then $\parr$ is associative and has identity 1-cells $X^\perp\colon X\to<125> X$, since

\begin{align*}
A\parr (B\parr C) &= A\parr (C^\perp\otimes  B^\perp)^\perp \cong  ((C^\perp\otimes  B^\perp)\otimes A^\perp)^\perp  \\
&\cong(C^\perp\otimes  (B^\perp\otimes A^\perp))^\perp \cong    (C^\perp\otimes  (A\parr B)^\perp)^\perp \\
&= (A\parr B)\parr C
\end{align*}

and  $X^\perp\parr A  \cong A\cong A\parr Y^\perp$.

To see that $\cal B$ is a linear bicategory, we will define
$$A\otimes(B\parr C)\to<125> (A\otimes B)\parr C$$
Since $A\otimes (B\parr C)\cong A\otimes  Z{\rm Mod}(B^\perp\!,C)$ and $(A\otimes B)\parr C \cong  Z{\rm Mod}({\rm Mod}Y(A,B^\perp\!),C)$, by Lemma <ref>; it suffices to define a 2-cell
$$A\otimes Z{\rm Mod}(B^\perp\!,C)\to<125>Z{\rm Mod}({\rm Mod}Y(A,B^\perp\!),C)$$
or equivalently,
$$({\rm Mod}Y(A,B^\perp\!)\otimes A)\otimes Z{\rm Mod}(B^\perp\!,C)\to<125>C$$
For this, we can use the evaluation maps
$$({\rm Mod}Y(A,B^\perp\!)\otimes A)\otimes Z{\rm Mod}(B^\perp\!,C)\to<125> B^\perp\! \otimes Z{\rm Mod}(B^\perp\!,C)\to<125>C$$

Similarly, we get a 2-cell $(A\parr B)\otimes C\to<125> A\parr (B\otimes C)$, as desired.

From this, we can claim the following examples.

* If $\cal V$ is a star-autonomous category, then ${\cal B}(\cal V)$ is a linear bicategory. If, in addition, $\cal V$ has equalizers and coequalizers, then so is $\bimod({\cal V})$. In particular, $\biquant$ and $\biloc$ are linear bicategories.

* If $\cal V$ is a star-autonomous category with set-indexed products and coproducts, then $\vmat$ is a linear bicategory. If, in addition, $\cal V$ has equalizers and coequalizers, then so is $\cal V$-$\prof$.  In particular,  $\qtld$ is a linear bicategory.

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§ APPENDIX: BICLOSED BICATEGORIES

In this section we review the notion of a biclosed bicategory $\cal B$.
We recall the definitions of  right extensions and right liftings (in the  sense of Street [20]) and
we present the examples of
biclosed categories which give us non-posetal examples of linear bicategories. Note that, since composition of 1-cells
$A\colon X\to<125>Y$ and $B\colon  Y\to<125>Z$ is often given by a tensor product in our examples (e.g., <ref>), we
write $A\otimes B$  for the composition.

A right extension of  $B\colon X\to<125> Z$  along $A\colon X\to<125> Y$ in $\cal B$  is a 1-cell which we denote $X{\rm Mod}(A,B)$, also
denoted by
$A -\!\!\!\!-\!\!\circ B$, together with a 2-cell
\ptriangle/->`->`-->/<400,400>[X`Y`Z;A`B`X{\rm Mod}(A,B)]
\morphism(220,250)<-180,0>[`;\varepsilon_{X,A}]
\efig$$

inducing a bijection between 2-cells $C\to<125> X{\rm Mod}(A,B)$ and $A\otimes C \to<125>B$, for all $C\colon Y\to<125> Z$.

A right lifting of $C\colon Z\to<125> Y$  through $A\colon X\to<125> Y$ is a right extension in ${\cal B}^{op}$, i.e., a 1-cell which we denote
${\rm Mod}Y(A,C)$ or $C \circ\!\!-\!\!\!\!- A$, together with a 2-cell

\dtriangle/<--`->`->/<400,400>[X`Z`Y;{\rm Mod}Y(A,C)`A`C]
\morphism(340,240)|l|<0,-170>[`;\varepsilon_{C,Y}]
\efig$$

inducing a  a bijection between 2-cells $B\to<125> {\rm Mod}Y (A,C)$ and $B\otimes A \to<125> C$, for all $B\colon Z\to<125> X$.

A bicategory $\cal B$ is called biclosed if it admits all right extensions and right liftings.

We have chosen to use the more traditional notation and terminology from bicategory theory. The authors of
[6] chose their terminology to be in agreement with Lambek's non-commutative linear logic rather than the theory of
bicategories. We note that our notion of right extension is the same as their notion of right hom and our notion of right lifting
is their left hom.

If $\cal B$ is a biclosed bicategory, then the induced 2-cells
$$X{\rm Mod}(A\otimes C,B)\to<125>Y{\rm Mod}(C,X{\rm Mod}(A,B))$$
are invertible and natural in $A\colon X\to<125> Y$,  $B\colon X\to<125> Z$,  and $C\colon Y\to<125> Z$.

This follows from the universal properties of the extensions and liftings.

Suppose $\cal V$ is a symmetric monoidal closed category, and let $\cal B(\cal V)$ denote the bicategory with one object $\star$ whose 1-cells  and 2-cells are the objects and morphisms of $\cal V$, respectively, with composition $\otimes$ and $\rm id_\star$ given by the tensor  and  unit $\top$ object of $\cal V$. Taking $A -\!\!\!\!-\!\!\circ B = {\rm Hom}(A,B)$ and $C  \circ\!\!-\!\!\!\!-  A =   {\rm Hom}(A,C)$, we get:

$\cal B(\cal V)$ is a biclosed bicategory.

For some of the details of this result, see [15]. We now discuss several important examples of biclosed bicategories.

Suppose $\cal V$ is a symmetric monoidal closed category with set-indexed products and coproducts. Recall [5] that a  $\cal V$-matrix  from a set $X$ to a set $Y$ is a function $A\colon X\times Y\to<125> |\cal V|$, also denoted by $A\colon X\tov Y$. A morphism $f\colon A\to<125> B$ of  $\cal V$-matrices $X\tov Y$ is a family
$$\{f (x,y) \colon A(x,y)\to<125>B(x,y) \mid (x,y)\in X\times Y\}$$
of morphisms of $\cal V$.  Let  $\vmat$ denote the bicategory of sets, $\cal V$-matrices, and  $\cal V$-matrix morphisms, where the composition of $A\colon X\tov Y$ and $B\colon Y\tov Z$ is given by matrix multiplication
$$(A\otimes_Y B)(x,z)=\coprod_y A(x,y)\otimes B(y,z)$$
the identity $X\tov X$ is

$${\rm id}_X(x,x')=
\begin{cases}
\top& x=x' \\
\qzero & \text{otherwise}\\
\end{cases}$$

where $\top$ is the unit for $\otimes$ and  $\qzero$ is initial in $\cal V$.

Given  $A\colon X\tov Y$, $B\colon X\tov Z$,  and   $C\colon Z \tov Y$ in $\vmat$, taking

$$X{\rm Mod}(A,B)(y,z)= \prod_x  \cV(A(x,y),B(x,z)) $$

$${\rm Mod}Y(A,C)(z,x)= \prod_y  \cV(A(x,y),C(z,y)) $$
we get:

If $\cal V$ is  symmetric monoidal closed category with set-indexed products and coproducts, then $\vmat$ is a biclosed bicategory.

Again, for some of the details of this result, see [15].

Suppose $\cal B$ is a biclosed bicategory with local equalizers and coequalizers stable under composition. Recall [5]
that a monad in $\cal B$ is a 1-cell $Q\colon X\to<125>X$ together with two 2-cells $e\colon {\rm id}_X\to<125> Q$ and
$m\colon Q\otimes Q \to<125> Q$  satisfying the usual associativity and identity axioms. Note that a monad $(X,Q)$ (denoted
by just $Q$ when $X$ is understood)  is  a monoid in the category ${\cal B}(X,X)$.
A $(Q,R)$-bimodule  $(X,Q) \tov (Y,R)$ is a 1-cell $A\colon X\to<125> Y$ together with 2-cells

\square<450,400>[X`X`Y`Y;Q`A`A`R]
\morphism(450,400)|m|<-450,-400>[X`Y;A]
\morphism(220,275)<-150,00>[`;\lambda]
\morphism(220,100)<150,0>[`;\rho]
\efig$$

(one-sided) associative and unit laws, and such that the  diagram

\square<600,400>[Q\otimes A\otimes R`Q\otimes A`A\otimes R`A;Q\otimes \rho`\lambda\otimes R`\lambda`\rho]
\efig$$

commutes. A 2-cell $f\colon A\to<125> B$ is a homomorphism of $(Q,R)$-bimodules if the diagrams

\square<450,400>[Q\otimes A`A`Q\otimes B`B;\lambda_A`Q\otimes f`f`\lambda_B]
\square(1400,0)<450,400>[A\otimes R`A`B\otimes R`B;\rho_A`f\otimes R`f`\rho_B]
\efig$$

Let $\bimod({\cal B})$ denote the bicategory of monads, bimodules, and homomorphisms; with the composition of $A\colon (X,Q) \tov (Y,R)$ and $B\colon (Y,R)\tov (Z,S)$ given by the coequalizer
$$A\otimes R \otimes B\two ^{\rho_A\otimes B}_{A\otimes \lambda_B}A \otimes B\to A\otimes_RB$$
in  ${\cal B}(X,Z)$, and  identity 1-cell  $Q\colon  (X,Q)\tov  (X,Q)$.

Note that ${\rm id}_X\colon X\to<125> X$ is a monad, every arrow $A\colon X\to<125> Y$ is an $({\rm id}_X,{\rm id}_Y)$-bimodule, and every  $f\colon A\to<125> B$ is a $({\rm id}_X,{\rm id}_Y)$-bimodule homomorphism. Moreover, $X\mapsto {\rm id}_X$ defines a pseudo functor which is a left pseudo adjoint to the forgetful (lax) functor $\bimod({\cal B})\to<125>{\cal B}$.

Given  $A\colon (X,Q)\tov (Y,R)$, $B\colon (X,Q)\tov (Z,S)$,  and   $C\colon (Z,S) \tov (Y,R)$, taking the local coequalizers

$$Q{\rm Mod}(A,B)\to<125>X{\rm Mod}(A,B)\two ^{\lambda^*_A}_{\hat \lambda_B} X{\rm Mod}(Q\otimes A,B)$$
where $\lambda^*_A = X{\rm Mod}(\lambda_A,B)$ and $\hat \lambda_B$ is the transpose of
$$Q\otimes A\otimes X{\rm Mod}(A,B)\to<180>^{Q\otimes \varepsilon} Q\otimes B\to<180>^{\lambda_B}B$$

and similarly,
$${\rm Mod}R(A,C)\to<125>{\rm Mod}Y(A,C)\two ^{\rho_A}_{\hat \rho_B} {\rm Mod}Y(A\otimes R,C\otimes R)$$

Note that, given $A\colon X\to<125>Y$ in $\cal B$, the  1-cells
$${\rm id}_X{\rm Mod}(A,B)\to<125>X{\rm Mod}(A,B) \quad  {\rm and} \quad {\rm Mod}\,{\rm id}_Y(A,B)\to<125>{\rm Mod}Y(A,B)$$
defining the extensions and liftings in $\bimod({\cal B})$, are invertible, and without loss of generality, we can take them to be equalities.

Moreover, one can show the following. For further details, see [15].

If $\cal B$ is a biclosed bicategory with local equalizers and coequalizers stable under composition, then $\bimod({\cal B})$  is a biclosed bicategory.

Suppose $\cal V$ is a symmetric monoidal closed category with equalizers and coequalizers. Then the bicategory $\bimod(\cal B(V))$, whose objects are monoids in $\cal V$ and also denoted by  $\bimod({\cal V})$, is a biclosed bicategory.  In particular, taking $\cal V=\rm Sup$, we see that the bicategory $\biquant$, of quantale, bimodules, and bimodule homomorphsims is biclosed, as is its full sub-bicategory $\biloc$ considered in Section <ref>.

Suppose $\cal V$ is a complete and cocomplete symmetric monoidal closed category. Then $\bimod(\vmat)= \cal V$-$\prof$, the bicategory of $\cal V$-categories, $\cal V$-profunctors, and $\cal V$-transformations, and so $\cal V$-$\prof$ is a biclosed bicategory.  In particular, taking $\cal V=\rm Sup$, we see that the bicategory $\qtld$ of small quantaloids, profunctors, and transformations is a biclosed bicategory.

Note that composition $A\otimes_RB$ of $A\colon Q\tov R$ and $B\colon R \tov S$ can also be described by the coend
$$(A\otimes_R B)(q,s)=\int^r A(q,r)\otimes B(r,s)$$
and $Q( -, -)\colon Q\tov Q$ is the identity 1-cell; and liftings and extensions by the ends

$$(A -\!\!\!\!-\!\!\circ B)(r,s)=\int_q {\rm Hom}(A(q,r),B(q,s))$$

for $A\colon Q\tov R$ and $B\colon Q\tov S$ and

$$(C \circ\!\!-\!\!\!\!- A)(s,q)=\int_r {\rm Hom}(A(q,r),B(s,r))$$

for $A\colon Q\tov R$ and $C\colon S\tov R$.


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