# An equational approach to enriched distributivity

Adriana Balan  University Politehnica of Bucharest, Romania
<EMAIL_ADDRESS>and  Alexander Kurz  Chapman University, Orange
California, USA<EMAIL_ADDRESS>
###### Abstract.

The familiar adjunction between ordered sets and completely distributive
lattices can be extended to generalised metric spaces, that is, categories
enriched over a quantale (a lattice of “truth values”), via an appropriate
distributive law between the “down-set” monad and the “up-set” monad on the
category of quantale-enriched categories. If the underlying lattice of the
quantale is completely distributive, and if powers distribute over non-empty
joins in the quantale, then this distributive law can be concretely formulated
in terms of operations, equations and choice functions, similar to the
familiar distributive law of lattices.

AMS 2010 Subject Classification: 18B35,18C05,06D10,18D20,06F07,08A65.

Keywords: quantale, enriched category, weighted (co)limit, (co)complete
enriched category, completely distributive quantale-enriched category.

##  1\. Introduction

###  1.1. The double dualization monad.

The contravariant adjunction

$[-,\mathbbm{2}]\dashv[-,\mathbbm{2}]:\mathsf{Ord}^{\mathsf{op}}\to\mathsf{Ord}$

produces a monad on $\mathsf{Ord}$, the category of antisymmetric ordered sets
and monotone maps, whose algebras are the completely distributive lattices
[23]. The latter are monadic also over $\mathsf{Set}$ [27], hence form an
(infinitary) variety. Recall that a completely distributive lattice $A$ is a
complete lattice satisfying

$\bigwedge_{k\in K}\bigvee A_{k}=\bigvee_{f\in\mathcal{F}}\bigwedge f(K)$

for every family of subsets $(A_{k})_{k\in K}$ of $A$, where
$\mathcal{F}=\\{f:K\to A\mid f(k)\in A_{k}\\}$ denotes the set of choice
functions. There is also a constructive definition of completely distributive
lattices available in arbitrary toposes [23]: if $\mathbbm{D}A$ denotes the
usual lattice of downsets of an ordered set $A$, and
$\mathsf{y}_{A}:A\to\mathbbm{D}A$ is the (Yoneda) embedding of principal
downsets, then $A$ being complete means that $\mathsf{y}_{A}$ has a left
adjoint ${\mathsf{sup}_{A}{}}$ which computes suprema of downsets, and being
completely distributive means that ${\mathsf{sup}_{A}{}}$ has itself a left
adjoint mapping each element to the downset of elements totally below it.111
An element $a\in A$ is totally below $b\in A$ if for every subset $S\subseteq
A$ such that $b\leq\bigvee S$ there is some $x\in S$ with $a\leq x$ [9].

As observed in [31], the dual adjunction between $\mathsf{Ord}$ and
$\mathsf{Ord}^{\mathsf{op}}$ exhibited above is in fact a special case of
$[-,\Omega]\dashv[-,\Omega]:\Omega\mbox{-}\hskip
0.5pt{\mathsf{cat}}_{s}^{\mathsf{op}}\to\Omega\mbox{-}\hskip
0.5pt{\mathsf{cat}}_{s}$, for $\Omega=\mathbbm{2}$. Here $\Omega\mbox{-}\hskip
0.5pt{\mathsf{cat}}_{s}$ is the category of skeletal categories enriched over
a commutative quantale $\Omega$. As in the ordered case, the category of
algebras for the induced monad can be equivalently described using a
distributive law between the enriched downset monad $\mathbbm{D}$ and the
enriched upset monad $\mathbbm{U}$ [23, 2, 38] and is monadic both over
$\Omega\mbox{-}\hskip 0.5pt{\mathsf{cat}}_{s}$ and over $\mathsf{Set}$ [28].

###  1.2. Quantales.

Historically, quantales were introduced in the 1980s [24] as a logical-
theoretic framework for studying certain spaces arising from quantum
mechanics. They can be perceived as lattices of “truth-values” or “distances”,
equipped with an extra operation expressing conjunction (logical
interpretation) or addition of distances (metric interpretation).
Alternatively, quantales are also complete idempotent semirings, hence
appearing in tropical and idempotent analysis.

###  1.3. Quantale-enriched categories.

These structures generalise both metric spaces and ordered sets [20] within
the realm of enriched category theory, making possible a theory of
quantitative domains [39, 37]. The category $\Omega\mbox{-}\hskip
0.5pt\mathsf{Sup}$ of (co)complete skeletal quantale-enriched categories and
cocontinuous functors, however, lives in the algebraic world: the objects are
just complete sup-lattices endowed with an action of the quantale, or, in
other words, complete semimodules over complete idempotent semirings [15, 26,
36]. The simplest example, namely cocomplete enriched categories over the two-
element quantale $\mathbbm{2}$, are nothing but complete sup-lattices. It is
the action of the quantale which enhances the path to the many-valued realm.

###  1.4. Completely distributive quantale-enriched categories.

Adding one more layer of structure finally brings us to the object of study of
the present paper: $\Omega\mbox{-}\hskip 0.5pt\mathsf{CD}$, the category of
completely distributive skeletal quantale-enriched categories and continuous
and cocontinuous enriched functors. There are (at least) three possible
approaches/motivations for studying $\Omega\mbox{-}\hskip 0.5pt\mathsf{CD}$:

  * •

Completely distributive quantale-enriched categories arise naturally in
quantitative domain theory and many-valued logics (see e.g. [39, 40, 11, 8]).

  * •

Categorically, they are (co)complete skeletal quantale-enriched categories for
which taking suprema distributes over limits, e.g. [17, 37], and they are also
the projective objects of the (infinitary) variety $\Omega\mbox{-}\hskip
0.5pt\mathsf{Sup}$.

  * •

Algebraically, completely distributive quantale-enriched categories are
precisely the algebras for the quantale-enriched version of the double
dualisation monad described above over $\Omega\mbox{-}\hskip
0.5pt{\mathsf{cat}}_{s}$ [2, 38]. Moreover, $\Omega\mbox{-}\hskip
0.5pt\mathsf{CD}$ is also monadic over $\mathsf{Set}$ [28]. Consequently,
$\Omega\mbox{-}\hskip 0.5pt\mathsf{CD}$ is equationally presentable [21]. But
Linton’s theorem does not help us to find a convenient equational
presentation.

The present paper contributes with an equational presentation for completely
distributive quantale-enriched categories under some natural requirements on
the quantale which, in particular, are satisfied in the case
$\Omega=\mathbbm{2}$.

##  2\. Preliminaires

In this section we gather the necessary technicalities from quantale enriched
category theory that will make this paper reasonably self-contained. Most of
the material is standard; for the general theory of enriched categories we
refer to Kelly’s monography [16], while for quantale/quantaloid-enriched
categories the reader might consult e.g. [39, 34, 35, 17, 36, 37, 28, 12].

###  2.1. Quantales and quantale-enriched categories.

A commutative quantale is a complete sup-lattice
$(\Omega,\lor,\land,\bot,\top)$ and a commutative monoid
$(\Omega,\otimes,e)$,222 The unit $e$ does not need to be the top element in
$\Omega$, but if it is the case, the quantale is called integral.  such that
$v\otimes-$ preserves arbitrary joins, for all $v\in\Omega$. In particular it
has a right adjoint $[v,-]$:

$v\otimes w\leq u\quad\iff\quad w\leq[v,u]\ \ .$

If we interpret $v\otimes w$ as a “conjunction” and $[v,w]$ as an
“implication”, then $v\otimes[v,u]\leq u$ becomes the usual “modus ponens” of
logic. Hence the elements of the quantale can be perceived as “truth values”.

######  Example 2.1.

  1. (1)

The simplest example of a (commutative) quantale is the two-element chain
$\Omega=(\mathbbm{2},\land,1)$, with meet as multiplication.

  2. (2)

The (extended) positive real numbers $\Omega=([0,\infty]^{\mathsf{op}},+,0)$,
with addition and zero, form a commutative quantale, with $[v,u]=u{-}v\mbox{
\sf{if} }v\leq u\mbox{ \sf{else} }0$.

  3. (3)

The three-element chain $\mathbb{3}=\\{0<\frac{1}{2}<1\\}$ supports two
quantale structures for which the tensor is idempotent [3]: Taking $\otimes$
to be the meet, one obtains the Heyting algebra $\Omega=(\mathbb{3},\land,1)$.

The second idempotent multiplication $\otimes$ on $\mathbb{3}$ has unit
$\frac{1}{2}$ and $[-,-]$ as indicated in the table on the right. In
particular, the resulting quantale $(\mathbb{3},\otimes,\frac{1}{2})$ is a
Sugihara monoid [25].

$\begin{array}[]{|c||c|c|c|c|}\hline\cr[-,-]&0&\frac{1}{2}&1\\\
\hline\cr\hline\cr 0&1&1&1\\\ \frac{1}{2}&0&\frac{1}{2}&1\\\ 1&0&0&1\\\
\hline\cr\end{array}$

Besides the two idempotent structures exhibited above, there exists only one
more quantale structure on $\mathbb{3}$ (non-idempotent and integral), given
by the Łukasiewicz truncated addition $v\odot w=\max(0,v+w-1)$.

  4. (4)

There are also non-distributive commutative quantales, among which we mention
the simplest, namely the lattices $M_{3}$ and the $N_{5}$, both with
idempotent tensor as indicated below

$\textstyle{\top\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$\top{\otimes}a=a$
---
$\textstyle{e\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$b=\top{\otimes}b$
---
$\textstyle{\bot=a{\otimes}b\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\top\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$b=\top{\otimes}b$
---
$\textstyle{e\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
$a=a{\otimes}b=a{\otimes}\top$
---
$\textstyle{\bot}$

For more examples we refer to [7].

An $\Omega$-category ${\mathscr{A}}$ consists of a set $A$, together with a
map333 In the sequel we shall refer to this map as the $\Omega$-hom,
$\Omega$-metric or $\Omega$-distance. ${\mathscr{A}}:A\times A\to\Omega$
satisfying

(1) $e\leq{\mathscr{A}}(a,a)\qquad\mbox{ and
}\qquad{\mathscr{A}}(a,b){\otimes}{\mathscr{A}}(b,c)\leq{\mathscr{A}}(a,c)$

for all $a,b,c\in A$. An $\Omega$-functor $f:{\mathscr{A}}\to{\mathscr{B}}$ is
a map between the underlying sets such that
${\mathscr{A}}(a,b)\leq{\mathscr{B}}(f(a),f(b))$ holds. Finally, an
$\Omega$-natural transformation $f\to g$ is given whenever
$e\leq{\mathscr{B}}(f(a),g(a))$ holds for all $a\in A$. Thus, there is at most
one $\Omega$-natural transformation between $f$ and $g$. We shall denote by
$\Omega\mbox{-}\hskip 0.5pt{\mathsf{cat}}$ the category of $\Omega$-categories
and of $\Omega$-functors (actually, it is a locally ordered category). As a
last piece of notation, let $[{\mathscr{A}},{\mathscr{B}}]$ be the
$\Omega$-category having as objects $\Omega$-functors
$f:{\mathscr{A}}\to{\mathscr{B}}$, with $\Omega$-homs
$[{\mathscr{A}},{\mathscr{B}}](f,g)=\bigwedge_{a}{\mathscr{B}}(f(a),g(a))$.

######  Example 2.2.

  1. (1)

The quantale $\Omega$ becomes an $\Omega$-category with $\Omega(v,w)=[v,w]$.

  2. (2)

Each set $A$ can be perceived as a an $\Omega$-category $\mathsf{d}A$ when it
is equipped with $\mathsf{d}A(a,a)=e$ and $\mathsf{d}A(a,b)=\bot$ for $a\neq
b$. Such an $\Omega$-category is called _discrete_. With the obvious action on
arrows, this extends to a functor
$\mathsf{d}:\mathsf{Set}\to\Omega\mbox{-}\hskip 0.5pt{\mathsf{cat}}$.

  3. (3)

Ordered sets $(A,\leq)$ are enriched categories over the two-element quantale
$\mathbb{2}$: the reflexivity and transitivity axioms correspond exactly to
(1) [20].

  4. (4)

Generalised metric spaces If $\Omega$ is the real half line
$([0,\infty]^{\mathsf{op}},+,0)$ as in Example 2.1.2, a small
$\Omega$-category ${\mathscr{A}}$ is a generalised metric space:

$0\geq{\mathscr{A}}(a,a),\quad{\mathscr{A}}(a,b)+{\mathscr{A}}(b,c)\geq{\mathscr{A}}(a,c)\
\ .$

The generalisation of the usual notion is three-fold. First,
${\mathscr{A}}(-,-):A\times A\to[0,\infty]$ is a pseudo-metric in the sense
that two distinct points may have distance $0$. Second, ${\mathscr{A}}(-,-)$
is a quasi-metric in the sense that distance is not necessarily symmetric.
Third, distances are allowed to be infinite, which has the important
consequence that the category of generalised metric spaces has colimits
(whereas metric spaces do not even have coproducts). An $\Omega$-functor is
then exactly a non-expanding map.

In view of the above examples, it is helpful to think of $\Omega$-categories
as $\Omega$-valued orders [12]. Actually, each $\Omega$-category
${\mathscr{A}}$ does carry an induced order444 That is, a reflexive and
transitive relation.

(2) $a\leq b\qquad\iff\qquad e\leq{\mathscr{A}}(a,b)\ .$

In particular, the underlying order induced by the $\Omega$-category structure
on $\Omega$ itself, as in Example 2.1.1, is the original order of the
quantale. An $\Omega$-category is said to be skeletal if its underlying order
is anti-symmetric: ${\mathscr{A}}(a,b)\wedge{\mathscr{A}}(b,a)\geq e$ implies
$a=b$. Let $\Omega\mbox{-}\hskip 0.5pt{\mathsf{cat}}_{s}$ denote the full
subcategory of skeletal $\Omega$-categories.

Notice that we use the same symbol $\leq$ for inequality in both $\Omega$ and
${\mathscr{A}}$, and rely on the context to tell them apart. We shall proceed
similarly for joins and meets in the underlying order of an $\Omega$-category,
whenever these exist.

###  2.2. A calculus of enriched downsets and upsets.

An $\Omega$-functor $\varphi:{\mathscr{A}}^{\mathsf{op}}\to\Omega$, usually
called a (contravariant) presheaf in category theory, can be perceived as an
$\Omega$-valued downset: the relation
${\mathscr{A}}(a,b)\leq[\varphi(b),\varphi(a)]$, equivalent to
${\mathscr{A}}(a,b)\otimes\varphi(b)\leq\varphi(a)$, reads in case
$\Omega=\mathbbm{2}$ as $(a\leq b$ and $b\in\varphi$ implies $a\in\varphi)$,
that is, $\varphi$ is a downset in the usual sense. Here we implicitly
identified a downset with its associated characteristic function. To preserve
this intuition, we shall denote by $\mathbbm{D}{\mathscr{A}}$ the
$\Omega$-category of contravariant presheaves
$[{\mathscr{A}}^{\mathsf{op}},\Omega]$. Notice that $\mathbbm{D}{\mathscr{A}}$
is skeletal, for its underlying order is pointwise, inherited from $\Omega$.
The $\Omega$-functor known as the Yoneda embedding,
$\mathsf{y}_{\mathscr{A}}:{\mathscr{A}}\to\mathbbm{D}{\mathscr{A}}$,
$\mathsf{y}_{\mathscr{A}}(a)={\mathscr{A}}(-,a)$, generalises the embedding of
an ordered set into the lattice of its downsets. The correspondence
${\mathscr{A}}\mapsto\mathbbm{D}{\mathscr{A}}$ extends to a functor
$\Omega\mbox{-}\hskip 0.5pt{\mathsf{cat}}\to\Omega\mbox{-}\hskip
0.5pt{\mathsf{cat}}$: for each $\Omega$-functor
$f:{\mathscr{A}}\to{\mathscr{B}}$, put
$\mathbbm{D}f(\varphi)=\bigvee_{a\in{\mathscr{A}}}{\mathscr{B}}(-,f(a))\otimes\varphi(a)$.
In fact, $\mathbbm{D}$ is a Kock-Zöberlein monad [23], with unit
$\mathsf{y}_{\mathscr{A}}:{\mathscr{A}}\to\mathbbm{D}{\mathscr{A}}$ and
multiplication
$\mu_{\mathscr{A}}(\Phi)=\bigvee_{\varphi\in\mathbbm{D}{\mathscr{A}}}\varphi\otimes\Phi(\varphi)$,
hence it satisfies
$\mathbbm{D}\mathsf{y}_{\mathscr{A}}\dashv\mu_{\mathscr{A}}\dashv\mathsf{y}_{\mathbbm{D}{\mathscr{A}}}$.

Dually, covariant presheaves, that is, $\Omega$-functors
$\psi:{\mathscr{A}}\to\Omega$, are the $\Omega$-enriched analogue of upsets;
put $\mathbbm{U}{\mathscr{A}}=[{\mathscr{A}},\Omega]^{\mathsf{op}}$ (the
$\mathsf{op}$ corresponding to the containment order for usual upsets), and
$\mathsf{y}\prime_{\mathscr{A}}:{\mathscr{A}}\to[{\mathscr{A}},\Omega]^{\mathsf{op}}$,
$\mathsf{y}\prime_{\mathscr{A}}(a)={\mathscr{A}}(a,-)$ the other Yoneda
embedding.

Given an $\Omega$-category ${\mathscr{A}}$, to each ordinary map
$f:A\to\Omega$ (an $\Omega$-subset555 For $\Omega=[0,1]$, this corresponds to
the usual notion of a fuzzy subset [43, 10]. of $A$), one can associate an
$\Omega$-downset $f^{\downarrow}:{\mathscr{A}}^{\mathsf{op}}\to\Omega$ by the
formula

(3) $f^{\downarrow}=\bigvee_{a\in A}f(a)\otimes{\mathscr{A}}(-,a)\ \ .$

Readers familiar with category theory will recognise in the above formula the
left Kan extension of $f$ along the inclusion of the discrete
$\Omega$-category $\mathsf{d}A$ into ${\mathscr{A}}$; in the particular case
$\Omega=\mathbbm{2}$, $f^{\downarrow}$ is the down-closure of a subset $f$.

######  Example 2.3.

Consider the Łukasiewicz chain $\Omega=\mathbb{3}$, together with the
singleton subset $\\{\frac{1}{2}\\}$ represented via its characteristic map as
the discrete $\Omega$-subset $f:\Omega\to\Omega$,
$f(\frac{1}{2})=1,f(0)=f(1)=0$.

$\begin{array}[]{|c||c|c|c|}\hline\cr a&0&\frac{1}{2}&1\\\ \hline\cr
f^{\downarrow}(a)&1&1&\frac{1}{2}\\\ \hline\cr\end{array}$

Then its ordinary downclosure is $\\{0,\frac{1}{2}\\}$, while the
$\Omega$-enriched $f^{\downarrow}$ is computed in the next table: in the
enriched context, the $\Omega$-valued downclosure $f^{\downarrow}$ of the
subset $\\{a\\}$ carries more information (it has a richer structure) than the
ordinary one.

###  2.3. Limits and colimits in $\Omega$-categories. (Co)completeness of
$\Omega$-categories.

In this subsection we shall briefly recall the usual notions of weighted
colimits and cocompleteness to be used in the sequel.

A colimit of an $\Omega$-functor $f:{\mathscr{K}}\to{\mathscr{A}}$, weighted
by an $\Omega$-functor $\varphi:{\mathscr{K}}^{\mathsf{op}}\to\Omega$,
consists of an object ${\mathsf{colim}_{\varphi}{f}}$ of ${\mathscr{A}}$, such
that

(4)
${\mathscr{A}}({\mathsf{colim}_{\varphi}{f}},a)=[{\mathscr{K}}^{\mathsf{op}},\Omega](\varphi,{\mathscr{A}}(f-,a))=\bigwedge_{k\in{\mathscr{K}}}[\varphi(k),{\mathscr{A}}(f(k),a)]$

holds, $\Omega$-natural in $a\in{\mathscr{A}}$.

An $\Omega$-category is called cocomplete if it has all $\Omega$-enriched
colimits; an $\Omega$-functor is cocontinuous if preserves all
$\Omega$-enriched colimits. Denote by $\Omega\mbox{-}\hskip 0.5pt\mathsf{Sup}$
the category of cocomplete skeletal $\Omega$-categories and cocontinuous
$\Omega$-functors.

It will be convenient for us to use the alternative characterisation of a
cocomplete $\Omega$-category as one having all colimits of the identity
functor [34]. That is, for each $\varphi:{\mathscr{A}}^{\mathsf{op}}\to\Omega$
there is an object ${\mathsf{colim}_{\varphi}{\mathsf{id}_{\mathscr{A}}}}$ in
${\mathscr{A}}$ such that

${\mathscr{A}}({\mathsf{colim}_{\varphi}{\mathsf{id}_{\mathscr{A}}}},a)=\mathbbm{D}{\mathscr{A}}(\varphi,\mathsf{y}_{\mathscr{A}}(a))=\bigwedge_{b\in{\mathscr{A}}}[\varphi(b),{\mathscr{A}}(b,a)]\
\ .$

Interpreting the above formula in the ordered case, we see that
${\mathsf{colim}_{\varphi}{\mathsf{id}_{\mathscr{A}}}}\leq a$ holds if and
only if $b\in\varphi$ implies $b\leq a$ for all $b$. Hence, intuitively,
${\mathsf{colim}_{\varphi}{\mathsf{id}_{\mathscr{A}}}}$ computes the
$\Omega$-suprema of the $\Omega$-contravariant presheaf
$\varphi:{\mathscr{A}}^{\mathsf{op}}\to\Omega$ (think again of $\varphi$ as an
”$\Omega$-valued downset”) and will be denoted
${\mathsf{sup}_{{\mathscr{A}}}{\varphi}}$. Then cocompleteness of
${\mathscr{A}}$ can be rephrased as the property of the Yoneda embedding of
having a left adjoint (namely,
${\mathsf{sup}_{{\mathscr{A}}}{}}:\mathbbm{D}{\mathscr{A}}\to{\mathscr{A}}$).

######  Example 2.4.

We list below two types of weighted colimits:

  1. (1)

If ${\mathscr{K}}$ is the unit $\Omega$-category666 Having only one object,
with corresponding $\Omega$-hom given by the unit of the quantale $e$.
$\mathbb{1}$, we may identify $f$ with an object $a$ of ${\mathscr{A}}$ and
$\varphi$ with an element $v$ of the quantale $\Omega$. The resulting colimit,
usually called the tensor (or copower) of $v$ with $a$, will be denoted $v\ast
a$ instead of ${\mathsf{colim}_{\varphi}{f}}$. Explicitly, the tensor $v\ast
a$ is uniquely determined by the relation ${\mathscr{A}}(v\ast
a,b)=[v,{\mathscr{A}}(a,b)]$ for all $a,b\in{\mathscr{A}}$.

  2. (2)

A conical colimit is a colimit weighted by a presheaf
$\varphi:{\mathscr{K}}^{\mathsf{op}}\to\Omega$ with constant value $e$, the
unit of the quantale; its defining property is therefore
${\mathscr{A}}({\mathsf{colim}_{\varphi}{f}},a)=\bigwedge_{k\in{\mathscr{K}}}{\mathscr{A}}(f(k),a)$
for all $a\in{\mathscr{A}}$. In particular, with respect to the induced order
on ${\mathscr{A}}$, ${\mathsf{colim}_{\varphi}{f}}\leq a$ holds if and only if
$f(k)\leq a$ holds for every $k$; that is, ${\mathsf{colim}_{\varphi}{f}}$ is
the join $\bigvee_{k\in{\mathscr{K}}}f(k)$ of the family
$(f(k))_{k\in{\mathscr{K}}}$ in the underlying ordered set of the
$\Omega$-category ${\mathscr{A}}$.

The importance of the previous two types of colimits resides in the
characterisation of a cocomplete $\Omega$-category as an $\Omega$-category
having all tensors and all conical colimits [16, 35]. For later use, we just
mention how the $\Omega$-suprema of a contravariant presheaf
$\varphi:{\mathscr{A}}^{\mathsf{op}}\to\Omega$ is obtained via joins and
tensors:

(5)
${\mathsf{sup}_{{\mathscr{A}}}{\varphi}}=\bigvee_{a\in{\mathscr{A}}}\varphi(a)\ast
a\ \ .$

In particular, a skeletal cocomplete $\Omega$-category is a complete lattice
with respect to the induced order.

######  Remark 2.5.

A careful analysis of the defining property of a weighted colimit (4) shows
that without loss of generality, the domain $\Omega$-category ${\mathscr{K}}$
of the weight can be chosen discrete. This is because the quantale in which we
enrich is actually a poset. The fact that weighted colimits can be discrete
means that we can treat them as operations, and this will be useful in the
sequel.

Dually, one can define weighted limits (in particular, cotensors $v\rhd a$,
meets $\bigwedge_{k}a_{k}$ and infima $\inf_{\mathscr{A}}\psi$ of
$\Omega$-valued upsets $\psi:{\mathscr{A}}\to\Omega$) and talk about
completeness of $\Omega$-categories and continuity of $\Omega$-functors. For
more details, we refer to e.g. [34, 35], and only recall here the formula
allowing to compute the limit of an $\Omega$-functor
$G:{\mathscr{K}}\to{\mathscr{A}}$ weighted by $w:{\mathscr{K}}\to\Omega$, by
meets and cotensors:

(6) $\mathsf{lim}_{w}G=\bigwedge_{k\in{\mathscr{K}}}w(k)\rhd G(k)\ .$

To deepen the analogy with order and lattice theory, we add that an
$\Omega$-category ${\mathscr{A}}$ is complete if and only if it is cocomplete
[34].777 Although the two notions are equivalent, we shall continue to refer
to “cocomplete” $\Omega$-categories, as we are mainly interested in suprema of
$\Omega$-downsets and in cocontinuous $\Omega$-functors. Alternatively,
${\mathscr{A}}$ is cocomplete if it is complete as a lattice and has all
tensors and cotensors [35]. This last result has the advantage of
characterising $\Omega\mbox{-}\hskip 0.5pt\mathsf{Sup}$, the category of
cocomplete skeletal $\Omega$-categories and cocontinuous $\Omega$-functors by
operations and equations, as we shall recall next. Formally, the forgetful
functor $\Omega\mbox{-}\hskip 0.5pt\mathsf{Sup}\to\mathsf{Set}$ is monadic.
This goes back to [15] and [26], but see also [13, 36].

###  2.4. Equational presentation of cocomplete $\Omega$-categories.

A cocomplete skeletal $\Omega$-category can alternatively be described as a
complete join-lattice $(A,\bigvee,\bot)$ endowed with an action of $\Omega$,
$(v,a)\mapsto v\ast a$ [35], satisfying the following equations:

(7) $\displaystyle\ e\ast a=a$ $\displaystyle v\ast(w\ast a)=(v{\otimes}w)\ast
a$ $\displaystyle\left(\bigvee_{i\in I}v_{i}\right)\ast a=\bigvee_{i\in
I}\left(v_{i}\ast a\right)\qquad$ $\displaystyle v\ast\left(\bigvee_{i\in
I}a_{i}\right)=\bigvee_{i\in I}\left(v\ast a_{i}\right)\ .$

Also, cocontinuous $\Omega$-functors are precisely the join-lattice
homomorphisms preserving the action of $\Omega$.

######  Remark 2.6.

Readers familiar with ring and module theory will certainly identify
cocomplete skeletal $\Omega$-categories as complete semimodules over $\Omega$,
$\Omega$ being perceived as a complete idempotent semiring, and cocontinuous
$\Omega$-functors as complete homomorphisms of $\Omega$-semimodules [4].

As mentioned above, a (skeletal) $\Omega$-category ${\mathscr{A}}$ is complete
if and only if it is cocomplete. In particular, cocompleteness implies
existence of arbitrary meets and cotensors. For example, the cotensor $v\rhd-$
is a unary operation on ${\mathscr{A}}$, right adjoint to $v\ast-$:

$v\ast a\leq b\quad\iff\quad a\leq v\rhd b\ .$

Also, meets and cotensors automatically satisfy equations dual to those in (7)
(formally deduced by adjunction), namely

(8) $\displaystyle\ e\rhd a=a$ $\displaystyle v\rhd(w\rhd a)=(v{\otimes}w)\rhd
a$ $\displaystyle\left(\bigvee_{i\in I}v_{i}\right)\ast a=\bigwedge_{i\in
I}\left(v_{i}\rhd a\right)\qquad$ $\displaystyle v\rhd\left(\bigwedge_{i\in
I}a_{i}\right)=\bigwedge_{i\in I}\left(v\rhd a_{i}\right)\ .$

The $\Omega$-category structure can be recovered from the equational
presentation above [26] as

(9) ${\mathscr{A}}(a,b)=\bigvee\\{v\mid v\ast a\leq b\\}=\bigvee\\{v\mid a\leq
v\rhd b\\}\ .$

For further use, let us also record the relation

(10) $\bigvee_{a\in{\mathscr{A}}}{\mathscr{A}}(a,b)\ast a=b$

which generalises the obvious statement that in an ordered set, the join of
all elements below some $b$ is $b$ itself.

######  Remark 2.7.

It will be beneficial in the next section to use an equivalent, but more
symmetric presentation of (the objects of) $\Omega\mbox{-}\hskip
0.5pt\mathsf{Sup}$: the objects are complete lattices (hence having both
arbitrary joins $\bigvee$ and meets $\bigwedge$), together with a family of
adjoint unary operators indexed by $\Omega$ (tensors $v\ast-$ and cotensors
$v\rhd-$), satisfying both (7) and (8), in addition to the usual complete
lattice equations. Notice that arrows in $\Omega\mbox{-}\hskip
0.5pt\mathsf{Sup}$ do not preserve all operations described above, but only
joins and tensors888 Consequently, meets and cotensors are only laxly
preserved..

######  Example 2.8.

  1. (1)

Perhaps the simplest example of a cocomplete $\Omega$-category is $\Omega$
itself, with the $\Omega$-category structure described in Example 2.1.1.
Trivially, the unit category $\mathbb{1}$ is also cocomplete.

  2. (2)

For any $\Omega$-category ${\mathscr{A}}$, the associated $\Omega$-category of
$\Omega$-downsets $\mathbbm{D}{\mathscr{A}}$ is cocomplete and skeletal. In
fact, $\mathbbm{D}{\mathscr{A}}$ is the free cocompletion of ${\mathscr{A}}$
[16, 34].999 In particular, taking the domain to be discrete, we obtain the
cocomplete $[\mathsf{d}A,\Omega]=\Omega^{A}$. The case $A=\emptyset$ produces
the cocomplete terminal $\Omega$-category $\mathbb{1}_{\top}$.  For later use,
recall how suprema and limits are computed in $\mathbbm{D}{\mathscr{A}}$:

(11)
$\mathsf{sup}_{\mathbbm{D}{\mathscr{A}}}\Phi=\mu_{\mathscr{A}}(\Phi)=\bigvee_{\varphi\in\mathbbm{D}{\mathscr{A}}}\Phi(\varphi)\otimes\varphi$

respectively

(12) $\mathsf{lim}_{w}G=\bigwedge_{k\in{\mathscr{K}}}[w(k),G(k)(-)]$

where $\Phi:(\mathbbm{D}{\mathscr{A}})^{\mathsf{op}}\to\Omega$,
$w:{\mathscr{K}}\to\Omega$ and $G:{\mathscr{K}}\to\mathbbm{D}{\mathscr{A}}$
are $\Omega$-functors.

  3. (3)

For each $\Omega$-category ${\mathscr{A}}$, there is an adjunction commuting
with both Yoneda embeddings

$\textstyle{{\mathscr{A}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathsf{y}_{\mathscr{A}}}$$\scriptstyle{{\mathsf{y}\prime}_{\mathscr{A}}}$$\textstyle{[{\mathscr{A}}^{\mathsf{op}},\Omega]\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\perp}$$\textstyle{[{\mathscr{A}},\Omega]^{\mathsf{op}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$

where the horizontal left adjoint maps an $\Omega$-downset $\varphi$ to

${\uparrow}\varphi=\bigwedge_{a\in{\mathscr{A}}}[\varphi(a),{\mathscr{A}}(a,-)]$

(the analogue of the (up)set of upper bounds), respectively an
$\Omega$-enriched upset $\psi:{\mathscr{A}}\to\Omega$ is mapped by the right
adjoint to the $\Omega$-downset of its lower bounds

${\downarrow}\psi=\bigwedge_{a\in{\mathscr{A}}}[\psi(a),{\mathscr{A}}(-,a)]\ \
.$

The fixed points of these adjunction determine a (complete and) cocomplete
skeletal $\Omega$-category into which ${\mathscr{A}}$ embeds continuously and
cocontinuously, known as the Isbell completion (the $\Omega$-categorical
analogue of the Dedekind-MacNeille completion of an ordered set) [39, 32, 34].
For example, in case $\Omega=([0,\infty],\geq_{\mathbb{R}},+,0)$, the Isbell
completion is known in the theory of metric spaces as the tight span [41].

##  3\. Completely distributive quantale enriched categories

There are several ways of introducing completely distributive
$\Omega$-categories. Here, we chose to take the perhaps simplest approach,
following [30]:

######  Definition 3.1.

A completely distributive $\Omega$-category101010 Also know as a totally
continuous $\Omega$-category. is a cocomplete $\Omega$-category
${\mathscr{A}}$, such that
${\mathsf{sup}_{{\mathscr{A}}}{}}:\mathbbm{D}{\mathscr{A}}\to{\mathscr{A}}$
has an $\Omega$-enriched left adjoint.

Completely distributive $\Omega$-categories have been studied in the past by
several authors, see e.g. [17, 36, 2, 38]. As the name suggests, for
$\Omega=\mathbbm{2}$ we recover the well-known completely distributive
lattices (the left adjoint to ${\mathsf{sup}_{{\mathscr{A}}}{}}$ mapping an
element to the downset of those totally below it). We shall denote by
$\Omega\mbox{-}\hskip 0.5pt\mathsf{CD}$ the category of completely
distributive skeletal $\Omega$-categories and continuous and cocontinuous
$\Omega$-functors.

######  Remark 3.2.

We shall see in Section 3.1 how the choice for arrows in $\Omega\mbox{-}\hskip
0.5pt\mathsf{CD}$ is imposed by the axiomatisation (15): all operations
encountered must be preserved. Actually, there is a deeper categorical reason
behind that: $\Omega\mbox{-}\hskip 0.5pt\mathsf{CD}$ is the category of
algebras for the lifting of the $\Omega$-downset monad $\mathbbm{D}$ to
$\Omega\mbox{-}\hskip 0.5pt\mathsf{Inf}$, the category of algebras for the
$\Omega$-upset monad $\mathbbm{U}$ [2, 38]. Consequently, the arrows in
$\Omega\mbox{-}\hskip 0.5pt\mathsf{CD}$ are simultaneously
$\mathbbm{U}$-homomorphisms and $\mathbbm{D}$-homomorphisms, that is,
continuous and cocontinuous $\Omega$-functors.

######  Example 3.3.

For any $\Omega$-category ${\mathscr{A}}$, $\mathbbm{D}{\mathscr{A}}$ is
completely distributive, the left adjoint of
${\mathsf{sup}_{\mathbbm{D}{\mathscr{A}}}{}}$ being
$\mathbbm{D}\mathsf{y}_{\mathscr{A}}$ [17, 37]. In particular, for any set
$A$, the $\Omega$-valued powerset $\Omega^{A}=[\mathsf{d}A,\Omega]$ is
completely distributive. Taking $A$ to be a singleton shows that the quantale
$\Omega$ is itself completely distributive as an $\Omega$-category, while
$A=\emptyset$ produces the completely distributive terminal $\Omega$-category
$\mathbb{1}_{\top}$. Also, the unit $\Omega$-category $\mathbb{1}$ is
completely distributive.

######  Remark 3.4.

  1. (1)

A cocomplete skeletal $\Omega$-category is completely distributive if and only
if it is a projective object in $\Omega\mbox{-}\hskip 0.5pt\mathsf{Sup}$ [36].

  2. (2)

Unlike the case for lattices, complete distributivity is no longer a self-dual
notion in general;111111 Neither is this the case when working in an arbitrary
topos [6]. in fact, in [17] it is proven that for an integral quantale
$\Omega$, every completely distributive $\Omega$-category is also completely
co-distributive if and only if $\Omega$ is a Girard quantale [42].

  3. (3)

Complete distributivity of a cocomplete $\Omega$-category does not necessarily
entail the complete distributivity of the underlying lattice. For example,
$\Omega$ itself is always completely distributive as an $\Omega$-category
[37], but not necessarily distributive as a lattice, as we have seen from
Example 2.1. However, there exists a positive result in this sense, due to
[17]: every completely distributive $\Omega$-category ${\mathscr{A}}$ is
completely distributive as a lattice if and only if $\Omega$ itself is a
completely distributive lattice.

Actually, if the reader is interested in cocomplete $\Omega$-categories which
are not completely distributive, there is a simple way of producing such
examples: take any complete lattice $A$ which is not completely distributive,
e.g. the diamond lattice $M_{3}$, and a quantale $\Omega$ which is completely
distributive as a lattice. Then the tensor product of $A$ and $\Omega$ in the
category of complete sup-lattices is a cocomplete $\Omega$-category – it is
the free cocomplete $\Omega$-category over the complete sup-lattice $A$ [15,
26], but not completely distributive as a lattice [33]. Therefore this tensor
product is neither $\Omega$-completely distributive.

###  3.1. The equational theory of completely distributive $\Omega$-categories

In [28] it is shown that the category $\Omega\mbox{-}\hskip 0.5pt\mathsf{CD}$
is monadic over $\mathsf{Set}$, hence a(n infinitary) variety [21]. In
particular, the free completely distributive $\Omega$-category over a set $A$
is $\mathbbm{D}\mathbbm{U}(\mathsf{d}A)$ [2, 38], similarly to the ordered
case [22].

We shall see in the sequel how $\Omega\mbox{-}\hskip 0.5pt\mathsf{CD}$ can be
described by a nice set of operations and equations, provided certain
assumptions on the quantale are fulfilled. We begin with a simple result,
which will turn out to be useful in computations:

######  Lemma 3.5.

Let ${\mathscr{A}}$ be a complete $\Omega$-category. Then

$[v,w]\ast a\ \leq\ v\rhd(w\ast a)$

holds for any $v,w\in\Omega$ and $a\in{\mathscr{A}}$. More generally, we have

$\left(\bigwedge_{i}[v_{i},w_{i}]\right)\ast a\ \leq\
\bigwedge_{i}\left(\vphantom{\bigwedge_{i}}v_{i}\rhd(w_{i}\ast a)\right)$

for all $v,w\in\Omega$ and all $a\in{\mathscr{A}}$.

###### Proof.

The following standard sequence of implications below proves the first
assertion:

$v\otimes[v,w]\leq w$ $v\ast([v,w]\ast a)=(v\otimes[v,w])\ast a\leq w\ast a$
$[v,w]\ast a\leq v\rhd(w\ast a)$

The second statement is an immediate consequence. ∎

###  3.2. Assumption.

In what follows, we shall consider $\Omega$ to be a commutative quantale, such
that

  1. (1)

$\Omega$ is completely distributive as a lattice.

  2. (2)

All cotensors $[v,-]:\Omega\to\Omega$, for $v\in\Omega$, preserve non-empty
joins, that is,

$[v,\bigvee_{i\in I}w_{i}]=\bigvee_{i\in I}[v,w_{i}]$

holds in $\Omega$ for non-empty $I$.

######  Remark 3.6.

  1. (1)

Observe that in any quantale, $[\bot,v]=\top$ holds for all $v$, hence in
particular $[\bot,\bot]=\top$.121212 Hence $[\bot,-]$ preserves the empty join
only if the quantale is trivial. Also, $[v,\bot]=\bot$ holds for all
$v\neq\bot$ in $\Omega$ if and only if the quantale has no zero
divisors.131313 That is, $v\otimes w=\bot$ implies $v=\bot$ or $w=\bot$.
Therefore extending Assumption 2 as to include the empty join, that is, the
bottom element of the quantale, would not be reasonable.

  2. (2)

The above assumptions are satisfied for all quantales in Example 2.1.1-3 and
also by all finite semilinear commutative residuated lattices (in particular
by the finite commutative MTL-algebras) [5, 14].

Using the symmetric presentation of cocomplete $\Omega$-categories, as
emphasised in Remark 2.7, we can now state the main result of this paper:

######  Theorem 3.7.

Let
${\mathscr{A}}=(A,\bigvee,\bigwedge,(v\ast-)_{v\in\Omega},(v\rhd-)_{v\in\Omega})$
be a cocomplete skeletal $\Omega$-category. If ${\mathscr{A}}$ is completely
distributive, then it satisfies

(13) ${\bigwedge_{k\in K}\psi(k)\rhd\left(\bigvee_{a\in A}G(k)(a)\ast
a\right)=\bigvee_{f\in\mathcal{F}}\bigwedge_{k\in
K}\psi(k)\rhd\left(\vphantom{\bigvee_{i}}G(k)(fk)\ast fk\right)}$

for every pair of functions $\psi:K\to\Omega$, $G:K\to\Omega^{A}$, where
$\mathcal{F}$ denotes the set of functions $K\to A$.

###### Proof.

Recall that a cocomplete $\Omega$-category ${\mathscr{A}}$ is completely
distributive if and only if
${\mathsf{sup}_{{\mathscr{A}}}{}}:\mathbbm{D}{\mathscr{A}}\to{\mathscr{A}}$
has an $\Omega$-enriched left adjoint, equivalently, if
${\mathsf{sup}_{{\mathscr{A}}}{}}:\mathbbm{D}{\mathscr{A}}\to{\mathscr{A}}$
preserves weighted limits. That is,

${\mathsf{sup}_{{\mathscr{A}}}{\left(\mathsf{lim}_{\psi}G\right)}}=\mathsf{lim}_{\psi}({\mathsf{sup}_{{\mathscr{A}}}{}}\circ
G)$

holds for every $\Omega$-functors $\psi:{\mathscr{K}}\to\Omega$ and
$G:{\mathscr{K}}\to\mathbbm{D}{\mathscr{A}}$. Expressing
${\mathsf{sup}_{{\mathscr{A}}}{}}$ by tensors and joins, and the weighted
limits above by cotensors and meets as in (5), (6), (11), (12), we obtain

(14) $\bigvee_{a}\left(\bigwedge_{k}[\psi(k),G(k)(a)]\right)\ast
a=\bigwedge_{k}\psi(k)\rhd\left(\vphantom{\bigvee_{i}}\bigvee_{a}G(k)(a)\ast
a\right)\ .$

As explained in Remark 2.5, we can replace ${\mathscr{K}}$ by a discrete
$\Omega$-category (a set) without loss of generality. Hence instead of an
$\Omega$-functor $\psi:{\mathscr{K}}\to{\mathscr{A}}$, we shall consider a
mere function $\psi:K\to A$.

The natural next step will then be to substitute the $\Omega$-functor
$G:{\mathscr{K}}\to\mathbbm{D}{\mathscr{A}}=[{\mathscr{A}}^{\mathsf{op}},\Omega]$
by a function $G:K\to\Omega^{A}=[A,\Omega]$. But in this case there is a price
to pay: the passage from the family of $\Omega$-downsets
$(G(k):{\mathscr{A}}^{\mathsf{op}}\to\Omega)_{k}$ to a family of
$\Omega$-subsets $(G(k):A\to\Omega)_{k}$ forces the appearance of the
$\Omega$-down-closure of each $\Omega$-subset $G(k)\in\Omega^{A}$ in (14),
namely

(15)
$\bigvee_{a}\left(\bigwedge_{k}[\psi(k),{G(k)^{{\downarrow}}(a)}]\right)\ast
a=\bigwedge_{k}\psi(k)\rhd\left(\bigvee_{a}G(k)^{\downarrow}(a)\ast a\right)$

But since “the supremum of an $\Omega$-subset is the supremum of its
$\Omega$-down-closure”, that is,

$\displaystyle\bigvee_{a}G^{\downarrow}(k)(a)\ast a$ $\displaystyle=$
$\displaystyle\bigvee_{a}\left(\bigvee_{b}G(k)(b)\otimes{\mathscr{A}}(a,b)\right)\ast
a$ $\displaystyle=$
$\displaystyle\bigvee_{b}G(k)(b)\ast\left(\bigvee_{a}{\mathscr{A}}(a,b)\ast
a\right)$ $\displaystyle=$ $\displaystyle\bigvee_{b}{G(k)(b)\ast
b\vphantom{\bigvee_{a}}}\ \mbox{\ \ (by~{}\eqref{eq:tensor-with-V-hom})}$

we see that the right hand side of Equation (15) above remains unchanged by
substituting $G(k)^{\downarrow}$ by $G(k)$ and that the complete
distributivity equation (14) becomes

(16)
$\bigvee_{a}\left(\bigwedge_{k}[\psi(k),{\bigvee_{b}G(k)(b){\otimes}{\mathscr{A}}(a,b)}]\right)\ast
a=\bigwedge_{k}\psi(k)\rhd\left(\bigvee_{a}G(k)(a)\ast a\right)\,.$

Notice that the left hand side involves now not only the tensor and the
internal hom of the quantale, but also the $\Omega$-category structure of
${\mathscr{A}}$, which, as expressed in Equation (9), leads beyond equational
logic. At this stage, the assumptions on $\Omega$ come into place for the
following sequence of computations:

$\displaystyle\bigvee_{a}\left(\,\bigwedge_{k}\
[\psi(k),\bigvee_{b}G(k)(b)\otimes{\mathscr{A}}(a,b)]\right)\ast a$
$\displaystyle=$ $\displaystyle\bigvee_{a}\left(\,\bigwedge_{k}\,\bigvee_{b}\
[\psi(k),G(k)(b)\otimes{\mathscr{A}}(a,b)]\right)\ast a$ (by Assumption.(2))
$\displaystyle=$ $\displaystyle\bigvee_{a}\left(\bigvee_{f}\bigwedge_{k}\
[\psi(k),G(k)(fk)\otimes{\mathscr{A}}(a,fk)]\right)\ast a$ (by Assumption.(1))
$\displaystyle\leq$
$\displaystyle\bigvee_{a,f}\bigwedge_{k}\psi(k)\rhd\left(\vphantom{\bigvee_{i}}\left(\vphantom{\bigvee_{i}}G(k)(fk)\otimes{\mathscr{A}}(a,fk)\right)\ast
a\right)$ (by Lemma 3.5) $\displaystyle=$
$\displaystyle\bigvee_{a,f}\bigwedge_{k}\psi(k)\rhd\left(\vphantom{\bigwedge_{k}}G(k)(fk)\ast\left(\vphantom{\bigwedge_{k}}{\mathscr{A}}(a,fk)\ast
a\right)\right)$ $\displaystyle\leq$
$\displaystyle\bigvee_{f}\bigwedge_{k}\left(\vphantom{\bigwedge_{k}}\psi(k)\rhd\left(\vphantom{\bigwedge_{k}}G(k)(fk)\ast(\bigvee_{a}{\mathscr{A}}(a,fk)\ast
a)\right)\right)$ $\displaystyle=$
$\displaystyle\bigvee_{f}\bigwedge_{k}\psi(k)\rhd\left(\vphantom{\bigwedge_{k}}G(k,fk)\ast
fk\right)$ (by (10))

Therefore $\displaystyle{\bigwedge_{k}\psi(k)\rhd\left(\bigvee_{a}G(k)(a)\ast
a\right)\leq\bigvee_{f}\bigwedge_{k}\psi(k)\rhd\left(\vphantom{\bigwedge_{k}}G(k)(fk)\ast
fk\right)}$ holds. The opposite inequality is trivial, as for each
$f\in\mathcal{F}$ we always have

$\bigwedge_{k}\psi(k)\rhd\left(\vphantom{\bigvee_{a}}G(k)(fk)\ast
fk\right)\leq\bigwedge_{k}\psi(k)\rhd\left(\bigvee_{a}G(k)(a)\ast a\right)$

hence

(17)
$\bigvee_{f}\bigwedge_{k}\psi(k)\rhd\left(\vphantom{\bigvee_{i}}G(k)(fk)\ast
fk\right)\leq\bigwedge_{k}\psi(k)\rhd\left(\bigvee_{a}G(k)(a)\ast a\right)$

and the proof is finished. ∎

######  Remark 3.8.

  1. (1)

We already know from [17] that assuming the quantale to be completely
distributive entails that each completely distributive $\Omega$-category is
also completely distributive as a lattice. To see Theorem 3.7 at work, we
shall recover the cited result of Lai and Zhang by choosing trivial weights
$\psi(k)=e$ and discrete $\Omega$-subsets $G(k)$ corresponding to a family of
ordinary subsets $(A_{k})_{k\in K}$ of $A$; that is,

$G(k)(a)=\begin{cases}e\mbox{ for }a\in A_{k}\\\ \bot\mbox{ otherwise
}\end{cases}\forall\,k$

Then Equation (13) becomes

${\bigwedge_{k}\bigvee A_{k}=\bigvee_{\\{f:K\to A\,\mid\,fk\in
A_{k}\\}}\bigwedge_{k}fk}\ .$

The reader will recognise the usual law of complete distributivity via choice
functions.

  2. (2)

Again by [17], each completely distributive $\Omega$-category is a homomorphic
image of a subobject of a product of copies of $\Omega$.141414This is the
$\Omega$-enriched generalization of Raney’s well-known result that a complete
lattice is completely distributive if and only if it is the homomorphic image
of a ring of sets [29]. This can be immediately seen as follows: first, any
completely distributive $\Omega$-category ${\mathscr{A}}$ is a retract of
$\mathbbm{D}{\mathscr{A}}$ via
${\mathsf{sup}_{{\mathscr{A}}}{}}:\mathbbm{D}{\mathscr{A}}\to{\mathscr{A}}$,
which is both continuous and cocontinuous, hence ${\mathscr{A}}$ is a quotient
of $\mathbbm{D}{\mathscr{A}}$ in $\Omega\mbox{-}\hskip 0.5pt\mathsf{CD}$.
Second, the inclusion functor
$\mathbbm{D}{\mathscr{A}}\to[\mathsf{d}A,\Omega]=\Omega^{A}$ is both
continuous and cocontinuous (with adjoints provided by Kan extensions), hence
$\mathbbm{D}{\mathscr{A}}$ is a subobject of $\Omega^{A}$ in
$\Omega\mbox{-}\hskip 0.5pt\mathsf{CD}$. That is, $\Omega\mbox{-}\hskip
0.5pt\mathsf{CD}=\mathsf{HSP}(\Omega)$. In particular, an equation holds in a
completely distributive $\Omega$-category ${\mathscr{A}}$ if and only if it
holds in $\Omega$. This is perhaps best illustrated by the following Corollary
(compare also with [18, Corollary 4.14]):

######  Corollary 3.9.

Let
${\mathscr{A}}=(A,\bigvee,\bigwedge,(v\ast-)_{v\in\Omega},(v\rhd-)_{v\in\Omega})$
be a cocomplete skeletal $\Omega$-category. Then ${\mathscr{A}}$ satisfies
(13) if and only if ${\mathscr{A}}$ is completely distributive as a lattice
and cotensors in ${\mathscr{A}}$ preserve non-empty joins:

(18) $v\rhd\bigvee_{i}a_{i}=\bigvee_{i}(v\rhd a_{i})$

###### Proof.

The complete distributivity of ${\mathscr{A}}$ as a lattice has been discussed
above. Take now $K=\\{0\\}$, $\psi(0)=v$, $G(k)(a_{i})=e$ and $G(k)(-)=\bot$
otherwise to obtain (18). Conversely, using that cotensors preserve non-empty
joins, that each complete $\Omega$-category is non-empty, and finally the
complete distributivity of the underlying lattice
$({\mathscr{A}},\bigvee,\bigwedge)$, we have that

$\displaystyle\bigwedge_{k}\psi(k)\rhd\left(\bigvee_{a}G(k)(a)\ast a\right)$
$\displaystyle=$
$\displaystyle\bigwedge_{k}\bigvee_{a}\left(\vphantom{\bigvee_{i}}\psi(k)\rhd\left(G(k)(a)\ast
a\right)\right)$ $\displaystyle=$
$\displaystyle\bigvee_{f}\bigwedge_{k}\left(\vphantom{\bigvee_{i}}\psi(k)\rhd(G(k)(fk)\ast
fk)\right)$

for each functions $\psi:K\to\Omega$, $G:K\to\Omega^{A}$, hence we recover the
complete distributivity law with obtained in Theorem 3.7. ∎

######  Remark 3.10.

Earlier we mentioned that the equational theory of $\Omega\mbox{-}\hskip
0.5pt\mathsf{CD}$ is generated by $\Omega$, regardless of Assumption. Hence
there is no hope for obtaining various variants of distributivity laws (like
the one discussed below) unless these already hold in $\Omega$.

For example, observe that the parentheses in expressions involving tensors and
cotensors, e.g. $v\rhd(w\ast a)$, do not change in the above law (13) of
complete $\Omega$-distributivity. One may then ask whether the distributivity-
like property

$v\rhd(w\ast a)=[v,w]\ast a$

could be of interest for $\Omega\mbox{-}\hskip 0.5pt\mathsf{CD}$. But the
corresponding relation for $\Omega$ itself, namely
$[v,w{\otimes}u]=[v,w]{\otimes}u$, implies that the quantale is trivial. To
see this, observe that $e\leq[v,v]=[v,e\otimes v]=[v,e]\otimes v\leq e$.
Therefore $\Omega$ is an ordered abelian group [19], hence it cannot be
bounded as a lattice unless it is trivial [1].

Consequently, completely distributive $\Omega$-categories are not so
distributive after all. ∎

Before ending the paper, we point out one more issue: comparing the
constructive versus the non-constructive equations of $\Omega$-completely
distributivity, namely (16) and (13), as recalled below, we see that the left
hand sides coincide.

(19) $\displaystyle\bigwedge_{k\in K}\psi(k)\rhd\left(\bigvee_{a\in
A}G(k)(a)\ast a\right)$ $\displaystyle=$ $\displaystyle{\bigvee_{a\in
A}\left(\bigwedge_{k\in K}[\psi(k),G(k)^{\downarrow}(a)]\right)\ast a}$
$\displaystyle\bigwedge_{k\in K}\psi(k)\rhd\left(\bigvee_{a\in A}G(k)(a)\ast
a\right)$ $\displaystyle=$
$\displaystyle{\bigvee_{f\in\mathcal{F}}\bigwedge_{k\in
K}\psi(k)\rhd(G(k)(fk)\ast fk)}\ .$

A careful inspection of the proof of Theorem 3.7 reveals that the inequality

$\bigvee_{a\in A}\left(\bigwedge_{k\in
K}[\psi(k),G(k)^{\downarrow}(a)]\right)\ast
a\leq\bigvee_{f\in\mathcal{F}}\bigwedge_{k\in K}\psi(k)\rhd(G(k)(fk)\ast fk)$

holds in $\Omega\mbox{-}\hskip 0.5pt\mathsf{CD}$ if Assumption holds. We shall
now see an example where the inequality is not true.

######  Example 3.11.

Take one of the two non-distributive quantales $M_{3}$ and $N_{5}$ from
Example 2.14. Then $\Omega$-distributivity (16) always holds in $\Omega$, but
(13), which implies lattice-distributivity, does not hold in the non-
distributive lattices $M_{3}$ and $N_{5}$. Let us consider for example
$\Omega=M_{3}$. Put $K=\\{0,1\\}$ and take the constant trivial weight
$\psi(0)=\psi(1)=e$. Also, consider the following $K$-indexed family of
$\Omega$-subsets of $\Omega$ itself, seen as an $\Omega$-category:

$G(k)(x)=\begin{cases}e&k=0,x=e\mbox{ or }k=1,x\in\\{a,b\\}\\\
\bot&\mbox{otherwise}\end{cases}\ .$

Equation (13) then becomes $e\wedge(a\vee b)=(e\wedge a)\vee(e\wedge b)$,
which interpreted in $\Omega=M_{3}$ is false.

During the evaluation the right-hand side of (13), observe that the ordered
downsets $G(0)^{\downarrow}$ and $G(1)^{\downarrow}$ are, respectively,
$\\{\bot,e\\}$ and $\\{\bot,a,b\\}$, hence their intersection is adequately
computed by $(e\wedge a)\vee(e\wedge b)=\bot$. But to compute the “enriched
meet”, we need to go back to the formula of the “weighted downsets” (3).

$\displaystyle\begin{array}[]{|c||c|c|c|c|c|}\hline\cr[-,-]&\bot&a&e&b&\top\\\
\hline\cr\hline\cr\bot&\top&\top&\top&\top&\top\\\ a&b&\top&b&b&\top\\\
e&\bot&a&e&b&\top\\\ b&a&a&a&\top&\top\\\ \top&\bot&a&\bot&b&\top\\\
\hline\cr\end{array}$ $\displaystyle\begin{array}[]{|c||c|c|c|c|c|}\hline\cr
x&\bot&a&e&b&\top\\\ \hline\cr\hline\cr
G(0)^{\downarrow}(x)&\top&b&e&a&\bot\\\ \hline\cr
G(1)^{\downarrow}(x)&\top&\top&\top&\top&\top\\\ \hline\cr\end{array}$

The table above on the left exhibits the internal hom for the quantale
structure of $M_{3}$ of Example 2.14. Therefore $G(0)^{\downarrow}(x)=[x,e]$
and $G(1)^{\downarrow}(x)=[x,a]\vee[x,b]$. We see that the enrichment does
change $G(1)^{\downarrow}$ significantly. One reason is – as for example the
calculation
$G(1)^{\downarrow}(e)=\left(G(1)(a)\otimes[e,a]\right)\vee\left(G(1)(b)\otimes[e,b]\right)=[e,a]\vee[e,b]=a\vee
b=\top$ shows – that due to the tensor products $e\otimes a=a$ and $e\otimes
b=b$ the element $e$ can be “drawn into” the downset $\\{\bot,a,b\\}$ even if
$e$ is not below $a$ or $b$. Another reason, is that while $[\top,a]$
evaluates to zero over $\mathbbm{2}$, it evaluates to $[\top,a]=a$ over
$M_{3}$, so that $G(1)^{\downarrow}(\top)=[\top,a]\vee[\top,b]=a\vee b=\top$.
In fact, the only difference to the right-hand side of (13) is the presence of
the $\Omega$-down-closure in the above calculation, but these, read in the
appropriate enriched way, do enlarge the downsets just enough to make the
equality work again.

## acknowledgements

The authors wish to thank the organisers of the special session Mathematical
Structures in Formal System Development and Analysis of the 9th Congress of
Romanian Mathematicians (2019) for inviting them to present their work at the
congress.

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