Hausdorff Coalgebras

Abstract

As composites of constant, finite (co)product, identity, and powerset functors, Kripke polynomial functors form a relevant class of $\textsf {Set}$-functors in the theory of coalgebras. The main goal of this paper is to expand the theory of limits in categories of coalgebras of Kripke polynomial functors to the context of quantale-enriched categories. To assume the role of the powerset functor we consider “powerset-like” functors based on the Hausdorff ${\mathcal {V}}$-category structure. As a starting point, we show that for a lifting of a $\textsf {Set}$-functor to a topological category $\textsf {X}$ over $\textsf {Set}$ that commutes with the forgetful functor, the corresponding category of coalgebras over $\textsf {X}$ is topological over the category of coalgebras over $\textsf {Set}$ and, therefore, it is “as complete” but cannot be “more complete”. Secondly, based on a Cantor-like argument, we observe that Hausdorff functors on categories of quantale-enriched categories do not admit a terminal coalgebra. Finally, in order to overcome these “negative” results, we combine quantale-enriched categories and topology à la Nachbin. Besides studying some basic properties of these categories, we investigate “powerset-like” functors which simultaneously encode the classical Hausdorff metric and Vietoris topology and show that the corresponding categories of coalgebras of “Kripke polynomial” functors are (co)complete.

Appendix

In this section we collect some facts about ${\mathcal {V}}$-categories and ${\mathcal {V}}$-functors, where ${\mathcal {V}}$ is a quantale; for more information we refer to [39, 53]. Furthermore, we present some useful properties of the reflector into the category of separated ${\mathcal {V}}$-categories that follow from standard arguments, but seem to be absent from the literature.

Definition A.1

Let ${\mathcal {V}}$ be a commutative and unital quantale. A ${\mathcal {V}}$-category is a pair (X, a) consisting of a set X and a map $a :X\times X\rightarrow {\mathcal {V}}$ satisfying

$$\begin{aligned} k\le a(x,x)&\text {and}&a(x,y)\otimes a(y,z)\le a(x,z), \end{aligned}$$

for all $x,y,z\in X$. Given ${\mathcal {V}}$-categories (X, a) and (Y, b), a ${\mathcal {V}}$-functor $f :(X,a)\rightarrow (Y,b)$ is a map $f :X\rightarrow Y$ such that

$$\begin{aligned} a(x,y) \le b(f(x),f(y)), \end{aligned}$$

for all $x,y \in X$.

In particular, the quantale ${\mathcal {V}}$ becomes a ${\mathcal {V}}$-category with structure $\hom :{\mathcal {V}}\times {\mathcal {V}}\rightarrow {\mathcal {V}}$. We refer to [53] for a list of examples of quantales ${\mathcal {V}}$ and the corresponding categories ${\mathcal {V}}\text {-}\textsf {Cat}$ of ${\mathcal {V}}$-categories and ${\mathcal {V}}$-functors.

For every ${\mathcal {V}}$-category (X, a), $a^\circ (x,y)=a(y,x)$ defines another ${\mathcal {V}}$-category structure on X, and the ${\mathcal {V}}$-category $(X,a)^\text {op}:=(X,a^\circ )$ is called the dual of (X, a). A ${\mathcal {V}}$-category (X, a) is called symmetric whenever $(X,a)=(X,a)^\text {op}$.

Clearly, ${\mathcal {V}}$-categories and ${\mathcal {V}}$-functors define a category, denoted as ${\mathcal {V}}\text {-}\textsf {Cat}$. The full subcategory of ${\mathcal {V}}\text {-}\textsf {Cat}$ defined by all symmetric ${\mathcal {V}}$-categories is denoted as ${\mathcal {V}}\text {-}\textsf {Cat}_\text {sym}$.

Remark A.2

Given ${\mathcal {V}}$-categories (X, a) and (Y, b), we define the tensor product of (X, a) and (Y, b) to be the ${\mathcal {V}}$-category $(X,a) \otimes (Y,b) = (X \times Y, a \otimes b)$, with

$$\begin{aligned} a \otimes b ((x,y),(x',y')) = a(x,x') \otimes b(y,y'). \end{aligned}$$

This operation makes ${\mathcal {V}}\text {-}\textsf {Cat}$ a symmetric monoidal closed category, where the internal $\hom$ of (X, a) and (Y, b) is the ${\mathcal {V}}$-category $[(X,a), (Y,b)] = ({\mathcal {V}}\text {-}\textsf {Cat}((X,a),(X,b)), [-,-])$, with

$$\begin{aligned} {[}f,g] = \bigwedge _{x \in X} b(f(x), g(x)). \end{aligned}$$

We note that [(X, a), (Y, b)] is a ${\mathcal {V}}$-subcategory of the X-fold product $(Y,b)^X$ of (Y, b).

The following propositions are particularly useful to construct ${\mathcal {V}}$-functors when combined with the fact that ${\mathcal {V}}\text {-}\textsf {Cat}$ is symmetrical monoidal closed.

Proposition A.3

For every set I, the assignments $f \mapsto \bigvee _{i \in I} f(i)$ and $f \mapsto \bigwedge _{i \in I} f(i)$ define ${\mathcal {V}}$-functors of type ${\mathcal {V}}^I\rightarrow {\mathcal {V}}$.

Proposition A.4

For every ${\mathcal {V}}$-category (X, a), the map $a :(X,a)^\text {op}\otimes (X,a) \rightarrow ({\mathcal {V}}, \hom )$ is a ${\mathcal {V}}$-functor.

The category ${\mathcal {V}}\text {-}\textsf {Cat}$ is well behaved regarding (co)limits.

Theorem A.5

The canonical forgetful functor ${\mathcal {V}}\text {-}\textsf {Cat}\rightarrow \textsf {Set}$ is topological. For a structured cone $(f_i :X\rightarrow (X_i,a_i))$, the initial lift (X, a) is given by

$$\begin{aligned} a(x,y) = \bigwedge _{i\in I}a_i(f_i(x),f_i(y)), \end{aligned}$$

for all $x,y\in X$. Moreover, ${\mathcal {V}}\text {-}\textsf {Cat}_\text {sym}$ is closed in ${\mathcal {V}}\text {-}\textsf {Cat}$ under initial cones; therefore the canonical forgetful functor ${\mathcal {V}}\text {-}\textsf {Cat}_\text {sym}\rightarrow \textsf {Set}$ is topological as well, and the inclusion functor ${\mathcal {V}}\text {-}\textsf {Cat}_\text {sym}\hookrightarrow {\mathcal {V}}\text {-}\textsf {Cat}$ has a left adjoint.

We also recall that ${\mathcal {V}}\text {-}\textsf {Cat}_\text {sym}\hookrightarrow {\mathcal {V}}\text {-}\textsf {Cat}$ has a concrete right adjoint which sends the ${\mathcal {V}}$-category (X, a) to its symmetrisation $(X,a_s)$ given by

$$\begin{aligned} a_s(x,y) = a(x,y) \wedge a(y,x), \end{aligned}$$

for all $x,y\in X$.

Every ${\mathcal {V}}$-category (X, a) carries a natural order defined by

$$\begin{aligned} x\le y \text { whenever } k\le a(x,y), \end{aligned}$$

which can be extended pointwise to ${\mathcal {V}}$-functors making ${\mathcal {V}}\text {-}\textsf {Cat}$ a 2-category. The natural order of ${\mathcal {V}}$-categories defines a faithful functor ${\mathcal {V}}\text {-}\textsf {Cat}\rightarrow \textsf {Ord}$. A ${\mathcal {V}}$-category is called separated whenever its underlying ordered set is anti-symmetric, and we denote by ${\mathcal {V}}\text {-}\textsf {Cat}_\text {sep}$ the full subcategory of ${\mathcal {V}}\text {-}\textsf {Cat}$ defined by all separated ${\mathcal {V}}$-categories. Tautologically, an ordered set is separated if and only if it is anti-symmetric.

Theorem A.6

${\mathcal {V}}\text {-}\textsf {Cat}_\text {sep}$ is closed in ${\mathcal {V}}\text {-}\textsf {Cat}$ under monocones. Hence, the forgetful functor ${\mathcal {V}}\text {-}\textsf {Cat}_\text {sep}\rightarrow \textsf {Set}$ is mono-topological and the inclusion functor ${\mathcal {V}}\text {-}\textsf {Cat}_\text {sep}\hookrightarrow {\mathcal {V}}\text {-}\textsf {Cat}$ has a left adjoint.

Let us describe the left adjoint $S :{\mathcal {V}}\text {-}\textsf {Cat}\rightarrow {\mathcal {V}}\text {-}\textsf {Cat}_\text {sep}$ of ${\mathcal {V}}\text {-}\textsf {Cat}_\text {sep}\hookrightarrow {\mathcal {V}}\text {-}\textsf {Cat}$. To do so, consider a ${\mathcal {V}}$-category (X, a). Then

$$\begin{aligned} x\sim y \qquad \text {whenever}\qquad x\le y\quad \text {and}\quad y\le x \end{aligned}$$

defines an equivalence relation on X, and the quotient set $X/{\sim }$ becomes a ${\mathcal {V}}$-category $(X/{\sim },{\widetilde{a}})$ by putting

$$\begin{aligned} {\widetilde{a}}([x],[y])=a(x,y); \end{aligned}$$

(A.1)

this is indeed independent of the choice of representants of the equivalence classes. Then the projection map

$$\begin{aligned} q_{(X,a)} :X \longrightarrow X/{\sim },\,x\longmapsto [x] \end{aligned}$$

is a ${\mathcal {V}}$-functor $q_{(X,a)} :(X,a)\rightarrow (X/{\sim },{\widetilde{a}})$, it is indeed the unit of this adjunction at (X, a). Furthermore, by (A.1), $q_{(X,a)} :(X,a)\rightarrow (X/{\sim },{\widetilde{a}})$ is a universal quotient and initial with respect to ${\mathcal {V}}\text {-}\textsf {Cat}\rightarrow \textsf {Set}$.

Lemma A.7

A cone $(f_i :(X,a)\rightarrow (X_i,a_i))_{i\in I}$ in ${\mathcal {V}}\text {-}\textsf {Cat}_\text {sep}$ is initial with respect to ${\mathcal {V}}\text {-}\textsf {Cat}_\text {sep}\rightarrow \textsf {Set}$ if and only if

$$\begin{aligned} a(x,y)=\bigwedge _{i\in I}a_i(f_i(x),f_i(y)), \end{aligned}$$

(A.2)

for all $x,y\in X$.

Proof

Clearly, if (A.2) is satisfied then $(f_i :(X,a)\rightarrow (X_i,a_i))_{i\in I}$ is initial with respect to ${\mathcal {V}}\text {-}\textsf {Cat}_\text {sep}\rightarrow \textsf {Set}$ since it is initial with respect to ${\mathcal {V}}\text {-}\textsf {Cat} \rightarrow \textsf {Set}$. Suppose now that $(f_i :(X,a)\rightarrow (X_i,a_i))_{i\in I}$ is initial with respect to ${\mathcal {V}}\text {-}\textsf {Cat}_\text {sep}\rightarrow \textsf {Set}$. Fix $x,y\in X$. Then

$$\begin{aligned} a(x,y) \le \bigwedge _{i\in I}a_i(f_i(x),f_i(y))=u \end{aligned}$$

because $f_i :(X,a) \rightarrow (X_i, a_i)$ is a ${\mathcal {V}}$-functor for every $i \in I$. It is left to show that $u\le a(x,y)$. This is certainly true if $u=\bot$; assume now that $\bot <u$. Let $2_u$ be the separated ${\mathcal {V}}$-category with underlying set $\{0,1\}$ and structure $a_u$ defined by

$$\begin{aligned} a_u(0,1)=u,\quad a_u(0,0)=a_u(1,1)=k,\quad \text {and}\quad a_u(1,0)=\bot . \end{aligned}$$

Consider $h :\{0,1\}\rightarrow X$ with $h(0)=x$ and $h(1)=y$. Then $f_i\cdot h$ is a ${\mathcal {V}}$-functor, for every $i\in I$. Hence, since $(f_i :(X,a)\rightarrow (X_i,a_i))_{i\in I}$ is initial, $h :2_u\rightarrow X$ is a ${\mathcal {V}}$-functor, which implies $u\le a(x,y)$. $\square$

Corollary A.8

The functor $\textsf {S}:{\mathcal {V}}\text {-}\textsf {Cat}\rightarrow {\mathcal {V}}\text {-}\textsf {Cat}_\text {sep}$ preserves initial cones with respect to the canonical forgetful functors.

Proof

Let $(f_i :(X,a)\rightarrow (X_i,a_i))_{i\in I}$ be an initial cone with respect to ${\mathcal {V}}\text {-}\textsf {Cat} \rightarrow \textsf {Set}$. Then, for every $[x],[y] \in \textsf {S}(X,a)=(X/{\sim },{\widetilde{a}})$, and with $\textsf {S}(X_i,a_i)=(X/{\sim },{\widetilde{a}}_i)$ for all $i\in I$,

$$\begin{aligned} {\widetilde{a}}([x],[y])= & {} a(x,y) = \bigwedge _{i\in I} a_i(f_i(x),f_i(y)) = \bigwedge _{i\in I} \widetilde{a_i}([f_i(x)],[f_i(y)]) \\= & {} \bigwedge _{i\in I} \widetilde{a_i}(\textsf {S}f_i([x]),\textsf {S}f_i([y])). \end{aligned}$$

Therefore, the claim follows by Lemma A.7. $\square$

Remark A.9

In [17] it is shown that $\textsf {S}:{\mathcal {V}}\text {-}\textsf {Cat}\rightarrow {\mathcal {V}}\text {-}\textsf {Cat}_\text {sep}$ preserves finite products. However, $\textsf {S}$ does not preserve limits in general, in particular, $\textsf {S}$ does not preserve codirected limits. For instance, consider the “empty limit” of [56] and equip every $X_i$ ($i\in I$) with the indiscrete ${\mathcal {V}}$-category structure $a_i$ where $a_i(x,y)=\top$ for all $x,y\in X_i$. Then $S(X_i,a_i)$ has exactly one element, for each $i\in I$; hence the limit of the corresponding diagram in ${\mathcal {V}}\text {-}\textsf {Cat}_\text {sep}$ has one element.