## [Dualities from Double Categories of Relations](https://alexhkurz.github.io/papers/Abstract-BLAST2022-duality-from-double-categories-of-relations-2.pdf)
<br>
###### BLAST 2022, Aug 10
<br>
###### Alexander Kurz
###### Chapman Univeristy
<br>
###### <font size="-1" color=grey>(layout optimized for Brave browser)</font>
---
<style type="text/css">
.reveal p {
text-align: left;
}
.reveal ul {
display: block;
}
.reveal ol {
display: block;
}
.reveal {
font-size: 36px;
}
</style>
### Stone Duality
<br>
Boolean Algebras --- Stone spaces
Distributive lattices --- Priestley spaces
Etc
Replacing functions by relations?
---
### Stone Duality for Relations
<br>
Relations as *arrows* (KMJ 2021):
- Rel(DL) --- Rel(Priestley)
- Rel(BA) --- Rel(Stone)
In this talk: *objects* with relational structure.
Method: A double category
$$ dom,cod:\mathbb C_1 \to \mathbb C_0$$
where *relations are the objects* of the category $\mathbb C_1$
(Grandis and Pare, 1999, 2004), (Shulman, 2008)
---
### Weakening Relations
<br>
**Definition:** A *relation* $R:A\looparrowright B$ is a sub-object
$$R\subseteq A\times B$$
and a monotone map
$$A^{op}\times B\to 2$$
where $2=\{0<1\}$.
**Remark:** Also called a *monotone* or a *weakening* (closed) relation.
---
### A Paradigmatic Example
<br>
A relation in $\sf DL$ (bounded distributive lattices) is similar to a sequent caclulus turnstile:
Taking subobjects in $\sf DL$ amounts to
$$
\frac{}{0 R 0}
\quad\quad
\frac{}{1 R 1}
\quad\quad
\frac{aRb \quad a'Rb'}{(a\wedge a') R (b\wedge b')}
\quad\quad
\frac{aRb \quad a'Rb'}{(a\vee a') R (b\vee b')}
$$
while the monotonicity condition amounts to weakening
$$\frac{a'\le a \quad a R b \quad b \le b'}{a'Rb'}$$
---
### Double Categories of Relations
<br>
Given a category $\mathcal C$ wsp, we can form the double category
$$\S\mathcal C = (\S\mathcal C)_1 \to \mathcal C$$
where $(\S\mathcal C)_1$ has relations as objects and "squares" as arrows $R\to R'$
$$
\begin{array}{ccc}
A & \stackrel R \to &B \\
{\downarrow} & \Downarrow & {\downarrow} \\
C & \stackrel {R'} \to & D \\
\end{array}
\quad\quad\quad\quad a\, R\, b \Rightarrow f(a)\,R'\,g(b)
$$
---
### Why Double Categories (1)
<br>
(co)limits in $\sf Rel(\mathcal C)$ are not well-behaved ...
$\quad$ <font color=grey>... but (co)limits in $\S \mathcal C$ are </font>
the dual of $\S\mathcal C = \mathbb C_1 \to \mathbb C_0$ we will need is $\mathbb C_1^{op} \to \mathbb C_0^{op}$ ...
$\quad$ <font color=grey>... direction of relations not reversed under duality</font>
$Pos^{op}\dashv DL$ lifts to $\S Pos^{op}\dashv \S DL$ ...
$\quad$ <font color=grey>... but not to $\sf Rel(\mathit{Pos})$ and $\sf Rel(\mathit{DL})$</font>
---
### Why Double Categories (2)
<br>
In (KMJ 2021): "vertical truncation" of $\S\mathcal C = \mathbb C_1 \to \mathbb C_0$ ...
... which is a category with **relations as arrows**.
<br>
Here: the sub-double-category of $\S\mathcal C = \mathbb C_1 \to \mathbb C_0$ which restricts to the endo-relations in $\mathbb C_1$ ...
... which has **relational structures as objects**.
---
### Stone Duality for Relations (1)
<br>
In (KMJ 2021) we give sufficient conditions for an adjunction
$$F\dashv G: \mathcal A \to \mathcal B$$
to extend to an adjunction
$$\overline F\dashv \overline G: \S \mathcal A \to \mathcal \S B$$
---
![](https://hackmd.io/_uploads/SyxvXWXlRc.png)
---
### Stone Duality for Relations (2)
<br>
**Theorem (KMJ 2021):** The double category of DL-relations is dually equivalent to the double category of Priestley-relations.
**Corollary:** The category determined by objects $(X,R)$ where $X$ is a Priestley space and $R$ is a Priestley-relation on $X$ is dually equivalent to the category determined by objects $(A,\prec)$ where $A$ is a bounded distributive lattice and $\prec$ is a DL-relation (aka subordination) on $A$.
---
### Corollaries
<br>
Rephrasing the corollary from the previous slide:
**Corollary:** The category of Priestley spaces with closed weakening relations is dually equivalent to (DL-based) subordination algebras.
We now look at further corollaries that can be obtained by adding conditions on either side of the duality, see [the BLAST 2022 abstract](https://alexhkurz.github.io/papers/Abstract-BLAST2022-duality-from-double-categories-of-relations-2.pdf).
---
### Conclusion
<br>
If you work with categories of relations: maybe double categories of relations can help.
If you encounter $R[a]\subseteq b$: maybe Stone duality for relations is lurking in the background.
If you are interested in duality for relational structures: maybe looking at relations as "horizontal arrows" of a double category reveals a new point of view.

{"title":"Dualities from Double Categories of Relations (Slides)","slideOptions":{"theme":"white","transition":"fade","transitionSpeed":"slow","spotlight":{"enabled":true},"controls":false}}