Dualities from Double Categories of Relations (Slides)

## [Dualities from Double Categories of Relations](https://alexhkurz.github.io/papers/Abstract-BLAST2022-duality-from-double-categories-of-relations-2.pdf) ###### BLAST 2022, Aug 10 ###### Alexander Kurz ###### Chapman Univeristy --- <style type="text/css"> .reveal p { text-align: left; } .reveal ul { display: block; } .reveal ol { display: block; } .reveal { font-size: 36px; } </style> ### Stone Duality Boolean Algebras --- Stone spaces Distributive lattices --- Priestley spaces Etc Replacing functions by relations? --- ### Stone Duality for Relations 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 **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 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 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) (co)limits in $\sf Rel(\mathcal C)$ are not well-behaved ... $\quad$ ... but (co)limits in $\S \mathcal C$ are the dual of $\S\mathcal C = \mathbb C_1 \to \mathbb C_0$ we will need is $\mathbb C_1^{op} \to \mathbb C_0^{op}$ ... $\quad$ ... direction of relations not reversed under duality $Pos^{op}\dashv DL$ lifts to $\S Pos^{op}\dashv \S DL$ ... $\quad$ ... but not to $\sf Rel(\mathit{Pos})$ and $\sf Rel(\mathit{DL})$ --- ### Why Double Categories (2) In (KMJ 2021): "vertical truncation" of $\S\mathcal C = \mathbb C_1 \to \mathbb C_0$ ... ... which is a category with **relations as arrows**. 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) 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) **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 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 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.