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
\[\overline F\dashv \overline G: \S \mathcal A \to \mathcal \S B\]

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.

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.