nLab Stone gamut

Idea

The Stone gamut is a unifying representation of several common categories in terms of Chu spaces. It is named after Stone duality. The study of Chu duality and the Stone gamut is known as chupology. (Pratt 2006)

Definition

Let $Chu_2$ be the category of Chu spaces over the Booleans, with $r : Chu_2 \to \mathbb{N}$ and $c : Chu_2 \to \mathbb{N}$ counting the number of rows and columns respectively, and let $FinChu_2$ be its full subcategory of finite Chu spaces. Let a skeleton of a Chu space be another Chu space with no repeated rows or repeated columns, and let the dual of a Chu space $A$, written $A^{\bot}$, be its transpose. Let a Chu space be skeletal if it is a skeleton. Let a discreteness be a function $\delta : FinChu_2 \to [-1, 1]$ such that:

• $\delta$ depends only on the number of rows and columns in its input’s skeleton, and
• $\delta(A^{\bot}) = -\delta(A)$

For our purposes, we will consider Pratt discreteness, defined as:

$$\delta(A) = \frac{P(A) - Q(A)}{P(A) + Q(A)}$$

Where $P(A) = |2^{r(A)} - c(A)|$ and $Q(A) = |2^{c(A)} - r(A)|$. Intuitively, $P$ and $Q$ count the number of “missing” rows and columns in the skeleton.

A property of a skeletal Chu space is a superset of its columns. (Def. 5, Pratt 1999)

Properties

Note that when $r(A) \gt c(A)$, $r(A) \leq 2^{c(A)}$; and dually, when $c(A) \gt r(A)$, $c(A) \leq 2^{r(A)}$, by the pigeonhole principle.

Pratt notes that their $\delta$ has the “odd property that the only discreteness possible in $[-\frac{1}{3}, \frac{1}{3}]$ is 0” (Pratt 1995).

Examples

Contravariant dualities

Category	$\delta$	Spaces $\cong$ Quantities	$\delta$	Category
CABA	-1	CABAs/overlap algebras $\cong$ sets	1	Set
SupLat	0	suplattices $\cong$ inflattices	0	InfLat
$Vect_{\mathbb{F}_2}$	0	vector spaces over $\mathbb{F}_2$ $\cong$ dual vector spaces over $\mathbb{F}_2$	0	$Vect_{\mathbb{F}_2}$

References

• Vaughan Pratt, The Stone gamut: a Coordinatization of Mathematics, Proceedings of Tenth Annual IEEE Symposium on Logic in Computer Science (1995) [doi:10.1109/LICS.1995.523278, pdf]
• Vaughan Pratt, Chu Spaces, School on Category Theory and Applications, University of Coimbra (1999) [notes (pdf)]
• Vaughan Pratt, dualities, Categories list mailing list (2006) [list archive]