nLab arity space

Idea

An arity space is a common generalization of coherence spaces, finiteness spaces, and totality spaces to an arbitrary set of “arities”.

This is an original and tentative definition. In particular, it’s not clear whether the allowed sets of arities should be restricted in some way. Should they be an arity class? The only previously studied examples appear to be the cases $\{0,1\}$ (coherence spaces), $\{0,1,2,3,\dots\}$ (finiteness spaces), and $\{1\}$ (totality spaces), which are all arity classes.

Definition

Let $\kappa$ be a set of cardinal numbers. Given two subsets $u,v\subseteq X$ of the same set $X$, we write $u\perp v$ if $|u\cap v|\in \kappa$. This relation defines a Galois connection in the usual way: for $\mathcal{U}\subseteq P(X)$ we have $\mathcal{U}^\perp = \{ v \mid \forall u\in \mathcal{U}. u\perp v \}$. Since $\perp$ is symmetric, $(-)^\perp$ is self-adjoint on the right.

We define a $\kappa$-arity space to be a set $X$ together with a $\mathcal{U}\subseteq P(X)$ that is a fixed point of this Galois connection, $\mathcal{U} = \mathcal{U}^{\perp\perp}$. We call the sets in $\mathcal{U}$ $\kappa$-ary and the sets in $\mathcal{U}^{\perp}$ co-$\kappa$-ary.

A morphism or relation between $\kappa$-arity spaces is a relation $R: X ⇸ Y$ such that

1. If $u\subseteq X$ is $\kappa$-ary, then $R[u] = \{ y \mid \exists x\in u, R(x,y) \}$ is $\kappa$-ary.
2. If $v\subseteq Y$ is co-$\kappa$-ary, then $R^{-1}[v] = \{ x \mid \exists y\in v, R(x,y) \}$ is co-$\kappa$-ary.

Examples

• If $\kappa=\{0,1\}$, then a $\kappa$-arity space is precisely a coherence space.

• If $\kappa = \omega = \{0,1,2,3,\dots\}$, then a $\kappa$-arity space is precisely a finiteness space.

• If $\kappa=\{1\}$, then a $\kappa$-arity space is (almost?) precisely a totality space.

Properties

Conjecture: For any $\kappa$, the category of $\kappa$-arity spaces is star-autonomous.

This might follow from constructing it using double gluing and orthogonality.

Construction as a Comma Double Category

We can define arity spaces by a variation on the double gluing construction.

Define a double category $Orth$ of orthogonalities 1. Objects are relations $\bot \subseteq X \times Y$ 2. A vertical morphism from $(X_1,Y_1,\perp_1)$ to $(X_2, Y_2, \perp_2)$ exists when $X_1$ and $Y_1$ are orthogonal subsets of $X_2$ and $Y_2$ respectively. 3. A horizontal morphism from $(X_1,Y_1,\perp_1)$ to $(X_2,Y_2,\perp_2)$ is a pair of a function $f_* : X_1 \to X_2$ and $f^* : Y_2 \to Y_1$ 4. A square from $f$ to $g$ exists when $f_*$ is the restriction of $g_*$ and similarly for $f^*$ and $g^*$.

Then the (2-)category of arity spaces can be defined as the comma double category (where $Rel$ and $Set$ are viewed as vertically discrete double categories:

Where $L_\kappa$ maps a set $X$ to the orthogonality $|U \cap V| \leq \kappa$ on $Subset(X) \times Subset(X)$ and a pair of sets $X, Y$ is given the trivial orthogonality $x \perp y = \bot$