nLab arity space

Arity spaces

Arity spaces


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}\{0,1\} (coherence spaces), {0,1,2,3,}\{0,1,2,3,\dots\} (finiteness spaces), and {1}\{1\} (totality spaces), which are all arity classes.


Let κ\kappa be a set of cardinal numbers. Given two subsets u,vXu,v\subseteq X of the same set XX, we write uvu\perp v if |uv|κ|u\cap v|\in \kappa. This relation defines a Galois connection in the usual way: for 𝒰P(X)\mathcal{U}\subseteq P(X) we have 𝒰 ={vu𝒰.uv}\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 κ\kappa-arity space to be a set XX together with a 𝒰P(X)\mathcal{U}\subseteq P(X) that is a fixed point of this Galois connection, 𝒰=𝒰 \mathcal{U} = \mathcal{U}^{\perp\perp}. We call the sets in 𝒰\mathcal{U} κ\kappa-ary and the sets in 𝒰 \mathcal{U}^{\perp} co-κ\kappa-ary.

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

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


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

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

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


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

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

Where L κL_\kappa maps a set XX to the orthogonality |UV|κ|U \cap V| \leq \kappa on Subset(X)×Subset(X)Subset(X) \times Subset(X) and a pair of sets X,YX, Y is given the trivial orthogonality xy=x \perp y = \bot

Last revised on August 18, 2022 at 12:48:55. See the history of this page for a list of all contributions to it.