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 (coherence spaces), (finiteness spaces), and (totality spaces), which are all arity classes.
Let be a set of cardinal numbers. Given two subsets of the same set , we write if . This relation defines a Galois connection in the usual way: for we have . Since is symmetric, is self-adjoint on the right.
We define a -arity space to be a set together with a that is a fixed point of this Galois connection, . We call the sets in -ary and the sets in co--ary.
A morphism or relation between -arity spaces is a relation such that
If , then a -arity space is precisely a coherence space.
If , then a -arity space is precisely a finiteness space.
If , then a -arity space is (almost?) precisely a totality space.
Conjecture: For any , the category of -arity spaces is star-autonomous.
This might follow from constructing it using double gluing and orthogonality.
We can define arity spaces by a variation on the double gluing construction.
Define a double category of orthogonalities 1. Objects are relations 2. A vertical morphism from to exists when and are orthogonal subsets of and respectively. 3. A horizontal morphism from to is a pair of a function and 4. A square from to exists when is the restriction of and similarly for and .
Then the (2-)category of arity spaces can be defined as the comma double category (where and are viewed as vertically discrete double categories:
Where maps a set to the orthogonality on and a pair of sets is given the trivial orthogonality
Last revised on August 18, 2022 at 12:48:55. See the history of this page for a list of all contributions to it.