Classical Distributive Restriction Categories
Robin Cockett, Jean-Simon Pacaud Lemay

TL;DR
This paper characterizes classical restriction categories within distributive restriction categories, showing that the existence of a classical product implies the category is a Kleisli category of an exception monad, linking structural properties to categorical constructs.
Contribution
It establishes that a distributive restriction category is classical if and only if it has a specific categorical product called the classical product, and connects this to being a Kleisli category of an exception monad.
Findings
Classical restriction categories have joins and relative complements.
A distributive restriction category is classical iff a certain product is categorical.
Such categories are equivalent to Kleisli categories of the exception monad.
Abstract
In the category of sets and partial functions, , while the disjoint union is the usual categorical coproduct, the Cartesian product becomes a restriction categorical analogue of the categorical product: a restriction product. Nevertheless, does have a usual categorical product as well in the form . Surprisingly, asking that a distributive restriction category (a restriction category with restriction products and coproducts ) has a categorical product is enough to imply that the category is a classical restriction category. This is a restriction category which has joins and relative complements and, thus, supports classical Boolean reasoning. The first and main observation of the paper is that a distributive restriction category is classical if and only if $A \& B := A…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Algebra and Logic · Logic, Reasoning, and Knowledge
