Classical Distributive Restriction Categories

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, $\mathsf{PAR}$, while the disjoint union $\sqcup$ is the usual categorical coproduct, the Cartesian product $\times$ becomes a restriction categorical analogue of the categorical product: a restriction product. Nevertheless, $\mathsf{PAR}$ does have a usual categorical product as well in the form $A \& B := A \sqcup B \sqcup (A \times B)$. Surprisingly, asking that a distributive restriction category (a restriction category with restriction products $\times$ and coproducts $\oplus$) has $A \& B$ 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 $A \& B := A \oplus B \oplus (A \times B)$ is a categorical product in which case we call $\&$ the ''classical'' product. In fact, a distributive restriction category has a categorical product if and only if it is a classified restriction category. This is in the sense that every map $A \to B$ factors uniquely through a total map $A \to B \oplus \mathsf{1}$, where $\mathsf{1}$ is the restriction terminal object. This implies the second significant observation of the paper, namely, that a distributive restriction category has a classical product if and only if it is the Kleisli category of the exception monad $\_ \oplus \mathsf{1}$ for an ordinary distributive category. Thus having a classical product has a significant structural effect on a distributive restriction category. In particular, the classical product not only provides an alternative axiomatization for being classical but also for being the Kleisli category of the exception monad on an ordinary distributive category.