Classical Distributive Restriction Categories
Robin Cockett, Jean-Simon Pacaud Lemay
TL;DR
The paper studies distributive restriction categories and shows that the existence of a classical product $A \& B := A \oplus B \oplus (A \times B)$ is equivalent to the category being classical (having joins and relative complements) and classified. It proves that such categories are precisely the Kleisli categories of the exception monad on ordinary distributive categories, linking classical reasoning with a canonical monadic construction. The results clarify how restriction products and coproducts interact to yield classical logic, with idempotent splitting providing the bridge to coproduct structures. Examples like $ extsf{PAR}$ and $k ext{-} extsf{CALG}^{op}_\bullet$ illustrate the theory, while TOP-type categories show limits of the approach and the role of clopen variants in achieving classicality.
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.