🤖 AI Summary
This paper addresses the algebraic characterization of “classicality” in restriction categories.
Method: It introduces and systematically studies the construction of the *classical product* $A & B coloneqq A oplus B oplus (A imes B)$, and investigates its categorical properties in distributive restriction categories.
Contribution/Results: It proves that a distributive restriction category is classical if and only if the classical product yields its categorical product; in this case, the category is necessarily equivalent to the Kleisli category of the exception monad $(− oplus 1)$ over an ordinary distributive category. This establishes, for the first time, a necessary and sufficient equivalence among three fundamental notions: the classical product, classical restriction categories, and Kleisli categories of the exception monad. The result unifies three complementary perspectives on classicality—syntactic (classical reasoning), algebraic (axiomatic characterization via products), and semantic (monadic models)—thereby providing a novel foundational paradigm for the logic of restriction categories.
📝 Abstract
In the category of sets and partial functions, $mathsf{PAR}$, while the disjoint union $sqcup$ is the usual categorical coproduct, the Cartesian product $ imes$ 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 $A &B := A sqcup B sqcup (A imes B)$. Surprisingly, asking that a distributive restriction category (a restriction category with restriction products $ imes$ 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 $A &B := A oplus B oplus (A imes B)$ 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 $A o B$ factors uniquely through a total map $A o B oplus mathsf{1}$, 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.
