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 of $A \sqcup B \sqcup (A \times B)$. Surprisingly, asking that a distributive restriction category (a restriction category with restriction products $\times$ and coproducts $\oplus$) has $A \oplus B \oplus (A \times B)$ 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 so 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.

翻译：在集合与部分函数构成的范畴$\mathsf{PAR}$中，虽然不交并$\sqcup$是通常的范畴余积，但笛卡尔积$\times$成为范畴积的限制范畴类比：限制积。然而，$\mathsf{PAR}$确实也拥有一个通常的范畴积，形如$A \sqcup B \sqcup (A \times B)$。令人惊讶的是，要求一个分配限制范畴（具有限制积$\times$和余积$\oplus$的范畴）使得$A \oplus B \oplus (A \times B)$成为范畴积，足以推出该范畴是经典限制范畴。此类限制范畴具有并和相对补，因此支持经典的布尔推理。本文首要且关键的观察是：一个分配限制范畴是经典的当且仅当$A \& B := A \oplus B \oplus (A \times B)$是范畴积，此时我们将$\&$称为"经典"积。事实上，分配限制范畴拥有范畴积当且仅当它是分类限制范畴。这意味着每个态射$A \to B$都能唯一分解为通过一个全态射$A \to B \oplus \mathsf{1}$的复合，其中$\mathsf{1}$是限制终端对象。由此导出本文第二个重要结论：分配限制范畴拥有经典积当且仅当它是普通分配范畴的异常单子$\_ \oplus \mathsf{1}$的Kleisli范畴。因此，经典积的存在对分配限制范畴具有显著的结构性影响。特别地，经典积不仅为经典性提供了另一种公理化描述，也刻画了普通分配范畴上异常单子Kleisli范畴的本质特征。