Classical distributive restriction categories (Q6593821)

Restriction categories were introduced in [\textit{J. R. B. Cockett} and \textit{S. Lack}, Theor. Comput. Sci. 270, No. 1--2, 223--259 (2002; Zbl 0988.18003)]. The concept of classified restriction categories was introduced in [\textit{J. R. B. Cockett} and \textit{S. Lack}, Theor. Comput. Sci. 294, No. 1--2, 61--102 (2003; Zbl 1023.18005)]. Classical restriction categories were explored in [\textit{R. Cockett} and \textit{E. Manes}, Math. Struct. Comput. Sci. 19, No. 2, 357--416 (2009; Zbl 1191.03049)]. The main result of this paper (Theorem 6.8) is that distributive restriction category is classical iff\N\[\NA\oplus B\oplus\left( A\times B\right)\N\]\Nis a categorical product.\N\NThe synopsis of the paper goes as follows.\N\N\begin{itemize}\N\item[\S 2] reviews the basics of distributive restriction categories.\N\N\item[\S 3] introduces the notion of classical products for distributive restriction categories.\N\N\item[\S 4] provides a review of classical restriction categories as well as some new observations about the complements of restriction idempotents.\N\N\item[\S 5] shows how restriction (co)products and categorical products are related via splitting restriction idempotent, and how in a classical restriction category, the categorical product must always be of the form \(A\oplus B\oplus\left( A\times B\right) \).\N\N\item[\S 6] establishes the main result of this paper.\N\N\item[\S 7] extends the main result, showing that a distributive restriction category is classical iff it is equivalent to the Kleisli category of the exception monad on a distributive category.\N\end{itemize}