Classical Distributive Restriction Categories

05/25/2023
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by   Robin Cockett, et al.
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In the category of sets and partial functions, 𝖯𝖠𝖱, while the disjoint union βŠ” is the usual categorical coproduct, the Cartesian product Γ— becomes a restriction categorical analogue of the categorical product: a restriction product. Nevertheless, 𝖯𝖠𝖱 does have a usual categorical product as well in the form of A βŠ” B βŠ” (A Γ— B). Surprisingly, asking that a distributive restriction category (a restriction category with restriction products Γ— and coproducts βŠ•) has A βŠ• B βŠ• (A Γ— 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 βŠ• B βŠ• (A Γ— B) is a categorical product in which case we call & the β€œclassical” 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 β†’ B factors uniquely through a total map A β†’ B βŠ•1, where 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 _βŠ•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.

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