关系学说中的商- comprehension 对偶性
Quotients-comprehensions duality in relational doctrines
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中文总结 AI 辅助
该研究在关系学说中建立商与 comprehension 的对偶性，将其扩展到2-范畴层面，还研究了二者与相对完备化的相互作用及2-monad可组合的充分条件。
中文摘要 AI 辅助
我们在关系学说（一类对关系演算（最小片段）进行建模的索引偏序集）中建立了商与 comprehension 的对偶性。我们证明每个关系学说都确定一个“对立”关系学说，且当且仅当其中一个具有商时，另一个具有 comprehension。该对偶性扩展到2-范畴层面，得到带商的关系学说2-范畴与带 comprehension 的关系学说2-范畴之间的2-同构，允许从商的对偶对应推导带 comprehension 的关系学说的结果，反之亦然。我们还研究了商与 comprehension 及其相对完备化的相互作用，给出诱导的2- monad 可组合的充分条件，即确保应用两次完备化时，第二个能保留第一个添加的性质的充分条件。
英文摘要
We establish a duality between quotients and comprehensions in relational doctrines, a class of indexed posets modelling (a minimal fragment of) the calculus of relations. We show that every relational doctrine determines an ``opposite" relational doctrine and that one has quotients if and only if the other has comprehensions. This duality extends to the 2-categorical level, yielding a 2-dual isomorphism between the 2-categories of relational doctrines with quotients and that with comprehensions, allowing to derive results on relational doctrines with comprehensions from their dual counterparts for quotients, and viceversa. We also study the interaction between quotients and comprehensions and their relative completions, giving sufficient conditions for the induced 2-monads to be composable, \ie sufficient conditions ensuring that applying both the completions, the second one preserves the properties added by the first one.
