Abstract Lawvere's hyperdoctrines mark the beginning of applications of category theory to logic. In particular, existential elementary doctrines proved essential to give models of non-classical logics. The clear connection between (typed) logical theories and certain Pos -valued functors is exemplified by the embedding of the category of elementary doctrines into that of primary doctrines, which has a right adjoint given by a completion which freely adds quotients for equivalence relations. We extend that result to show that, in fact, the embedding is 2-functorial and 2-comonadic. As a byproduct we apply the result to produce an algebraic way to extend a first order theory to one which eliminates imaginaries, discuss how it relates to Shelah's original, and show how it works in a wider variety of situations.

中文翻译：

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作为代数的基本教义

摘要 Lawvere 的hyperdoctrines 标志着范畴论在逻辑中的应用的开始。特别是，存在主义基本学说被证明对于给出非经典逻辑模型是必不可少的。（类型化的）逻辑理论与某些 Pos 值函子之间的明确联系可以通过将基本学说的范畴嵌入到基本学说的范畴中来举例说明，该范畴有一个右伴随词，由完成式给出，可以自由地为等价关系添加商。我们扩展该结果以表明，实际上，嵌入是 2-functorial 和 2-comonadic。作为副产品，我们应用结果来产生一种代数方法，将一阶理论扩展到消除想象的理论，讨论它与 Shelah 的原始理论的关系，并展示它如何在更广泛的情况下发挥作用。