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Elementary doctrines as coalgebras
Journal of Pure and Applied Algebra ( IF 0.770 ) Pub Date : 2020-05-18 , DOI: 10.1016/j.jpaa.2020.106445
Jacopo Emmenegger; Fabio Pasquali; Giuseppe Rosolini

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.
更新日期:2020-05-18

 

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