/sci/ - Science & Math

Hello /sci/. I was hoping a category bro could give me some insight here. If Heyting categories are exactly those categories with first-order internal logics, and Heyting categories are also exactly those coherent categories with a right adjoint to the base change functor that acts as universal quantification, then does removing the latter restriction mean that coherent categories are exactly those categories with existential first-order internal logics?

I am asking because I have been reading Ronald Fagin's work in descriptive complexity theory, and how P corresponds to first-order logic with a fixed-point operator, and NP corresponds to existential second order logic (Fagin's theorem). I am interested in what properties characterize categories whom's internal logic correspond to given complexity classes, and if there is a cleaner, more natural, and more concise way of expressing this relationship.

Again, any insight would be most appreciated. Thanks!

>pic mostly unrelated, I think I posted it in a handwriting thread a while back or something