Christian Espíndola: Recovering an infinitary first-order theory from its category of models

Abstract: Makkai’s reconstruction result allows to recover a first-order theory, up to pretopos completion, by equipping the category of models with appropriate structure and considering set-valued functors preserving this structure. One of the difficulties in extending this result to the infinitary case is that the Keisler–Shelah isomorphism theorem, which asserts that elementary equivalent models have isomorphic ultrapowers, does not hold for infinitary logic. However, we will see that the theory of accessible categories can provide extra structure, under appropriate large cardinal assumptions, to recover the theory up to a notion of infinitary pretopos-completion.