On Makkai's Strong Conceptual Completeness Theorem

Jacob Lurie, Institute for Advanced Study
Fine Hall 314

One of the most fundamental results of mathematical logic is the celebrated Godel completeness theorem, which asserts that every consistent first-order theory T admits a model. In the 1980s, Makkai proved a sharper result: any first-order theory T can be recovered, up to a suitable notion of equivalence, from its category of models Mod(T) together with some additional structure (supplied by the formation of ultraproducts). In this talk, I'll explain the statement of Makkai's theorem and sketch a new proof of it, inspired by the theory of "pro-etale sheaves" developed by Scholze and Bhatt-Scholze.