On Makkai's Strong Conceptual Completeness Theorem
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 firstorder theory T admits a model. In the 1980s, Makkai proved a sharper result: any firstorder 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 "proetale sheaves" developed by Scholze and BhattScholze.