Title: Homotopical Ideas in First-Order Logic
Abstract: The interaction between logic and homotopy theory is dominated by higher category theory and type theories. However, recent developments in traditional first-order logic are being interpreted in an increasingly homotopical paradigm, leading to new Stone-duality-esque theorems by Makkai, Butz, Moerdijk, Lurie, and others. These results show that---while type theories have an internal homotopy theory---there exists an external homotopy theory of first-order theories. This talk is a survey of the history of categorical logic and its applications for homotopy theory, as well as a discussion on why these homotopical results are showing up in the first place.
