Universes and Higher Algebra in Synthetic ∞-Category Theory

Explore the construction of synthetic ∞-categories and their applications to higher algebra in this advanced mathematics seminar. Delve into how categories axiomatize function composition and discover the natural weakening of this concept to composition defined only up to homotopy, illustrated through examples from algebraic topology including fundamental groups and fundamental ∞-groupoids. Learn about the challenges of constructing universes in set-based mathematics and examine a novel approach using modal extensions of simplicial homotopy type theory. Investigate the synthetic ∞-category of ∞-groupoids and the synthetic ∞-category of ∞-categories, understanding how ∞-groupoids form an ∞-category with morphisms that weakly preserve composition. Discover immediate applications to higher algebra and explore future research perspectives in this emerging field, based on collaborative work with Daniel Gratzer and Ulrik Buchholtz that addresses fundamental questions in category theory and homotopy type theory.