Distillation Systems as Models of Homotopy Colimits

This talk presents joint research by Kristine Bauer, Kathryn Hess, Brenda Johnson, and Julie Rasmusen exploring distillation systems as models for homotopy colimits. Dive into the fundamental challenges of working with colimits in topology when spaces undergo continuous deformation, and discover how homotopy colimits address these issues despite not satisfying universal properties in the homotopy category. Learn how the researchers identified essential properties for homotopy limits in homotopical categories and developed the concept of distillation systems—structures that relate two actions of the category of small categories through a lax linear functor. The presentation explains when these systems successfully model homotopy colimits and when they don't, providing valuable insights for applications in functor calculus. This 59-minute Topos Institute Colloquium was delivered on May 1, 2025.