How to Enhance Categories and Why - Localization and Homotopical Enhancement

Explore the concept of enhancing categories in this 54-minute lecture by Dmitry Kaledin from HSE. Delve into the accepted wisdom of localizing categories with respect to morphism classes, resulting in categories with "homotopical enhancement." Examine the limitations of traditional "infinity-categories" or "quasicategories" approaches, and discover an alternative technique based on Grothendieck's "derivator" concept. Gain insights into this conceptually optimized method for understanding enhanced categories in mathematical theory.