We discuss reflectivity of colocalizing subcategories of triangulated categories under suitable set-theoretical assumptions. In earlier joint work with Gutierrez and Rosicky, we proved that if K is any locally presentable category with a stable model category structure, then Vopenka’s principle implies that every full subcategory L of the homotopy category of K closed under products and fibres is reflective. Moreover, if L is colocalizing, then the reflection is exact. Using recent progress in large-cardinal theory, we show that the statement that every full subcategory closed under products and fibres is reflective is, in fact, equivalent to the so-called weak Vopenka principle. Hence this statement cannot be proved using only the ZFC axioms.
This video is part of the New Directions in Group Theory and Triangulated Categories seminar series.
