Derivator theory, initiated by Grothendieck and Heller in the ’90s to correct the shortcomings of triangulated categories, motivated a lot of research regarding the foundation of (∞,1)-category theory, and its applications to algebraic geometry/topology.
For a 2-category theorist, a (pre)derivator is a familiar object – (a suitably co/complete) prestack on the category cat of small categories – and yet still little is known about the formal properties of the 2-category PDer. The present talk is motivated by the belief that time is ripe for a more conceptual look into the foundations of derivator theory, and that far from being a mere exercise in style, such a conceptualization yields many practical advantages.
After briefly outlining the essentials of “formal category theory” (2-categories can be used to organize the theory of “categories with structure” just as category theory organizes the theory of “sets with structure”), I will report on a conjecture regarding the possibility to provide a “yoneda structure” or a “proarrow equipment” to the 2-category of pre/derivators. Under suitable assumptions, these are equivalent ways to equip PDer with a calculus of Kan extensions, and building on prior work of Di Liberti and myself, this allows to speak about “locally presentable” and “accessible” objects (showing that Adamek-Rosický and Renaudin’s definitions eventually coincide); the overall goal is to provide a suitable form of special/general adjoint functor theorem for a morphism of prederivators (such a theorem would simplify a lot the life of the average algebraic geometer).
This video is part of the New Directions in Group Theory and Triangulated Categories seminar series.
