The language of triangulated categories has countless applications in Algebraic Topology, Geometry and Algebra. Nevertheless, with the passing of time the theory outlined by Grothendieck and Verdier [Ver] proved itself to be full of serious drawbacks (choices have to be made, there is no universal property for the derived category of an abelian category A, there is no way to retain essential homotopical informations after localization).

The language of stable infty-categories aims at repairing this deficiency, by conveying the idea [Lur,SSh, Kuz] that triangulated categories arise as 1-dimensional shadows of highly-structured geometric objects whose investigation is the primary task of Homological Algebra.

The aim of the talk is to present some of these ideas, trying to convince the audience already familiar with the menagerie of triangulated categories that they are already "experts" also in the theory of stable infty-categories, and yet they don't know.

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[Kuz] Kuznetsov, Alexander, and Valery A. Lunts. "Categorical resolutions of irrational singularities." International Mathematics Research Notices (2014): rnu072.

[Lur] Lurie, Jacob. "Higher algebra. 2012." Preprint, available at http://www.math.harvard.edu/~lurie.

[SSh] Schwede, Stefan, and Brooke Shipley. "Stable model categories are categories of modules." Topology 42.1 (2003): 103-153.

[Ver] Verdier, Jean-Louis, Des Catégories Dérivées des Catégories Abéliennes, Astérisque (Paris: Société Mathématique de France) 239.