in the 90s in his manuscript on the topic. Essentially, these are a system of homotopy categories given by the diagram categories for a category with a class of weak equivalences . These categories are then related by the morphisms of diagrams . This formalism has the advantage of being able to recover the homotopy limits and colimits, which replaces the cone construction. Stable ∞-categories Another alternative built is the theory of stable ∞-categories. The homotopy category of a stable ∞-category is canonically triangulated, and moreover mapping cones become essentially unique (in a precise homotopical sense). Moreover, a stable ∞-category naturally encodes a whole hierarchy of compatibilities for its homotopy category, at the bottom of which sits the octahedral axiom. Thus, it is strictly stronger to give the data of a stable ∞-category than to give the data of a triangulation of its homotopy category. Nearly all triangulated categories that arise in practice come from stable ∞-categories. A similar (but more special) enrichment of triangulated categories is the notion of a dg-category. In some ways, stable ∞-categories or dg-categories work better than triangulated categories. One example is the notion of an exact functor between triangulated categories, discussed below. For a smooth projective variety X over a field k, the bounded derived category of coherent sheaves comes from a dg-category in a natural way. For varieties X and Y, every functor from the dg-category of X to that of Y comes from a complex of sheaves on by the Fourier–Mukai transform. By contrast, there is an example of an exact functor from to that does not come from a complex of sheaves on . In view of this example, the "right" notion of a morphism between triangulated categories seems to be one that comes from a morphism of underlying dg-categories (or stable ∞-categories). Another advantage of stable ∞-categories or dg-categories over triangulated categories appears in algebraic K-theory. One can define the algebraic K-theory of a stable ∞-category or dg-category C, giving a sequence of abelian groups for integers i. The group has a simple description in terms of the triangulated category associated to C. But an example shows that the higher K-groups of a dg-category are not always determined by the associated triangulated category. Thus a triangulated category has a well-defined group, but in general not higher K-groups. On the other hand, the theory of triangulated categories is simpler than the