The definition of an extriangulated category was given by Nakaoka and Palu. This notion simultaneously generalises those of an exact category and a triangulated category. Examples of extriangulated categories which need not be exact nor triangulated appear, for example, when studying pure-exact sequences in compactly generated triangulated categories.
In this talk I will discuss what are called extriangulated functors, from joint work with myself and Amit Shah. As above, the definition unifies that of exact functors between exact categories, and triangulated functors between triangulated categories. Other examples of extriangulated functors include the delta functor from an exact category to its derived category, and the restricted Yoneda functor from a compactly generated triangulated category to the appropriate functor category. Time permitted, I will explain the role of a category of extensions.