A first class on category theory typically introduces this rather silly category whose objects are natural numbers, and morphisms \(p\to q\) are \(q\times p\) matrices – this is perhaps to drive home the point that anything and everything under the sun can be a category – and then proceeds to show that it is equivalent to the category of vector spaces (an “honest” category by anyone’s definition). Continuing this rather silly exercise, we will examine a category whose objects are all finite matrices, and then proceed to take off its guise.

An object \(G\) in a triangulated category \(\mathscr{T}\) is a classical generator if it is not contained in a proper triangulated subcategory of \(\mathscr{T}\). It is a generator if \(\{G\}\) is a spanning class, i.e. \(\text{Hom}^\bullet(G,x) = 0\) if and only if \(x=0\). Every classical generator spans; I discuss two geometric counterexamples to the converse — based on Dmitrii Pirozhkov’s exposition which attributes them to de Jong.