Abelian subcategories of triangulated categories induced by simple minded systems

If \(k\) is a field, \(A\) a finite dimensional \(k\)-algebra, then the simple \(A\)-modules form a simple minded collection in the derived category \(\operatorname{D}^\mathrm{b}(\operatorname{mod}A)\). Their extension closure is \(\operatorname{mod}A\); in particular, it is abelian. This situation is emulated by a general simple minded collection \(S\) in a suitable triangulated category \(\mathcal{C}\). In particular, the extension closure \(\langle S\rangle\) is abelian, and there is a tilting theory for such abelian subcategories of \(\mathcal{C}\). These statements follow from \(\langle S\rangle\) being the heart of a bounded \(t\)-structure.

It is a defining characteristic of simple minded collections that their negative self extensions vanish in every degree. Relaxing this to vanishing in degrees \(\{-w+1,…,-1\}\) where \(w\) is a positive integer leads to the rich, parallel notion of \(w\)-simple minded systems, which have recently been the subject of vigorous interest within negative cluster tilting theory.

If \(S\) is a \(w\)-simple minded system for some \(w\geq2\), then \(\langle S\rangle\) is typically not the heart of a \(t\)-structure. However, it is possible to prove by different means that \(\langle S\rangle\) is still abelian and that there is a tilting theory for such abelian subcategories. We will explain the theory behind this, which is based on Quillen’s notion of exact categories.