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.