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.