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 $\mathscr{D}^{\rm b}(\operatorname{mod}\,A)$. Their extension closure is $\operatorname{mod}\,A$; in particular, it is abelian. This situation is emulated by a general simple minded collection $\mathscr{S}$ in a suitable triangulated category $\mathscr{C}$. In particular, the extension closure $\langle \mathscr{S} \rangle$ is abelian, and there is a tilting theory for such abelian subcategories of $\mathscr{C}$. These statements follow from $\langle \mathscr{S} \rangle$ 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 $\{ -w+1, \ldots, -1 \}$ only, 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. If $\mathscr{S}$ is a $w$-simple minded system for some $w \geqslant 2$, then $\langle \mathscr{S} \rangle$ is typically not the heart of a $t$-structure. Nevertheless, using different methods, we will prove that $\langle \mathscr{S} \rangle$ is abelian and that there is a tilting theory for such abelian subcategories. Our theory is based on Quillen's notion of exact categories.