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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$\operatorname{D}^b( \operatorname{mod} A )$. Their extension closure is$\operatorname{mod} A$; in particular, it is abelian. This situation isemulated by a general simple minded collection $\mathcal{S}$ in a suitabletriangulated category $\mathcal{C}$. In particular, the extension closure$\langle \mathcal{S} \rangle$ is abelian, and there is a tilting theory forsuch abelian subcategories of $\mathcal{C}$. These statements follow from$\langle \mathcal{S} \rangle$ being the heart of a bounded $t$-structure.  It is a defining characteristic of simple minded collections that theirnegative self extensions vanish in every degree. Relaxing this to vanishing indegrees $\{ -w+1, \ldots, -1 \}$ where $w$ is a positive integer leads to therich, parallel notion of $w$-simple minded systems, which have recently beenthe subject of vigorous interest.  If $\mathcal{S}$ is a $w$-simple minded system for some $w \geqslant 2$, then$\langle \mathcal{S} \rangle$ is typically not the heart of a $t$-structure.Nevertheless, using different methods, we will prove that $\langle \mathcal{S}\rangle$ is abelian and that there is a tilting theory for such abeliansubcategories. Our theory is based on Quillen's notion of exact categories, inparticular a theorem by Dyer which provides exact subcategories of triangulatedcategories.  The theory of simple minded systems can be viewed as "negative clustertilting theory". In particular, the result that $\langle \mathcal{S} \rangle$is an abelian subcategory is a negative counterpart to the result from (higher)positive cluster tilting theory that if $\mathcal{T}$ is a cluster tiltingsubcategory, then $( \mathcal{T} * \Sigma \mathcal{T} )/[ \mathcal{T} ]$ is anabelian quotient category.

Abelian subcategories of triangulated categories induced by simple minded systems