A null system in a triangulated category is a triangulated subcategory whose objects may consistently be regarded as being equivalent to the zero object. Null systems give a convenient means for encoding and computing localization of triangulated categories.
A null system of a triangulated category $C$ is a full subcategory $N \subset C$ such that
$N$ is saturated: every object $X$ in $C$ which is isomorphic in $C$ to an object in $N$ is in $N$;
the zero object is in $N$;
$X$ is in $N$ precisely if $T X$ is in $N$;
if $X \to Y \to Z \to T X$ is a distinguished triangle in $C$ with $X, Z \in N$, then also $Y \in N$.
The point about null systems is the following:
for $N$ a null system, let $N Q$ be the collection of all morphisms in $C$ whose “mapping cone” is in $N$, precisely: set
Then $N Q$ admits a left and right calculus of fractions in $C$.
David Roberts: Would Serre class?es fit in here? Perhaps that’s one step back.
For instance section 10.2 of