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.
is saturated: every object in which is isomorphic in to an object in is in ;
the zero object is in ;
is in precisely if is in ;
if is a distinguished triangle in with , then also .
The point about null systems is the following:
for a null system, let be the collection of all morphisms in whose “mapping cone” is in , precisely: set
Then admits a left and right calculus of fractions in .
For instance section 10.2 of