nLab null system

Idea

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.

Definition

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 $Y \in N$.

Properties

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

$$N Q := \{ X \stackrel{f}{\to} Y | \exists dist. tri. X \to Y \to Z \: in \: C \: with \: Z \in N\} \,.$$

Then $N Q$ admits a left and right calculus of fractions in $C$.

Examples

• For $K(C)$ the category of chain complexes modulo chain homotopy of an abelian category, the full subcategory of $K(C)$ of chain complexes $V$ whose homology vanishes, $H(V) \simeq 0$ is a null system. Then$D(C) := K(C)/N$ is the derived category of $C$.

David Roberts: Would Serre class?es fit in here? Perhaps that’s one step back.

References

For instance section 10.2 of

• Kashiwara–Schapira, Categories and Sheaves