Simple object in derived category or stable model category? - MathOverflow

Simple object in derived category or stable model category?

2011-07-11T16:38:15Z

Exist any common definition of simple objects in derived categories, or even better, in stable model categories? I was only able to find definition for abelian categories.

Thanks.

Answer by Neil Strickland for Simple object in derived category or stable model category?

2011-07-11T17:55:46Z

In an arbitrary category, you can define subobjects of $X$ to be equivalence classes of pairs $(A,f)$, where $f:A\to X$ is a monomorphism, and $(A,f)$ is equivalent to $(B,g)$ if there is an isomorphism $p:A\to B$ with $gp=f$. You can then say that an object in an additive is simple if it has only the two obvious subobjects (and they are different). In a triangulated category, any monomorphism $f:A\to X$ is split, because there is a distinguished triangle $W\xrightarrow{m}A\xrightarrow{f}X\xrightarrow{n}\Sigma W$, and $fm=0$ so $m=0$. This means that the simple objects are precisely those that are indecomposable under direct sums.

Answer by Sasha for Simple object in derived category or stable model category?

2011-07-11T20:52:07Z

Sometimes an object $E$ of a $k$-linear category $C$ is called simple if $Hom_C(E,E) = k$. This notion is frequently used in derived categories.