# Is there a name for this kind of subcategory of an abelian category?

I have encountered a full subcategory $$\mathcal D$$ of an abelian category $$\mathcal C$$ which satisfies the following property:

If

$$0 \to M' \to M \to M'' \to 0$$

is a short exact sequence in $$\mathcal C$$, then:

$$(1)$$ If at least one of $$M',M''$$ is contained in $$D$$, then $$M$$ is contained in $$\mathcal D$$.

$$(2)$$ If $$M$$ is contained in $$\mathcal D$$, then at least one of $$M', M''$$ is contained in $$\mathcal D$$.

I wonder if such a subcategory $$\mathcal D$$ has a name in the literature of abelian categories?

• ncatlab.org/nlab/show/Serre+subcategory – Ivan Di Liberti Feb 25 at 13:03
• I don't think that this is a Serre subcategory, because one of my condition is weaker and the other is stronger... – the L Feb 25 at 13:05
• Condition $(1)$ looks weird. It implies that if your subcategory contains the zero object then it is the whole category $\mathcal{C}$... – Arnaud D. Feb 25 at 15:07
• Your definition is equivalent to a subcategory $\mathcal{D}$ such that the subcategory $\mathcal{D}'$ of objects that are not isomorphic to an object of $\mathcal{D}$ is a Serre sucategory. – Arnaud D. Feb 25 at 15:25
• Incidentally, my category does not contain the zero object. But I do like your second observation! – the L Feb 25 at 15:46