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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?

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

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