I am not even sure of the definition of extension. I have read that given a morphism $f:X\to Y$ in a category $\mathcal{C}$, a extension along a monomorphism $i:X\hookrightarrow Z$ is a morphism $\overline{f}:Z\to Y$ such that $f=\overline{f}\circ i$.

If this definition is correct, when do we say that a subcategory is closed under extensions? Do I need the morphism $f:X\to Y$ to be in the subcategory or just X?

  • $\begingroup$ Where did you see the terminology? $\endgroup$
    – Kenta S
    Jun 24, 2022 at 8:44
  • $\begingroup$ Here: ncatlab.org/nlab/show/extension. I thought that what I needed was the first notion, the one labeled as "extension of morphisms", but know I have seen that it is not. $\endgroup$ Jun 28, 2022 at 9:36

1 Answer 1


This is a quite different and unrelated use of the term "extension." Generally the ambient category is an abelian category (although generalizations are possible), and in this context an extension of an object $B$ by an object $A$ is an object $C$ fitting into a short exact sequence

$$0 \to A \xrightarrow{f} C \xrightarrow{g} B \to 0$$

meaning that $f$ is a monomorphism, $g$ is an epimorphism, and $\text{ker}(g) = \text{im}(f)$ (see the link for definitions).

A subcategory is then said to be closed under extensions if whenever $A$ and $B$ lie in that subcategory, then so does any extension $C$. For example, inside the abelian category of abelian groups, the following (full) subcategories are closed under extensions (some of these are nice little exercises):

  • Finite abelian groups
  • Finitely generated abelian groups
  • Torsion abelian groups
  • Torsion-free abelian groups
  • Divisible abelian groups

However, the following subcategories are not closed under extensions:

  • Cyclic abelian groups
  • $p$-torsion abelian groups
  • $\begingroup$ I am new here, do I have to do something once my question gets a good answer? $\endgroup$ Jun 28, 2022 at 9:37
  • $\begingroup$ @Jorge: you can upvote the answer and accept it by clicking the little check mark. Thanks! $\endgroup$ Jun 28, 2022 at 16:30

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