Doubt about meaning of a subcategory "being closed under extensions"

Question

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?

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