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

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?

• Where did you see the terminology? Jun 24, 2022 at 8:44
• 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. Jun 28, 2022 at 9:36

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