Isomorphism-closed subcategory

In category theory, a branch of mathematics, a subcategory ${\displaystyle {\mathcal {A}}}$ of a category ${\displaystyle {\mathcal {B}}}$ is said to be isomorphism closed or replete if every ${\displaystyle {\mathcal {B}}}$-isomorphism ${\displaystyle h:A\to B}$ with ${\displaystyle A\in {\mathcal {A}}}$ belongs to ${\displaystyle {\mathcal {A}}.}$ ^[1] This implies that both ${\displaystyle B}$ and ${\displaystyle h^{-1}:B\to A}$ belong to ${\displaystyle {\mathcal {A}}}$ as well.

A subcategory that is isomorphism closed and full is called strictly full. In the case of full subcategories it is sufficient to check that every ${\displaystyle {\mathcal {B}}}$-object that is isomorphic to an ${\displaystyle {\mathcal {A}}}$-object is also an ${\displaystyle {\mathcal {A}}}$-object.

This condition is very natural. For example, in the category of topological spaces one usually studies properties that are invariant under homeomorphisms—so-called topological properties. Every topological property corresponds to a strictly full subcategory of ${\displaystyle \mathbf {Top} .}$