Isomorphism-closed subcategory

In category theory, a branch of mathematics, a subcategory [math]\displaystyle{ \mathcal{A} }[/math] of a category [math]\displaystyle{ \mathcal{B} }[/math] is said to be isomorphism closed or replete if every [math]\displaystyle{ \mathcal{B} }[/math]-isomorphism [math]\displaystyle{ h:A\to B }[/math] with [math]\displaystyle{ A\in\mathcal{A} }[/math] belongs to [math]\displaystyle{ \mathcal{A}. }[/math] This implies that both [math]\displaystyle{ B }[/math] and [math]\displaystyle{ h^{-1}:B\to A }[/math] belong to [math]\displaystyle{ \mathcal{A} }[/math] 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 [math]\displaystyle{ \mathcal{B} }[/math]-object that is isomorphic to an [math]\displaystyle{ \mathcal{A} }[/math]-object is also an [math]\displaystyle{ \mathcal{A} }[/math]-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 [math]\displaystyle{ \mathbf{Top}. }[/math]