Isomorphism-closed subcategory

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