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