In category theory, a branch of mathematics, a subcategory of a category is said to be **isomorphism closed** or **replete** if every -isomorphism with belongs to ^{[1]} This implies that both and belong to 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 -object that is isomorphic to an -object is also an -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

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*This article incorporates material from Isomorphism-closed subcategory on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.*