# Isomorphism-closed subcategory

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 This implies that both and belong to as well.

A subcategory which is isomorphism-closed and full is called **strictly full**. In the case of full subcategories it is sufficient to check that every -object which is isomorphic to an -object is also an -object.

This condition is very natural. E.g. in the category of topological spaces one usually studies properties which are invariant under homeomorphisms – so called topological properties. Every topological property corresponds to a strictly full subcategory of

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