Isomorphism-closed subcategory

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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.