Isomorphism-closed subcategory

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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 which 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 which is isomorphic to an \mathcal{A}-object is also an \mathcal{A}-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 \mathbf{Top}.


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