Isomorphism closed subcategory

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