Isomorphism-closed_subcategory

Isomorphism-closed subcategory

A subcategory mathcal{A} of a category mathcal{B} is said to be isomorphism-closed or replete if every mathcal{B}-isomorphism h:Ato B with Ainmathcal{A} belongs to mathcal{A}. This implies that both B and h^{-1}:Bto 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 we usually study properties which are invariant under homeomorphisms - so called topological properties. Every topological property corresponds to a strictly full subcategory of mathbf{Top}.


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