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 $h:A\to B$ with $A\in 𝒜$ belongs to $𝒜.$ This implies that both $B$ and $h^{-1}:B\to A$ belong to $𝒜$ as well.

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

This condition is very natural. For example, in the category of topological spaces one usually studies properties that are invariant under homeomorphisms—so-called topological properties. Every topological property corresponds to a strictly full subcategory of $𝐓𝐨𝐩.$

References

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