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 $\mathbf{𝐓}\mathbf{𝐨}\mathbf{𝐩}.$

References

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