Isomorphism-closed subcategory

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 $\mathbf {Top} .$

References

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

This article is issued from Wikipedia - version of the 8/12/2015. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.