Isomorphism-closed subcategory

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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 ${\mathcal {A}}.$ ^[1] This implies that both $B$ and $h^{-1}:B\to A$ belong to ${\mathcal {A}}$ 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 ${\mathcal {B}}$ -object that is isomorphic to an ${\mathcal {A}}$ -object is also an ${\mathcal {A}}$ -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 $\mathbf {Top} .$

References

^ https://www.cs.cornell.edu/courses/cs6117/2018sp/Lectures/Subcategories.pdf

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