isomorphism-closed subcategory

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In category theory, a branch of mathematics, a subcategory $\backslash mathcal\{A\}$ of a category $\backslash mathcal\{B\}$ is said to be isomorphism closed or replete if every $\backslash mathcal\{B\}$-isomorphism $h:A\backslash to\; B$ with $A\backslash in\backslash mathcal\{A\}$ belongs to $\backslash mathcal\{A\}.$ https://www.cs.cornell.edu/courses/cs6117/2018sp/Lectures/Subcategories.pdf This implies that both $B$ and $h^\{-1\}:B\backslash to\; A$ belong to $\backslash mathcal\{A\}$ 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 $\backslash mathcal\{B\}$-object that is isomorphic to an $\backslash mathcal\{A\}$-object is also an $\backslash mathcal\{A\}$-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 $\backslash mathbf\{Top\}.$