Definitions

# Isomorphism-closed subcategory

A subcategory $mathcal\left\{A\right\}$ of a category $mathcal\left\{B\right\}$ is said to be isomorphism-closed or replete if every $mathcal\left\{B\right\}$-isomorphism $h:Ato B$ with $Ainmathcal\left\{A\right\}$ belongs to $mathcal\left\{A\right\}.$ This implies that both $B$ and $h^\left\{-1\right\}:Bto A$ belong to $mathcal\left\{A\right\}$ 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 $mathcal\left\{B\right\}$-object which is isomorphic to an $mathcal\left\{A\right\}$-object is also an $mathcal\left\{A\right\}$-object.

This condition is very natural. E.g in the category of topological spaces we usually study properties which are invariant under homeomorphisms - so called topological properties. Every topological property corresponds to a strictly full subcategory of $mathbf\left\{Top\right\}.$

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