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

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