Isomorphism-closed subcategory explained

l{A}

of a category

l{B}

is said to be isomorphism closed or replete if every

l{B}

-isomorphism

h:A\toB

with

A\inl{A}

belongs to

l{A}.

This implies that both

and

h^-1:B\toA

belong to

l{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

l{B}

-object that is isomorphic to an

l{A}

-object is also an

l{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

Top.