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closure operator; hereditary closure operator; injective object; factorization pair
A notion of hereditarity of a closure operator with respect to a class of monomorphisms is introduced. Let $C$ be a regular closure operator induced by a subcategory $\Cal A$. It is shown that, if every object of $\Cal A$ is a subobject of an $\Cal A$-object which is injective with respect to a given class of monomorphisms, then the closure operator $C$ is hereditary with respect to that class of monomorphisms.
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