# Concrete balanced category

A category is called balanced if every bimorphism is an isomorphism.

Consider a concrete category such that every bijective morphism is a isomorphism. Does the category is balanced? Does converse is true?

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In the category of rings, every bijective morphism is an isomorphism, but the inclusion of $\mathbb{Z}$ in $\mathbb{Q}$ is a bimorphism that is not an isomorphism, so the category of rings is not balanced.

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The converse is true, of course: a bijective morphism in a concrete category must be a bimorphism and hence if a concrete is balanced, all its bijective morphisms are isomorphisms. –  Rob Arthan Aug 5 '13 at 16:58
Is there a general name for concrete categories where isomorphisms are exactly bijective morphisms? –  Corvus Aug 5 '13 at 17:40
–  Martin Brandenburg Aug 5 '13 at 18:00